Canadian Tidal Manual - Fisheries and Oceans Canada Library

CANADIANTIDAL

MANUAL

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CANADIAN TIDAL MANUAL

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CANADIAN TIDAL MANUAL

Prepared under contract byWARREN D. FORRESTER, PH.D.*

DEPARTMENT OF FISHERIES AND OCEANSOttawa 1983

*This manual was prepared under contract (FP 802-1-2147) funded by The Canadian Hydrographic Service. Dr. Forrester was Chiefof Tides, Currents, and Water Levels (1975-1980). Mr. Brian J. Tait, presently Chief of Tides, Currents, and Water Levels, CanadianHydrographic service, acted as the Project Authority for the contract.

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©Minister of Supply and Services Canada 1983Available by mail from:

Canadian Government Publishing Centre,Supply and Services Canada,Hull, Que., Canada KIA OS9

or through your local bookseller

or from

Hydrographic Chart Distribution Office,Department of Fisheries and Oceans,

P.O. Box 8080, 1675 Russell Rd.,Ottawa, Ont.

Canada KIG 3H6

S.B. MacPhee

H.R. Blandford

B.J. Tait

Director General

Director, Navigation Publicationsand Maritime Boundaries Branch

Chief, Tides, Currents and Water Levels

The Canadian Hydrographic Service produces and distributes Nautical Charts, Sailing Directions, Small CraftGuides, Tide Tables, and Water Levels of the navigable waters of Canada.

Canada $20.00Other countries $24.00

Correct citation for this publication:

FORRESTER, W.D. 1983. Canadian Tidal Manual.Department of Fisheries and Oceans, CanadianHydrographic Service, Ottawa, Ont. 138 p.

Cat. No. Fs 75-325/1983EISBN 0-660-11341-4

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Chapter 1. Tides as Waves ........................................................................................................... 11.1 What is the tide? ............................................................................................................ 11.2 Waves ............................................................................................................................ 11.3 Surface gravity waves .................................................................................................... 21.4 Long and short waves of small amplitude ........................................................................ 41.5 Particle motions in long waves ........................................................................................ 51.6 Basin Oscillations ........................................................................................................... 71.7 Internal waves ............................................................................................................... 81.8 Coriolis acceleration ..................................................................................................... 201.9 Inertial currents ............................................................................................................ 22

1.10 Amphidromic systems .................................................................................................. 221.11 Tides and tidal streams ................................................................................................. 231.12 Shallow water effects ................................................................................................... 24

Chapter 2. The Tide-Raising Forces .......................................................................................... 212.1 Introduction .................................................................................................................. 212.2 Sun's tide-raising force ................................................................................................. 212.3 Moon's tide-raising force .............................................................................................. 232.4 Tidal potential and the equilibrium tide ........................................................................... 242.5 Semidiurnal and diurnal equilibrium tides ....................................................................... 252.6 Long-period equilibrium tides ........................................................................................ 292.7 Mathematical analysis of the equilibrium tide ................................................................ 292.8 Spring and neap tides .................................................................................................... 302.9 Classification of tides .................................................................................................... 31

Chapter 3. Tidal Analysis and Prediction ................................................................................... 353.1 Introduction .................................................................................................................. 353.2 The Fourier Theorem ................................................................................................... 353.3 Harmonic analysis of tides ............................................................................................ 363.4 Nineteen-year modulation of lunar constituents ............................................................. 383.5 Shallow-water constituents ........................................................................................... 393.6 Record length and sampling interval .............................................................................. 393.7 Harmonic analysis of tidal streams ................................................................................ 403.8 Harmonic method of tidal prediction ............................................................................. 413.9 Prediction of tidal and current extrema ......................................................................... 42

3.10 Cotidal charts ............................................................................................................... 423.11 Numerical modelling of tides ......................................................................................... 48

CONTENTS

Preface ....................................................................................................................................... ix

Part ITheory and Concepts

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Chapter 4. Meteorological and Other Non-Tidal Influences ................................................... 504.1 Introduction .................................................................................................................. 504.2 Wind-driven currents .................................................................................................... 504.3 Wind set-up .................................................................................................................. 514.4 Atmospheric pressure effect ........................................................................................ 524.5 Storm Surges ................................................................................................................ 534.6 Seiches ........................................................................................................................ 534.7 Precipitation, evaporation, and runoff ............................................................................ 544.8 Effect of Coriolis force on currents ............................................................................... 554.9 Estuarine circulation ..................................................................................................... 56

4.10 Melting and freezing ..................................................................................................... 584.11 Tsunamis ...................................................................................................................... 59

Chapter 5. Datums and Vertical Control .................................................................................... 615.1 Vertical datums ............................................................................................................ 615.2 Equi-geopotential or level surfaces ................................................................................ 615.3 Geopotential, dynamic, and orthometric elevations ......................................................... 615.4 Geodetic datum ............................................................................................................ 625.5 International Great Lakes Datum (1955) ...................................................................... 635.6 Hydrographic charting datums ...................................................................................... 635.7 Special tidal surfaces .................................................................................................... 655.8 Land levelling and water transfers ................................................................................ 665.9 Purpose and importance of benchmarks ...................................................................... 67

Chapter 6. Establishment of Temporary Water Level Gauge .................................................. 706.1 Introduction .................................................................................................................. 706.2 Stilling wells ................................................................................................................. 706.3 Gauge shelters ............................................................................................................. 726.4 Float gauges ................................................................................................................. 736.5 Pressure gauges - diaphragm type ................................................................................ 746.6 Pressure gauges - bubbler type ..................................................................................... 756.7 Pressure gauges - deep sea .......................................................................................... 756.8 Staff gauges ................................................................................................................. 786.9 Sight gauges (electrical type gauges) ............................................................................ 79

6.10 Data recorders ............................................................................................................. 806.11 Selection of gauge site .................................................................................................. 806.12 Benchmarks - general .................................................................................................. 816.13 Benchmarks - standard type ......................................................................................... 816.14 Benchmarks - special types .......................................................................................... 826.15 Benchmarks - description ............................................................................................. 836.16 Levelling - general ........................................................................................................ 846.17 Levelling - method and terminology .............................................................................. 856.18 Levelling - equipment ................................................................................................... 856.19 Levelling - instruments ................................................................................................. 866.20 Levelling - instrument adjustments ................................................................................ 87

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6.21 Levelling - observation and recording procedures .......................................................... 916.22 Levelling - accuracy ..................................................................................................... 916.23 Setting gauge zeros ..........................................................................................................

Chapter 7. Gauge Operation and Sounding Reduction ............................................................. 937.1 Introduction ........................................ ....................................................................... 937.2 Sounding datum trom existing BMs .................... ......................................................... 937.3 Sounding datum by water transfer—tidal waters ....... .................................................. 937.4 Sounding datum by water transter—lakes .............. ..................................................... 947.5 Sounding datulll by water transfer—rivers ........... ........................................................ 957.6 Daily gau•e inspection .............................. .................................................................. 967.7 Documentation of gauge records ...................... .......................................................... 997.8 Datum notes on field sheets ......................... ............................................................. 1007.9 Submission of records and documents ................. ...................................................... 100

7.10 Sounding reduction—general .......................... .......................................................... 1017.11 Sounding reduction—cotidal charts ................... ........................................................ 1017.12 Sounding reduction—non-tidal waters ................. ...................................................... 1037.13 Sounding reduction—offshore gauging ................. ..................................................... 103

Chapter 8. Current Measurement .............................. .......................................................... 1058.1 Introduction ........................................ ..................................................................... 1058.2 Preparatory investigation ........................... ............................................................... 1058.3 Location and depth of current measurement ........... ................................................... 1058.4 Time and duration of measurements ................... ...................................................... 1068.5 Observation methods—general ......................... ........................................................ 1068.6 Selt:contained moored current meters ................ ....................................................... 1078.7 Over-the-side current meters ........................ ............................................................ 1088.8 Suspended current cross ............................. ............................................................. 1098.9 Drift poles—general ................................. ............................................................... 110

8.10 Drift poles—tethered ................................ ................................................................ 1118.11 Drit‘t poles—free-floating .......................... ............................................................... 1118.12 Current drogues ..................................... ................................................................... 1118.13 Continuity method ................................... ................................................................. 1128.14 Hydraulic method .................................... ................................................................ 1138.15 Long wave method .................................... .............................................................. 1148.16 Electromagnetic method .............................. ............................................................. 1148.17 Geostrophic method .................................. ............................................................... 1158.18 Current surveys—general remarks ..................... ...................................................... 116

Bibliography ................................................ .................................................................................. 117

Appendix A. Major Tidal Harmonic Constituents ............... ........................................................ 119Appendix B. Form TWL-502, Temporary Gauge Data .............. ................................................. 120

Index ....................................................... ....................................................................................... 124

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List of Plates

FRONTISPIECE. Wake behind Turret Island at maximum ebb tidal stream in Nakwakto Rapids at theentrance to Seymour and Belize inlets, British Columbia ........... ............................... ii

PLATE 1. Hopewell Rocks (“the flowerpots”) at Cape Hopewell, New Brunswick ...... ......... 18PLATE 2. Fishing weir at low water near Saint John, New Brunswick ............... ................... 18PLATE 3. Jetty and “mattress” at low water, Parrsboro, Nova Scotia ............. ...................... 18PLATE 4. MV Theta resting on wooden “mattress” beside jetty at low water, Parrsboro,

Nova Scotia ........................................................... ............................................. 19PLATE 5. Jetty at Parrsboro, Nova Scotia, at extreme low water and high water .. ................ 19PLATE 6. Reversing Falls at Saint John, New Brunswick, at the mouth of the St. John River .. 20PLATE 7. Tidal bores on the Petitcodiac River at Moncton, New Brunswick and the

Salmon River near Truro, Nova Scotia .................................. ............................... 20PLATE 8. Temporary water level gauge structures ................................ ............................... 72PLATE 9. Diaphragm-type pressure gauge and strip-chart recorder; drum type recorder;

strip-chart recorder and sight gauge .................................. ................................... 76PLATE 10. Multiple staff gauges at Frobisher Bay, Northwest Territories ......... ....................... 79PLATE 11. Benchmark photographs ................................................. ..................................... 84

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Fluctuations in the water level along theshores of any body of water are of obvious interestand concern to those who inhabit those shores -interest in the cause and the nature of thefluctuations, and concern over the possibility offlooding, dried out jetties, exposed water intakes,increased erosion, reduced irrigation, etc. Peoplewho work on or in the water or travel upon it arealso concerned with fluctuations in water level,although they may think of them more asfluctuations in depth, being involved in such tasksas navigating a vessel in shallow water, drillingfrom an oil rig, or setting a lobster pot. Of equalimportance to some is the horizontal flow, orcurrent, the fluctuations in which are frequentlyrelated to those in the water level.

Along ocean coasts and in bays andestuaries connected to the ocean the tide is usuallythe major cause of fluctuations in water levels andcurrents, but non-tidal phenomena such as windstress, storm surges, and atmospheric pressuremay play important roles as well. In lakes, even inlarge lakes such as the Great Lakes, the tide hasno significant effect on the water levels. There is,however, one place in the Great Lakes where thecurrent is significantly influenced by the tide. InLittle Current Channel, the narrow and shallowconnection between Georgian Bay and the NorthChannel of Lake Huron, a reversing tidal stream ofmore than one knot in amplitude may be observedin the absence of wind and other non-tidaldisturbances. This is an exceptional situation, andusually in lakes and rivers the fluctuations incurrents, as well as in water levels, result fromvariations in precipitation, evaporation, runoff,atmospheric pressure and wind, and from basinoscillations called seiches. In some inland systemsthe water level and flow may, of course, becontrolled by dams, or temporarily backed up byice or log jams.

The hydrographer’s interest in water levelfluctuations relates to his responsibility to provideaccurate depth information on charts. Since theactual depths at a particular time depend on thewater level at that time as well as on thebathymetry, the depths shown on a chart must bereferred to a standard reference surface, or datum.This chart datum is chosen as a surface belowwhich the water level will seldom fall, so that only

rarely could the actual depth be less than thecharted depth. Choice of a suitable chart datumrequires a knowledge of the nature of the waterlevel fluctuations over the region being charted: thisknowledge can usually be obtained only fromlengthy observations of the water level. During thehydrographic sounding survey the height of thewater surface above the chart datum must bedetermined, to permit each sounding to be reducedby that amount to provide the chart depth belowchart datum. After the chart is produced and inservice. mariners, surveyors, or engineers may stillrequire to know the water level above chart datumso they may add it to the charted depths to obtainthe actual depths.

Water level information for soundingreduction is usually provided from temporarygauges established in the immediate vicinity of thesurvey by the hydrographic field party. Permanentwater level gauges are maintained at major portsand other selected sites around the coast and onmajor inland waterways to provide continuouswater level information for these localities. Thedata accumulated over the years from thepermanent and temporary gauges provide the basisfor interpretation of the water level characteristics,selection of chart datums, and prediction orestimation of future water levels. These data areavailable to the public in a variety of formatsthrough a central data bank at the MarineEnvironmental Data Service (MEDS), throughvarious bulletins and publications (see Bibliographyunder Canadian Hydrographic Service), and insome cases as real-time data through directtelephone communication with teleannouncinggauges. For tidal waters, predicted times andheights of high and low water are providedannually in the Canadian Tide and Current Tables.A tidal block on the navigation chart tabulates theextreme and average heights of high and lowwater for various locations on the chart. Onnavigation charts of non-tidal waters a hydrographis usually provided to show the average and theextreme monthly mean water levels that have beenobserved. No predictions are made for waterlevels in non-tidal waters, but a Monthly WaterLevel Bulletin for the Great Lakes and MontrealHarbour gives, along with statistics of pastobservations, a forecast of the monthly

PREFACE

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mean water levels for the next 6 months within anenvelope of error.

The hydrographer is also required to provideinformation on currents where they may be ofconcern to navigation, particularly in restricted anddifficult passages. The gathering and publicationof information are more complicated and difficultfor currents than for water levels. This is becausecurrents may vary in direction as well as intensity,they may change in a very short distance under theinfluence of the bathymetry, they may differconsiderably between the surface and the bottom,and their observation usually involves offshoremoorings. Current information is provided on thenavigation charts where this is feasible. For manyimportant tidal passages the times and speeds atmaximum flood and ebb and the times of slackwater are predicted and published annually in theCanadian Tide and Current Tables. Currentinformation is also published in Sailing Directionsand Small Craft Guides in more descriptive form,particularly when the information is difficult toquantify or is based only on superficialobservations or reports. In regions where thecurrents display great temporal and spatial

variation, and where these variations areunderstood, publication of a Current Atlas may berequired to represent the information adequately.

The material contained in this Tidal Manualis designed to provide the theoretical background(Part I) and the technical instruction (Part II)necessary for the effective performance of thetasks involved in gathering and using tide, current,and water level information on hydrographic fieldsurveys. In treating instrumentation andtechniques the emphasis is mainly on principles,with reference to manuals or other documentationfor the specifics of particular instruments orroutines. It is hoped that this will retard the adventof the Manual’s obsolescence in the face ofadvancing technology. A minimum of mathematicshas been employed, and an attempt has been madeto discuss the results in simple terms following anymathematical development. A reader who is notcomfortable with mathematics is advised to readbetween the formulae and to try to absorb thebasic ideas, but not to skip the sections completely.

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PART ITheory and Concepts

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CHAPTER 1

TIDES AS WAVES

1.1 What is the Tide?

Every reader of this book will have somenotion of what is meant by the word "tide" asapplied to the ocean. Some will think of the dailyor twice-daily rise and fall of the water on the faceof a cliff or around the pilings of a pier, others ofthe advance and retreat of the water over ashallow foreshore, and still others may think of thevariable horizontal flow of the water that carriestheir ship off course, sometimes in one direction,sometimes in another. The tide is all of thesethings, but more generally we will define the oceantide as “the response of the ocean to the periodicfluctuations in the tide-raising forces of the moonand the sun”. This response is in the form of longwaves that are generated throughout the ocean.They propagate from place to place, are reflected,refracted, and dissipated just as are other longwaves. Thus it is that the tide observed at aparticular place is not locally generated, but is thesum of tide waves arriving from all over the ocean,each modified by its experiences along the way.To better understand the tide it will therefore bedesirable to consider the characteristics of longwaves as well as those of the tide-raising forcesthat produce them.

1.2. Waves

Wave motion in or along a medium ischaracterized by:

(a) periodic vibration but no net transport of theparticles in the medium,

(b) propagation of energy along or through themedium. and

(c) a restoring force that opposes thedisplacement of the particles of themedium

When a sound wave travels through air, theparticles experience a to-and-fro vibration. and therestoring force is provided by the pressuregradient. When a sound wave travels through a

solid, the particles also experience a to-and-frovibration, and the restoring force is provided by theelasticity of the material. When a wave travelsalong a taut string, the particles experience atransverse vibration, and the restoring force isprovided by the tension in the string. When awave travels along the surface of a body of water,the particles experience both a to-and-fro and anup-and-down vibration, and the restoring force isprovided by a combination of gravity (actingthrough the hydrostatic pressure) and surfacetension. Surface tension is the dominant restoringforce only for ripples with 2 cm or less betweencrests, and these are called "capillary waves." Forall longer water waves the dominant restoringtorce is gravity, and for this reason they are called"gravity waves." Surface chop, sea, swell,tsunamis, and tides are all gravity waves.

The terminology used to describe waves isillustrated in Fig. I a and Fig. I b. In Fig. I a thewave form is viewed perpendicular to its directionof travel at an instant in time.

Fig. I b depicts the variation in water level at afixed location over an interval of time as the wave

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passes. The wavelength ( λ) is the distancebetween successive crests or successive troughs.The range (R) is the vertical distance of the crestabove the trough, or of the high water (HW) abovethe low water (LW). The terms crest and troughare more commonly used in connection with wavesthat are short enough to reveal their wave form tothe eye. The terms HW and LW are morecommonly used only in connection with the tidewaves, which are much too long to reveal theirtorm to the eye. The amplitude (H) is one half ofthe range. The period (T) is the interval betweenthe passage of two successive crests, or betweenthe occurrence of two HWs: successive troughs orany other identifiable parts of the wave form couldequally well be used to define the period. Thefrequency (f) is the number of periods (or cycles)occurring per unit time; therefore f = 1/T. Thewave speed (c ) is the horizontal rate of advanceof all parts of the wave form (crests, troughs, etc.).Since a travelling wave advances one wavelengthin one period, c = λ/T.

A sinusoidal wave form, such as that in Fig.1, can be generated as the product of the amplitudetimes the sine or cosine of a continuouslyincreasing angle, called the phase. The angle bywhich the phase of a wave lags behind the phaseof a reference wave is calledthe phaselag. In tidal work, the cosine form ismost commonly used, so that in Fig. I b the heightabove mean water level (MWL) would beexpressed as:

h(t) = H cos(2πft - [π/2])

With respect to a wave with phase 2 πft,h(t) would be said to have a phaselag of π/2. Therate at which the phase increases is called theangular speed (ω), and ω = 2πf radians per unittime. In tidal literature the angular speed is usuallyquoted in degrees per hour and given the symbol“n”. The wave number (k) is the rate at which thephase changes with distance, and k = 2 π/λ radiansper unit distance .

1.3. Surface Gravity Waves

It would admittedly be a rare occasion onwhich the actual sea surface could be adequatelyrepresented by a simple sinusoidal wave as inFig. 1. However, quite complicated sea states maybe represented as a composite of many suchcomponent waves, each with its own amplitude,wavelength, and direction of propagation. A longswell running on an otherwise calm sea closelyresembles a single such component wave.Because the tide can usually be adequatelyrepresented as the superposition of a manageablenumber of these component waves, we will restrictour investigation of surface gravity waves to thoseof sinusoidal form.

A wave that is moving across the surface asa train of parallel crests and troughs is called aprogressive wave . If it is moving in the positive x-direction, the height at distance x and at time t isgiven by:

(1.3.1) h1(x,t) = H cos 2π([t/T]-[x/λ])

= H cos (ωt- kx)

This expression may be verified by considering thatfor an observer travelling with the wave speed c =λ/T the phase would remain constant, because theincrease due to the increase in t is offset by thedecrease due to the increase in x. If the wavetrain is moving in the negative x-direction, theheight is:

(1.3.2) h2(x,t) = H cos 2π([t/T]+[x/λ])

= H cos (ωt+ kx)

The superposition of two progressive wavesthat have the same amplitude and frequency butare travelling in opposite directions produces whatis called a standing wave. Adding equations(1.3.1) and ( 1.3.2) and invoking sometrigonometric relations give the followingexpression for the standing wave form:

(1.3.3) hs(x,t) = h1 + h2

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= (2H cos 2π [X/λ])(cos 2π [t/T])Figure 2 illustrates the formation of a

standing wave from two oppositely directedprogressive waves. From equation 1.3.3 andFig. 2 it is clear that the period and wavelength ofthe standing wave are the same as those of thecomponent progressive waves, that the amplitudeof the rise and fall of the surface varies from zeroto 2H according to the value of cos kx, and that thephase of the rise and fall is everywhere the sameor opposite, according to the sign of cos kx. Theplaces in the standing wave form at which theamplitude is zero are called nodes, and those atwhich it is maximum are called anti-nodes. Thespace between nodes is called a loop. Within each

loop the phase is the same, but is different by 180°from that in adjacent loops. A standing wave isfrequently formed by the reflection of aprogressive wave back upon itself, which is whythe tide usually displays the character of a standingwave in coastal bays and inlets. In practice wewill never encounter a pure progressive or standingwave; every wave will have some of thecharacteristics of each. The tide in the Strait ofBelle Isle is an example of a regime that is neitherpurely standing nor progressive. The tidepropagating out from the Gulf of St. Lawrencecombines with the tide propagating in from theAtlantic, but, since the two do not have the sameamplitude, only a partially standing wave is formed.

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At a true node there should be zero amplitude anda reversal of phase on either side: in the Strait ofBelle Isle there is a degenerate node, exhibitingreduced amplitude and rapid spatial change inphase. When a standing wave is formed byreflection, the standing character is most nearlyperfect near the reflecting barrier, because awayfrom the barrier the incident wave has a largeramplitude than the reflected wave as a result ofattenuation along their paths.

1.4 Long and Short Waves of SmallAmplitude

In our theoretical consideration of waves wewill implicitly assume that the amplitude is smallwith respect both to the wavelength and to thedepth. The amplitude of a tide wave is alwayssmall with respect to its wavelength, but notalways with respect to the depth, so we mustexpect some distortion of our results in shallowwater. It can be shown that the wave speed of asinusoidal wave in water of total depth D is:

(1.4.1) c = ([g/k] tan h kD) 1/2

where g is the acceleration due to gravity, and thatthe horizontal particle motion (wave current) and

the pressure associated with the passage of thewave both decrease exponentially with the depthby the factor exp (-kz), z being the depth from thesurface. The magnitude of kD ( = 2 πDλ) thusprovides a criterion by which to categorize waves.Short (or deep-water) waves are those for whichthe wavelength is much less than the depth, andlong (shallow-water) waves are those for whichthe wavelength is much greater than the depth. Itmust be remembered that the terms are relative,not absolute, and that a short wave may become along wave on entering shallower water. For shortwaves kD is very large, and tanh kD is close tounity, so that the wave speed becomes:

cs = (g/k)1/2.For long waves the value of kD is very small, andtan(h) kD is approximately equal to kD, so that thewave speed becomes

cL = (gD)1/2.Because for short waves the speed depends on thewavelength, they experience dispersion, the longercomponent waves travelling faster and becomingdispersed from the shorter component waves.This is why the long swells (forerunners) from adistant storm arrive first. Long waves do notexperience dispersion, their wave speed dependingonly on the water depth. They do, however,experience refraction if one part of the wave frontis travelling in shallower water than the others.The part of the wave front in the shallower waterslows down, allowing the rest of the front to pivotaround, changing the direction of propagation ofthe wave. As illustrated in Fig. 3, refraction isresponsible for orienting waves parallel to beachesbefore they break on the shore. Short waves donot experience refraction; but, of course, they maybecome long waves on entering shallow water andthen be refracted as in Fig. 3.

The particle motion and the pressurefluctuation associated with the passage of a shortwave decrease rapidly with depth, being only about4% of their surface values at a depth of half awave length. The particle motion and pressurefluctuation associated with the passage of a longwave are, however, virtually uniform over thedepth (except for the frictional effect near thebottom). These are important facts to considerwhen planning subsurface pressure or current

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measurements.Since surface tides are long waves even in thedeepest parts of the ocean, their signal may bedetected by sensors at any depth, whereas thesignal from short waves is effectively filtered outbelow a depth of a half a wavelength. Theproperties of long and short waves are summarizedin Table 1. There are, of course, waves that areintermediate between the long and the shortwaves, and their wave speed is given by equation1.4.1. However, since tides are always longwaves, we shall confine our further considerationsto long waves.

1.5. Particle Motions in Long Waves

In this section we will develop expressionsfor both the wave speed and the particle speed in along surface wave. This is being done partly todemonstrate the physical principles, and partly toemphasize the relation between these two speedsand between the particle motion and the waveform. We assume that the particle speed is

uniform over depth and is a to-and-fro motion withthe same period, but not necessarily the samephase, as the surface rise and fall. Consider aprogressive wave moving from left to right in Fig.4, with surface amplitude H. Let the particlespeed have amplitude U and phase lag withrespect to the surface elevation. Therefore

(1.5.1) h(x,t) = H cos (ωt - kx)u(x,t) = U cos (ωt - kx - q)

In Fig. 4, MNOP is one side of a rectangularprism of unit thickness perpendicular to the page,with length dx, and height D + h. Its volumeincreases at the rate (δh/δt)dx. By the principle ofcontinuity (conservation of matter) this must beequal to the rate at which water is entering minusthe rate at which it is leaving through the sides ofthe prism. Neglecting the small height, h, withrespect to the large depth, D, the rate at whichwater is accumulating inside the prism is

D[u(x,t)] - D[u(x,t) + (δu/δx)dx] = - D (δu/δx)dx

Equating these two rates gives

(1.5.2.) δh/δt = - D(δu/δx)

Differentiating the expressions in 1.5.1 andsubstituting in 1.5.2 yields

-H ω sin (ωt - kx) = - DkU sin (ωt—kx - θ)whence

(I.5.3) θ = 0, and U = (H/D) (ω/k) = (H/D) c

Consider now a particle of water on thesurface and assume that its acceleration equals thelocal acceleration, δu/δt (a reasonable assumptionfor waves of small amplitude). The force per unitmass acting on the particle is the component ofgravity parallel to the surface,-g(δh/δx). By Newton’s law of motion these twoquantities must be equal, whence, upondifferentiating the expressions in (1.5.1) and usingthe relations given in 1.5.3,

δu/δt = - g (δh/δx)or- ω(ω/δt)(H/D) sin (ωt - kx)

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Table 1. Properties of long and short waves.

Short waves (deep water) Long waves (shallow water)

Definition l < D l > D

Wave speed (g/k)1/2 (gD)1/2

Particle motion Decreases with depth. Uniform with depth.

Wave pressure Decreases with depth. Uniform with depth.

Dispersion Yes No

Refraction No Yes

= - gkH sin (ωt - kx)so

(1.5.4) (ω/k)2 = gD = c2

From 1.5.3, since θ = 0, we have the veryimportant result that in a progressive wave theparticle motion is in phase with the surface waveform; so that the particle speed is greatest in thedirection of the wave travel at the crest, greatest inthe direction opposite to the wave travel at thetrough, and zero midway between crest andtrough. The particle speed beneath any point in thewave is in fact the wave speed multiplied by theratio of the wave height to the depth. The wavespeed is given by 1.5.4 asc = (gD)1/2

as previously deduced from the more accurateexpression in 1.4.1.

We will now examine the relation betweenparticle motion and wave form in a standing wave.Just as we obtained the expression 1.3.3 for theform of a standing wave by adding the forms oftwo oppositely directed progressive waves, wemay obtain the expression for the particle motionby adding the particle motions of two oppositelydirected progressive waves. The particle speed ina wave travelling to the left is in phase with thewave form, and so must be given a negative sign.The addition gives the standing wave particlemotion as

( 1.5.5) us,(x,t) = Ucos(ωt - kx) - Ucos(ωt + kx).

or us,(x,t) = 2U sin ωt sin kx

From equation 1.3.3 we had the standing waveheight as

(1.5.6) hs,(x,t) = 2H cos ωt cos kx

Recalling that the sine of an angle is 90° outof phase with the cosine, we see from acomparison of 1.5.5 and 1.5.6 that in a standingwave the particle speed has maximum amplitudewhere the surface rise and fall has zero amplitude(i.e. at the nodes) and has zero amplitude at theanti-nodes. We also see that the particle speedachieves its local maximum everywhere when thewave form is flat, and is everywhere zero whenthe surface has its maximum distortion (i.e. at HWand LW). Figure 2 illustrates the relations betweenparticle motion and wave form in progressive andstanding waves.

To demonstrate that tide waves are indeedlong waves (λ > D) and to emphasize the relationbetween particle speed (tidal stream) and wavespeed, Table 2 lists the wavelength, wave speedand particle speed of a tide wave of one-metreamplitude and 12-h period travelling in variousdepths of water. Comparison of the values in thelast two columns shows that the wave speed iseverywhere much greater than the particle speed,but that while the wave speed decreases, theparticle speed increases with decreasing depth ofwater. This is one reason that tidal streams aremuch more evident in coastal waters than in the

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open ocean.1.6. Basin Oscillations

Almost every physical system has a naturalfrequency at which it will oscillate when disturbedfrom its rest position or shape, until friction brings itonce more to rest. The most obvious example isthe pendulum (or the hair-spring or the quartzcrystal) in a clock, whose natural period ofoscillation is the time unit that is summed by theclock to record the passage of time. If a system isleft undisturbed to oscillate at its natural frequency,it is said to be in tree oscillation; if it is forced tooscillate at the frequency of an imposed force, it issaid to be in forced oscillation. When thefrequency of the driving force is equal to thenatural frequency of the system, a large amplituderesponse may be obtained with the input of verylittle energy. This phenomenon is called resonance,and is explained by the fact that the driving forceand the restoring force within the system actmostly in unison at forcing frequencies close to thenatural frequency of the system, and mostly inopposition to each other at forcing frequencies farfrom the natural frequency. A simple example of aresonant system is a child seated on a swing that isbeing pushed by a friend; since the swing is pushedonly at the end of each cycle, the frequency of thedriving force is automatically matched to thenatural frequency of the swing. The clockpendulum is another resonant system, operating onthe same principle as the swing. A person singingin the shower may notice that a particular notecauses a delightful reverberation; this is becausethe column of air in the shower stall is resonant atthe frequency of that note.

The free oscillation of the water in a closedbasin (bathtub, lake, etc.) takes the form of astanding wave with an anti-node at each end of thebasin and one or more nodes between (Fig. 5a). Ifthere is only one node, the length of the basin ishalf a wavelength, and the natural period ofoscillation is given approximately as

(I.6.1) Tn = (2L)/c = (2L)/(gD)1/2

where L is the length of the basin and D is anaverage depth. Although it is less common, aclosed basin could oscillate across its width as wellas along its length. The free oscillation of thewater in a basin open at one end (harbour, bay,inlet, etc .) takes the form of a standing wave witha node at the open end and an anti-node at theclosed end (Fig. 5b). If there are no other nodesbetween the ends, the length of the open basin is aquarter wavelength, and the natural period of

Table 2. Characteristics of a tide wave of 12-hour period and 1 metre amplitude in various depths.

Depth (m) Wavelength (km) Wave speed (m/s) Particle speed amplitude (m/s)

5,000 9,600 220 0.04500 3,000 70 0.14

50 960 22 0.44

* 5 * 300 * 7 * 1.40* Since this depth is not very large w.r.t. the wave amplitude, the wave would be distorted, and these values

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oscillation is given approximately as

(1.6.2) Tn = (4L)/c = (4L)/(gD)1/2

Free oscillations of water in basins (open or closed)are called seiches. Much of the early study ofseiches was done on lakes in Switzerland, and theequations 1.6.1 and 1.6.2 are called Merian’sformulae after one of the Swiss workers in this field.

Figure 6 illustrates how the tide at theentrance to an inlet off a large body of water drivesa forced oscillation in the inlet in the form of astanding wave with an anti-node at the head of theinlet. If, as is usually the case, the inlet is shorterthan a quarter of the tidal wavelength, the standingwave will have a virtual node outside the entrance.It is apparent that the amplitude of the tidal oscillationat the head of the inlet is greater than at theentrance, and that the amplification would begreatest if the node fell right at the entrance; thelatter situation corresponds to the condition forresonance. If L is the length of the inlet, D its meandepth, c = (gD)1/2 the wave speed in the inlet, and Tthe tidal period, then the tidal wavelength is cT andthe portion of a wavelength within the inlet is L/cT.This represents a phase angle along the x-axis fromthe head of the inlet of kx = 2 π L/cT. Thus, if H2 isthe amplitude at the head and H 1 that at the entranceof the inlet, by 1.5.6,

(1.6.3) H1 = H2 cos (2π L)/cT

or H2/H1 = sec (2π L)/cT

The amplification factor, H 2/H1, in 1.6.3 is seen tobe infinite for 4L = cT, which is the resonancecondition (tidal period, T, equal natural period, 4L/c). However, friction, which we have neglected,becomes very important near resonance, andformula 1.6.3 should not be used for systems nearresonance. The Saguenay fjord provides anexample of tidal amplification in a system that isnot near resonance. The length from the entranceoff the St. Lawrence Estuary at Tadoussac to thehead of the fjord at Port Alfred is 95 km (L), themean value of the long-wave speed in the fjord is40 m/s (c), and the tidal period is 12.4 h (T). Fromthis, 1.6.3 gives the amplification factor as

H2/H1, = sec(0.33 rad.) = sec 19° = 1.06.

The actual amplification of the tide range at PortAlfred over that at Tadoussac is 1.16. The extraamplification over that predicted is probably causedby shoaling (see section I.12) of the tide wave inthe shallow water near the head. An example of asystem that is nearly in resonance with thesemidiurnal (T = I/2-d) tide is the systemcomprising the Gulf of Maine and the Bay ofFundy.

The ocean basins themselves have naturalperiods of oscillation, but their modes of oscillationare much too complicated to be revealed by thesimple considerations above. However, calculationof the natural period of east-west oscillation of theAtlantic and Pacific oceans from Merian’sformula, 1.6.1, provides the interesting informationthat the Atlantic Ocean is more closely tuned tothe semidiurnal and the Pacific Ocean moreclosely tuned to the diurnal tidal frequencies.Taking eastwest widths of 4,500 and 8,000 km,respectively, for the Atlantic and Pacific and amean depth of 4,000 m for both, 1.6.1 gives 12.6and 22.3 h, respectively, as the natural periods ofthe Atlantic and Pacific for east-west oscillation.Pacific tides are indeed observed to have muchmore diurnal character in general than the Atlantictides.

1.7. Internal Waves

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(I 7.3) U2 = = ciHi/h2

where Hi is the amplitude of the internal wave atthe interface. If we had wished, we could havetreated surface waves as special cases of internalwaves at the air-water interface, taking p 1 and p2as the densities of air and water, h 1 as thethickness of the atmosphere, and h 2 as the depthof the water. Putting p1 < p2 and h2 < h1 tocomply with this, reduces 1.7.1 approximately to

Ci2 = g(p2) / [p2/h2] = gh2

in agreement with our previous expression 1.5.4.If, as is always the case for stratified water, p 2and p1 are nearly equal, 1.7.1 simplifies to

(1.7.4) Ci2 = g (∆p/p) [(h1h2)/(h1 + h2)]

and if it is further assumed that the upper layer ismuch thinner than the lower layer, as is frequentlythe case, this simplifies further to

(1 7.5) Ci2 = gh1(∆p/p)

Admittedly the two-layer system of Fig. 7,with its discontinuity in density and particle velocityat the interface, could never occur in a naturalbody of water. However, the equations 1.7.1, 2and 3 reveal the following important characteristicsof internal waves:

I ) Their wave speeds, and hence theirwavelengths, are much less than those ofsurface waves of the same frequency.

2) The particle velocities of internal waves, unlikethose of surface waves, may reverse phaseand have different amplitudes at differentdepths.

3) They can exist only in stratified water.4) They may have very large amplitudes (tens of

metres) because the restoring force is sosmall.

5) Although their vertical amplitude is zero at thefree surface, their particle velocities areusually greatest there, because of a thinsurface layer of less-dense water.

These are waves that occur below thesurface at the interfaces between layers of waterof different densities (i.e. in “stratified” water).They may exist independently from any surfacewave, but are sometimes induced as a secondaryeffect of surface waves. Internal tides arefrequently formed by the partial reflection of asurface tide wave at a sudden rise in bottomtopography. The simplest case to consider is thatof a wave at the interface in a two-layer system asshown in Fig. 7. The subscripts, 1 and 2, refer tothe upper and lower layers, respectively, and p isthe density, h the layer thickness, and u the particlevelocity (with amplitude U). The wave form at theinterface is that of a long progressive wavetraveling from left to right with wave speed c i.The restoring force in this internal wave is not thefull force of gravity, g, per unit mass, but is thebuoyancy force g∆p/p, where ∆p is the densitydifference p2 - p1 and p is the mean density.

By reasoning that is just a little more difficultthan that in section 1.5 for a surface wave. it canbe shown that

(l 7 l) Ci2 = [g(p2 - p1)] / [p1/h1 + p2/h2]

(1.7.2) - U1 = ciHi/h1

and

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Internal tides are internal waves of tidalfrequency, and these have been observed in the St.Lawrence Estuary. Semidiurnal internal tideswere observed in the estuary below Tadoussacwith wavelengths of about 60 km. Their presencehelped to explain the tidal streams in the area,which could not be satisfactorily accounted for bythe surface tide alone. The water column in theestuary can be very crudely represented as twolayers with h1 = 50 m, h2 = 250 m, and ∆p/p =0.003. Substitution of these values, along with g =9.8 m/s2 into 1.7.4 gives a wave speed of 1.1 m/s,or 4.0 km/h, corresponding to a wavelength of 49km for the semidiurnal tidal period of 12.4 h.

1.8. Coriolis Acceleration

Newton’s classical laws of motion applyonly when all measurements are made withrespect to an inertial coordinate system, that is, onethat is neither accelerating nor rotating. Thus,when measurements are made relative to acoordinate system fixed in the earth, allowancemust be made for the rotation of the earth about itsaxis.

This is done by providing two “fictitiousforces,” the centrifilgal force and the Coriolisforce, in addition to the apparent forces that causeacceleration of a body relative to the surface of theearth. A mass resting on the earth’s surface isactually revolving about the earth’s axis on alatitude circle once each day, and so is acceleratingtoward the centre of that circle. The inertia of themass resists this centripetal acceleration, and, to anearth-bound observer, the mass appears to bepulled away from the axis by what he calls thecentrifugal force. Since it varies only with latitudeand not with time, the centrifugal force (CF) isconveniently combined with the earth’sgravitational attraction (C) in what we know as“gravity” (g). Figure 8 depicts the vector additionof the two forces to give gravity, with the relativesize of the centrifugal force vector greatlyexaggerated for clarity. The centrifugal force isobviously greatest at the equator and zero at thepoles, contributing to the fact that gravity is less atthe equator than at the poles.

A body in motion relative to the surface ofthe earth experiences an acceleration to the rightof its horizontal direction of travel in the NorthernHemisphere (to the left in the SouthernHemisphere), an acceleration that is proportional toits velocity and to the sine of the latitude. Thisacceleration is also a result of the earth’s rotation,and is allowed for in the Coriolis force. Figure 9attempts to illustrate the origin of this force. Thereis a Coriolis force on objects moving vertically anda vertical component of Coriolis force on objectsmoving horizontally, but we will consider only thehorizontal component of the Coriolis force onobjects moving horizontally. Imagine the earth tobe covered with a frictionless film, the surface ofwhich conforms to that of a level surface, i.e. iseverywhere normal to the direction of gravity. Nand S are the north and south poles, and Ω is theearth’s angular velocity. As a body moves tohigher latitude, the easterly velocity of the earth’ssurface decreases, and so the easterly velocity ofthe body relative to the earth increases. This isseen as an acceleration to the right in the NorthernHemisphere and to the left in the SouthernHemisphere.

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Special cases of this are shown for a bodyprojected south from N and for one projected northfrom S. Viewed from an inertial coordinate systemboth of these bodies would travel along the greatcircle NS, with no east-west velocity. Relative tothe earth, however, they appear to follow the pathsNM and ST, acquiring westerly velocitycomponents as they move to lower latitudes, andso accelerating to the right in the NorthernHemisphere, and to the left in the SouthernHemisphere.

The Coriolis force due to the east-westvelocity component arises from the fact that aneasterly moving body experiences centrifugal forcein excess of that included in gravity, and thisexcess centrifugal force has a component thataccelerates the body along the level surfacetoward the equator. Similarly, a westerly movingbody experiences a centrifugal force less thanthat in gravity and is accelerated along the levelsurface toward the pole. These accelerations areagain seen to be to the right of the velocity in theNorthern Hemisphere and to the left in theSouthern Hemisphere. This effect is illustrated atpoints A and D for easterly velocity and at points Band C for westerly velocity. The velocity vector ateach point is directed into the page. The vectors drepresent a deficit and vectors e an excess ofcentrifugal force over that allowed for in gravity

for a body at rest on the surface. Their horizontalcomponents are the horizontal Coriolis forces.Points E and É show that for an east-west velocityat the equator the Coriolis force has only a verticalcomponent. There is no horizontal Coriolis forcefor a north-south velocity at the equator becausethe rate of change of the earth’s surface velocitywith latitude is zero there. This can all be summedup in the statement that the horizontal componentof the Coriolis force acting on a body moving withvelocity v over the earth’s surface acts to the rightof the velocity in the Northern Hemisphere and tothe left in the Southern Hemisphere, and hasmagnitude 2Ωv sinϕ, where ϕ is the latitude. 2Ωsinϕ is called the Coriolis parameter, usuallydenoted as f.

The Coriolis force is rarely noticeable inlaboratory-scale measurements, but is verysignificant in large-scale geophysical motions suchas winds, ocean currents, and tides. It is this forcethat imparts the cyclonic and anti-cycloniccirculation to the atmosphere around low and highpressure regions and turns the ocean currentsystems into large circular gyres. It also acts onthe tidal streams, changing the direction ofpropagation and the shape of the tide waves.When the tide propagates as a progressive wavealong a channel in the Northern Hemisphere (NH),the range of the tide is observed to be greater onthe shore to the right of the direction ofpropagation. This is because the tidal streams atHW are in the direction of propagation, and theCoriolis force acting on them moves water to theright until a slope of the surface is created tobalance it. This raises the HW on the right shoreand lowers it on the left. At LW the tidal streamsare in the opposite direction, and the surface slopecreated to balance the Coriolis force lowers theLW on the right shore and raises it on the left.When the channel width is small compared to thetidal wavelength, only insignificant cross-channeltidal streams are required to create the surfaceslopes referred to above. Most channels are muchnarrower than the half wavelength of the surfacetide required for resonance, but many may have awidth comparable to the half wavelength of aninternal tide. This is the case in the St. LawrenceEstuary, where the Coriolis force acts on an

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internal tide propagating seaward to produce strongcrosschannel tilting of the interface betweendensity layers, with correspondingly strong cross-channel tidal streams oppositely directed in the twolayers.

1.9. Inertial Currents

The Foucault pendulum is one of the fewlaboratory experiments that can demonstrate theeffect of earth rotation (Coriolis force) on a bodyin motion. It consists of a heavy mass suspendedon a long single filament swinging freely through asmall arc. The vertical plane of the oscillation isobserved to rotate through 360° in a period of(24/sinϕ) hours, which period is referred to as thependulum day. It is easiest to visualize thisphenomenon for the special case of a pendlllumsuspended directly over the North or South Poleand swinging back and forth in a plane fixed inspace, while the earth rotates once in 24 h beneathit. lf it were possible to design such a pendulum tohave a period of oscillation equal to one pendulumday, it would be observed to travel around in acircle once each half pendulum day. Again it iseasiest to visualize this at one of the earth’s poles:the pendulum would start tracing a circle at thecentre of its swing and complete the circle when itreturned again to the centre a half period later; bythis time the earth would have rotated 180°, so thecircle traced by the pendulum in the next halfperiod would fall on top of the first circle. Toexplain this circular motion in a coordinate systemfixed to the earth it would be necessary to invoke,in addition to gravity, the centrifugal force due tothe circular motion and the Coriolis force due tothe motion relative to the surface.

The above thoughts are pertinent to theconsideration of what are called inertial currents.Water in the ocean that has been set in motion andis now drifting under its own inertia could beexpected to keep deflecting to the right (NH) orleft (SH) until it is moving in a circle (clockwise inthe NH, counterclockwise in the SH) such that thecentrifugal force away from the centre of thecircle just balances the Coriolis force toward thecentre. This is called the inertial circle. The timetaken to complete the circle is the inertial period,

and will be seen to equal one half pendulum day. lfthe water is moving at speed v in a circle of radiusr, the centrifugal force away from the centre is v 2/r. Let f be the Coriolis parameter at the latitude ϕ(f= 2Ω sinϕ), so that the Coriolis force toward thecentre is fv. The balance of forces is therefore

(1.9.1) fv = v2/r, whence r = v/f

The circumference of the inertial circle isthus 2πr = 2πv/f, and the time taken to travelaround the circumference is the inertial period, T 1,so

(1.9.2) T1 = 2π(r/v) = 2π/f = 2π/(2Ω sinϕ)

Since Ω = 2π/24 hours, T1 = (12/sinϕ) hours

The inertial period is seen to be the halfpendulum day as anticipated. This is a period thatis frequently detected in ocean currentmeasurements. At 45° latitude it is 17 h, at 30° itis 24 hours, and at 75° it is 12.4 hours (the same asthe semidiurnal tidal period). From 1.9.1 the radiusof the inertial circle is seen to be proportional to thecurrent speed for a given latitude. At 45° latitudethe radius for a I km/h current is 2.7 km, and at thepole it is 1.9 km. At the equator the radius isinfinite, meaning there are no inertial circles theresince the Coriolis force is zero. It should be notedthat the motion in an inertial circle is not that of aneddy, and that all parts of the water are moving inthe same direction at the same time in inertialmotion. For those versed in carpentry a helpfulanalogy might be that of the movement of anorbital sanding plate (cf. inertial motion) versus themovement of a rotary sanding disk (cf. eddymotion).

1.10 Amphidromic Systems

The word amphidrome is from the Greek for“a round race course,” and describes a system inwhich wave crests propagate like the spokes of awheel around a central amphidromic point, withwave amplitude increasing outward from zero atthe centre. Figure 10 illustrates the formation ofan amphidromic system in a coastal embayment bythe action of the Coriolis force on what would

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otherwise be a simple standing wave. Let theembayment be greater than a quarter wavelengthlong, so that in the absence of earth rotation therewould be a nodal line across the embayment atBB’, with high water (HW) at A coinciding withlow water (LW) at A’ and vice versa. Let us nowadd earth rotation and follow the oscillation throughone period (T), starting with HW at A at time t = 0.From t = 0 to t = T/2, there is an axial flow throughP from A to A’, and the resulting Coriolis force tothe right sets up a cross flow component from B toB’ sufficient to create a surface slope thatbalances the Coriolis force. Since the axial flow isgreatest at t = T/4, the surface slope across theembayment is also greatest then, giving a HW atB’ and a LW at B at t = T/4. At t = T/2 HW is atA ‘ and LW at A, as in the ordinary standing wave.At t = 3T/4 the axial flow through P from A ‘ to Ais maximum, and HW will be at B with LW at B’,to provide the surface slope necessary to balancethe Coriolis force on the outflowing water. Thus,in the Northern Hemisphere earth rotation canconvert a simple standing wave in a basin into anamphidromic system (or amphidrome), in which thecrest travels counterclockwise around theperimeter of the basin about a pivotal point, P,called the amphidromic point. The vertical

amplitude is zero at P and the particle velocityreaches its maximum there, but now the particlevelocity vector rotates counter-clockwise, tracingout an ellipse. The amplitudes of the wave at Band B’ and of the particle velocity across the basindepend on the geometry and size of the basin andthe length of the period of oscillation relative to thatof the half pendulum day. The origin and nature ofamphidromes in the open ocean are less simplethan those described above, and sometimes thesense of rotation is opposite to that in anembayment. Figure 29 shows the amphidromicsystem of the semidiurnal tide wave in the Gulf ofSt. Lawrence, and Fig. 30 shows an amphidromeof the diurnal tide wave in the Atlantic Ocean offNova Scotia.

1.11 Tides and Tidal Streams

Since the tide propagates as a set of longwaves in the ocean, much of the character of itsvertical and horizontal motion has been revealed inthe preceding consideration of long waves. Theterms defined in section 1.2 to describe thecharacteristics of a wave are also applied to tides,but some special tidal terms are used as well. Thedefinitions given here conform as closely aspossible to common usage in Canadian tidalliterature. In a tide wave the horizontal motion, i.e.the particle velocity, is called the tidal stream. Thevertical tide is said to rise andfall, and the tidalstream is said to flood and ebb. If the tide isprogressive, the flood direction is that of the wavepropagation: if the tide is a standing wave, the flooddirection is inland or toward the coast, i.e.“upstream.” The flow is the net horizontal motionof the water at a given time from whatevercauses. The single word “current” is frequentlyused synonymously with “flow”,” but the termresidual current is used for the portion of the flownot accounted for by the tidal streams. A tidalstream is rectilinear if it flows back and forth in astraight line, and is rotary if its velocity vectortraces out an ellipse. Except in restricted coastalpassages, most tidal streams are rotary, althoughthe shape of the ellipse and the direction of rotationmay vary. The ellipse traced out by a tidal streamvector is called the tidal ellipse. Slack water refers

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to zero flow in a tidal regime. The stand of the tideis the interval around high or low water in whichthere is little change of water level: this need notcoincide with slack water.

In a purely progressive surface tide,maximum flood occurs at HW, maximum ebboccurs at LW, and slack water occurs at mid-tiderising and falling. In a purely standing surface tide,the slack waters occur at HW and at LW,maximum flood occurs at mid-tide rising, andmaximum ebb occurs at mid-tide falling. Thisfollows from the discussion in section 1.5, and isillustrated in Fig. 5 for long waves in general.Except for some frictional effect near the bottom,the tidal streams associated with a surface tide arethe same from top to bottom. If tidal streams areobserved to vary in speed, phase or direction overthe water column, the presence of an internal tideis indicated. The average tidal stream in such acase belongs to the surface tide, and thedepartures at various depths from this average arethe tidal streams belonging to the internal tide.This situation presents the possibility for slackwater to occur at different times at differentdepths. Figure 11 illustrates various flow patternsthat may result from the vector addition of aresidual current and a rectilinear or a rotary tidalstream. It is seen that a rectilinear tidal streamexperiences slack water twice during each period(Fig. 11 a) unless it is accompanied by ( 1 ) aresidual current in the same direction but withspeed greater than the tidal stream amplitude, inwhich case the flow is unidirectional with varyingspeed (Fig. 11b), or (2) a residual current in adifferent direction from that of the tidal stream, inwhich case the flow changes direction through asmall angle (Fig. 11c). It is also apparent from Fig.11 that a rotary tidal stream rarely experiencesslack water, but that its direction changes through360° in each cycle (Fig. 11d) unless the speed ofthe residual current exceeds or equals theamplitude of the tidal stream in that direction (Fig.11e and f). In the latter cases, the direction of theflow swings back and forth through an angle lessthan or equal to 180°.

Since the observed tide consists not of asingle wave, but of the superposition of many tide

waves of different frequency and amplitude, it willnever fit exactly any of our simple descriptions.Because of this, we cannot expect the heights ofsuccessive HWs or of successive LWs to beidentical, even when they occur in the same day.Thus, the two HWs and two LWs occurring in thesame day are designated as higher and lowerhigh water (HHW and LHW), and higher andlower low water (HLW and LLW). It is likewiseonly the tidal stream associated with a singlefrequency tide wave that traces a perfect tidalellipse. The composite tidal stream each daytraces a path more closely resembling a doublespiral, with no two days’ patterns identical. Also,no tide is ever a purely progressive or a purelystanding wave, so that slack water should not beexpected to occur at the same interval before HWor LW at all locations.

1.12. Shallow-water Effects

One of our assumptions in the discussion oflong waves of sinusoidal form was that theamplitude was much less than the depth. When atide propagates into shallow water, this assumptionmay no longer be valid, and, as might be expected,the wave form is distorted from its sinusoidal form.In such shallow water the crest is found topropagate faster than the trough, producing asteeper rise and a more gradual fall of the waterlevel as the tide wave passes. Figure 12demonstrates this effect on the St. LawrenceRiver tide between Neuville and Trois Rivières.The outflow of the river and the bottom frictioncontribute to the distortion of the wave. The tide inthis part of the St. Lawrence River is attenuatedby friction as it progresses upstream, and is notreflected to produce a standing wave.

Tides in the open ocean are usually of muchsmaller amplitude than those along the coast. Asmentioned earlier, this is partly due to amplificationby reflection and resonance. It is, however, moregenerally the result of shoaling. as the wavepropagates into shallower water, its wave speeddecreases and the energy contained betweencrests is compressed both into a smaller depth anda shorter wavelength. The tide height and the tidal

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stream strength must increase accordingly. If, inaddition, the tide propagates into an inlet whosewidth diminishes toward the head, the wave energyis further compressed laterally. This effect, calledfunneling, also causes the tide height to increase.

Sometimes the front of the rising tidepropagates up a river as a bore, a churning andtumbling wall of water advancing up the river notunlike a breaking surf riding up a beach. Creationof a bore requires a large rise of tide at the mouthof the river, some sandbars, or other restrictions atthe entrance to impede the initial advance of thetide, and a shallow and gently sloping river bed.Simply stated, the water cannot spread uniformlyover the vast shallow interior area fast enough tomatch the rapid rise at the entrance. Friction atthe base of the advancing front, plus resistancefrom the last of the ebb flow still leaving the river,causes the top of the advancing front to tumbleforward, sometimes giving the bore the appearanceof a travelling waterfall. There are spectacularbores a metre or more high in several rivers andestuaries of the world. The best known bore inCanada is that in the Petitcodiac River nearMoncton, N.B., but there is another in theShubenacadie River and in the Salmon river nearTruro, N.S., all driven by the large Bay of Fundytides. These are impressive (about a metre) onlyat the time of the highest monthly tides, and maybe no more than a large ripple during the smallesttides.

The Reversing Falls near the mouth of theSt . John River at Saint John, N.B. is also causedby the large Bay of Fundy tides and theconfiguration of the river. A narrow gorge at Saint

John separates the outer harbour from a largeinner basin. When the tide is rising most rapidlyoutside, water cannot pass quickly enough throughthe gorge to raise the level of the inner basin at thesame rate, so on this stage of the tide the waterraces in through the gorge, dropping several metresover the length of the gorge. When the outsidetide is falling most rapidly, the situation is reversed,and the water races out through the gorge in theopposite direction, again dropping several metres insurface elevation. Twice during each tidal cycle,when the water levels inside and out are the same,the water in the gorge is placid and navigable. Thesurface of the water in the gorge near the peakflows is violently agitated and the velocity of flowis too rapid and turbulent to permit navigationthrough the gorge. This phenomenon is called atide race in other less notorious situations.

A tide rip or overfall is an area of breakingwaves or violent surface agitation that may occurat certain stages of the tide in the presence ofstrong tidal flow. They may be caused by a rapidflow over an irregular bottom, by the conjunction oftwo opposing flows, or by the piling up of waves orswell against an oppositely directed tidal flow. Ifwaves run up against a current, the wave form andthe wave energy are compressed into a shorterwavelength, causing a growth and steepening ofthe waves. If the current is strong enough, thewaves may steepen to the point of breaking, anddissipate their energy in a wild fury at sea. Violenttide rips may be formed in this way.

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Plate 1.Hopewell Rocks ("theflowerpots") at CapeHopewell, New Brunswick,on Chignecto Bay at the innerend of the Bay of Fundy, atlow water. The Rocks havebeen eroded and formed intounusual shapes by water andsand suspended in the strongtidal stream. (Photo courtesyof the Canadian GovernmentOffice of Tourism.)

Plate 2. Fishermen checking salmon fishing weirat low water near Saint John, New Brunswick.Fish are carried into and trapped by the weirbecause of the strong tidal flow: they are thenfished out of the weir at low water. (Photo by R.Brooks, NFB Phototeque, 1964.)

Plate 3. View of jetty and "mattress" at low water,Parrsboro, Nova Scotia, on the north shore of Minas Basin,at the inner end of the Bay of Fundy. (Photo by R.Belanger, Bedford Institute of Oceanography.)

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Plate 4. MV Theta resting on wooden "mattress" beside a jetty at low water. Parrsboro, Nova Scotia.(Photo by Canadian Hydrographic Service, 1960)

Plate 5. Corresponding views of jetty at Parrsboro, Nova Scotia, at extreme low water (left) and highwater (right). (Photos by C. Blouin, NFB Phototeque. 1949.)

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(Plate 6.) Left shows the inflow through the gorge at high water in the outer harbour (7.6 m above chartdatum at time of photo). Right shows the outflow through the gorge at low water in the outer harbour(0.9 m above chart datum at time of photo). (Photos by D.G. Mitchell, Canadian Hydrographic Service,1963.)

Plate 6. The Reversing Falls at SaintJohn, New Brunswick, at the mouthof the St. John River. The photo is anaerial view at slack water, showingthe inner basin, the outer harbour, andthe bridge over the gorge thatseparates them. The recordedextreme high and low waters at SaintJohn are 9.0 and -0.4 m,respectively,above chart datum, and at these timesthe flows would have beencorrespondingly greater. (Photo byLockwood Survey, NFB Phototeque,1966.)

Plate 7. (Left) Tidal bore on the Petitcodiac River at moncton, New Brunswick. (Photo by D.G. Mitchell,Canadian Hydrographic Service, 1960.); (Right) Tidal bore on the Salmon River, near Truro, Nova Scotia.(Photo by F.G. Barber, Ocean Science and Surveys, DFO, 1982.)

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CHAPTER 2

THE TIDE-RAISING FORCES

the centrifugal force due to rotation of the earth onits axis, determines the shape of level surfaces andhence the shape of the mean level of the sea; but itdoes not contribute to the tide-raising forcesbecause it does not vary with time. Although, aswe shall see later, the moon has more effect on thetide than does the sun, it will be convenient toconsider the sun’s contribution first, since theorbital parameters are easier to envisage for theearth-sun system.

2.2. Sun’s Tide-Raising Force

In this section we require Newton’s laws ofmotion and of universal gravitation, and anunderstanding of centripetal acceleration. The lawof motion states that the acceleration of a bodyequals the force acting on it per unit mass, or

(2.2.1 ) acceleration = force / mass

The law of universal gravitation states that a bodyof mass M exerts a gravitational attraction on aunit mass at a distance r of

(2.2.2) Fg = GM / r2

in which G is the universal gravitational constant.The centripetal acceleration is the acceleration of abody toward the centre of curvature of the pathalong which it is moving, and for a body withvelocity v along a path with radius of curvature r,it is:

Ac = v2 / r

Let us now compare the gravitationalattraction of the sun on the earth to that of themoon on the earth. The mass of the sun is 27million times that of the moon, and the distance ofthe sun from the earth is 390 times that of themoon. Using this information in equation 2.2.2gives

Fg (sun) / Fg (moon) = 27x106 / 3902 = 178

2.1 Introduction

It was explained in Chapter 1 that the localtide results from the superposition of long wavesof tidal frequencies generated throughout theocean by the tide-raising forces of the moon andthe sun. It remains to investigate these forces,particularly with a view to determining thefrequencies that characterize their fluctuations. Ithas been reasonably assumed, and later establishedby experience, that these are also the frequenciesof the tide waves generated in the ocean, and soare the main frequencies present in thefluctuations of the local tide. Shallow waterdistortion, however, may be expected to addmultiples and combinations of these frequencies(over-tides) to the spectrum of a coastal tide.Fluctuation in the tide or in the tidal force at aparticular frequency is called the harmonicconstituent at that frequency. The amplitudes andphaselags of the constituents are the harmonicconstants of the tide, the phaselag usually beingreferred to the phase of the correspondingconstituent in the tide-raising force at Greenwich.While it may be expected that the harmonicconstituents present in the spectrum of the tide-raising forces will be present in the spectrum ofthe local tide, it should not be expected that theywill be present in the same proportion or with thesame phase relation. This is because ocean basinsand coastal embayments are more nearly resonantat some tidal frequencies than at others, becausenodes and amphidromes occur at differentlocations for constituents of different frequencies,and because processes such as the transfer ofenergy from surface tide to internal tide may befrequency selective in different situations.

The tide-raising forces are simply theportions of the moon’s and the sun’s gravitationalattraction that are unbalanced by the centripetal(centrally directed) acceleration of the earth in itsorbital motions. At the centre of mass of the earth,and only at this point, there is an exact balancebetween the gravitational attractions and thecentripetal accelerations, this being the conditionfor orbital motion. Earth gravity, which includes

22

so the gravitational attraction of the sun on theearth is 178 times that of the moon. This may atfirst seem surprising since we know the moon tobe more effective in producing tides; but it is onlythe portion of the gravitational force not balancedby the centripetal acceleration in the earth’s orbitalmotion that produces tides. This unbalancedportion will shortly be shown to be proportional tothe inverse cube rather than the inverse square ofthe distance from the earth, but still proportionalto the mass as in equation 2.2.2. Thus, the tide-raising forces of the sun are about 178/390 = 0.46times those of the moon.

Figure 13 depicts a portion of the earth’sorbit around the sun, with the cross sectionthrough the earth greatly exaggerated with respectto the sun’s size and distance. Since theacceleration related to the earth’s axial rotation isalready accounted for in earth gravity, the earthshould be thought of here as maintaining a fixedorientation in space during its revolution about thesun; thus each part of the earth experiences thesame centripetal acceleration toward the sun. Inparticular, the centripetal acceleration at the centre

of the earth, O, is exactly equal to the sun’sgravitational attraction at that point, this being thecondition for orbital motion. The centripetalacceleration, being everywhere constant, istherefore everywhere equal to the gravitationalattraction at the centre, GS / r 2, where S is thesun’s mass and r its distance from the centre of theearth. At a point such as A, that is closer to thesun, the gravitational attraction is greater than atthe centre, O, and so has an unbalancedcomponent that attempts to accelerate a mass at A,away from O and toward the sun. At a point suchas A’, that is farther from the sun, the gravitationalattraction is less than at O, and the unbalancedcomponent attempts to accelerate a mass at A’away from O and away from the sun. At B and B’in Fig. 13, the gravitational attraction has almostthe same magnitude as at O, but is directed towardthe sun along a slightly different line, so that theunbalanced components are both acting toward O.These unbalanced components of gravitationalattraction are the sun’s tide-raising forces. At A,A’, B, and B’ they are vertical, but at intermediatepoints they are inclined to the vertical . At four of

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the intermediate points the forces are entirelyhorizontal. The horizontal components of the tide-raising forces are called the tractive forces since itis they that accelerate water away from B and B’toward A and A’ in an attempt to bring the surfaceeverywhere normal to the vector sum of gravityand the tide-raising force. This ideal surface,referenced to the mean sea level defined by gravityalone, is called the equilibrium tide. To picturethe sun’s equilibrium tide in three dimensions,imagine the shapes traced out by revolving Fig. 13about the axis AA’. Ocean tides are significantmostly because the water moves relative to thesolid surface of the earth. If the earth weresufficiently pliable, it too would change shape toconform to the equilibrium tide surface, and therewould be little or no relative movement of thewater. The earth is not perfectly rigid, and doeschange shape slightly in response to the tidalforces, but these earth tides are small enough toneglect in this mostly qualitative discussion.

We will now estimate the magnitude of thetide-raising forces. As already stated, the sun’sgravitational attraction at O in Fig. 13 is GS / r 2.

At A it is GS / (r-e)2, at A’ it is GS / (r+e)2, and at

B and B’ it is GS / (r2+e2), where e is the earth’sradius. All the attractions are directed from thepoint toward the sun’s centre H. Since the tide-raising force at a point is the difference betweenthe sun’s local attraction and its attraction at thecentre of the earth, we have the sun’s tide-raisingforce, F1, at A as

(2.2.3) F1(A) = [GS / r2] [1 + 2e/r + ... -1]

= 2GSe / r3

In 2.2.3 and 2.2.4 we use the binomialexpansion for (1 - e/r) -2 and (1 + e/r)-2 andneglect squares and higher powers of e/r, since it isso small. At A’,

(2.2.4) - F1(A’) = [GS / r2] [1 - 2e/r + ... -1]

= - 2GSe / r3

In 2.2.3 and 4, and in what follows, we haveadopted the sign convention that a force directedvertically upward is positive. This explains theminus signs on the left side of 2.2.4 and 2.2.5. AtB and B’, the vector subtraction of the sun’sattraction at O from that at B and B’ gives, withinthe same approximation as above, only acomponent of the tide-raising force directedtoward O, and

(2.2.5) - F1(B) = - F1(B’)

= [GS/r2 (1 + e2/r2)-1] sinβ = GSe/r3

where β = angle OHB = angle OHB’. In 2.2.5 we

neglected e2/r2 and approximated sin β as (e/r).

From the above expressions we see that thetidal forces are proportional to the mass of the sunand to the inverse cube of its distance, and that thecompressional forces around the great circle BB’,midway between A and A’, are one half thestrength of the expansional forces at the points Aand A’, at which the sun is in the zenith and thenadir, respectively.

2.3 Moon’s Tide-Raising Force

In the previous section we spoke of theearth as orbiting around the sun, but actually theearth and the sun are both orbiting around acommon centre of mass, which is less than 500 kmfrom the centre of the sun. Similarly, the moonand the earth are orbiting about a common centreof mass, which is inside the earth, about 1 700 kmbeneath the surface. It is the revolution of theearth in this small orbit that is the counterpart ofits revolution about the sun, which was consideredin section 2.2. With this in mind, and with r as themoon’s distance and M, the moon’s mass,replacing S, we may apply the logic of section 2.2directly to the earth-moon system (with H in Fig.13 now being the moon’s centre). This permits usto write down immediately expressions for themoon’s tide-raising forces. The expansionalforces at the points for which the moon is in the

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zenith and the nadir are

(2.3.1) F1(A) = F1(A’) = 2 GMe / r3

and the compressional forces on the great circlearound the earth’s surface midway between thesetwo points are

(2.3.2) F1(B) = F1(B’) = - GMe / r3

We have already noted in section 2.2 thatthe tide-raising forces of the sun are only about ahalf of those of the moon. It may be of someinterest to compare the moon’s tide-raising forceto the force of gravity at the earth’s surface.Neglecting the centrifugal force due to axialrotation, the surface gravity is

(2.3.3) g = GE / e2 so G = ge2 / E

where E is the earth’s mass. The maximum lunartidal force is that expressed in 2.3.1 which, withthe help of 2.3.3 may be rewritten as

(2.3.4) F1(A) = 2g(M/E )(e/r)3

M/E = 1/80 and e/r = 1/60, which, on substitutioninto 2.3.4 give the maximum lunar force as 10 -7g.So the tidal force is at most one ten-millionth ofthe earth’s surface gravity. These are small forcesindeed, but they act on every particle of waterthroughout the depth of the ocean, acceleratingthem toward the sublunar (or subsolar) point onthe near side of the earth and toward its antipodeon the far side. The undulations thus set up in thedeep ocean are in fact quite gentle, and onlybecome prominent when their energy iscompressed horizontally and vertically as they rideup into shallow and restricted coastal zones.

2.4. Tidal Potential and the Equilibrium Tide

Many force fields can be expressed as thenegative gradient of a scalar field, called thepotential field. Such force fields are said to beconservative, since the work done against theforce in moving from a point A to a point Bdepends only on the positions of the two points,

and not at all upon the path followed in movingbetween them. This constant amount of workrequired to move unit mass (or unit charge, etc.)from A to B is the difference in potential betweenA and B. The earth’s gravity field is aconservative field, whose potential is given thename geopotential. The difference in geopotentialbetween points is the work done against gravity inmoving a unit mass from one point to the other.Equi-geopotential surfaces are the familiar levelsurfaces, to which free water surfaces wouldconform in the absence of forces other thangravity. The lunar and solar tide-raising forces arealso conservative, and can be expressed as thenegative gradient of the tidal potential. Since thesum of one or more conservative force fields canbe expressed as the negative gradient of the sum oftheir potentials, we may add the tidal potential tothe geopotential and interpret equipotentialsurfaces in the combined field as “level” surfacesin the combined gravity and tidal force fields. Inparticular, one of these equipotential surfaceswould be the surface of the equilibrium tide, thesurface to which water would conform if it couldrespond quickly enough to the changing tidalforces. Because the geopotential does not varywith time and because we are interested in thetime-variable tides. we need only consider thetidal potential, and interpret its variations asvariations in the total potential at the mean sealevel.

The tidal potentials, p t at the point P(Fig. 14) are given very closely by the expressions

(2.4.1 ) -p1(moon) = (GMa2 / 2rm3)(3 cos2αm - I)

-p1(sun) = (GSa2 / 2rs3)(3 cos2αs - I)

where rm and rs are the moon’s and the sun’sdistances from the earth, the angles αm, and αsare their zenith angles (co-altitudes), and a is thedistance from the centre of the earth to the point P(equals earth’s radius, e, if P is at the surface).The other symbols are as previously defined. Theminus signs in front are required to conform withthe convention that the force is the negativegradient of the potential. Differentiation of 2.4.1with respect to a gives the vertical component ofthe tidal force.

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With cos a = 1, this reproduces the expression2.2.5 for the tidal force at A (Fig. 13), and withcos a = 0, it reproduces the expression 2.2.5 forthe tidal force at B.

The equilibrium tide surface must be anequipotential surface in the combined tidal andgravity field, and so any increase in the tidalpotential must be matched by a decrease in thegeopotential (i.e. a fall in the surface), and anydecrease in tidal potential must be matched by anincrease in geopotential (i.e. a rise in the surface).Using this fact we can calculate the height of theequilibrium tide above the mean sea level. Let theheight of the equilibrium tide be ∆h, whichcorresponds to an increase of g ∆h in geopotential.To maintain an equi-potential surface, thisincrease must be equal and opposite to the tidalpotential, pt so

(2.4.2) ∆h = - pt/g

Substituting the expressions for G from 2.3.3 andfor pt, from 2.4.1 into 2.4.2 gives the heights ofthe lunar and solar equilibrium tides as

(2.4 3) ∆h(moon) = (Me4 / 2Erm3) (3 cos2αm - 1)

∆h(sun) = (Se4 / 2Ers3)(3 cos2αs - 1)

In substituting from 2.4.1 we put a equal to ebecause the equilibrium tide is for points at theearth’s surface. The extreme values of ∆h occurfor α = 0° and α = 90°. Using

e = 6,400 km, M/E = 0.012

e/rm, = 0.017 , S/E = 3.3x10 5

e/rs = 4.3 x 10-5,

2.4.3 gives the extreme equilibrium tide heights as

(2.4.4) ∆h(moon) = 0.38 m, and -0.19 m∆h(sun) = 0.17 m, and -0.08 m.

The ratio of the solar to the lunar values in 2.4.4 is0.46, the same as the ratio of the extreme solarand lunar tide-raising forces (see section 2.2). Infact, the equilibrium tide reflects all the importantcharacteristics of the tide-raising forces, and,being a scalar instead of a vector, is a much moreconvenient reference for local tidal observationsand predictions.

2.5 Semidiurnal and Diurnal EquilibriumTides

The equilibrium lunar and solar surfacesdefined by the expressions in 2.4.3 are ovals ofrevolution centred at the earth’s centre and withaxes directed toward the moon and the sun. Theyappear to rotate from east to west as the earthrotates daily on its axis with respect to the moonand the sun. The inclination of their axes northand south of the equator changes with thedeclination of the moon and of the sun, in amonthly cycle for the moon and an annual cyclefor the sun. The ovals also change in shape as theorbital distances, rm and rs change in monthly andannual cycles, respectively. In formal tidal studythe characteristics of the equilibrium tide aredetermined from mathematical analysis ofexpression 2.4.3 and the known astronomicalparameters. It is, however, useful to obtain anintuitive understanding of how the various tidalharmonic constituents arise, and that is how wewill now proceed.

Figure 15 depicts the sun’s equilibrium tidesuperimposed on the equi-geopotential surface ofmean sea level, (a) for the sun on the equator, (b)for the sun, at maximum north declination, and (c)for the sun at maximum south declination.

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From (a) it is seen that with the sun at zerodeclination an observer on the equator rotates withthe earth once each solar day with respect to thesun’s equilibrium tide, passing through LW atpoints B and B’ (B’ is on the opposite side of theearth from B), and through HW at points A andA’. In fact, an observer at any latitude wouldexperience equilibrium LW as his meridian passedthrough B and B’, and HW as it passed through Aand A’, although the heights of HW woulddecrease with increasing latitude north or south ofthe equator. This is the origin of the semidiurnalsolar constituent of frequency two cycles per day(30°/h); it is designated S 2. If we simply replacethe sun with the moon in the above discussion, wehave the explanation for the origin of thesemidiurnal lunar constituent (M 2). Its frequencyis two cycles per lunar day (28.98°/h). The lunarday is about 50 min longer than the solar day,because the moon advances about 12.5° in its orbiteach day with respect to the sun’s position.

When the sun is north or south of theequator, one centre of HW for its equilibrium tideis north and the other is south of the equator, asshown in Fig. 15b and c. In these cases, anobserver moving around with the earth at theequator would still experience two equal HWs andtwo equal LWs per day, although the HWs wouldnot be as high as in case (a). An observer at anorthern latitude would experience HHW at noonand LHW at midnight in (b) while an observer at asouthern latitude would experience HHW atmidnight and LHW at noon. In case (c), therewould be the same inequality in the two HWs forobservers away from the equator, but the northernobserver would now experience HHW atmidnight, etc. In the equilibrium tide the two LWswould have the same height (but not necessarily soin an actual tide). The difference in heightbetween HHW and LHW is called the diurnalinequality, and it increases with the declination ofthe sun and with the observer’s latitude (north orsouth) for an equilibrium tide. In fact, if the sun’sdeclination is d, the band of low water around theearth in the equilibrium tide extends no farthernorth and south than latitude 90° - δ, andobservers at higher latitudes than this would seeonly a distorted diurnal tide, with one true HWand an extended period of low water. Asemidiurnal tide with a diurnal inequality can beconsidered as the sum of a semidiurnal and diurnaltide. This is illustrated in Fig. 16, which showsthe combination of the semidiurnal and diurnalcontributions to produce the equilibrium tide ofthe sun at 23° declination, for an observer atlatitude (a) 15°, (b) 35°, (c) 55°, and (d) 75°.

Since the diurnal tide must reinforce thenoon HW when the sun and the observer are onthe same side of the equator, fall to zero when thesun is on the equator, and reinforce the midnightHW when the sun is on the opposite side of theequator, it is clear that more than a single diurnalconstituent is required. From trigonometry wehave the relation:(2.5.1) cos (n1 + no)t+cos (n1 - no)t = 2(cos not)(cos n1t)

If n1 is the angular speed 360° per solar day(15°/h) and no is 360° per year (0.04°/h), the rightside of 2.5.1 is seen to be a diurnal oscillation offrequency n1 whose amplitude is modulated at the

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annual frequency n 0 falling to zero at n 0t = 90° and270°, and having maximum amplitude but withopposite phase at n 0t = 0° and 180°. Figure 17shows a plot of 2.5.1 for a few cycles of n 1, aroundn0t = 90° to illustrate the change in amplitude andshift in phase. This is the origin of the two solardeclinational diurnal constituents P 1 withfrequency 14.96°/h, (n 1 - n0) and K, withfrequency 15.04°/h, (n 1 + n0). Because the earth’srate of rotation on its axis with respect to the“fixed stars” is equal to its rate of rotation withrespect to the sun plus its rate of orbital revolutionabout the sun, the frequency of K, is seen to beone cycle per sidereal day.

The lunar equilibrium tide changes with thedeclination of the moon over a period of a monthin the same manner as the solar tide changes withthe sun’s declination over a year. This, then, givesrise to two lunar declinational diurnal constituents

with frequencies of one cycle per lunar day plusand minus one cycle per lunar month. Thefrequency of the earth’s rotation with respect tothe moon plus the moon’s frequency of orbitalrevolution about the earth is also equal to onecycle per sidereal day, so that one of the moon’sconstituents has the same frequency as the

28

corresponding one for the sun, K 1 . Because ofthis, K1 serves double duty, and is called the luni-solar declinational diurnal constituent. The otherlunar constituent of the pair is O 1. with angularspeed 13.94°/h.

The orbits of the moon about the earth andof the earth about the sun are not circles, but areellipses, with the earth and the sun occupying oneof the foci in the respective orbits. Thus, thedistances of the moon and the sun from the earthchange within the period of the particular orbit, 1month for the moon and 1 year for the sun. Theorbital points at which the moon is closest to andfarthest from the earth are called perigee andapogee, respectively. The corresponding points inthe earth’s orbit about the sun are called perihelionand aphelion. The dependence of the tidalpotential on the inverse cube of these distances(rm, and rs in 2.4.1 ) causes the shape of the solarequilibrium tide (see Fig. 15 ) to lengthen along itsaxis and compress in the middle at perihelion, andto shorten along its axis and expand in the middleat aphelion. The shape of the lunar equilibriumtide changes similarly at perigee and apogee,respectively, but the change is much morepronounced for the lunar than for the solar tide.The effect of this is to modulate the amplitudes ofthe solar constituents with a period of a year andof the lunar constituents with a period of a month.But there is a further complication; the earth andthe moon do not travel at constant angularvelocities around their orbits, but travel fasterwhen they are closer to the central body. Theeffect of this is to modulate the phases of the lunarconstituents over a period of a month, and those ofthe solar constituents over a period of a year. Thecombined effect of the amplitude and phasemodulations can be imitated mathematically byadding to each constituent two satelliteconstituents with frequencies equal to that of themain constituent plus and minus the orbitalfrequency, but with the amplitude of one satellitegreater than that of the other. Figure 18 attemptsto illustrate how the amplitude and phasemodulations are accomplished. Tidal constituentsmay be regarded as rotating vectors, since theyhave a fixed amplitude and a uniformly increasingphase angle. A vector sum is obtained graphicallyby placing all of the participating vectors head to

tail and joining the tail of the first to the head ofthe last. OR is the main constituent, with angularspeed n, RS is the first satellite constituent, withspeed n1 = n + n’, and ST is the second satelliteconstituent, with speed n 2 = n - n’ . The sum ofthe three constituents is OT, and relative to therotating vector OR, the point T traces out anellipse with centre at R. Its semi-major axisequals the sum of the satellite amplitudes, and itssemi-minor axis equals their difference. It may beseen from Fig. 18 that as T moves around theellipse once for each cycle of the main constituent,the amplitude of the vector sum, OT, oscillatesbetween OP and OP’, and the phase oscillatesabout that of OR through the angle QOQ’. If thesatellite amplitudes are equal, the ellipse collapsesto a line, and there is amplitude modulation only.This is the origin of the larger and smaller lunarelliptic semidiurnal constituents N 2 (speed 28.44°/h) and L2 (speed 29.53°/h), respectively, and alsoof the larger and smaller solar elliptic semidiurnalconstituents T2 (speed 29.96°)/h) and R 2 (speed30.04°/h). R2 is so small it is usually neglected.

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2.6 Long-period Equilibrium Tides

Here we will discuss tidal constituentswhose periods are comparable to the sun’s or themoon’s orbital period. It is important todistinguish between a long-period constituent anda long-period modulation of a short-periodconstituent. The long-period modulation changesthe range of the tide over the long period, but doesnot change the mean water level, whereas thelong-period constituent does not change the rangeof the tide, but introduces a long-periodfluctuation in the mean water level. Todemonstrate the origin of the long-period tidalconstituents we look again at Fig. 15. It isapparent that an observer near the North or SouthPole will experience a lower daily averageequilibrium tidal elevation when the tide-raisingbody (sun or moon) is on the equator as in (a) thanwhen it is north or south of the equator as in (b) or(c). Although most easily envisaged for highlatitudes, this effect is present at other latitudes aswell. It results in the introduction of the lunarfortnightly constituent M f (speed I .10°/h), into thelunar equilibrium tide, and the solar semi-annualconstituent Ssa (speed 0.08°/h) into the solarequilibrium tide. Mf and Ssa are thus related tothe moon’s and the sun’s cyclic changes indeclination. There is also a lunar monthlyconstituent, Mm (speed 0.54°/h), and a solarannual constituent, Sa(speed 0.04°/h), and theseare related to changes in the lunar and solardistance over a month and a year, respectively.

2.7 Mathematical Analysis of theEquilibrium Tide

In the preceding discussions we haveconsidered the equilibrium tide as the envelope ofequal tidal potential surrounding the earth at agiven time. We will now express it as a time-varying function at a fixed location on the earth.To do this, we must express cos a in 2.4.3 in termsof the local latitude and of the declination andhour angle of the sun and moon. The hour angleof a celestial object is its longitude angle west ofthe observer’s longitude.

In Fig. 19, PSN is a spherical triangle on asphere surrounding the earth, and with its centre atthe earth’s centre, O. P is the projection of theobserver’s position from the centre of the earthonto the sphere, N is the projection of the NorthPole onto the sphere, and S is the intersection withthe sphere of the line joining the centre of theearth to that of the celestial object (sun or moon).Angle PON is the co-latitude of P (90° - ϕ), angleSON is the co-declination of the celestial object(90°- δ), POS is its zenith angle ( α) with respect toP, and PNS is its hour angle (H) with respect to P.From a formula of spherical trigonometry we have

(2.7.1) cos α = sin ϕ sin δ + cos ϕ cos δ cos H

Substituting 2.7.1 into 2.4.3, and using sometrigonometric relations to simplify it, gives theequilibrium tide at P as

(2.7.2) ∆h = BCO(cos 2δ- 1/3) +

BC1(sin 2δ) cos H + BC2 (cos 2δ + 1) cos 2H,

where B = 3Me4 / 2Erm3 for the moon, 3Se4 /

2Ers3 for the sun,and

CO = 3/8 (cos 2ϕ- 1/3)C1 = 1/2 sin 2ϕC2 = 1/8 (cos 2ϕ + 1).

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H and δ are the local hour angle and declination ofthe moon or the sun, as appropriate. The C i areconstants for a given latitude.

In discussing 2.7.2 we will refer only to themoon, but the same logic applies for the sun andits equilibrium tide. The first term on the right of2.7.2 introduces the long-period species ofconstituent, since B varies over a period of amonth and cos 2δ varies over a period of half amonth. The second term introduces the diurnalspecies of constituent since the hour angle (H)advances at a frequency of one cycle per lunar day.Multiplication by sin 2δ splits the species intoconstituents whose frequencies differ by twocycles per month, as shown in 2.5.1 and Fig. 17from a different approach. The third term on theright of 2.7.2 introduces the semidiurnal species ofconstituent, since 2H advances at a frequency oftwo cycles per lunar day. Multiplication by Bmodulates the species at a frequency of one cycleper month, giving rise to constituents N 2 and L2as defined in section 2.5. The factor cos 2 δ alsomodulates a portion of the semidiurnal species at afrequency of two cycles per month, introducing apair of lunar declinational semidiurnal constituentsnot previously discussed. Their frequencies aretwo cycles per lunar day plus and minus twocycles per month. The constituent with the higherfrequency is also the larger of the pair, and has thesame frequency as the corresponding solarconstituent, both being equivalent to two cyclesper sidereal day. It is thus called the lunisolardeclinational semidiurnal constituent or K 2 (speed30.08°/h). Many other constituents could bediscovered by examining the modulation of thedeclinational constituents by B and by treating thedeparture of some of the factors from truesinusoidal functions of time. The relativeamplitudes of the constituents in the equilibriumtide can also be determined from analysis of 2.7.2and substitution of the parameter values. Thepurpose of this section, however, is simply todemonstrate some of the techniques employed inidentifying the important tidal frequencies.Numerical analysis on fast electronic computersnow provides tools for investigation of theequilibrium tide that were not available during theearly development of tidal theory. It is now quitefeasible to generate from 2.7.2 the combined

equilibrium tide for the sun and the moon as atime series of elevations covering many years, andto analyse it numerically into its constituents,identifying their frequency, phase, and amplitude.In Appendix A are listed a few of the equilibriumtidal constituents, along with their frequencies (asangular speed) and their amplitudes relative to thatof M2.

2.8 Spring and Neap Tides

It cannot be stressed too much that at noplace on the earth is the actual tide the same as theequilibrium tide at that place. Nevertheless, manyof the characteristics of the two are similar exceptfor magnitude and timing. The equilibrium springtide occurs on the day that the sun’s and themoon’s HWs fall on the same meridian, which, asshown in Fig. 20, occurs at new and full moon.The HWs occur near local noon and midnight andare higher than average because of thereinforcement of the two. The two LWs alsoreinforce, but in the opposite sense, making themlower than average. The result, then, is a largerthan average range of equilibrium semidiurnal tideat spring tide. The equilibrium neap tide occurson the day that the sun’s and the moon’s HWsmost closely coincide with the other’s LWs,which, as shown in Fig. 20, occurs at the moon’sfirst and last quarter. The result is a smaller thanaverage range of tide at neap tide. The range ofthe combined solar and lunar tide does not, ofcourse, change suddenly at spring and neap, but ismodulated sinusoidally over the half-month periodbetween successive springs or neaps. From thestandpoint of tidal constituents. it is theinteraction of M2 and S2 as they come in and outphase with each other that produces the springsand neaps. This fortnightly modulation of thesemidiurnal tide is prominent in actual tide recordsas well as in the equilibrium tide, so much so infact that in many parts of the world HW and LW atsprings are taken as standards of extreme high andlow waters. This is not invariably the case,however, because in other parts of the world theremay be another characteristic tide that dominatesover the spring tide. In the Bay of Fundy, forexample, the perigean tides (the large semidiurnal

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tides associated with the moon’s perigee) areequally as significant as the spring tides. Inregions such as Canada’s Pacific coast and parts ofthe Gulf of St. Lawrence it is the diurnalinequality that renders the simple spring tideheights unsatisfactory as standards of extremehigh and low water.

2.9 Classification of Tides

Tides are frequently classified according tothe diurnal inequality they display, as a means ofproviding a simple description of the character ofthe tide in various regions. The formalclassification is usually made on the basis of theratio of some combination of the diurnal harmonicconstituents over a combination of the semidiurnalconstituents. A criterion that is commonly used is

the ratio of the amplitude sum of O 1 and K1 overthe amplitude sum of M1 and S2.

The ratio that is used in Canadian tidalclassification is more complicated than this, butthe principle is the same - the larger the numericalvalue of the ratio, the larger the diurnal inequality.The purpose of defining a ratio is to automate theclassification once the constituents are known,avoiding the need to scan long periods of recordvisually. Regardless of the method used, the intentis to classify tides into four groups, qualitativelydescribed as follows:

Semidiurnal (SD): two nearly equal HWs and twonearly equal LWs approximately uniformly spacedover each lunar day.Mixed, mainly semidiurnal (MSD) : two HWs andtwo LWs each lunar day, but with marked

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inequalities in height and irregularities in spacing.Mixed, mainly diurnal (MD): sometimes twounequal HWs and LWs at irregular spacing over alunar day, and sometimes only one HW and oneLW in a day.Diurnal (D): only one HW and one LW each lunarday.

Since the equilibrium tide is the same for allpoints at the same latitude, the earth could bedivided into bands of latitude conforming to theabove classification, with equilibrium tides atlatitudes less than 10° being SD, those betweenlatitudes 10° and 40° being MSD, those between40° and 60° being MD, and those at latitudeshigher than 60° being D. The actual tides, ofcourse, may reflect the character of tide wavespropagated from far away, and should not beexpected to conform to the same classifications

within latitude bands. Figure 21 shows a sampletidal curves for one month at four Canadianlocations to illustrate the four classes definedabove. It is noted that all four locations lie withinthe same three-degree band of latitude. Figure 22indicates on a map of Canada the regions in whichthe various types of tide dominate. Although theEast Coast is predominantly a region of mainlysemidiurnal tide, we find the only examples of thediurnal tide in the Gulf of St. Lawrence. This isbecause these points lie near an amphidromicpoint of the semidiurnal tide. We also note thatthe tide observed in the Arctic archipelago isheavily semidiurnal in character, unlike theequilibrium tide for these latitudes. This isbecause much of the tide in the Canadian Arctichas propagated in through the passages from theNorth Atlantic Ocean.

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CHAPTER 3

TIDAL ANALYSIS AND PREDICTION

3.1 Introduction

We have demonstrated in Chapter 2 how toidentify the fundamental tidal frequencies presentin the spectrum of the equilibrium tide, and havespeculated that these should be the fundamentalfrequencies present in any actual tidal record.Basically, tidal analysis consists of identifying in atidal record the amplitudes of all the importantharmonic constituents and their phaselags withrespect to the phases of the correspondingconstituents in the equilibrium tide. Tidal predictionconsists of recombining these constituents in theproper phase relation to the correspondingconstituents in the equilibrium tide at the desiredtimes. Both analysis and prediction require aknowledge of the phases of the harmonicconstituents in the equilibrium tide at specifiedtimes, which may be extracted from tables orcalculated by formulae involving the astronomicalparameters. Although it is usually easier to takethings apart than it is to put them back togetheragain, the principles of tidal prediction are muchsimpler than are those of tidal analysis.Nevertheless, we will start with a consideration oftidal analysis, the breaking down of a tidal recordinto its component parts. Initially, a look at theFourier theorem will be helpful.

3.2 The Fourier Theorem

This theorem states that if F(t), a function ofthe variable t, is defined over the range fromt = -T/2 to t = T/2, then F(t) can be expressed as aconstant plus an infinite series of sinusoids offrequencies (or wave numbers, if t is thought of asa distance) 1/T, 2/T, 3T, . . . i/T . . . etc. Thesinusoidal oscillation with frequency 1/T is calledthe fundamental, and the oscillations whosefrequencies are multiples of 1/T are calledharmonics of the fundamental. Every harmonicthat is added to the series increases the precisionto which it reproduces F(t) in the range -T/2 to T/2. Outside this range, the series of sinusoids wouldproduce the same image of F(t) between T/2 and3T/2, and repeat it again and again for every

interval of length T. Unless F(t) is believed to beperiodic with period T over all values of t, theserepetitions are simply ignored, and the series ofsinusoids is used only to reproduce values of F(t)within the defined range -T/2 to T/2. Instructionsare also included in the theorem for the evaluationof the terms in the series. The constant term issimply the average value of F(t) over the definedrange. Evaluation of the fundamental or one of theharmonics involves multiplying every point in F(t)by the sine and by the cosine of that harmonicfrequency times t, and forming the average ofthese products over T. The Fourier theorem isstated much more compactly in mathematical formas follows:

If F(t) is defined between -T/2 and T/2, then

It would be convenient indeed if the tidalcycle repeated itself exactly at regular intervalssuch as a month or a year, because then a Fourierseries could be formed as described above toprovide predictions for all time, with no regard tofurther tidal theory. Certainly a Fourier series canbe formed to reproduce any finite tidal record toany desired accuracy, but since tides do not repeatexactly after any known interval, the series couldnot be used to predict values for any time outsidethe record, and would be of little value. So what isthe pertinence of the Fourier theorem to tidalanalysis? If, in 3.2.1, we allow T to becomeinfinite, the fundamental frequency becomesinfinitesimal, and the frequency interval betweenharmonics also becomes infinitesimal; i.e. allfrequencies become candidates for inclusion in theFourier series. Our knowledge of the frequenciespresent in the equilibrium tide tells us to look onlyat these frequencies out of the Fourier spectrum ofall possible frequencies. The last two equations of3.2.1 are then used to estimate the amplitudes andphaselags of the tidal constituents from observed

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tidal records. The tidal records are not infinitelylong, however, and some ingenuity is required informing meaningful averages for the expressions in3.2.1 from short tidal records. This is thechallenge of tidal harmonic analysis.

3.3 Harmonic Analysis of Tides

The harmonic method of analysis treatsevery tidal record as consisting of a sum ofharmonic constituents of known frequency plusnon-tidal “noise.” This may be expressed as

in which h(t) is the instantaneous height, H i’ is theamplitude of the “i”th constituent, E i’ is the phaseof the equilibrium constituent at Greenwich atGreenwich Mean Time (GMT) numerically equalto the local observation time, and g i is the phaselagof the constituent behind the Greenwich phase, E i’.It is important to note that the Greenwich phaseused here is the actual phase of the equilibriumconstituent at Greenwich only if the observationsare recorded in GMT. If they are recorded in atime zone z hours west of Greenwich, then E i’ isthe phase of the constituent at Greenwich z hoursearlier. This may seem confusing at first, but itavoids the need to convert observation times intoGMT, and the choice of a reference phase can bequite arbitrary as long as it advances at the properspeed and is applied consistently in all calculations.It is, however, necessary to record the time zonecarefully along with the results of any tidal analysisto assure consistency in phase reference. Thesignificance of the primes on H i’ and Ei’ willbecome apparent later. The subscript, i, is simply acounter for the n harmonic constituents considerednecessary to represent the tidal signal adequately;Ho’ represents the average water level during therecord (zero frequency, E o’ = go = 0).

The aim of the harmonic analysis is todetermine the values of all the H i’ and the gi. Ho’is simply the average of all the observations, and isusually denoted as Zo in tidal terminology. If eachvalue of h(t) is multiplied by cos E i’ and the

average taken over the length of the record, 3.3.1gives(3.3.2) avg. [(cos E1')h(t) = H1' cos g1(cos2E1')

+ H2' cos g2 (cos E1' cos E2')+ H1' sin g1 (cos E1' sin E1')+ H2' sin g2 (cos E1' sin E2')+ etc. + (cos E1')(“noise”)]

and, if each value of h(t) is multiplied by sin E 1',the average gives(3.3.3) avg. [(sin E1')h(t) = H1' sin g1(sin2E1')

+ H2' sin g2 (sin E1' sin E2')+ H1' cos g2 (sin E1' cos E1')+ H2' cos g2 (sin E1' cos E2')+ etc. + (sin E1')(“noise”)]

The “noise” terms are considered to average tozero, their signal being assumed to be random withrespect to tidal frequencies. If all n constituentscould complete an exact number of cycles in thesame length of record, all of the averagedcoefficients in 3.3.2 and 3.3.3 would be zero

except for cos2E1' and sin2E1', whose averagevalues would be 0.5. This would give the simpleFourier solution of 3.2.1, namely

(3.3.4) H1' sin g1 = avg. of 2h(t) sin E1'H1' cos g1 = avg. of 2h(t) cos E1'

Of course the tidal constituents cannot allcomplete an exact number of cycles in the samelength of record, and we must contend with theresidual terms in 3.3.2 and 3.3.3 instead ofassuming the enticingly simple solution of 3.3.4.Repeating the above process of multiplication bythe sine and cosine of the E i’ and averaging for theother n-l constituents completes a set of 2nequations in the 2n unknowns (the H i’ sin gi andthe Hi’ cos gi). As a matter of interest, these arethe same 2n “normal equations” that would havebeen produced if the problem had been tackled bythe “method of least squares,” so their solution forthe Hi’ and gj gives a best fit to the data in a “leastsquares” sense.

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While the generation of the 4n 2 coefficientsand the solution of the 2n simultaneous equations in2n unknowns is feasible with today’s high-speed,large-memory electronic computers, such was notalways so in the days of manual computation.

One simplification that is frequently used formanual computation is to replace the sine andcosine multipliers by the “box-car” functions thatequal + I when the corresponding sine or cosineare positive, equal - I when they are negative, andequal zero when the sine or cosine are zero.Figure 23 illustrates how multiplication by the box-car equivalent of the sine or cosine functionproduces an average contribution of 2/ π times thecorresponding (sine or cosine) component of a

constituent of the same frequency, zero times thecomplementary (cosine or sine) component of aconstituent of the same frequency, and zero timesboth the cosine and sine components of aconstituent of exactly twice the frequency. Theprinciple by which the pure sine and cosinemultipliers sort out the coefficients of 3.3.3 and3.3.4 is the same as that by which their box-caranalogues worked in Fig. 23. The box-carmultipliers, however, are most effective inseparating one species of tide from another(diurnal, semidiurnal, etc.) but for separation ofconstituents of the same species, the pure sine andcosine multipliers produce a better-behaved set ofcoefficients.

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3.4 Nineteen-year Modulation ofLunar Constituents

The moon’s orbit about the earth is in aplane always inclined at approximately 5° to theplane of the earth’s orbit about the sun (theecliptic). but the line of intersection of the two planesrotates once every 18.6 years about the pole of theecliptic. Figure 24 shows a celestial sphere (a sphere ofinfinite radius, with the earth’s centre as its centre) ontowhich are projected the earth’s equator and its polaraxis, the plane of the ecliptic and its axis. and the planeof the moon’s orbit and its axis. The spherical screen ofa planetarium is a model of a portion of a celestialsphere. The line NN’ is the intersection of the moon’sorbital plane with the ecliptic, and N and N’ are referredto as “nodes.” N is the “ascending node,” since themoon is moving from south to north of the ecliptic as itpasses N. As the axis of the moon’s orbit rotates aboutthe axis of the ecliptic, tracing out the 5° cone onceevery 18.6 yr, the point N moves around the eclipticfrom east to west in the same period. Since the moontravels its orbit in the opposite sense (west to east), thisis referred as the “regression of the moon’s ascendingnode,” and it has a major influence on the moon’sdeclination over the approximate l9-year period. Theinclination of the ecliptic to the equator is 23 1/2°, soover the course of a year the sun changes declinationbetween 231/2° north in summer and 231/2° south inwinter. Since the moon’s orbit is inclined at 5° to theecliptic, the moon’s declination may change over the

course of a month between 281/2° (i.e. 231/2° + 5°)north and south during one part of the l9-year cycle,and between only 181/2° (i.e. 231/2°—5°) 91/2 yearslater in the cycle. The moon’s maximum monthlyswing in declination occurs when its ascendingnode, N, coincides with the vernal equinox, and itsminimum swing occurs when its ascending nodecoincides with the autumnal equinox. The vernalequinox (γ) is the point at which the sun crossesthe equator on its way north, and the autumnalequinox is the point at which it crosses it on its waysouth.

The regression of the moon’s ascendingnode has the effect of modulating both theamplitude and the phase of the lunar tidalconstituents in a l9-year period. Because theperiod is so long, it is assumed that the modulationof the constituents in the real tide is the same asthat of the equilibrium constituents. The amplitudemodulation is represented by a nodal factor, f,which varies about a mean value of unity over theperiod of 18.6 years. The phase modulation isrepresented by a nodal correction, u, which variesabout a mean value of zero over the same period.There is no nodal modulation of the solarconstituents, and the f and u values are differentfor each lunar constituent. Values of the nodalparameters are tabulated, and may also becomputed from formulae involving the astronomicalvariables. f of K2 varies between about 0.75 and1.30, while f of M varies only between 0.96 and1.04. u of K2 varies between plus and minus 17°,while u of M2 varies only between plus and minus2°.

To conform with the above, the equilibriumphase, Ei’, used in the harmonic analysis must bethe phase of the mean equilibrium constituent, E i,plus the nodal correction, U i, for that constituent atthat time. The amplitude, H i’, that results from theanalysis will be the amplitude of the meanconstituent,Hi, times the nodal factor, f i, for that constituent atthat time. Thus, in section 3.3

Ei’ = Ei + ui, and Hi’ = fiHi

The tidal constants that are retained fromthe analysis are the amplitudes of the meanconstituents (Hi = Hi’/fi) and the phaselags of themean constituents (gi).

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3.5 Shallow-water Constituents

Section 1.12 discusses, and Fig. 12illustrates, the distortion of a tide wave as it travelsin shallow water. The Fourier theorem suggeststhat this distortion could be represented by addingharmonics of the fundamental tidal frequencies, aprocedure that is attractive because it is compatiblewith the methods of harmonic analysis andprediction. This, then, is the origin of the shallow-water constituents (sometimes called ‘`over-tides”). They are introduced as a mathematicalconvenience to represent distortion of the tidewave, and do not arise directly from the tidalforces. As an example, M 6, is the secondharmonic of the fundamental tidal constituent M 2,with three times its angular speed. The mostcommon shallow-water constituents are thequarter-diurnals M4 and MS4, with frequenciestwice that of M2 and the sum of those of M 2, andS2, respectively. Their combination produces aquarter-diurnal tide whose amplitude is modulatedat their difference frequency, which is the same asthe modulation frequency of the semidiurnal tideproduced by the combination of M 2 and S2. Thispart of the quarter-diurnal tide is thus able toincrease and decrease in the same spring-neapcycle as the part of the semidiurnal tide producedby the combination of M2 and S2. Figure 25shows graphically the distortion produced in afundamental constituent by the addition of a firstharmonic. This may be compared to the distortionof the semidiurnal tide in the St. Lawrence River,shown in Fig. 12. The shallow-water constituentscan be included in the harmonic analysis proceduredescribed in section 3.3.

3.6 Record Length and Sampling Interval

Theoretically, if a water level recordcontained nothing but a pure tidal signal consistingof the contributions fromthe amplitudes andphaselags of the n constituents could bedetermined from almost any set of 2n observationpoints. Of course, nothing in this world is so pure,and tidal records are contaminated with “noise,”both of meteorological and observational origin.This is why it is necessary to rely on statistical

averaging and filtering of long tidal records, asdescribed in section 3.3, to resolve the tidal signalfrom the background noise and to distinguish theindividual harmonic constituents. In practice, thelowest frequency constituent that could possibly bedistinguished in a tidal record is one whose periodequals the length of the record, and the highestfrequency constituent is one whose period is twicethe sampling interval. Whether two neighbouringconstituents can be separated from each other inan analysis depends both on the difference in theirfrequency and the length of the record. The“Rayleigh criterion” for the separation of twoconstituents requires that they should change phasewith respect to each other by at least 360° duringthe record period. If n1 and n2 are theconstituents’ angular speeds in degrees per hourand T is the record length in hours, the Rayleighcriterion for separation is

(3.6.1) (n1 - n2)T is greater than, or = 360°

Sometimes, if a record seems relatively free ofnoise, this criterion might be relaxed, and the rightside of 3.6.1 made 180°. Diurnal constituentscould be separated from semidiurnal constituents

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record, the greater is the number of constituentsthat can be analysed, and the higher is theaccuracy of their determination. For temporarywater level gauges installed in tidal waters, aminimum recommended length of record is Imonth.

Tidal analyses are usually carried out ondata that has been read at hourly intervals from theoriginal record. A sampling interval of I h is shortenough to detect the highest frequency constituentsof interest in most tidal work. However, if higherfrequencies are seen to be present, even thoughthey may not be of direct interest, they should besmoothed out of the record before the hourlysamples are taken. This is to avoid the processknown as aliasing, by which frequencies higherthan the sampling frequency may masquerade aslower frequencies. Figure 26 illustrates theprinciple of aliasing. It is usually more important tosmooth the samples from a short record than froma long one, because the aliased signal is likelyrandom with respect to the tidal signal, and so canbe adequately eliminated in the averaging andfiltering of the long analysis. Today’s computerprograms for tidal analysis are sufficiently flexibleto accept data that has been sampled at irregularintervals, and even data that has gaps of severaldays in it. In fact, two records taken at differenttimes may be more valuable than a singlecontinuous record of their combined length. Forexample, 1 day of record at spring tide and I day ofrecord at neap tide could permitseparation of M2 and S2, whereas 2 days ofcontinuous record could not. A single continuousrecord covering the whole period from spring toneap tide would, of course, be more valuable still.

3.7 Harmonic Analysis of Tidal Streams

The only difference between the analysis ofa water level record and the analysis of a currentmeter record is that the current speed and directiondata must first be resolved into two mutuallyperpendicular horizontal vector components. Twoseparate harmonic analyses are then performed,one on each time series of component speeds.The choice of the two component directions isarbitrary, as long as it is well recorded what they

on the basis of a single day’s record, but toseparate constituents of the same species requiresa much longer record. As an example, considerthe length of record required to permit theseparation of constituents M 2 and S2. FromAppendix A we see that their difference in speedis 1.016°/h, so that 3.6.1 gives the length of recordnecessary to separate them as T = 360/1.016 =354 h, or 14.8 days. But to separate N 2 from M2would, by the same reasoning, require a recordlength of T = 360/0.544 = 662 h, or 27.6 days; andto separate K2 from S2, would requireT = 360/0.082 = 4390 h, or 183 days. It should notbe surprising that the separation periods turn out tobe the basic astronomical periods or tractions ofthem, since it was from these that the constituentsinherited their frequencies.

When a record is too short to allowseparation of all the constituents that are known tocontribute significantly to the tide in that region,relationships must be assumed between theinseparable constituents. These are called regionalrelations, because the justification for their adoptionis that waves of such nearly equal frequenciesmust behave very similarly, and so maintain verynearly the same relation to each other over a largeregion. The ratio of amplitudes and the differenceof phaselags of the two constituents are thereforeassumed to be the same as those observed at thenearest comparable location for which a morecomplete analysis is available. In any tidalanalysis, every harmonic constituent must be (a)included directly in the analysis, (b) allowed forthrough regional relations with other constituents,or (c) omitted because it is known to be negligiblein the region of observation. The longer the tidal

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are. A common choice, and one that isrecommended, is that of the true azimuths Northand East. If the current observations were takenin a well defined channel or passage, the directionsalong and across the channel might be chosen forthe components. The result of the analysis of acurrent record is two sets of harmonic constants(amplitudes and phaselags), one for eachcomponent direction. Even though the twodirections are known to be perpendicular to eachother, it is im portant to quote both of them, toavoid a possible 180° uncertainty. Sometimes theconstants for each harmonic constituent are usedto calculate that constituent’ s tidal ellipse (seesection 1.11). The ellipse may be described by thedirection of its major axis, the amplitude andphaselag of the tidal stream component in thatdirection, the tidal stream amplitude in the minoraxis direction, and a statement that the sense ofrotation of the stream is clockwise orcounterclockwise. In tidal ellipse form, thephaselag on the minor axis is always 90° differentfrom that on the major axis. The sense of rotationof the stream tells how the difference should beapplied. There is no objection, of course, to statingexplicitly the direction of the minor axis and thephaselag as well as the amplitude for that direction.The tidal ellipse form for the results of a tidalstream analysis has some advantage if it is wishedto display the individual harmonic constituentsgraphically, but it has the significant disadvantagethat the stream components are resolved indifferent directions for each harmonic constituent.

Tidal streams reflect the presence ofinternal tides as well as of surface tides, and.because of the properties of internal wavesdiscussed in section 1.7, tidal streams may vary indepth, may vary seasonally with changes instratification, and may vary spatially in a patterndifferent trom that of the surface tide. For thesereasons, the coherence between tidal streamrecords taken on different moorings at differenttimes is usually poor, and the records should not becombined into a single analysis. Even if a longcurrent record is obtained from a single mooring, itis usually wise to break the record up into onemonth lengths for separate analysis. Care shouldalso be used in applying regional relations tor theconstituents in tidal stream analysis, because of the

greater variability in tidal streams than in surfacetides in the same region .

3.8 Harmonic Method of Tidal Prediction

Prediction of the tidal height at any desiredtime, t, involves summing the contributions from allthe important harmonic constituents in their properphase for that time, and adding their sum to themean water level, Zo. Expressed mathematically,this is

where the symbols have the same meanings as insections 3.3 and 3.4. The nodal parameters, f andu, and the Greenwich phase of each equilibriumconstituent, E, may be obtained for time t eitherfrom tables or from formulae involving knownastronomical parameters. According to theconvention explained in section 3.3, the values of Emust be obtained for GMT numerically equal to t,for t in the same time zone to which the phaselags,g, refer. Values for the constituent amplitudesand phaselags, H and g, come from a harmonicanalysis of tidal data previously recorded at thatlocation, or, perhaps, from interpolation betweenknown values at nearby locations. There is nomodel of world tides that is accurate or detailedenough to provide local tidal predictions from firstprinciples. Previous observation of the tide in aregion is a prerequisite to its prediction. The word“prediction” as used here includes calculation oftidal heights for past times (hindcasting) as well asfor future times (forecasting).

The prediction of tidal streams is basicallythe same as the prediction of tides, except thatpredictions must be made separately in the twocomponent directions, and then combinedvectorially to give speed and direction. In 3.8.1, Z owould be the component of the steady current inthat direction. Many requirements for current andtidal stream predictions are for channels in whichthe flow is nearly rectilinear along the axis of thechannel. In these cases it is possible to limit thepredictions to one component direction, that alongthe axis of the channel .

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3.9 Prediction of Tidal and Current Extrema

The prediction of the times and heights ofhigh and low water (HW and LW) is usually doneby generating a series of predicted heights athourly intervals and scanning it for changes intrend. When a change in trend (e.g. fromincreasing to decreasing height) occurs,intermediate predictions are inserted until theextreme height at which the trend reverses islocated within a narrow interval. The time andheight of the extremum (HW or LW) are theninterpolated in the interval. Ingenious mechanicalanalogue apparatus was used in the past togenerate the time series, and the scanning wasdone visually. Today, the task is accomplishedalmost exclusively by electronic digital computersprogrammed to perform all the steps.

Most locations for which current predictionsare made are in channels or passages where thecurrent is nearly rectilinear along the axis of thechannel. Predictions are usually published only forthe times and speeds of maximum flood and ebband for the times of slack water. Using the set ofharmonic constants for the component directionalong the channel, a time series of current speedsis generated, with the positive sign for the flooddirection and the negative sign for ebb. This seriescan be scanned to identify maxima and minima inthe absolute value of the speed (i.e. withoutregard to sign). When the maxima are found, thesign of the speed at those times identifies them asfloods ( + ) or ebbs (—). The minima usuallyoccur at zero speed, and thus give the times ofslack water. However, when there is a largeresidual current, the tidal streams may not be largeenough to cause a reversal in the flow, and theminima may not be zero. In these cases theminima are identified as minimum flood or ebbaccording to the sign of the speed, and there is noslack water and no “time of turn.”

For locations in which the currents are notrectilinear, a time series of current vectors may beproduced from the two component sets ofharmonic constants and scanned for maxima andminima in the magnitude (speed) of the vector.Since the concept of flood and ebb is inexact whenthe current is not rectilinear, the extrema should beidentified by time, speed, and direction. In quoting

the direction of a current the convention used isopposite to that used for wind direction: thedirection of a current is the direction toward whichit is flowing. Figure 11 illustrates some of themany patterns that may result from thecombination of a steady current with a rectilinearor rotary tidal stream. If there is a large diurnalinequality in the tidal stream, the patterns could beeven more variable.

3.10 Cotidal Charts

Cotidal charts of the major harmonicconstituents of the tide are frequently constructedto illustrate their different propagation patterns andto locations where no observations exist. A cotidalchart consists of a set of co-phase lines and a setof co-range lines drawn on a suitable chart. Eachco-phase line traces out the locus of points atwhich the constituent has a particular phaselag,and each co-range line traces out the locus ofpoints at which it has a particular amplitude.Provided sufficient intormation is available, there isno limit to how large an area may be covered by acotidal chart for a single harmonic constituent, andsome have been constructed covering the wholeworld ocean. Figure 27 and Figure 28 show cotidalcharts of Hudson and James bays for theconstituents M 2 and K1 respectively. The markeddifference in the two reflects the fact that thebasin responds differently to waves of differentfrequencies. Charts of constituents within thesame species of tide usually resemble each otherquite closely over fairly large regions, and may becombined into composite cotidal charts for thediurnal and semidiurnal species separately.Figure 29 and Figure 30, respectively, show cotidalcharts for the semidiurnal and diurnal tides of theEast Coast and Gulf of St. Lawrence. Figure 31and Figure 32 show the same thing for the WestCoast. Large differences are apparent betweenthe semidiurnal and diurnal charts, with respect tolocation of amphidromes and other characteristics.Clearly, where the tide is of the mixed type (MSDor MD), a cotidal chart that attempted to combinethe two species to represent the total tide couldcover only a very small region.

As will be discussed more fully in Part II,

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cotidal charts must often be prepared for use inreduction of soundings to datum on offshoresurveys. Since they usually attempt to representthe total tide, the extent of the region they maycover depends upon the extent to which the tide isof the mixed type. In cotidal charts for soundingreduction, times and heights are most oftenreferred to the tide at a nearby reference port.The region is commonly divided into two sets ofoverlapping zones, one set being zones in which therange is considered to bear a constant ratio to thatat the reference port. and the other being zones inwhich the arrival time of the tide is considered todiffer by a constant from that at the referenceport.

3.11 Numerical Modelling of Tides

Numerical modelling is becoming more andmore common in modern tidal studies, beingencouraged by the ever-increasing capabilities ofthe electronic digital computer. Modelsincorporate the physical principles of the equationsof motion and of continuity, a description of theshape and the bathymetry of the basin, and a set ofboundary conditions that must be preserved. Theboundary conditions consist of the harmonicconstants at all gauge and current meter sites forwhich analyses exist, and a statement of thecharacter of the tide along any open boundaries.

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The region is divided into a set of grid points,closely enough spaced to define the characteristicsbeing investigated. The computer is programmedto commence with an initial set of elevations at allgrid points and to change the elevationsprogressively in accordance with the physicalprinciples and the boundary conditions. In thisway, the progression of the tide can be modelledover as many cycles as desired. The tidal streamsare also modelled in the same operation, through

the application of the principle of continuity. Thecotidal charts of Figure 27 and Figure 28 are theresult of numerical modelling of the tides in HudsonBay. Figure 33 shows current vectors for onestage of the tide in Chignecto Bay and MinasBasin, as deduced by numerical modelling, andFigure 34 shows a similar result from numericalmodelling in the Strait of Georgia and Juan de FucaStrait.

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CHAPTER 4

Meteorological and Other Non-Tidal Influences

4.1 Introduction

Because the tide usually dominates thespectrum of water level and current fluctuationsalong the ocean coasts, it is common to think ofnon-tidal fluctuations mostly in connection withinland waters. The tide in the deep ocean,however, can be quite insignificant, and, as shownin section 1.5 and Table 2, the tidal streamscompletely negligible from the standpoint ofnavigation. Wind-driven surface currents in thedeep ocean, on the other hand, are of majorimportance to navigation. Water levels along oceancoasts are just as surely affected by atmosphericpressure and wind as are water levels along theshores of inland bodies of water. The range,however, is generally small compared to that of thetide on the coast, and the importance may not befully realized until an extreme of the non-tidalfluctuations coincides with a correspondingextreme (high or low) of the tidal fluctuation. Inusing tidal predictions, such as those in theCanadian Tide and Current Tables, it should beborne in mind that they contain no allowance fornon-tidal effects, other than for the averageseasonal change in mean water level. The non-tidal influences are discussed below with referenceboth to ocean and inland waters.

4.2 Wind-driven Currents

The major current systems of the ocean aredriven by the wind stress acting on the surface.The direct effect of the wind stress is transmittedonly to a limited depth by viscosity and turbulence,but the pressure gradients resulting from theinduced surface slopes can set up deep flows indirections different from those of the surfaceflows. The main surface current systems of theAtlantic and Pacific oceans are in the form oflarge gyres that occupy most of the width of theocean and are clockwise in the NorthernHemisphere and counter-clockwise in the SouthernHemisphere. The Coriolis acceleration isresponsible for these circular patterns, deflecting

both the winds and the currents driven by thewinds. It may seem surprising that these“permanent” large-scale features of the oceancirculation could be the result of something ascapricious as the wind. But, while the wind doesvary from day to day with the passage of weathersystems, it has a fairly consistent average patternover much of the ocean, as witnessed by theDoldrums near the equator, the Trades in thetropics, and the Westerlies at mid-latitudes.Changes in the ocean current systems associatedwith seasonal changes in the average wind fieldare not well documented, except in certain areassuch as the northern Indian ocean, where there is amarked difference between the current patternduring the southwest Monsoons of northernsummer and that during the northeast Monsoons ofnorthern winter.

Ekman, a Swedish mathematician andoceanographer, demonstrated that, in the absenceof constraining boundaries, the surface currentshould flow in a direction 45° to the right of thewind stress in the Northern Hemisphere, and thatover the whole water column there should be a nettransport of water 90° to the right of the windstress . Observations indicate that the wind-drivensurface current flows at about 20° to 25° to theright of the wind as measured ten metres abovethe surface, and with a speed about two per centof that of the wind. If, in the NorthernHemisphere, the wind blows parallel to a coastlineon its right, the Ekman transport piles water againstthe coast until a surface slope is created to balancethe Coriolis force; the current then flows parallel tothe coast in the direction of the wind. If thecoastline is to the left of the wind, the surfacewater is displaced away from the coast and deeperwater rises to replace it. This “upwelling’’ is ofbiological importance in that it brings chemicalnutrients back up into the euphotic zone (depthpenetrated by sunlight), where they can be utilizedin the growth of marine vegetation.

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4.3 Wind Set-up

The term wind set-up refers to the slope ofthe water surface in the direction of the windstress. The slope perpendicular to a wind blowingalong a coast, mentioned in the previous section,balances the Coriolis force on the along-shorecurrent driven by the wind. It is an indirect effectof the wind, but is not usually thought of as windset-up. When a wind commences to blow acrossthe water surface, the wind stress is initiallyoccupied in accelerating the water. When a steadystate has been achieved, and the water is no longeraccelerating, the balance of forces must bebetween the pressure gradient force due to thesurface slope and the surface and bottom stress onthe water (due to wind and bottom drag). Figure35 illustrates the balance of forces in the directionof the wind stress for a wind blowing towardshore, or along the axis of a lake or coastalembayment. The wind is assumed to have beenblowing long enough for a steady state to havebeen reached, so the currents and the surfaceslope are not changing with time. The wind stressis just sufficient to maintain the surface currentagainst the surface slope and the drag of theslower moving water beneath it. The pressuregradient caused by the surface slope is the same atall depths, and below a certain depth it drives areturn flow. This flow is opposed by the bottomstress, or drag. The total drag force on a columnof water of unit cross section and spanning theentire depth, D, is τa, + τb, where τa is the surfacewind stress and τb is the bottom stress. The

horizontal pressure gradient is ρwgi, where ρw isthe water density, g is gravity, and i is theinclination of the surface in the wind direction.The pressure gradient force on the total volume ofthe same column of water is therefore ρwgiD. Thebottom stress is usually considered small andproportional to the wind stress, and the balance offorces is approximately

(4.3.1) τa = ρwgiD

The wind stress is equal to the drag coefficient forair on water, C, times the density of air, ρa, timesthe square of the wind speed, W. We may thuswrite 4.3.1 as

(4.3.2) i = (CρaW2)/(ρwgD)

The drag coefficient is not a precise constant. buthas a value of approximately 2 x 10 -3, ρa/ρw, is 1.3x 10-3 , and g is 10 m/s2. Thus 4.3.2 becomes

(4.3.3) i = 2.6 x 10 -7 (W2/D)

for W in metres per second and D in metres.

Observations on lakes Ontario, Erie, andHuron, the Gulf of Bothnia and elsewhere indicatethat the constant in 4.3.3 is too low, and should bebetween 4 x 10 -7 and 5 x 10-7. This is probablybecause of the neglect of the bottom stress in 4.3.1and because of some funneling effects toward theends of the lakes. Taking the larger experimentalvalue for the constant, and expressing i as ∆h/L,where ∆h is the difference in water level over thelength L,4.3.3 gives

(4.3.4) ∆h = 4.5 x 10-7 (W2L/D)

with all dimensions in metres and seconds.

We see from 4.3.4 that the difference inelevation between two ends of a lake, caused by awind blowing along its length, is proportional to thesquare of the wind speed and the length, butinversely proportional to the depth. Wind set-up isthus of particular importance in shallow bodies of

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water with large horizontal extent. This isdemonstrated by comparison of the values of ∆hfor Lakes Erie and Ontario given in Table 3 forvarious wind speeds along the lengths of the lakes.While the values in Table 3 have been derivedfrom substitution in 4.3.4. examination of wind andwater level data on these lakes confirms therelationship very closely. Because of the action ofthe Coriolis force, the axis of greatest slope wouldbe slightly to the right (NH) of the wind direction.The line of flow of the surface current and thereturn deep current (Fig. 35) would also beoriented to the right of the wind.

4.4 Atmospheric Pressure Effect

The depression of the water surface underhigh atmospheric pressure, and its elevation underlow atmospheric pressure, is frequently referred toas the “inverted barometer” effect. In a standardmercury-in-glass barometer, one atmosphere ofpressure supports 0.76 m of mercury; if waterwere used instead of mercury in the barometer, theheight of the column supported would be 10metres. Since one atmosphere is approximately100 kilopascals (kPa), we have the barometricequivalence of 10 cm of water and I kPa (or1millibar of pressure and I centimetre of water).Of course, if the water level is to rise in one place,it must fall in another; clearly, the level in a glass ofwater does not drop by 10 cm when theatmospheric pressure rises by 1 kPa. It is theslope of the water surface that adjusts to theatmospheric pressure gradient along the surface,so that in the absence of other forces, if theatmospheric pressure at A exceeds that at B by I

kPa, the water level at B will exceed that at A by10 cm.Stated another way, other forces again beingabsent, the water level at any location on a body ofwater differs from the mean surface level by anamount equivalent (but in the opposite sense) to thedifference between the local and the averageatmospheric pressure over the same body ofwater. The ocean is sufficiently large that it isfairly safe to assume that the average atmosphericpressure over its surface is constant, and that theinverted barometer effect is therefore fullyexperienced at each location. On lakes, however,a constant average pressure cannot be assumed,and water level differences from place to placemust be treated instead of changes at one locationonly.

The change in water level caused bypressure change cannot easily be separated fromthat caused by wind set-up, because the winds aredriven by the pressure gradients, and the two areclosely correlated. It is usually best to assume thatthe pressure compensation is complete, and tocredit the wind with the remaining surface slope.The justification for the assumption that thepressure compensation is complete is that thesurface disturbance travels at the speed of a longwave, (gD)1/2, which is usually fast enough to keeppace with moving weather systems. An interestingand useful result of the inverted barometer effect isobserved in records from self-contained pressuregauges moored on the ocean bottom (section 6.7).Since these gauges are not compensated foratmospheric pressure, they record the sum ofatmospheric and hydrostatic pressure. Thecompensation for changes in atmospheric pressureby changes in hydrostatic pressure, provided by the

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inverted barometer effect, is so nearly completethat most of the ‘’noise” usually found in a tidalrecord disappears.Because there is no significant tidal signal in thewater level fluctuations imposed by variations inatmospheric pressure, the loss of this part of thewater level record simply leaves a cleaner recordfor tidal analysis. This is not, of course, a desirablefeature if it is wished to record actual water levelsfor navigation or charting.

4.5 Storm Surges

As the name suggests, storm surges arepronounced increases in water level associatedwith the passage of storms. Much of the increaseis the direct result of wind set-up and the invertedbarometer effect under the low pressure area nearthe centre of the storm. There is. however,another process by which the surge may becomemore exaggerated than would be anticipated fromthese two effects alone. As the storm depressiontravels over the water surface, a long surfacewave travels along with it. If the storm path issuch as to direct this wave up on shore, the wavemay steepen and grow as a result of shoaling andfunneling, as discussed for long waves in general insection 1.12. The term ‘’negative surge” issometimes used to describe a pronounced non-tidaldecrease in water level. These could beassociated with offshore winds and travelling highpressure systems, and are not usually as extremeas storm surges. Negative surges may, however,

be of considerable concern to mariners, since theycan create unusually shalIow water if they occurnear the low tide stage.

4.6 Seiches

A seiche is the free oscillation of the waterin a closed or semi-enclosed basin at its naturalperiod. They were discussed in section 1.6, andthe formulae for the natural period of closed andopen basins were given in equations 1.6.1 and 2.Seiches are frequently observed in harbours, Iakes,bays. and in almost any distinct basin of moderatesize. They may be caused by the passage of apressure system over the basin or by the build-upand subsequent relaxation of a wind set-up in thebasin. Following initiation of the seiche, the watersloshes back and forth until the oscillation isdamped out by friction. Seiches are not apparentin the main ocean basins, probably because there isno force sufficiently co-ordinated over the ocean toset a seiche in motion. The tides are not seiches,being forced oscillations at tidal frequencies. If thenatural period, or seiche period, is close to theperiod of one of the tidal species, the constituentsof that species (diurnal or semidiurnal) will beamplified by resonance more than those of otherspecies. The constituent closest to the seicheperiod will be amplified most of all, but theresponse is still a forced oscillation whereas aseiche is a free oscillation.

A variety of seiche periods may appear inthe same water level record because the main

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body of water may oscillate longitudinally orlaterally at different periods, it may also oscillateboth in the open and closed mode if the open end issomewhat restricted, and bays and harbours offthe main body of water may oscillate locally attheir particular seiche periods. Seiches generallyhave half-lives of only a few periods, but may befrequently regenerated. The largest amplitudeseiches are usually found in shallow bodies ofwater of large horizontal extent, probably becausethe initiating wind set-up can be greater underthese conditions. Table 4 lists a few seiche periodsthat have been observed and/or calculated forsome Canadian waters. The list is by no meansexhaustive, since seiches can be identified onalmost any water level record; the entries in thetable have been chosen simply to illustrate some ofthe principles mentioned above. The ‘typicallarge” ranges listed are not extremes, but aretypical of perhaps the largest 10 or 20% ofobserved seiches. The last two entries in Table 4are included not because of actual seiche activityin the Bay of Fundy, but because of interest in thepart played by resonance in the large Bay of Fundysemidiurnal tides. For a long time study wasconcentrated on the resonant (or seiche) period ofthe Bay of Fundy alone, whereas it is now believedthe important resonance is that of the tide with theoscillation of the combined Bay of Fundy - Gulf ofMaine system.

4.7 Precipitation, evaporation, and runoff

The precipitation and evaporation consideredhere are those that occur at the water surface, notthose that occur elsewhere in the drainage basin.The runoff is all the water that flows into the watersystem in question, and thus is the net result ofprecipitation, evaporation, and absorption of waterover the land portion of the drainage basin. In thewater budget of a system, precipitation is a positiveterm, evaporation a negative term, and runoffusually a positive term. In very arid regions, runoffcould be negative by virtue of absorption of waterinto the parched soil along the shores; we arefortunate that in Canada this would be a rarityindeed. If the rate of input to the system from thesum of the three terms exceeds the rate of outflowat its mouth, the water levels within the system

must rise, and, conversely, if the rate of outflowexceeds the rate of input, the water levels mustfall. If there is no control on the outflow of asystem, such as might come from dams, log jamsor ice jams, the outflow would increase ordecrease steadily with the rise or fall of the waterlevel until an equilibrium was achieved betweeninput and outflow. This is the basis for establishing“stage-discharge relations” from which the flowcan be judged from the water level; they are validonly at locations below which there are no controlstructures or barriers.

There are seasonal variations in precipitationand evaporation that reflect in seasonal variationsin water level and outflow of inland water systems,but the most dramatic changes are thoseassociated with changes in runoff. Runoff reflectsprecipitation over the whole drainage basin, whichmay cover many times the area of the actual watersurface. During a heavy sustained rainfall, only aportion of the water can be absorbed into theground, and the runoff from a large drainage areacan cause “flash flooding” of a water system. TheGreat Lakes system is unusual in that it has a smalldrainage area in relation to its large water surface,so no dramatic changes in level or outflow occur inthat system. The water storage capacity of theland area of a drainage basin is greatly increased inwinter, when much of the surface and groundwater is locked up as ice, and the precipitationaccumulates as snow cover. As the ice and snowmelt in the spring, the runoff can increase rapidly,resulting in the spring “freshet” in streams andrivers, and in raised water levels throughout thesystem. Sea level along open coasts is notnoticeably affected by precipitation, evaporation,and runoff because their net average for the wholeocean is close enough to zero not to affect theelevation of such a large surface area. Water levelrecords from a harbour at the mouth of a rivermay, however, reflect fluctuations in runoff. Whilethe average water budget for the ocean isessentially zero, the local budgets are not, andwater must be moved from place to place tominimize the occurrence of bumps and hollows onthe surface. Significant currents are thus set up inthe ocean to disperse water away from regions ofhigh precipitation and/or runoff, and to divert watertoward regions of excess evaporation (the ocean“deserts”). These currents do not flow directly

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from regions of budget surplus to regions of budgetdeficit, but are deflected, as are all currents, by theCoriolis force.

4.8 Effect of Coriolis Force on Currents

The Coriolis force was introduced anddiscussed in section 1.8. Its effect on tidepropagation was considered in sections 1.8 andI.10, and its effect on wind and wind-drivencurrents was mentioned in section 4.2. The freshwater runoff from land provides another exampleof the Coriolis force in action. Instead ofcontinuing to flow directly away from the coast,this water is deflected to the right (NH) and formsa coastal current flowing along the coast to theright, for an observer facing seaward in theNorthern Hemisphere (left in SH). In large lakes,bays and gulfs the runoff and the Coriolis forcecontribute to a cyclonic circulation, and around alarge island they contribute to an anticycloniccirculation. The term “cyclonic” refers to rotationin the same sense as the earth’s rotation on itsaxis. It is counter-clockwise when viewed fromthe Northern Hemisphere, and clockwise when

viewed from the Southern Hemisphere. The term“anti-cyclonic” has just the opposite meaning. Useof these terms avoids continual reference to thereader’s chosen hemisphere. The main winddriven ocean current gyres are, in this terminology,anti-cyclonic. The Coriolis effect on currents isillustrated in surface current charts of the northAtlantic and Pacific (Fig. 36), the Gulf of St.Lawrence and East Coast (Fig. 37), and the Straitof Georgia (Fig. 38) between Vancouver Islandand the B.C. mainland.

It must be remembered that the Coriolisforce does not generate currents, nor does it speedthem up: its action is always perpendicular to themotion, and so can only change the direction offlow. The main anti-cyclonic gyres in both oceanssouth of latitude 45° N are driven by the anti-cyclonic wind stress and are reinforced in theiranti-cyclonic pattern by the Coriolis force. Thesmaller cyclonic gyres north of latitude 45° N havethis pattern partly because of the shape of thebathymetry and the coastline, and partly becausethe winds have a more cyclonic pattern at higherlatitudes. The cyclonic circulations in the Gulf ofSt. Lawrence and Hudson Bay result from runoff

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and a slightly cyclonic average wind stress,assisted by Coriolis force, which likes to pile waterup on the right-hand shore in the North. Thecirculation pattern in the Strait of Georgia fits thepattern of coastal runoff deflected to the right bythe Coriolis force. In all of the above cases, thebathymetry of the basins also plays an importantrole in shaping the circulation patterns, but we shallnot pursue that aspect here.

Inertial currents were the subject of section1.9, but are mentioned here again because of theirfrequent appearance in current records from oceanmoorings. They may be recognized in a record bytheir characteristic frequency, (12/sinj) hours,where j is the latitude. Inertial oscillations are notcontinuous in most records, but may appear andreappear several times, somewhat in the manner ofa seiche in a water level record. The inertialsignature is almost always present in currentrecords from the deep ocean, but is rarely seen inrecords from shallow coastal regions. The senseof rotation is always anti-cyclonic around theinertial circle.

4.9 Estuarine Circulation

An estuary is, for our purposes, any semi-enclosed body of water that has free access to the

.sea, a significant intrusion of sea water, and aninflow of fresh water. The mouths of rivers thatflow into the sea are estuarine as far upstream asthe limit of salt penetration, and most coastalharbours receive enough fresh water runoff toqualify as estuaries. The St. Lawrence system isestuarine from the limit of salt penetration atQuebec City through the Gulf to Cabot Strait andthe Strait of Belle Isle. Estuarine circulation is asystem of oppositely directed surface and deepcurrents driven by the mixing of the outflowingfresh water with the underlying salt water. Figure39a illustrates in a simplified manner the principlesof estuarine circulation; the fresh water is shownas if it entered only at the head of the estuary. Thesurface of the estuary slopes down toward the sea,and the fresher surface water flows seawarddown this pressure gradient, becoming saltier as itmixes with the underlying water along its way.How much mixing takes place between the freshand the salt water depends to a large extent uponthe wind and the tidal action, but, qualitatively atleast, the result is to tilt the isohalines (surfaces ofequal salinity) down toward the head of the estuaryas shown. The average density, which is roughlyproportional to the salinity, is seen to be less for acolumn of water at the head of the estuary than fora column at the mouth. Because of this, thehydrostatic pressure increases more rapidly withdepth at the mouth than at the head of the estuary,and the seaward-sloping pressure gradient near thesurface may be replaced by a pressure gradient in

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the opposite direction at greater depths. This deephorizontal pressure gradient drives a return flow ofsaltier water into the estuary beneath the fresheroutflow.If conditions remain unchanged for a sufficientlylong time, a steady state would be reached inwhich as much salt is being transported into theestuary in the deep layer is being transported out inthe surface layer, and the volume outflow of waterexceeds the volume inflow by exactly the amountof the fresh water input. Without furtherconsideration of the dynamics (i.e. the forcesinvolved). we may find an interesting relationsimply from the principles of continuity (i.e. theconservation of matter and the continuous natureof a liquid).

Consider a vertical cross-section of theestuary across the flow, and let S o, and Si be theaverage salinities of the outflowing and inflowingwater respectively, V o and Vi be their volumetransports, and R be the rate of volume input offresh water. To conserve a steady state for thevolume of water inside the estuary,

(4.9.1) Vo - Vi = R

and to conserve a steady state for the amount ofsalt inside,

(4.9.2) SoVo = SiVi

Solution of 4.9.1 and 2 for V o and Vi gives

(4.9.3) Vo = R [Si / (Si - So)]

and Vi = R [So / (Si - So)]

The volume transports in and out of the estuarydepend critically on their salinity difference as wellas on the fresh water input rate. The amount ofmixing between the fresh and salt water along theestuary is therefore very important since itdetermines the salinity difference. If we imaginethe unrealistic situation in which there is no mixing,and the fresh water simply flows out over theundisturbed salt water beneath, S o would be zeroand 4.9.3 would give V o = R and Vi = 0. A morerealistic example is the St. Lawrence estuary nearRimouski, Que., for which S o is approximately 30

parts per thousand, S i is 34 parts per thousand, andR is about 10 000 m 3/s. From these values, 4.9.3gives Vo as 85 000 m3/s and Vi as 75 000 m3/s.Taking the width of the estuary at Rimouski as 45km, the depth of the upper layer as 50 m, and thatof the lower layer as 250 m, we may calculate thecross-sectional areas through which these volumetransports are flowing, and so convert thetransports into mean velocities in the two layers.This gives the mean outflow velocity in the upperlayer as 0.04 m/s, and the mean inflow velocity inthe lower layer as 0.01 m/s. These values agreewell with observation, but they must be recognizedas average values only. There is usually a strongvertical shear in the velocity, with the largestvalues near the surface, and both the outflow andinflow are usually stronger on their respectiveright-hand sides because of the Coriolis force.

The effect of earth rotation (i.e. Coriolisforce) on a simple estuarine circulation is shown inFig. 39b, a cross-sectional view of the estuary.The surface slopes up across the channel towardthe right side of the outflow, and the isohalinesslope down toward the same side. The strongestoutflow is at the surface on the right side (NH)facing out. The strongest inflow is on its right sidefacing in, but is not always strongest at the bottom.The outflowing Gaspe Current (Fig. 37) is at leastpartly the result of intensification of the surfaceestuarine outflow along the Gaspe shore becauseof earth rotation.

The Mediterranean Sea is an example of abody of water in which the evaporation exceedsthe sum of the precipitation and the runoff. Such abody of water does not qualify by our definition asan estuary, but the term “negative estuary” issometimes applied to it, and the formulae in 4.9.3may be used with the negative value of R tocalculate what is now a surface volume transportinward and a deep volume transport outward.What happens physicalIy in this case is thatevaporation lowers the level of the surface,causing surface water from the ocean outside toflow inward down the slope. The evaporation alsoraises the salinity of the inside water, making theaverage density of a column of water insidegreater than that of a corresponding column ofwater outside. Below a certain depth this reversesthe direction of the pressure gradient, and drives a

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deep flow of high salinity water from the inland seato the ocean. The effect of earth rotation is to tiltthe surface up on the right side of the inflow and totilt the isohalines in the opposite sense.

Figure 40a and b illustrates the situationwhen R is negative. The picture is in every waysimilar to that in Fig. 39a and b, but with thedirections of flow reversed. The warm and saltywater that flows out from the Mediterraneanthrough the Strait of Gibraltar can be detected atintermediate depths far out in the mid-Atlantic.

In the above discussion, no mention hasbeen made of temperature as a factor indetermining the density of seawater. In the openocean, where salinity differences are small,temperature is in fact the controlling factor fordensity, and colder water almost invariablyunderlies warmer water. In estuaries and othercoastal regions, however, large salinity variations

are common, and it is they that usually determinethe density, with temperature inversions frequentlyoccurring.

4.10 Melting and Freezing

When seawater freezes, it is only the waterthat forms into ice crystals. The salt becomestrapped between the crystals in a concentratedbrine that eventually leaches out, leaving mostlypure ice floating on the surface, surrounded by seawater of increased salinity and density. Since theice displaces its own weight in this denser water, itdoes not displace as much volume as it occupiedbefore freezing. Because of this, freezing has aneffect similar to that of evaporation - it lowers thewater level and increases the surface salinity anddensity. Surface water must therefore flowtoward a region of freezing, while the cold saltywater that is formed must sink and flow away

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from the region. In the polar regions, particularlyin the Antarctic. freezing produces cold saltywater that sinks and flows along the ocean bottomfor thousands of kilometres. When sea ice melts,mostly fresh water is released, and this decreasesthe salinity and the density of the surroundingwater. Melting thus has an effect similar to that ofprecipitation - it raises the water level anddecreases the surface salinity and density.Surface water must therefore flow away from aregion of melting ice. The speed of currentsassociated with freezing and melting in the oceanare never great.

4.11 Tsunamis

A tsunami is a disturbance of the watersurface caused by a displacement of the sea-bedor an underwater landslide, usually triggered by anearthquake or an underwater volcanic eruption.

The surface disturbance travels out from thecentre of origin in much the same pattern as do theripples from the spot where a pebble lands in apond. In some directions the waves may almostimmediately dissipate their energy against a nearbyshore, while in other directions they may be free totravel for thousands of kilometres across the oceanas a train of several tens of long wave crests.Being long waves, they travel at the speed (gD) 1/2,giving them a speed of over 700 km/h (almost 400knots) when travelling in a depth of 4,000 m. Theperiod between crests may vary from a fewminutes to the order of 1 h, so that in a depth of4,000 m the distance between crests might rangefrom less than a hundred to several hundredkilometres. The wave heights at sea are only theorder of a metre, and over a wavelength of severalhundred kilometres this does not constitute asignificant distortion of the sea surface. Whenthese waves arrive in shallow water, however,their energy is concentrated by shoaling and

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possibly funneling (section I.12) causing them tosteepen and rise to many metres in height. Notonly are the tsunami waves high, but they are alsomassive when they arrive on shore, and arecapable of tremendous destruction in populatedareas. Because of the relative gentleness oftsunamis in deeper water, ships should alwaysleave harbour and head for deep offshore safetywhen warned of an approaching tsunami. Theorigin of the word is, in fact, from the Japaneseexpression for “harbour wave.” This name hasbeen adopted to replace the popular expression“tidal wave,” whose use is to be discouraged sincethere is nothing tidal in the origin of a tsunami.Another expression sometimes used for thesewaves is “seismic sea wave,” suggesting theseismic, or earthquake, origin of most tsunamis.

A tsunami warning system for the Pacifichas been established by the United States, with itsheadquarters in Honolulu, Hawaii. Othercountries, including Canada, that border on thePacific have since been recruited into the system.

Canada’s direct contribution consists of twoautomatic water level gauges programmed torecognize unusual water level changes that couldindicate the passage of a tsunami, and to transmitthis advice to Honolulu. The gauges are at Tofinoon the west coast of Vancouver Island, and atLangara Island off the northwest tip of the QueenCharlotte Islands group. The tsunami warningcentre at Honolulu receives immediate informationfrom seismic recording stations around the Pacificof any earthquake that could possibly generate atsunami; it calculates the epicentre and intensity ofthe quake and the arrival time of the as yethypothetical tsunami at the water level sensingstations in the network; it initiates a “tsunamiwatch” at all water level stations in the path, for agenerous time interval around the ETA of thehypothetical tsunami; and it issues tsunamiwarnings to the appropriate authorities inthreatened locations if the water levelinterpretation indicates that a tsunami has indeedbeen generated.

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CHAPTER 5

Datums and Vertical Control

5.1 Vertical Datums

It is apparent that the elevation of somethingcan only be expressed relative the elevation ofsomething else, whether the reference elevation bethat of the centre of the earth, the mean surface ofthe sea, the orbit of a satellite, or simply a bolt setin bedrock. The chosen zero to which otherelevations are referred is called a datum of verticalreference or simply a vertical datum. The latterterm is most commonly used. but it must not bemisinterpreted to mean that the datum is vertical.The plural of “datum” is “datums” in this context,to distinguish it from the word “data” that signifiesany set of observed values of a parameter. If thedatum is defined over an area, it is called a datumsurface.

A distinction must usually be made betweenthe concept chosen to define a vertical datum andthe realization of the concept in practice. Forexample, two agencies may both choose mean sealevel as their reference surface, and they mayindependently determine a value for the elevationof the same benchmark. Because of differencesin technique and errors of observation, the twovalues would almost certainly differ. The twodatums would then be said to differ by that amountat the location of the benchmark, even though theyprofess to have the same ideal reference surface.It is not unusual to have elevations assigned to thesame benchmark by several survey organizations.

5.2 Equi-geopotential orLevel Surfaces

These are surfaces along which no work isdone by or against gravity in moving from one pointto another. The concept of a force potential wasintroduced in section 2.4, and equi-geopotential orlevel surfaces are simply surfaces of constantpotential in the earth’s gravity field. Gravity actseverywhere perpendicularly to level surfaces, andthey are the surfaces to which all water levelswould eventually conform in the absence of allforces other than gravity. The geoid is the levelsurface that most closely fits the mean surface of

the world’s oceans. The term “Mean Sea Level(MSL)” is frequently loosely used, without cleardefinition of its intended meaning; for our purposes,the surface of MSL will be defined as identical to thegeoid. By this definition it is clear that the meanelevation of the sea surface at a particular locationneed not be the same as the elevation of MSL, sincethe elevation of MSL (the geoid) could bedetermined only by fitting a level surface toobservations of the mean level of the sea surfaceover the whole of the ocean. The local mean waterlevel (MWL) departs from MSL in the oceanbecause of surface slopes caused by prevailing windstress patterns, persistent anomalies in thedistribution of precipitation, evaporation, freezing,melting, heating and cooling, and by the deflection ofocean currents by the Coriolis force.

5.3 Geopotential, Dynamic, and OrthometricElevations

The difference in geopotential between twolevel surfaces equals the work done in raising a unitmass from the lower to the higher surface. Sincethis amount of work equals the vertical distancebetween the two surfaces times the average gravityalong the vertical path, division by a standard valueof gravity (e.g. the average value of gravity at sealevel for a specified latitude) gives a number equal tothe linear vertical separation of the two surfaces at alocation where gravity equals the standard gravity.This number is the difference in geopotentialelevation, and is quoted in units such as geopotentialmetres. All points with the same geopotentialelevation above the geoid (MSL) must, by definition,lie on the same level surface, since the geoid is itselfa level surface. To perform geopotential levelling inthe field requires a knowledge of the value of gravityalong the path of the levelling. To obtain differencesin geopotential elevation, all instrumentally observeddifferences in elevation along the line must bemultiplied by the ratio of the local gravity to thestandard gravity.

The concept of dynamic elevation is preciselythe same as that of geopotential elevation, the onlydifference being that geopotential levelling uses an

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observed value of local gravity in correcting theinstrumental differences in elevation, whereasdynamic levelling uses a value calculated from agravity formula involving latitude and altitude only.Local gravity anomalies can thus introduce localerrors into dynamic levelling, in addition to anyinstrumental errors committed. The errorsintroduced by the anomalies tend to cancel out asthe anomalies are passed, and do not accumulateover a long line of levelling.

The name given to the concept of verticallinear distance above the geoid is orthometricelevation. Although this may at first seem to bethe most straightforward definition of elevation, itwill be seen to have some drawbacks. Theaverage shape of a level surface on the earth isthat of an oblate spheroid, with its centre at theearth’s centre and its axis of revolution along theearth’s axis. The family of level surfaces areconcentric, but they are not all the same shape;their spheroidal shape becomes progressivelyflatter with distance from the centre of the earth.This progressive flattening is illustrated in Fig. 41,and simply follows from the definition ofgeopotential and the fact that gravity is greatest atthe poles and decreases with latitude, being least atthe equator. The decrease of gravity with latituderesults from the increasing outward centrifugalforce near the equator and from the flattenedshape of the earth itself, which is probably alsoattributable to centrifugal force during the earth’sformative period. The oblateness of all surfaces inFig. 41 is greatly exaggerated to illustrate theprinciple, since if drawn to scale at this reduction,all level surfaces would be indistinguishable fromspheres. Local gravity anomalies cause localincreases or decreases in the separation betweenneighbouring level surfaces, but no attempt hasbeen made to illustrate this in Fig. 41.

To demonstrate the correction toinstrumental differences in elevation that isrequired to obtain orthometric differences, considera line of levelling run from south to north along alevel surface from A to B in Fig. 41. Since thelevelling is along a level surface, there would be nodifference in elevation detected instrumentally;however, because the level surfaces convergetoward the north, B is at a lower orthometricelevation than is A. The orthometric correction to

the instrumental difference observed in levellingfrom A to B is the height BB’, where AB’ liesalong a surface that is parallel to the geoid. Theamount of the correction is calculated fromformulae involving latitude, altitude, and the north -south extent of the line. The formulae for thedynamic and the orthometric corrections are basedon the same model of earth gravity, making thetwo systems mutually convertible. Local gravityanomalies introduce local errors into orthometriclevelling, but, as in dynamic levelling, the errorstend to cancel out as the anomalies are passed, anddo not accumulate over a long line of levelling.

The greatest objection to an orthometricsystem of elevations is that points on the samelevel surface are not given the same elevation ifthey are at different latitudes. This is particularlydisturbing for hydraulic and hydrodynamic studieson lakes and rivers. As an example, theorthometric elevation of the level surface of LakeWinnipeg (no wind, etc.) would be about 0.08 mless at the north end than at the south end. Whenworking at sea level there is no orthometric ordynamic correction to apply, because bothcorrections are approximately proportional to thealtitude. In small local surveys, such as betweenthe control benchmarks and a water level gauge,there would probably never by any need to correctthe instrumental differences to either the dynamicor orthometric system, because the ranges ofelevation and latitude would be too small togenerate a significant correction.

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5.4 Geodetic Datum

Geodetic Datum (GD) is the referencesurface to which the Geodetic Survey of Canadarefers elevations. It is referred to as a sea-leveldatum because it professes in concept to be thegeoid, which is also called Mean Sea Level. Inpractice, of course, it can only be an approximationto the geoid, and its physical location is preciselydefined only with reference to Geodeticbenchmarks in a region. Geodetic Datum existsonly as a concept in regions not yet included in theGeodetic network of vertical control. Elevationsabove the Geodetic Datum are always quoted inthe orthometric system. Geodetic Datum (i.e. theelevations of the benchmarks in the network) isbased on a 1928 adjustment of the Canadianlevelling network, in which the mean water levelsat the gauging stations of Halifax, Yarmouth,Pointe-au-Pere, Vancouver, and Prince Rupertwere all held fixed at zero. Since the mean seasurface is known not to be a level surface (section5.2), Geodetic Datum is seen to have departedimmediately from the precise concept of the geoid.It was realistically reasoned, however, that theerrors introduced by equating MWL to MSL wereless than those incurred in long lines of landlevelling.

5.5 International Great Lakes Datum(1955)

International Great Lakes Datum (1955), orIGLD, is a datum established by the Canada - U.S.Coordinating Committee on Great Lakes BasicHydraulic and Hydrological Data, to provide aunified datum for use in hydraulic and hydrologicalstudies on both sides of the border along the GreatLakes and St. Lawrence River. Theestablishment of IGLD also met the need for arevision of datums in the Great Lakes regioncaused by the cumulative effect of crustalmovement over the years. Crustal tilting in theGreat Lakes basin appears capable of raising oneend of a lake with respect to the other end by asmuch as one metre in three hundred years. IGLDmay be referred to as a sea-level datum, but, inrecognition of the fact that sea level varies from

place to place, it was defined as the level surfacepassing through the mean water level at the outletof the system. In practice, this was determined asthe mean level at Pointe-au-Père over the periodfrom 1941 to 1956, and extended throughout thesystem by a network of over-land levelling andwater level transfers. Dynamic elevations areused in the IGLD system because it was wished tohave the same elevation quoted for all points on thesame level surface, since water surfaces seeklevel surfaces, not surfaces equidistant above thegeoid. The geopotential system would have beenmore desirable still, but insufficient gravityinformation was available at the time. Thestandard gravity used to convert geopotentialnumbers into dynamic elevations is the averagesea-level value of gravity at 45° latitude. Thelength of a dynamic metre at a particular locationtherefore equals the length of a linear (ororthometric) metre times the ratio of the standardto the local gravity, a ratio that is very close tounity in the region served by IGLD. Thedistinction in units can usually be ignored whendealing with the small instrumental differences inelevation encountered in small local surveys.

Because the physical location of a datum isdefined locally by the elevations of benchmarksthat move with the earth’s crust, crustal movementhas continued to distort IGLD with respect to levelsurfaces since its establishment (as it has alldatums in the region). In the interest ofconsistency within the system, the elevation of anew benchmark should be established only bytransfer from a nearby original benchmark, on theassumption that the difference in crustal movementbetween two nearby locations is small. Eventually,the distortion within the system may becomeintolerable, and a complete re-levelling andreadjustment of the network undertaken. Newelevations would then be assigned to allbenchmarks, and the new datum would beidentified by its date of adjustment.

5.6 Hydrographic Charting Datums

Depths and elevations shown onhydrographic charts must be below and abovespecified datum surfaces. For purposes of

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navigational safety, depths are referenced to a lowwater datum and elevations to a high water datum,so only rarely could there be less depth or lessclearance than that charted. Water level gaugemeasurements and tide height predictions must alsorefer to specified datums. It is universal practiceto reference the water levels and the tidepredictions to the same datum as that used forcharted depths, so addition of the observed orpredicted water height to the charted depth willgive the appropriate total depth.

Chart datum (CD) is the datum to whichdepths on a published chart. all tide heightpredictions, and most water level measurementsare referred. It was agreed in 1926 by memberstates of the International HydrographicOrganization that chart datum “should be a planeso low that the tide will but seldom fall below it.”The wording indicates that the resolution wasformulated with only tidal waters in mind, and,since the word “seldom” was left undefined, itprovides but a qualitative instruction for the choiceof chart datum. The following three criteria placesomewhat more restriction on its choice: chartdatum should be:

I ) so low that the water level will but seldomfall below it,

2) not so low as to cause the charted depths tobe unrealistically shallow, and

3) it should vary only gradually from area toarea and from chart to adjoining chart, toavoid significant discontinuities.

On most Canadian coastal charts the surface oflower low water, large tide, or LLWLT (seesection 5.7), has been adopted as chart datum, butthe term “lowest normal tide,” or “LNT,” has beenretained on the charts since it encompasses avariety of other choices for chart datum on someolder charts. On United States charts, chart datumis taken as mean lower low water (MLLW), asurface somewhat higher than LLWLT. It hasbeen agreed by the two countries that on chartscovering both Canadian and U.S. waters Canadianchart datum is to be used on the Canadian side ofthe boundary, and U.S. datum on the U.S. side,regardless of which country publishes the chart.This policy causes a discontinuity in chart datum

along the international boundary, but preserves theprinciple that charts of the same waters should allhave the same chart datum, and that it should bethe same datum as used for tidal predictions inthose waters.

The choice of a chart datum is usually moredifficult on inland waters than on coastal watersbecause inland waters lack the stabilizing influencethe huge ocean reservoir exerts on the mean waterlevel. Whereas a 2-month water level record at acoastal location provides sufficient tidal informationto determine a reasonably accurate chart datum,many years of record may be necessary to providethe information on seasonal and secularfluctuations in mean water level required todetermine chart datum on lakes and rivers.Shorter term fluctuations, such as those due toseiches and wind set-up may also be considered insetting chart datum, but information on these canbe obtained over a fairly short record period. Dryand wet periods in many drainage basins (e.g. theGreat Lakes basin) seem to occur in almost regular‘ cycles” of several years, causing correspondingperiods of low and high water levels in thedrainage systems. The chart datums must be setwith low stage years in mind, and may appear tobe pessimistically low over several years of highstage. There are a fortunate few inland waters forwhich chart datum is easily chosen - those inwhich the minimum water level is controlled duringthe navigation season. A guideline sometimes usedin setting inland chart datums is that the water levelmay fall below the datum 5% of the time, but thismay not be severe enough if the water levelundergoes large fluctuations. A preferred guidelineis that the daily mean water level should never fallmore than 0.2 m below the chart datum during thenavigation season.

It should by now be apparent that chartdatum need not be a level surface even over theextent of a single chart. Along a river, chart datummust slope with approximately the slope of thewater surface of the river at low stage. Evenalong the coast, where there is no appreciableslope of the mean water level, the surface of chartdatum must slope down from regions of small tidalrange toward regions of larger tidal range toaccommodate the lower low waters. On mostlakes, however, it is common to adopt a single level

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surface as chart datum over the whole lake.Sounding datum is simply the datum to

which soundings are reduced when compiling a“field sheet” during a hydrographic survey. It mayor may not remain as the chart datum. While asounding datum may be chosen rather arbitrarily tofacilitate an immediate start for a sounding survey,it is imperative that its elevation and the elevationof the zero for any water level records bereferenced to permanent benchmarks on shore.This is required to permit adjustment of thesoundings to the final chart datum, and to permitrecovery of the chart datum in future surveys ofthe same region.

The datum for elevations on a chart is thesurface to which the charted elevations ofprominent targets (lights, beacons, steeples,chimneys, etc.) and clearances under obstacles(bridges, power lines, etc.) are referred. It isusually the same high water datum used to definethe shoreline on a chart. On most Canadiancoastal charts the surface adopted as datum forelevations is higher high water, large tide, orHHWLT (see section 5.7). On Canadian charts ofnon-tidal inland waters, however, for reasons thatare no longer apparent, the low water chart datumis also used as the datum for elevations, while ahigh water surface is used to define the shoreline.

5.7 Special Tidal Surfaces

It is found useful to define and name severalaverage tidal elevations that can be used incomparing tidal characteristics from place to place.Some of these have already appeared in thepreceding text. From a single gauge site only oneelevation can be determined for each definition, butit is proper to think of each elevation as only onespot on a continuous tidal surface over the wholeocean. The tidal surfaces presently in vogue inCanada are listed and discussed below.

MWL - mean water level - average of all hourlywater levels over the available period of record.HHWLT - higher high water, large tide - averageof the highest high waters, one from each of 19years of predictions.HHWMT - higher high water, mean tide - average

of all the higher high waters from 19 years ofpredictions.LLWMT - lower low water, mean tide - averageof all the lower low waters from 19 years ofpredictionsLLWLT - lower low water, large tide - average ofthe lowest low waters, one from each of 19 yearsof predictions.LNT - lowest normal tide - in present usage it issynonymous with LLWLT, but on older charts itmay refer to a variety of low water chart datums.

Of the above tidal surfaces, MWL is the only onewhose elevation is determined in practice bystraightforward application of the definition. Theothers are at present calculated from semi-empirical formulae involving the harmonicconstants of the major tidal constituents. Today’shigh speed computers, however, possess thecapability of generating nineteen years ofprediction and applying the definitions directly withno great difficulty. Another possibility is togenerate only one year of predictions, that being ayear in which the moon experiences its averageexcursions in declination (nodal factors, f, nearunity), and to take the appropriate averages andextremes from that year of predictions only. Theseoptions are under consideration, but in themeantime the semi-empirical method of calculationgives values that have been shown to simulate thedefinitions very well. Figure 42 illustrates thesetidal surfaces and their relation to the chartingdatums and other charted features.

A variety of other tidal surfaces are definedand used by hydrographic agencies in differentcountries. Chart datum for United States charts onboth the Atlantic and Pacific coasts is nowdetermined as mean lower low water (MLLW),which is defined as the average of all the lowerlow waters over a specified 19-year period.Previous to 1980 the chart datum for U.S. Atlanticcoast charts was defined as mean low water(MLW), the average of all the low waters over anearlier specified l9-year period. Because of thesmall diurnal inequality on the east coast andbecause of a difference in sea level between thetwo l9-year periods, the change from MLW toMLLW has made only a minor change in eastcoast chart datums, one that is not reflected in the

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charted depths. The discontinuity in datumsbetween adjoining U.S. and Canadian chartsremains. Chart datum for British charts is nowdefined by the tidal surface of lowest astronomicaltide (LAT), which is the lowest water levelpredicted in a l9-year period. Partly because this isdifficult to determine, and partly to accommodateolder charts, the definition of chart datum isrelaxed to permit it to be 0.1 m above a rigorousLAT. A tidal surface used to define chart datumon many older British Admiralty charts, includingsome in Canadian waters, is mean low watersprings (MLWS). It is the average of all availablelow water observations at the time of spring tide,and applies only where the diurnal inequality issmall. While it is no longer in general use, MLWSprovides a simple example of how the harmonicconstants may be used to approximate theelevations of tidal surfaces. MLWS isapproximated as the height of MWL above chart

datum minus the sum of the amplitudes of thesemi-diurnal lunar and solar constituents, or,symbolically,

MLWS = Z0 - (M2 + S2)

The semi-empirical formulae used to approximateLLWLT, etc. are much more complicated, and willnot be given here.

5.8 Land Levelling andWater Transfers

Anyone who installs or operates a waterlevel gauge must ensure that the zero of the gaugeis always accurately referenced to localbenchmarks. This is done by standard spiritlevelling around the closed network of benchmarksand gauge. It is convenient if one of the local

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benchmarks is a Geodetic benchmark, but if noGeodetic BM is accessible within a kilometre ofthe gauge, the task of tying the gauge and its BMsinto Geodetic Datum is left to the Geodetic Surveyof Canada, the experts in long-distance overlandlevelling.

Water transfers were mentioned in section5.5 in connection with the establishment of IGLD.They provide a means of transferring elevationsacross large expanses of water, on the assumptionthat the slope of the water surface can beestimated from the hydraulic and meteorologicalfactors. Installation of water level gauges isusually essential for accurate water transferbecause reasonably long records are required toaverage out seiche activity and to span a variety ofmeteorological conditions, whose effects may thenbe evaluated. Water transfers are used mostfrequently over large lakes, because on theaverage their surfaces approximate very closely tolevel surfaces. Water transfers of chart datumalong the sloping surface of a river may also becarried out, but since the transfer is not along alevel surface, elevations may be determined onlywith respect to the sloping chart datum, not withrespect to sea level. In performing a watertransfer along a river, interpolation should be madebetween two reference gauges, because the slopeof the river may not be the same with respect tothe slope of the chart datum at all stages. Watertransfer of sounding datum from gauge to gaugealong the sea coast is also a common practice, asdescribed in Chapter 6. This is also the transfer ofa sloping rather than a level datum, and is based onthe assumption that the tide curves at theneighbouring station have the same shape, but thatone may lag the other in time and have a differentvertical scale (i.e. different range). Only wherethe tide has a small diurnal inequality are theassumptions likely to be valid.

It is partly for this reason that it is referredto here as a transfer of sounding datum, ratherthan of chart datum, because the final chart datumwould almost certainly be based on an analysis ofthe full tidal record available at the end of thesurvey, rather than on the preliminary watertransfer.

5.9 Purpose and Importance ofBenchmarks

The purpose of permanent Hydrographicbenchmarks is to identify locally the elevation ofthe physical surface that is chart datum. Since allother charting datums and tidal surfaces arereferred to chart datum, the Hydrographic BMsare the fundamental references for vertical controlin charting and water level gauging on navigablewaters. While other agencies, such as theGeodetic Survey, frequently tie the HydrographicBMs into their networks and provide elevations forthem on their own datums, it remains the elevationof the BM above chart datum that is basic tocharting and gauging procedures. Only theresponsible Hydrographic agency may assign oralter the elevation quoted for a BM above chartdatum. Although it is not necessary for chartingpurposes, it is desirable that chart datum bereferenced to Geodetic Datum, so that theGeodetic elevation of chart datum can be suppliedto engineers and surveyors and documented on thecharts. Subsequent readjustment of the Geodeticnetwork could provide new Geodetic elevations forthe Hydrographic BMs and chart datums, butwould not affect the quoted elevations of the BMsabove chart datum. On the Great Lakes, wherechart datum is defined on each lake as a fixedelevation above IGLD, it is necessary that theHydrographic BMs be tied in to IGLD as part ofthe procedure for establishing datum for charting.Ultimately an adjustment will need to be made toall BMs in the IGLD network, to correct for thecrustal movement since 1955 and to incorporatenew levelling. This will provide new IGLDelevations for Hydrographic BMs and chartdatums, but will not change the BM elevationsrelative to chart datum. These can be changedonly if a new chart datum is defined, and the chartsrevised accordingly.

As part of the installation procedure of anywater level gauge, a minimum of three BMs areestablished in the immediate vicinity (1/2 km) ofthe gauge, with no two in the same feature orstructure. The elevation difference between thepreliminary gauge zero and each of the BMs isthen determined by accurate spirit levelling. Whenthe elevation of chart datum is finally chosen with

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respect to the preliminary gauge zero, the BMelevations are converted and recorded in the BMdescriptions as elevations above chart datum. Ifthe water level gauge is to continue in operation, itspermanent zero would be set to chart datum. TheBMs provide for the recovery of chart datum infuture surveys and for consistency in the setting ofgauge zero for all water level measurements at thesame site.

One BM at a site is insufficient becausethere would be no comparison by which to test itsstability over the time since its installation. TwoBMs are insufficient because if one is found tohave moved with respect to the other, there wouldbe no way to know whether one, the other, or bothwere unstable. Three BMs provide the possibilityof identifying one unstable member of the group.This is why three is the minimum required numberof control BMs at each gauge site. More thanthree BMs is, of course, desirable because there isno guarantee that two BMs may not be found tohave been unstable. When a BM is found to beunstable, it must be destroyed and replaced by anew one in a different location. The elevation of

the new BM above chart datum is determined bylevelling from the remaining stable BMs. Theelevation and description of the new BM arerecorded, along with notice of the destruction ofthe unstable BM.

It is worth noting that chart datum may beprecisely (a few millimetres) related to otherdatums, such as GD and IGLD, only at gauge siteswhere those datums have been tied in to theHydrographic BMs. This is true for the followingreasons: firstly, because away from the gauge sitesthe chart datum is determined only to the accuracyto which the soundings are observed andcorrected, and secondly, because chart datum ateach sounding site is determined in effect by watertransfer from the gauge site along the watersurface, and the shape of the water surface withrespect to the geoid is not determined as part ofthe sounding survey. It is thus not possible todefine the continuous surface of chart datum interms of its accurate elevation above Geodetic orother survey datums, the relation being accuratelyknown only at gauge sites.

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PART IIInstruments and Procedures

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CHAPTER 6

Establishment of Temporary Water Level Guage

6.1 Introduction

From the viewpoint of a field hydrographer,the immediate function of a temporary water levelgauge is to provide the information necessary forthe transfer or establishment of sounding datum(and, eventually chart datum) and for the reductionof soundings to this datum. If properly recordedand documented, however, the information fromsuch gauges may serve other functions as well,such as provision of harmonic constants for tidalprediction and of information on short-term waterlevel fluctuations. Since the hydrographer may notknow to what use the water level and benchmarkinformation may be put in the future, the careexpended in installation and operation of a gaugeshould not be limited to that necessary to achievethe required accuracy of sounding reduction. Thisis why some of the accuracy standardsrecommended below may at first appear to beunnecessarily severe: they should, however, bereadily achievable with the exercise of moderatecare. As discussed in section 3.6, every effortshould be made on tidal waters to obtain at leastone month of water level record, to permit propertidal analysis of the data. One of the fringebenefits that may accrue to the field hydrographeras a result of added care in the installation andoperation of gauges is the improvement of cotidalcharts for use in future surveys of the same ornearby areas.

6.2 Stilling Wells

A stilling well is a vertical enclosure withonly limited access to the outside water; itspurpose is to damp out most of the rapid verticaloscillation of the water surface whose elevation isbeing measured. A stilling well is always requiredfor use with a float-type water level gaugebecause rapid rise and fall of the float may causeits suspension cable to slip over, or even jump off,the pulley wheel. Other problems that may becured by installation of a stilling well are excessive“chatter” in a pen-on-paper record, and excessivescatter among readings taken at fixed intervals by

a digital recorder. A small portable stilling well isuseful when it is wished to level to the watersurface, as is required in checking the zero settingof a submerged pressure gauge. Such a portablewell could be simply a length of metal or plasticpipe sealed at one end except for a small intakehole far enough above the sealed end to avoidobstruction. With the well set vertically in theshallow water near shore (secured by rocks orother temporary supports), a levelling rod maymore easily and accurately be held on the waterlevel inside the well than on that outside.

A much more substantial stilling well thanthat described above is required for use with anautomatic recording gauge. It may be constructedfrom wooden planks, metal or plastic pipe, sectionsof culvert, etc. It must be vertical and havesufficient cross-sectional area to accommodate thefloat and counter-weight clear of the sides of thewell at all water level stages; it must extend frombelow the lowest to above the highest water levelsanticipated (including wave action outside thewell); except for the small intake hole, it must bewater-tight over the portion of its length that maybe submerged; and it must be sturdily constructedand mounted to withstand wave action withoutsignificant motion of the well. Particular careshould be given to strengthening the bottom of thewell, since a sudden surge (up or down) in thewater level outside the well creates a pressure (inor out) against the bottom of the well of 0.1atmospheres per metre of surge.

The intake opening should be so placed as tobe submerged at all times, but should not be soclose to the bottom of the well that it could becomeblocked by the accumulation of silt inside oroutside. It is sometimes difficult to find a locationthat is both convenient and suitable for constructionof a stilling well. For example, the vertical side ofa pier provides a convenient surface to which toattach a stilling well, and the pier can provide easyaccess to the gauge; but in regions with large tidalrange (e.g. Bay of Fundy), the area around the piermay dry out at some low water stages. In such acase, it may be feasible to dig the bottom of thewell down below the low water stage and feed itthrough the siphon action in a hose running from

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deeper water off shore to the inside of the well;the hose or pipe may pass through the wall of thewell at any convenient spot near the bottom of thewell, but it must be assured that the end of thehose inside as well as that outside the well remainsubmerged at all times. A similar arrangementmay be used when silting is a problem near thebottom of the well, feeding the well through a hoseor pipe whose outside end is secured in deeperwater where silting is not a problem .

The damping action of a stilling well is afunction of the ratio of the cross-sectional area ofthe inside of the well to that of the intake. thelarger the ratio the greater the damping. Figure 43,shows for different intake ratios, the rate ofadjustment of the water level inside a well to asudden and sustained surge in the outside level.While the decay rate of the level difference is notstrictly exponential, it is nearly enough so overmuch of the curves to make the concept of aresponse time meaningful. Taking the responsetime of a well as the time required for the insidewater level to adjust half way to a sudden andsustained surge in outside level, Fig. 43 providesthe following response times for wells of variousintake ratio, R:

6 seconds for R = 50

11 seconds for R = 10022 seconds for R = 20045 seconds for R = 40090 seconds for R = 800

The data for Fig. 43 was obtained by measurementin a section of plastic pipe whose cross-sectionalarea was 900 cm2 and whose wall thickness was 3mm. The wall thickness and the roughness of theintake surface may influence the response timessomewhat. It is recommended that the area of theintake be 1/1 00th that of the well (R = 100), orthat the diameter of the intake be 1/10th that of thewell . If a long intake pipe or hose is employed,the pertinent intake area is the smallest cross-sectional area along its length. Since friction in along intake pipe increases the response time of thewell, the cross section of a very long pipe or hosemay need to be greater than that indicated by R =100. Figure 44 illustrates the damping effect of awell with R= 100 on waves of 1/2 metre amplitudeand periods of 12 hours, 6 minutes, and 6 seconds,respectively. These three periods were chosen torepresent a possible tide wave, harbour seiche, and

Fig. 43 Response of water level in stilling well to suddenand sustained surge of outside water level, for variousintake ratios.

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surface swell. The well is seen to damp out thehigh frequency waves effectively, while passingmost of the intermediate frequency seiche, and allof the low frequency tide signal.

Detailed instructions for the installation ofstilling wells are not given here because eachsituation presents its own challenges. and someingenuity may be required to assure that the well isvertical, rigid and motionless, accessible, protectedfrom damage by boats berthing nearby, and freefrom excessive silting, while still being deep enoughto have its intake below the lowest water level.Figure 45 illustrates schematically the use of astilling well in conjunction with a float-actuatedwater level gauge.

6.3 Gauge Shelters

Most installations will requireconstruction ofsome sort of protective shelter for the gaugeagainst the weather and interference from curiouspassers-by. At permanent gauge sites, a smallwalk-in gauge house is usually provided, but at

Plate 8. Various makeshift structures to supporttemporary water level gauges. The gauge sheltersshown house float gauges mounted over stilling wellsconstructed of wooden planks. (Photos by CanadianHydrographic Service.)

temporary sites, an enclosure large enough toaccommodate the gauge itself is sufficient. It isconveniently constructed from plywood, with adoor hinged at the bottom so it can be droppeddown out of the way or secured horizontally byhooks and chains to form a working surface. Ifthere is a stilling well, the gauge shelter may befastened securely to the top of the well for support,with a weather-tight connection between the welland the shelter. Holes drilled in the floor of theshelter should be no larger than necessary toaccommodate passage of the leads or cables fromthe sensor in the well to the gauge recorder. Aninspection hatch should be provided near the top ofthe well to give access for cleaning, repairing, orreplacing the float or other sensor mechanisms.Large holes in the floor of the shelter arediscouraged, because of the propensity for loosearticles to fall through them into the well, possiblyfouling the sensor mechanism. Figure 45 illustratesa typical gauge shelter in conjunction with a stillingwell and float gauge. When a gauge site is in anarea inhabited or travelled by the public, particularcare should be given to neatness of construction,

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including the painting of the gauge shelter and anysupporting framework. When unattended, theshelter should be securely padlocked. A noticeattached to the shelter identifying the installation asa water level gauge and briefly describing itsfunction will satisfy most people’s curiosity, and, itis believed. decrease the likelihood of meddling.

6.4 Float gauges

The float gauge has long been the standardinstrument for the precise measurement of waterlevels. It provides a direct measurement of thewater level, and so does not require calibrationover its range of operation, although its zeroadjustment must be regularly monitored. Its majordrawback is the rigid requirement for a stilling well,provision of which may be a problem at manylocations. Figure 45 illustrates in principle theoperation of a float-actuated water level gauge(float gauge). The illustration does not pretend torepresent any particular make or model ofinstrument, and the drives, linkages, etc.encountered on actual equipment may differconsiderably from the simple ones shown.

The water level information is transmittedfrom the float to the recorder by the thin cablewhich is attached at one end to the float, passesover the pulley on the recorder, and is attached atthe other end to the counterweight. The float isusually cylindrical in the centre and spherical at theends; it is hollow, and the level at which it floatsmay be adjusted by the addition or removal of leadshot. When deployed, the water line on the floatshould come about half way up the cylindricalsection, to assure a linear change in buoyancy withchange in depth. Increasing the cross-sectionalarea of the float increases the sensitivity of thegauge to changes in water level, but a practicallimit is set by the size of the well and the inspectionhatch. The counterweight must be heavy enoughto keep sufficient tension in the float cable toprevent if from slipping on the pulley, but not soheavy as to lift the float too high up in the water.It should be solid and made all or mostly of lead, tominimize its loss of weight to buoyancy if it issubmerged over part of the range. Since the floatwill ride slightly lower in the water when thecounterweight is submerged than when it is in air, a

small but systematic error can thus be introducedinto the readings near high water with respect tothose near low water. At permanent gaugeinstallations an effort is made to avoid this sourceof error, either by mounting the gauge high enoughabove the highest water level that thecounterweight need never reach the water, or byproviding a separate water-tight dry well toaccommodate the counterweight. Theserefinements are not required for temporaryinstallations, but if the range of water levels issmall, it should be a simple matter to mount thegauge high enough and to cut the float cable to alength that would keep the counterweight out ofthe water. The float cable should be strong butlight. It is particularly important that it be lightwhen the range of the tide is large, because theeffective weight of the counterweight is increasedby the weight of cable on its side of the pulley, anddecreased by the weight of cable on the float side.Further details concerning the installation of aparticular model of float gauge should be obtainedfrom the instrument manual accompanying it orfrom the agency issuing the equipment. A variety

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of types of recorder may be used in conjunctionwith a float gauge, and some of these arediscussed below; in Fig. 45 a drum-type paperchart recorder is shown.

6.5 Pressure gauges—diaphragm type

The hydrostatic pressure at depth h in acolumn of water is pgh, where p is the meandensity of the water in the column above the depthh, and g is the acceleration due to gravity. Gravitymay be considered constant for the purpose ofwater level measurement, but differences in waterdensity from place to place may be important.particularly if large differences in salinity occur.The difference in density of ocean water at 0°Cand of fresh water at 25°C is about 3%, 2½ %being due to the salinity difference, and ½ % due tothe temperature difference. Clearly, if water levelsare to be interpreted from hydrostatic pressuremeasurements, different calibration scales wouldbe required for fresh and salt water. Use ofpressure sensors instead of float gauges tomeasure water levels at temporary locations forthe control of hydrographic surveys has becomealmost standard practice. This is becauseinstallation of the pressure gauge is much simpler,especially if no wharf is available. A stilling well isnot normally required with a pressure sensor, anydamping that is required usually being supplied bythe design of the sensor head itself and by thenatural damping of the pressure signal of shortwaves with depth (see section 1.4). Figure 46illustrates the damping of the pressure signal froma 6 second and a 12-second sinusoidal wave atdepths of 10, 20, and 30 metres. The vertical scalein Fig. 46 is shown in metres, after conversionfrom pressure units. There is, of course, nodamping of the pressure signal from long waves(tides, seiches, etc. ).

Figure 47 shows schematically a diaphragmtype pressure gauge assembly. The pressuresensor is a flexible rubber diaphragm that formsone face of a hollow air chamber; the outside ofthe diaphragm is exposed to the water pressurethrough holes in a protective housing. Adjustingthe size of the holes controls the damping of theresponse much as in a stilling well. The air

chamber behind the diaphragm has an air-tightconnection through a small ( 1-2 mm insidediameter) capillary tube to the inside of a Bourdontube, bellows or aneroid chamber at the recordersite on shore. These devices translate changes indifferential pressure (inside vs. outside) into amotion which can be conveyed by various linkagesto a recorder. A Bourdon tube uncoils slightly asits internal pressure increases, a bellows extendslengthwise, and an aneroid lid becomes moreconvex. The gauge depicted in Fig. 47 has abellows linked to the pen arm of a drum typerecorder, but other arrangements are possible.The diaphragm housing must be securely mountedface down on some supporting structure below thelowest water level, such that it may not move(especially not vertically) during the recordingperiod. If a wharf is available, the attachment maybe to one of its pilings, but since the sensor may be200 m or more from the recorder, a small rock cribor other support can usually be constructed in areasonably protected location on firm bottom offshore. The principle of operation of this type ofpressure gauge is that the static air pressure isuniform within any closed system (except for the

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negligible weight of the vertical column of air).Thus, as long as all seals are tight, the pressureinside the bellows equals the pressure at thediaphragm, which consists of the hydrostaticpressure due to the column of water plus theatmospheric pressure at the water surface. Sincethe pressure outside the bellows is the atmosphericpressure, the pressure difference to which thebellows responds is the hydrostatic pressure due tothe column of water above the diaphragm.

6.6 Pressure gauges - bubbler type

Bubbler gauges (also called gas purgegauges) are not as frequently used as diaphragmgauges at temporary gauge sites in Canada, butthey have many of the same advantages, e.g. theydo not require a stilling well and the pressuresensor can be installed a considerable horizontaldistance from the recorder. Figure 48 showsschematically a bubbler-type pressure gaugeassembly. The pressure sensor is simply theorifice at the underwater end of a long flexible air-tight tube. The tube may be larger than that of adiaphragm gauge because the volume of gascontained in the system is not a limiting factor; aninside diameter of 5 mm is recommended. At therecorder site on shore compressed air or nitrogenis continuously introduced into the system from acylinder (A) through a reduction valve (B) thatlowers the gas pressure to the working range ofthe recorder and other equipment. The pressure atthe high and low side of the reducing valve is

displayed on the needle gauges (C and D). Thepressure at the low side (gauge D) must alwaysexceed the greatest hydrostatic pressure that couldbe experienced at the orifice. The gas then passesthrough a flow control valve (E), which is a valvelike that used in underwater breathing apparatus tomaintain a steady flow of air regardless of changesin pressure at the downstream end. At this pointthe tube branches, one branch going to therecorder and the other continuing on to feed theorifice (F). At a convenient spot in the systembelow the flow control valve a bubble chamber (G)containing oil or water is inserted so the rate of gaspurging can be monitored and controlled to aboutone bubble per second. As long as gas is issuingfrom the orifice at about this rate, the pressurethroughout the system below the flow control valvewill be sensibly uniform, and equal to the pressurein the water at the orifice. Again, variousrecorders could be used, but Fig. 48 depicts aBourdon tube linked to the pen arm of a drum-typerecorder. Since the outside pressure on theBourdon tube is atmospheric, the pressuredifference to which it responds is equal to the hydrostatic pressure dueto the column of water above the orifice. If theflow of air is allowed to stop, water will flowthrough the orifice up the tube, giving a faultyreading; and if the flow of air is too rapid, there willbe a slight pressure drop along the tube toward theorifice, giving slightly too high a reading. Onecylinder of gas should operate the gauge for fouror five weeks.

6.7 Pressure gauges - deep sea

These are gauges that are self-contained intheir own protective case, which can be anchoredon the sea bottom to record changes in pressurefor periods up to a year and in depths up to severalkilometres. The type most commonly used atpresent senses pressure by means of a quartzcrystal which forms part of an electrical oscillatorcircuit. The resonant frequency of the crystal, andhence of the oscillator, depends upon the pressureapplied to the crystal (and to a lesser extent uponthe temperature of the crystal). By exposing thecrystal to the external pressure through a pressureport in the case, the frequency of the oscillation ismade to depend upon the pressure outside the

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Plate 9. (Lower) Diaphragm-type pressure gauge,showing diaphragm housing, coiled-up capillarytubing, and strip-chart recorder. (Top right) Drum-type recorder for use with float gauge. (Top left)Strip-chart recorder for use with float gauge withsight (electrical tape) gauge. (Photos by CanadianHydrographic Service.)

case. The frequency response is not linear with thepressure, however, and the instruments must becalibrated to relate frequency to pressure. Theoscillator frequency is recorded regularly at a pre-selected sampling interval. Most present modelsrecord data on magnet- ic tape, but solid-statememory banks may largely replace tapes in thenear future, thus reducing the power requirementand eliminating a source of trouble in the movingtape drive assembly. Data storage space is usuallythe limiting factor determining the maximum recordlength attainable, the shorter the sampling interval,the shorter the record. In deep sea operation thegauge is un- attended from the time of mooring tothe time of recovery, and the record is notavailable for use until it is removed from the gaugeand electronically translated. For use at shallowerlocations, however, most models have an acoustictransducer which may be engaged to transmit thereadings as sound pulses that can be received byhydrophone in real time. Alternatively, if the

mooring is not too deep, the readings can betransmitted by electrical cable from the gauge toan auxiliary recorder in a moored buoy, or even onshore. A system may soon be available whereby agauge can be interrogated acoustically from avessel and made to play back all or part of itsstored data. Such features may give these gaugesa role in the reduction of hydrographic soundings,but at present they are mostly used for the study oftides at offshore locations.

From the standpoint of water levelmeasurement, there are two major difficulties withthe self-contained gauges moored far from shore:the first is that they sense total pressure(hydrostatic plus atmospheric), and hence do notreflect the true water level unless the atmosphericpressure is separately measured and subsequentlysubtracted (see section 4.4); the second is that it isdifficult or impossible to relate the zero setting ofthe gauge accurately to benchmarks on shore. Forthe de- termination of tidal constants at offshore

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locations, however, the self-contained gauges aremost adequate, because neither a slight drift in thegauge zero nor the inclusion of the atmosphericpressure signal is likely to contribute any significantenergy at the tidal frequencies. In fact, the totalpressure displays a much cleaner tidal signal thandoes the hydrostatic pressure alone. This isbecause in the ocean the fluctuations in localatmospheric pres- sure are largely offset bycorresponding fluctuations of the opposite sense in

the water level (in- verted barometer effect;section 4.4). In spite of their shortcomings in waterlevel measurement, the deep-sea gauges may beuseful in improving the accuracy of soundingreductions over shallow banks far from shore (e.g.the Grand Banks).

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6.8 Staff gauges

A staff gauge is simply a graduated staff(usually marked in metres, decimetres, andcentimetres) mounted vertically in the water withits zero below’ the lowest anticipated water level.It is usually constructed from prefabricated metalor wooden sections I metre long; the sections areplaced end to end and fastened to a straightwooden plank or pole to achieve the lengthrequired to cover the local range of water level. Ifa wharf is nearby, the staff may be attached to oneof the vertical wharf pilings. If there is no existingstructure to which to attach the staff, it may benecessary to construct a stone-filled wooden cribor tripod with a wide base to provide a rigidsupport. On some types of bottom the polesupporting the staff may be driven into the bottomuntil it is firm, and secured vertically by at leastthree guy wires fastened to anchors. Sometimes,when the tidal range is large and the bottom slopeis small, the intertidal zone is so broad that two

staff gauges may be required, one near shore to beread during the upper part of the range, and onefarther off shore to be read during the lower partof the range. It is even possible that more than twostaff gauges could be required under rare circum-stances (e.g. the tidal flats of the upper Bay ofFundy or Ungava Bay). When more than one staffgauge is used to cover the range of water levels,they should be related to each other so that there isa slight overlap in the part of the range that theycover, and so that they give the same reading inthe region of overlap. Figure 49 sketches severalpossible staff gauge installations, but thehydrographer’s ingenuity may produce others; allare satisfactory as long as the staff is rigid andsteady and is convenient to read.

A staff gauge is required at every gaugingsite. It may serve as the only gauge for a brieflocal survey at a location where the tidal constantsare already known, or on non-tidal waters, although

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it would then require continuous monitoring duringsounding. Where it is feasible to install anautomatic gauge (float, diaphragm, etc.), a staffgauge is still required, against which to makechecks of the accurate operation of the automaticgauge.

6.9 Sight gauges (electrical tape gauges)

A sight gauge is used to make spot readingsof the water level inside a stilling well, usually as acheck on the operation of the automatic gauge. Itprovides a more accurate check than can beobtained from a staff gauge, but may not beconsidered as a replacement for the staff gauge.This is because both the sight gauge and theautomatic gauge read the level inside the well, anda comparison of their readings tells nothing aboutpossible blockage of the well intake. At permanentgauge installations, comparisons are always madeof the automatic gauge with both the staff andsight gauge. Use of a sight gauge at temporaryinstallations is optional, but it does offer aconvenient and accurate means of checking gaugezero and referring it to benchmarks.

A sight gauge is mounted on the floor of thegauge shelter, and consists of a graduated metaltape spooled onto a metal drum, with a plumb bob(or plummet) fastened to the running end so thatthe lower end of the plummet forms the zero pointfor the tape graduations. As shown in Fig. 50, thecore of the metal drum is electrically connected inseries with a low-voltage battery, a needlegalvanometer, and with the water in the well(either through the wall of a metal well or througha wire on the inside of a non-metallic well). A flat-topped peg, called a “gnomon,” is set alongside the

Plate 10. Use of multiple staff gauges at Frobisher Bay, Northwest Territories, to span the large range of tide betweenlow and high waters. (Photos by Canadian Hydrographic Service.)

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tape on the shelter floor. To read the verticaldistance of the water level below the gnomon, thetape is unrolled from the drum through a small holein the shelter floor until the tip of the plummettouches the water surface, completing theelectrical circuit through the water and causing thegalvanometer needle to jump. At this instant thelength of tape out at the level of the top of thegnomon is read. The tape is then slowly raised untilthe galvanometer needle drops back again, andanother reading of the tape against the gnomontaken. The mean of the two readings should betaken as the distance of the water level in the wellbelow the top of the gnomon at the central time ofthe operation. The gnomon must be mounted so asto be clearly in view through the open shelter door,to permit its elevation, and hence that of the gaugezero, to be referenced to benchmarks.

6.10 Data recorders

The type of recorder most commonly usedat present with automatic water level gauges attemporary locations provides a continuous pen-on-paper trace of the water level on a chart driven ata constant speed by a spring-wound clock. Themovement of the pen is mechanically controlled bythe movement of the float pulley or the pressureelement (bellows, etc.). The drum-type recorder(Fig. 45, 47,48) uses a single sheet of chart paperthat fits exactly once around the drum, which isdriven to rotate once per day. In non-tidal watersthe chart on the drum recorder must be changedeach day, to avoid confusing overlaps of thetraces. In tidal waters, however, the chart may beused over several days because the daily advanceof the tide (50 minutes per day) distinguishes oneday’s record from another. The strip-chartrecorder uses a long strip of chart paper that feedsfrom a supply spool over a recording plate and ontoa take-up spool. The chart is long enough tocontain a full month of record, but segments maybe documented for identification and cut off foruse during the sounding survey, provided that thegauge is the responsibility of the sounding party.Segments of record may not be removed fromgauges at permanent stations, whose records servepurposes other than sounding reduction.

At many permanent gauging stations dataare digitally recorded on punched paper tape or insolid-state memory core that can be read remotelyby telephone or locally in the gauge house. Thereare also tele-announcing gauges that can be

interrogated by telephone to give the present waterlevel and the trend (rising or falling) in plainlanguage. There are obvious advantages to theapplication of similar technology to the temporarygauges; for example, to telemeter water level databy radio link to the survey vessel, either in realtime or in blocks of specified length. Equipment ofthis type will probably soon replace the traditionalequipment, and the hydrographer should keepabreast of such developments.

6.11 Selection of gauge site

The first consideration in choosing gaugesites for vertical control of hydrographic soundingsurveys should be given to how well the waterlevel fluctuations at the gauge sites reflect those inthe survey area. This will depend not only upon thedistance between the gauge and the survey area,but also upon the rate at which the tidal charactermay change in the region, the change in slope of ariver along its length, the response of a lake towind set-up and seiches, etc. If a survey is tocover a long stretch of coast-line, it may bedesirable to have two gauges in simultaneousoperation and to leap-frog them along the coast asthe survey progresses. Two gauges may also berequired in the survey of a long tidal inlet, since thetidal character can change significantly betweenthe entrance and the head. More than one gauge isoften required along a strait that joins two bodiesof water of different tidal character because thetide must change character rapidly along the strait.The surface slope along a river may be different atdifferent stages of flow, and so may not always beparallel to the slope of the low-water stage chosenas chart datum: for this reason, two gauges mayagain be necessary, one at the upstream end andone at the downstream end of the survey area,with no rapids, waterfalls, locks, or other datumdiscontinuities between them. The approximatelocation of temporary gauges should be plannedbefore entering the field, and the Regional TidalOfficer should be consulted when advice orassistance is required.

The detailed local selection of a gauge siteshould be made with the following considerationsin mind:(1) Ease of installation: the existence of ready-

made structures to which the automatic gaugeand the staff gauge may be attached (wharf,fish stakes, bridge pilings, etc.); presence offirm bottom on which to construct support for

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gauges if no ready-made structure is available;presence of sufficiently deep water near shoreto assure that the gauge does not “dry out” orthe surrounding water become impounded in atidal pond at low water; availability of materialsfrom which to construct support structures;accessibility of the site by water and/or land;and suitability of nearby ter- rain or structuresfor establishment of benchmarks.

(2) Ease of maintenance and operation: naturalprotection provided against full impact ofwaves and current; likelihood of silting aroundgauge intake or sensor; possibility of damagefrom or obstruction to marine traffic; andaccessibility of the gauge and recorder both bylaunch and by foot.

6.12 Benchmarks - general

Benchmarks are the fixed elevation markersagainst which the zero setting of the gauge ischecked during its operation, from whichhydrographers may recover chart datum for futuresurveys, and through which surveyors andengineers may relate their surveys and structuresto chart datum. The function and importance ofbenchmarks, and the reason that a minimum ofthree is required at each gauge site, has beendiscussed in section 5.9. The benchmarks (BMs)should be in place by the time the gauge is to beset in operation. To minimize the length of thelevelling lines, an attempt should be made to keepall three BMs within a radius of half a kilometre ofthe gauge. The primary consideration, however, isthat they be solidly set in stable structures,bedrock, or firm ground. No two BMs should be inthe same structure or within 70 m of each other inhorizontal distance, to minimize the likelihood oftwo of them experiencing the same instability.

Outcrops of bedrock provide the most stablesetting for BMs, but structures with substantialfoundations that extend below the frost line (publicbuildings, water towers, bridges, etc.) are usuallyalso excellent. Permission should, of course, beobtained before placing a BM in a privatestructure. A BM should never be placed in ahollow or depression in which water might collectand freeze, and care should be given to the finalappearance of the BM, since this reflects indirectlyon the credibility of the other aspects of the

survey. As an aid to future recovery, BMs set inbedrock should, whenever possible, be set close toa distinctive feature in the rock or to some easilydescribable landmark. Existing BMs that mayhave been established by other agencies in thevicinity of the gauge may serve as reference BMsfor the gauge if they meet the stability andaccessibility standards. Indeed. the use of suchBMs is encouraged, since it determines the localrelation between chart datum and the other surveydatum. Enquiries about existing BMs should bemade to pertinent survey agencies before enteringthe field.

6.13 Benchmarks - standard type

The standard Canadian HydrographicService BM tablet is illustrated in Fig. 51. It ismade of bronze, and has a cap 5.6 cm in diameterwith a shank 6.4 cm long and 1.5 cm in diameter.In the lower end of the shank there is a slot about1.5 cm deep to receive a bronze wedge, whichspreads the prongs of the shank against the sidesof the hole when the tablet is set in rock orconcrete. On the face of the cap there is a grooveabout 3.0 cm in length. When the tablet is set intoa vertical face (i.e. with the shank horizontal), thegroove marks the BM elevation, and the tabletmust be set so that the groove is horizontal.During levelling, a benchmark chisel inserted intothe groove provides the horizontal surface onwhich to rest the levelling rod. When the tablet isset in a horizontal face (i.e. with the shankvertical), the BM elevation is the highest point onthe slightly convex surface of the cap, and the flatbottom of the levelling rod may be rested directlyon the face of the tablet.

The standard method for setting the tabletinto a horizontal or vertical face of rock orconcrete is to drill a hole with a rock chisel, slightlydeeper than the length of the shank of the tabletand slightly wider than its diameter. If the hole hasthe proper dimensions, the underside of the cap willcome flush with the surface of the rock orconcrete and the wedge will spread the prongs onthe shank to grip the sides of the hole when thetablet is driven in. If the hole has been madeslightly too deep, a small pebble may be placed in it

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to bear on the wedge. Under no circumstancesmay the shank be shortened to fit into a hole that istoo shallow. Before finally setting the tablet, thehole should be cleaned out and filled with sufficientcement mix to squeeze into all the spaces aroundthe shank and under the cap where it does not seatperfectly against the rock or concrete when it iswedged into the hole. Excess cement should becleaned off, leaving the tablet neatly sealed againstthe penetration of water and frost. It is stressedthat the purpose of the cement is to seal the smallhollows left when the underside of the cap of thetablet is as nearly as possible flush with the originalsurface: cement is not to be used to fill in a spaceleft because the hole was not drilled deep enough.

6.14 Benchmarks - special types

Benchmarks may be placed in suitable soilby fastening them to an iron pipe (called a “soilpost”), when no rock or concrete structure is

Fig. 51 Standard Canadian Hydrographic Servicebenchmark tablet.

available. Figure 52 illustrates the setting of a soilpost BM. A standard bronze tablet is welded tothe top of a smooth iron pipe that is long enough toreach below frost level (2 or 3 metres). Holes aredrilled through both sides of the pipe at its base,through which are fitted steel rods about 20 cmlong to help anchor the pipe. The pipe must havean inside diameter large enough to accommodatethe shank of the tablet, and an outside diameter nogreater than that of the cap of the tablet. A hole ofabout 20 cm in diameter must be dug deep enoughthat only about 15 cm of the pipe protrudes. Apost hole auger should be of assistance in this.Sufficient cement is placed in the bottom of thehole to encase the anchoring rods and seal thebottom of the pipe against the intrusion of groundwater. Soil is then tamped in firmly around thepipe to fill the hole, with the surface soil mounded

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up around the pipe to allow for consolidation andshrinkage of the back-filled soil. The outsidesurface of the pipe should be smooth, so that frostheaving of the surface soil is less likely to affectthe pipe. Soil posts should be placed on highground with good drainage. Sandy soil is usuallyexcellent for the installation of soil posts, but claysoil should be avoided, being too subject to frostheaving. If it is anticipated that soil post BMs maybe required, a supply of pipes should be preparedbefore leaving for the field, having the tabletswelded to the tops and the anchor-pin holes drillednear the bottoms. Rather than preparing a varietyof lengths of pipe, one could prefabricate the pipesin short sections of three types - top sections,threaded on the inside at one end, and with thetablet welded to the other end; bottom sections,threaded inside at one end, and with anchor holesdrilled at the other end; and insert sections,threaded inside at both ends. In the field, thesections could be fastened together with threadedplugs to form the desired length. The sectionsshould be joined on the inside by threaded plugs,rather than on the outside by threaded collars, topreserve a smooth exterior.

Special techniques for setting BMs in areasof perma-frost, muskeg, and spongy soil do exist,but they are often laborious, and sometimes requirespecial motori7.ed equipment. If at all possible,gauge sites should be chosen to avoid the need forsuch difficult BM installations. The GeodeticSurvey of Canada has a great deal of equipmentand experience related to the installation of specialtypes of BM. Sometimes, if a request is madebefore the final planning of their field season, aGeodetic benchmark party may be able to visitproblem gauge sites to install special BMs. TheRegional Tidal Officer should be consulted if BMinstallation problems are anticipated, and requestsfor assistance from the Geodetic Survey or otheragencies should be made through him.

6.15 Benchmarks - descriptions

The face of each BM tablet must bestamped with the BM number and the year of itsestablishment (see Fig. 51). The stamping is donewith metal dies before the tablet is set. Because

Hydrographic BMs do not form a continuousnetwork, but exist in discrete clusters around gaugesites, it is acceptable to repeat the same set ofconsecutive numbers ( I, 2, 3, etc. ) for the BMsat separate sites. If a BM has been lost ordestroyed, its number must be retired, and the BMthat is installed to replace it should receive the nextnumber that has not been previously used from thenumbering sequence at that gauge site. To enablea BM to be recovered and used at a future date, adescription of its appearance and location must berecorded on the Temporary Gauge Data form(Appendix B), copies of which will be retained andupdated by the Regional Tidal Officer. In allrecords, the BM name should appear exactly asstamped on the tablet, e.g. BM 1, 1983. The BMdescription consists of three elements:(a) a verbal description,(b) a sketch of the immediate vicinity, and(c) photographs of the BM and surrounding area

The verbal description should tell the type ofBM, its number, how and in what it is set, itsdistance and direction from any easily identifiablemarks (e.g. the corner of a building), and any otherinformation that might assist in its recovery. Thesketch should be kept as simple as possible, butmust include at least the following basicinformation: true north direction, distance scale,high water line, drying areas and their type (e.g.rock, sand, shingle), prominent structures orfeatures and their names, and distances betweenBMs and from BMs to structures or features.Two photographs should be taken, one a close-upto show the tablet and the surface or structure inwhich it is set, and the other a more distant view ofthe BM in relation to its surroundings, particularlyidentifiable features such as buildings, boulders, theshoreline, or even trees. In the second photograph,someone should point to or hold a levelling rod onthe BM so there is no doubt as to its location.

If a previously occupied gauge site isrevisited and the original BMs are located, theircatalogued descriptions should be checked foraccuracy and for possible changes that may haveoccurred at the site. Necessary revisions to thedescriptions are to be noted on the TemporaryGauge Data form. If an old BM is found to havebeen destroyed, it is to be replaced with a new one,

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bearing a different number from the one that wasdestroyed. If an old BM is determined to beunstable, as demonstrated by a shift in its elevationwith respect to the surrounding BMs. it is to bedestroyed and replaced by a new BM with a newnumber. Notice of the destruction of BMs and thedescriptions of BMs planted to replace them mustbe reported on the Temporary Gauge Data form,so that this information can be kept up to date inthe benchmark catalogues.

6.16 Levelling - General

Levelling is an essential part of the gaugeinstallation procedure, whose function is toestablish the elevations of the BMs with respect toeach other and with respect to the staff gauge andthe automatic gauge zeros. When sounding datum

Plate 11. Benchmark photographs, used to aid in recovery of the benchmarks on future surveys. Location of thebenchmark may be indicated by someone pointing to it or holding a levelling rod on it when photographed. Photo inlower right is a close-up of the benchmark whose location relative to its surroundings is shown in the upper rightphoto. (Photos by Canadian Hydrographic Service.)

is decided upon relative to the gauge zero, itselevation can thus be referenced to the BMs.When chart datum is ultimately confirmed, thecatalogued elevations of the BMs above chartdatum will depend for their accuracy upon thelevelling done during the gauge installation, andfrom time to time during its operation. Because ofthe importance of levelling, the basic principles andprocedures are discussed here. but a novice maywish to refer to a manual of surveying or civilengineering for a more detailed treatment. Thethree systems for calculating elevations above sealevel that were discussed in section 5.2(geopotential, dynamic, and orthometric) should beunderstood by the hydrographer, but need notconcern him in connection with the local gaugesite levelling. This is because there would be nosignificant difference in the three systems over theshort distances and the small elevation differences

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involved, and it is acceptable to use the observedvalues directly. If the Hydrographic BMs arelater tied by another agency into a more extensivelevelling network, their elevations above thatagency’s datum will be measured in one of thethree systems of section 5.2, but this will notchange their quoted elevations above chart datum,for which the Canadian Hydrographic Servicealone is responsible .

6.17 Levelling - method and terminology

The traditional method of levelling employs alevelling instrument (or simply a “level”) with aviewing telescope, which is set up on a tripodbetween two points whose difference in elevationis to be determined. With the optical axis of thetelescope horizontal, a reading is made on agraduated rod resting vertically on one of thepoints; the rod is then placed vertically on thesecond point, the telescope is swung around in thesame horizontal plane, and another reading is takenon the rod. The difference in rod readings givesthe difference in elevation between the points, thelarger rod reading corresponding to the lower point.If the distance between the two points is large, ifthe difference in their elevations is greater than thelength of the rod, or if they are not both visiblefrom a single instrument set-up, the process mustbe repeated in steps, using intermediate pointscalled turning points between the ends of the line.This is illustrated in Fig. 56. If the levelling isproceeding from BM1 to BM2, the sightings on therod when it is closer to 1 are called backlights, andthose when it is closer to 2 are called foresights.The difference in elevation from 1 to 2 shouldalways be measured twice, once by running theline from 1 to 2, and again by running it from 2 to 1. The discrepancy between the values obtained iscalled the closing error, or the closure. Thismethod of levelling has long been referred to as“spirit levelling” because the horizontality of thetelescope was set with reference to the bubble in avial containing alcohol, called a spirit level. Manyof today’s levelling instruments have a system ofoptics suspended on fibres to eliminate the need forthe literal spirit level, but the name persists. Amore apt descriptive name for the method isdifferential levelling.

6.18 Levelling - equipment

In addition to the levelling instrument itself,the following items are required for theperformance of proper differential levelling:instrument tripod, levelling rod, rod level,benchmark chisel, and portable turning point. Thetripod usually is mated to the particular levellinginstrument. Most have telescoping legs, and careis required to see that all wing nuts are tight afterthe tripod is set up. The base plate on which theinstrument sits sometimes has a small sphericallevel, whose bubble should be approximatelycentred when the tripod is set up. If there is nospherical level, the rough levelling can beaccomplished by sighting along the surface of thebase plate at the horizon. On uneven terrain it ispermissible to clamp the legs of the tripod atdifferent lengths to obtain a more stable set-up.

The levelling rod may be made of wood,with a metal foot plate and with a graduated invarmetal scale mounted on its face, the zero of thegraduations being at the base of the foot plate.There is on most rods a sliding section that can beextended and clamped, to double the usable lengthof the rod; in using this, one should be certain thatthe extended section is fully seated and securelyclamped against its stops. While it may be wise tocarry a spare rod, the same rod should be usedover a complete line of levelling. This is so that ifthe base of the foot plate does not coincide withthe zero of the graduations, this zero error willcancel out in the difference between foresights andbacksights.

It is important that the levelling rod be heldno more than a few degrees off the vertical whenit is being read. To aid the rodman in thisendeavour, a rod level is supplied. It is a piece ofmetal about 10 cm long, with a right-angle groovealong its full length, and a small spherical levelmounted at one end. The rodman holds the rodlevel with one hand against the corner edge of therod, just below eye level, and swings the rod untilthe spherical level bubble is centred; the groove inthe rod level, and hence the rod itself, should thenbe vertical. The observer can tell from the verticalcross-hair in the telescope if the rod is off verticalto one side or the other, but he can not tell if it isoff vertical fore-andaft: this is why rodmen are

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sometimes asked to sway the rod slightly back andforth so the observer can take the lowest reading,which should be the reading obtained when the rodis vertical. Use of the rod level provides a mucheasier solution, especially when three-wire levellingis called for, as described below

A benchmark chisel is a flat piece of metalabout 3 cm wide and 30 cm long, with a knife-edgeat one end to fit into the horizontal groove of a BMtablet that has been set in a vertical face. Theother end of the chisel is bent in a right-angle toform a handle, and a small level vial is mounted onthe face of the chisel to indicate when it ishorizontal. When the chisel is set into the grooveof the tablet and is horizontal, it forms the base onwhich the levelling rod is rested while being read.If the rod is not extended and if it is not windy, therodman alone can usually perform this operation;otherwise, two men may be required, one to holdthe rod and one to hold the chisel.

A portable turning point is a metal plate thatis placed on the ground to provide a rest for thelevelling rod at intermediate points (turning points)in the levelling between terminal points. It isusually an aluminum plate about 20 cm in diameter,with three pointed feet on the bottom and a smallraised pedestal at the centre on the top, on whichto rest the rod. The purpose of the feet is not toraise the plate off the ground, but to prevent iffrom slipping sideways, and on soft ground the feetshould be pressed down so that the underside ofthe plate is bearing on the ground over most of itsarea.

6.19 Levelling - instruments

Two types of levelling instrument are ingeneral use today, the “spirit level” and the“automatic (or self-levelling) level.” We willexamine first the characteristics they have incommon. Both have three toot-screws and a smallspherical level on the base of the instrument, bywhich to level it on the tripod. Both have a reticlewith one vertical crosshair and three equally-spaced horizontal crosshairs (or fine lines etchedon glass). Their telescopes have an objective lensto focus the image of the rod onto the plane of thereticle, and an eyepiece through which to view the

reticle and rod image. Some instruments haveazimuth locking screws, which must be loosenedwhen the instrument is being swung through alarge horizontal angle, and tightened when theazimuth tangent screw is to be used for preciseaiming or when a reading is being taken. Otherinstruments may have only the azimuth tangentscrew, the horizontal rotation being tight enoughnot to require a locking screw.

The spirit level has a long sensitive level vialmounted to the telescope tube, with adjustingscrews by which the horizontal axis of the vial maybe set parallel to the optical axis of the telescope.The departure of the two axes from parallel iscalled the collimation error. On some spirit levelsthe telescope and level assembly can be flippedthrough 180° about its longitudinal axis; this featureis used only in the procedure for removing thecollimation error, as described below. On olderinstruments the bubble was viewed in a tiltingmirror, and was centred between graduationsetched on the surface of the level vial. On newerinstruments the bubble is seen through a viewerwith a split optical path, so that half of one end ofthe bubble is juxtaposed alongside half of the otherend of the bubble in the field of view. Thetelescope is leveled before each reading by turningthe telescope tilting screw until the two halves ofthe bubble ends match up alongside each other inthe field of view, as shown in Fig. 53. Illuminationof the level vial may be by a built-in battery light, orby natural light directed onto the vial by a movablemirror.

The automatic level has no precise level vial.Instead, it has as part of its optical train aningenious combination of prisms and/or mirrors, bywhich every ray of light that enters the systemhorizontally is deflected until it is parallel to the axisof the telescope. It might be said that with thespirit level the telescope axis is set parallel to thehorizontal light rays, whereas with the automaticlevel the horizontal light rays are set parallel to thetelescope axis. In both systems the horizontal raysare focused at the centre of the reticule. Theprinciple of the optical compensator is illustrated inFig. 54. All components are shown here asmirrors, but in practice, prisms with one facesilvered may be used. The two upper mirrors arefixed to the telescope tube, while the lower mirror

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is suspended by tour filaments from the top of thetube. The geometry of the suspension is such thatwhen the telescope axis is tilted up or down by anangle θ from the horizontal, the suspended mirrorwill be tilted up or down by exactly the angle 3θ/2.In Fig. 54 the path of a horizontal light ray throughthe system is shown (a) with the telescope axishorizontal, (b) with the telescope axis tilted up, and(c) with the telescope axis tilted down. The anglesof tilt in Fig. 54 b and c are greatly exaggerated forpurposes of demonstration, the actual operativerange of the compensator being only about plus orminus 10 minutes of arc. This operative range issufficient as long as the spherical level on theinstrument base has been centred by use of thefoot-screws. Note, however, that levelling with thefoot-screws may not be done between the readingof a backsight and a toresight, because it canchange the height of the instrument.

6.20 Levelling - instrument adjustments

The adjustment of the spherical level on theinstrument base should be checked first. Toaccomplish this, the telescope is positioned overone of the foot-screws. and the bubble is centredin the spherical level by means of the foot-screws.The telescope is then rotated 180° in azimuth, and.if the bubble has not remained in the centre of thescribed circle, it is brought half way back to thecentre by means of the foot-screws. Theinstrument should now be level, and the sphericallevel can be adjusted by centering the bubble therest of the way by means of the spherical leveladjusting screws. The adjustment may be checkedby returning the telescope to its original azimuth;

the bubble should remain centred. The reasons forpositioning the telescope over a foot-screw at thestart are to provide a reference for its reversal inazimuth, and to facilitate the centering of thebubble with the footscrews.

The collimation of the instrument should nextbe tested, and adjusted if necessary. For a spiritlevel this involves setting the axis of the main levelvial parallel to the optical axis of the telescope; foran automatic level it involves matching the cross-hairs to the axis of the optical compensator. Thefirst step is to establish a truly horizontal line ofsight. Some spirit levels are designed so the

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telescope can be rotated 180° about its longitudinalaxis, permitting rod readings to be taken in both theupright and the inverted position. Since thecollimation error appears with the opposite sign inthe two readings, their average is the trueinstrument height above the reference point onwhich the rod is resting. With the telescope backin the upright position. it is moved with the tiltingscrew to read the average of the two previousreadings on the rod. The telescope should nowhave its optical axis horizontal, and it remains onlyto adjust the axis of the precise level vial bymoving its adjusting screw until the bubble iscentred. The distance from the instrument to thereference point should be about 40 m. Since onlyone point is required, and since a peg driven intothe ground is frequently used as the referencepoint, this procedure is often called the “one-pegtest.”

Instruments that do not have the reversingfeature for the telescope require what is called a“two peg test” for adjustment of the collimation;some spirit levels and all automatic levels fall intothis class. The true difference in elevationbetween two reference points (pegs) is firstestablished by setting the instrument up exactlyequidistant from the two points and reading the rodon each of them. Since the collimation error willappear in both readings with the same sign andmagnitude, the difference between the readingsshould give the true difference in elevation of thepoints. The instrument is then set up as close toone point as is possible while still retaining theability to focus on the rod, and a reading is takenon the rod at that point. Applying the knowndifference in elevation of the two points to thisreading gives the reading that should be obtainedon the rod at the distant point if there is no error.The instrument is pointed at the distant rod and, if itis an automatic level, the reticule is adjusted inposition until the centre horizontal cross-hair is atthe required rod reading. If the instrument is aspirit level, the telescope is moved with the tiltingscrew to the required rod reading, and the preciselevel vial is adjusted until the bubble is centred.The distance between the two points in the testshould be about 50 m. When taking the readingvery close to the rod, only a few divisions on therod may be visible, and it is wise to have the

rodman verify the approximate reading by pointingto it with a pencil when the observer calls it out.

An instrument should receive a collimationtest at the start of a season, after it has beentransported, and whenever it is suspected that itmay have received a jolt or been exposed to roughhandling. The effect of any collimation error thatmay not have been detected can be greatlyreduced by keeping the total distance of theforesights as close as possible to that of thebacksights in each line of levelling. Detaileddescriptions of the locations and methods ofmanipulation of the various parts of a particularinstrument will be found in the instruction manualaccompanying the instrument.

6.21 Levelling - observation and recordingprocedures

Suppose levelling is to be run from BM 1 toBM 2. The instrument is set up on its tripod at apoint not more than about 30 m from BM 1 and inthe general direction of BM 2. The eyepiece isfocused to give a sharp image of the cross-hairs, ifpossible against a blank field of view such as thesky. The rod is then held to rest vertically on BM1, and the telescope is directed at and focused onthe rod by means of the objective lens focusingscrew. The test for proper focus of the objectivelens is that there should be no parallax between thecross-hairs and the image of the rod; i.e. when theobserver moves the viewing position of his eye upand down slightly, there should be no relativemotion of the image and the cross-hairs. Whenthis is achieved, no further adjustment of theobjective lens should be made during the reading,but the image and the cross-hairs may besharpened up by further adjusting the eyepiece ifnecessary. Readings are made on the rod at thepositions of the three horizontal cross-hairs (upper,middle, and lower) and are recorded as the A, B,and C entries in the ‘Backsights” column of thestandard levelling torm, a sample of which isshown in Fig. 55. If a spirit level is being used, thebubble must be centered at the time of eachreading. using the telescope tilting screw to do so.If the image of the rod as viewed through theeyepiece is inverted. the “upper” crosshair is the

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one that gives the highest rod reading, even thoughit appears in the lower part of the field of view.

As soon as the recorder determines that thedifferences in the pairs of readings, A-B and B-C.agree to within 2 mm. the rodman is directed to asuitable turning point, TP 1, in the general directionof BM 2, and no more than about 30 m from theinstrument. Readings are taken on the rod at TP 1,in the same manner as before, and recorded in the“Foresights” column of the levelling form, tocomplete the readings for set-up # 1 . It isessential that the height of the instrument not bechanged between reading of backsight andtoresight, which means that neither the tripod northe foot-screws may be adjusted during thisinterval. After assuring thatA-B and B-C agree.the instrument is moved to a new position in thegeneral direction of BM 2, up to 30 m from TP 1 .The process of leapfrogging the rod and instrumentalong is continued until the final foresight readingsare taken on BM 2. The series of set-ups isillustrated in Fig. 56. Calculation of the remainingquantities (F to L) on the levelling form, andcompletion of the summary on the opposite side ofthe form finishes the forward running of the line.The line must, however, be run again in theopposite direction (i.e. from BM 2 to BM 1). Ifsatisfactory agreement is obtained between theforward and backward running, the mean value isaccepted as the difference in elevation betweenBM 1 and BM 2.

The levelling form shown in Fig. 55 mayappear at first glance to be unnecessarilycomplicated, but the safeguards against error thatare provided by three-wire levelling and the checkcalculations in the levelling torm more than pay forthemselves by reducing the need tor re-levelling.Although the form is laid out so it can becompleted mechanically step-by-step, a little timespent in understanding the logic of the steps canmake the task more satisfying, and perhaps moreefficient. The check on the mean of threereadings (G = J) comes from the simple algebraicidentity

(A + B + C)/3 = B + ( (A-B) - (B-C) )/3 .

The difference between the upper and lowerreading (A-C) is the “stadia interval,” and isproportional to the distance of the rod from theinstrument. Multiplication of this interval by the“stadia factor” for the instrument would give theactual distance, but since our interest is simply toequalize the total distance of the foresights to thatof the backsights, the stadia interval serves just aswell. The length of the backsight and foresight ateach set up should as nearly as practicable equaleach other, but the recorder should keep an eye onthe running sums of the stadia intervals and advisethe observer, so that any imbalance can becorrected by the end of the line. The check on thestadia interval (K=L) is simply the identity

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A-C = (A-B) + (B-C).

Use of a pocket-size electronic calculatorcan be of great assistance in completing thelevelling torm. There are available programmablepocket calculators which, once the three readings(A, B, and C) are entered, will carry out theremaining calculations successively at the press ofa button. All values must, of course, still beentered on the torm. Levelling form sheets withoutthe summary section on the reverse are alsoprovided, to supplement that shown in Fig. 55when more than three instrument set-ups arerequired between primary points (BMs, gaugegnomon, staff gauge zero). One summary pagemust, however, be completed for each section oflevelling between primary points. Thesesummaries provide the input for the benchmarkand datum information that must also be shown onthe Temporary Gauge Data form.

6.22 Levelling—Accuracy

The accuracy demanded of hydrographiclevelling exceeds that necessary strictly forsounding reduction because the information may beused as well for other purposes, because theincreased accuracy may help to identify anunstable mark more quickly, and because the smallextra investment in time and effort required doesnot add significantly to that expended at lowerstandards.

The collimation error may never becompletely removed from an instrument. If theerror is found to be no more than 20 seconds ofarc, which corresponds to a reading error of 3 mmover a distance of 30 m, no further adjustmentneed be made to the collimation, provided that careis taken to balance the sums of foresight andbacksight distances to within 10 m over eachsegment of line between primary points.

The precision of the rod readings is judgedby the agreement between the two halves of thestadia interval, i.e. the upper stadia reading minusthe middle wire reading. and the middle wirereading minus the lower stadia reading. If thedifference is greater than 2 mm, the readings

should be repeated.The most revealing test of levelling precision

is the closing error, the disagreement between theelevation differences determined on the forwardand backward running of the line. When BMs areso close together that only one instrument set-up isrequired, the backward running of the segmentmay amount simply to moving the instrument to adifferent location and repeating the measurements.Some errors are constant or systematic, and maybe made to cancel out of the calculations;examples are the zero error on a levelling rod,which cancels out in subtracting foresight frombacksight, and a small collimation error, which canbe made to cancel out by equalizing foresights andbacksights. There are also systematic errors thatmay not cancel out, such as might be caused by arod with an expanded or contracted scale.Fortunately, most systematic errors would notaccumulate a large error in the short distances andsmall elevation differences usually involved inhydrographic levelling. The criterion on which theagreement between forward and backwardrunnings is judged assumes that the errors involvedare random errors, so that their effect would beexpected to accumulate as the square root of thedistance covered in the levelling. It is, of course,reasonable to hope that if the random errors havebeen kept small, so also have the systematic ones.The criterion is that the difference betweenforward and backward values must not exceed thegreater of

3 mm or 8(K)½ mm,

where K is the length of the levelling line (oneway) in kilometres. Following are the values of8(K)½ mm up to about I km: 190 m (or less ) 3 mm I 91 m to 316 m 4 mm 317 m to 472 m 5 mm 473 m to 660 m 6 mm 661 m to 878 m 7 mm 879 m to 1129 m 8 mm

6.23 Setting gauge zeros

There are usually two gauge zeros to set,

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that of the staff gauge and that of the automaticgauge. The zero of the staff gauge is normally setfirst, and the zero of the automatic gauge then setto agree with it. If, however, it is felt that the staffgauge zero has been set too high, the zero of theautomatic gauge may be set lower to avoidnegative readings; if this is done, the difference inzeros should be some simple amount such as 20 or30 cm. The gauge zeros ought not to be changedonce they have been referenced to the BMs andwater level recording commenced.

If the gauge site has been previouslyoccupied, it should be possible to recover the oldBMs, whose elevations above chart datum areknown. By standard levelling from one of theBMs, the height of the levelling instrument abovedatum is found for a set-up from which the staffgauge is visible. The staff gauge is then juggledand secured in position when the instrumentreading on the staff equals the instrument heightabove datum. Since this operation cannot beperformed perfectly, the actual zero of the staffmust still be referenced to the BMs by repeatlevelling, and the information recorded on theTemporary Gauge Data form as well as on thelevelling sheets. If an electric sight gauge isinstalled in conjunction with an automatic gaugeand stilling well, the elevation of the gnomon on thefloor of the gauge shelter is found by standardlevelling from the BMs. (Note: a short length oflevel rod or metre stick is required to fit in the doorof the shelter for the reading on the gnomon, andthe zero of its scale must be matched to that of themain level rod .) The elevation of the gnomonabove datum should then be marked on a card andprominently displayed inside the gauge shelter.The length of tape required to reach from the topof the gnomon to the water surface in the well issubtracted from the gnomon elevation to give theheight of the water above chart datum at thatinstant. The automatic gauge is then made to readthat water level, thus setting its zero to chartdatum. Setting the water level on a float gauge isdone by holding the float wire and slipping thepulley past it until the desired reading is obtained.Pressure recorders usually have an adjustingscrew with which to make the setting, but theappropriate instrument manual should be consultedfor particulars. Use of a sight gauge wherever a

stilling well is available is encouraged, but is notmandatory. If there is no sight gauge, the settingof the automatic gauge must be done againstreadings taken on the staff gauge. In either case,the setting should be done in two stages, a coarsesetting against the first water level reading,followed by a fine setting against a second waterlevel reading. It should then be checked against athird reading.

In some regions the relation between chartdatum and Geodetic Datum or IGLD may be wellenough established that BMs in one of those netscan be used to set gauge zeros, even when therehas been no gauge at the site before.

When it is not possible to set the gaugezeros from known BM elevations, the zero of thestaff gauge may be set rather arbitrarily, providedonly that it be safely (about half a metre) below thelowest water level anticipated. A few observationson a temporary staff may be compared withobserved or predicted water levels at nearbylocations to help in making this judgement. Thestaff gauge zero must then be related to the BMsby standard levelling. If a sight gauge is beingused, the elevation of its gnomon above the zero ofthe staff gauge is determined by levelling, and thevalue is displayed in the gauge shelter. The zero ofthe automatic gauge may be set to agree with thatof the staff gauge, and may be checked from timeto time, by using the sight gauge as describedabove. If there is no sight gauge, the setting of theautomatic gauge must be done against readingstaken on the staff gauge.

In all of the above cases, the elevations ofthe gauge zeros with respect to each other andwith respect to the BMs must be entered onto theTemporary Gauge Data form in the appropriateslots.

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CHAPTER 7

Gauge Operation and Sounding Reduction

7.1 Introduction

This chapter treats the general procedures thatshould be followed in operating a temporary waterlevel gauge, and in applying the information fromthat gauge to the accurate reduction of soundingstaken in the vicinity. Detailed operating instructionsfor a particular model of gauge (paper loading, penfilling, etc. ) may be obtained from the instructionmanual accompanying the instrument. Althoughseveral methods by which to determine asatisfactory sounding datum are described,situations will almost certainly arise for which nosingle method is exactly suitable. In such asituation, it is hoped that the principles explainedhere may be combined with a little common senseto suggest a solution. It is highly desirable that thesounding datum chosen be as close as possible tothe final chart datum, but it is recognized that therewill be occasions when sounding must be startedbefore sufficient water level information isavailable to permit an elegant choice of soundingdatum. The sounding datum should not be alteredduring the survey, since this would be more likelyto give rise to error in the final reduction to chartdatum than would consistent use of a poorlychosen sounding datum. Cotidal charts for thereduction of soundings in tidal waters not in theimmediate vicinity of a gauge are treated herefrom the standpoint of the user in the field, notfrom that of the tidal officer who must preparethem. It is important, however, that the fieldhydrographer advise the tidal officer of hisrequirements for cotidal charts well in advance ofthe survey, and discuss them with him, to obtainthe most suitable presentation of the cotidalinformation .

7.2 Sounding datum from existing BMs

Usually when an area is being re-surveyed,it is possible to recover BMs from the previoussurvey, whose elevations have been determinedwith respect to chart datum. When this is the case,the gauge zeros are set to chart datum asdescribed in section 6.23, and sounding datum at

the gauge site is gauge zero. Presumably soundingdatum and chart datum would be identical in such acase, but until confirmed by the Regional TidalOfficer, sounding datum remains just that. If thereare Geodetic or IGLD BMs in thc immediatevicinity of the gauge, and if the local relationbetween chart datum and these datums is knownapproximately, then the gauge zeros and thesounding datums may be set by reference to theseBMs, as in section 6.23. Sounding datum andgauge zero should then be close, but not identical,to chart datum.

7.3 Sounding datum by water transfer—tidalwaters

When sounding datum cannot be determinedfrom existing BMs, it may be obtained by watertransfer from a site nearby where a gauge hasbeen or is in operation, and for which chart datumhas been established. The method described herefor datum transfer in tidal waters requires three orfour days of high and low water heights measuredwith respect to gauge zero at the new gauge site,and a corresponding set of high and low waterheights (observed or predicted) with respect tochart datum at the reference gauge site. Themethod is more accurate if the sets of heights canbe obtained when the range of tide is large, usuallyaround the spring tide. The assumptions are madethat the mean water levels at the two sites overthis short period are the same as they would beover the long-term average, and that the tidecurves at the two places have the same shape,although they may differ in range and arrival time.Since the distance of chart datum below meanwater level is determined by the range at large tide,it follows from the assumptions that the ratio ofthese distances at the two sites should be the sameas the ratio of the ranges of the tide at the twoplaces. If the tide at the two sites is semidiurnal ormixed-semidiurnal, all of the HWs and LWs areaveraged to give a mean HW and a mean LW foreach site. The mean water level (MWL) is takento be the same as the mean tide level (MTL), i.e.half way between MHW and MLW. The average

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range over the short interval being treated is themean HW minus the mean LW. The distance ofsounding datum below the observed mean level atthe new site is then taken as the distance of chartdatum below the mean level at the reference sitetimes the range ratio (new/reference). Subtractingthe observed height of the mean level from thisvalue gives the distance of the sounding datumbelow the gauge zero at the new site. Thesecalculations and entries are all to be entered on theTemporary Gauge Data forlll. The gauge zeroshould not be changed to agree with soundingdatum, since there is less chance of error if allwater levels are recorded on the same gauge zerothroughout the survey .

The method is less accurate, and theassumptions less easy to justify, when thecharacter of the tide is mixed-diurnal or diurnal.The calculation is nevertheless carried out in muchthe same way as described above for thesemidiurnal case, with the major exception thatonly the HHWs and LLWs in each set are includedin the averages. It is important that there be a one-to-one correspondence between the HHWs at thenew site and those at the reference site (and forthe LLWs as well). Recognition of matching pairsat the two sites is not always easy, particularly ifthe reference site is far away. A little care inmaking the selection and carrying out thecalculations should, however, produce asatisfactory sounding datum, although a furtheradjustment to chart datum will probably be requiredlater when the complete water level record isavailable. When there is a choice of acceptablereference sites, it is best to choose the one with thegreater tidal range.

Tables 5 and 6 show examples of thedetermination of sounding datum by water transferin tidal regimes with small and large diurnalinequalities, respectively. R and r are the ranges atthe reference and new sites, respectively; M is theMWL at the reference site, above its chart datum;m is the MWL at the new site, above its gaugezero; and m’ = Mr/R is the calculated distance ofsounding datum at the new site below its MWL.The height of the gauge zero above the soundingdatum at the new site is thus d = m’—m = (Mr/R)- m. Provision is made on the Temporary GaugeData form for the entries and calculations shown in

the tables to be made directly on the form. There isalso provision for the calculation of the high waterdatum to which elevations are referred. This datumat the reference site is HHWLT, and its heightabove chart datum, which we will call H, should beavailable either from the Tide Tables or from theRegional Tidal Officer. The height of the datum forelevations at the new site is calculated as h = Hr/R, above sounding datum, and therefore is h - dabove the gauge zero. Just as sounding datum isprovisional, pending final determination of chartdatum, the datum for elevations is also provisional,pending a final determination when all data isavailable.

7.4 Sounding datum by water transfer—lakes

Chart datum on lakes is usually chosen as alevel surface, whose elevation above one of thesurvey datums (e.g. Geodetic or IGLD) is known.Only when sounding datum cannot be found bylevelling from one of these BMs or from apreviously established Hydrographic BM would thewater transfer method be used. To transfer datumby water transfer on a lake requires that there be areference gauge in actual operation at a location onthe lake where chart datum has been established:there is no equivalent to the predicted values thatmay be used in tidal waters. The data inputrequired is a set of hourly (or more frequent) waterlevels above chart datum for a period of two orthree days at the reference gauge, and asimultaneous set of water levels above gauge zeroat the new gauge. The data should be gathered ondays when the wind is light and the seiche action issmall. The wind is usually the more importantfactor, since averaging over 2 or 3 days will filterout most seiche effect, whereas it will not removethe effect of a steady wind set-up. The assumptionis made that the mean surface of the lake over thesampled period is a level surface, and so is thesame distance above chart datum at all locations.If M is the mean water level at the reterencegauge above chart datum, and m is the mean waterlevel at the new gauge above gauge zero, then thedistance of sounding datum below the gauge zeroat the new site is d = M - m. If a high water datumfor elevations has been established at the

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reference site, then the datum for elevations at thenew site should be set the same distance abovesounding datum as that at the reference site isabove chart datum.

7.5 Sounding datum by water transfer - rivers

Once again, if chart datum is known

precisely or even approximately with respect tolocal BMs, sounding datum should be establishedby levelling from a BM. If this cannot be done,datum may be established by water transfer froman operating reference gauge, much as describedfor lakes in section 7.4. The reference gauge mustnot only be on the same river, but must be on thesame stretch of the river, with no locks, dams, ormajor changes in cross-section between it and the

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new gauge. This is because the assumption will bemade that the water level is the same distanceabove chart datum at both gauges, for any givenriver discharge (i.e. that the “stage-dischargerelation” of the river is the same at both locations).Because of this assumption, the mechanics of thedetermination of sounding datum on a river bytransfer from a single reference gauge are thesame as on a lake, and, in the terminology ofsection 7.4, d = M-m. Unfortunately, the surfaceslope of a river is unlikely to be the same at all highand low stages, and so cannot legitimately beexpected to be exactly the same as the slope ofchart datum, except when the river is at the lowwater stage chosen to define chart datum. For thisreason it is desirable to have two referencegauges, one above and one below the new gauge,and to interpolate the datum transfer betweenthem. Let Lu and Ld be the distances of theupstream and downstream reference sites from thenew site, Mu and Md be the mean water levelsabove their respective chart datums for the periodof interest, and m be the mean water level abovegauge zero for the same period at the new site. Ifm' is the distance of sounding datum below themean water level at the new gauge site, we have

m' = (LuMd + LdMu) / (Lu + Ld)

This follows simply from weighting thecontributions from the two reference gauges ininverse proportion to their distances from the newgauge. The distance of sounding datum below thegauge zero is therefore d = m'-m.

Figure 57 attempts to illustrate the principlesand implied assumptions involved in the transfer ofsounding datum on tidal waters, lakes and rivers.CD denotes chart datum at the reference gauge,and SD denotes sounding datum at the new gauge.Situations will undoubtedly arise where there areno existing BMs and also no suitable referencegauges. The hydrographer must then examine thelimited water level data and whatever otherinformation he has been able to gather beforesoundings are to be commenced, and choose whatappears to be a reasonable sounding datum.Conscientious operation of the gauge for as long aspossible before, during, and after the soundingsurvey will greatly assist in the eventual selectionof chart datum.

7.6 Daily gauge inspection

Every water level gauge should be inspectedat least once each day. In tidal waters an attemptshould be made to inspect the gauge near highwater and near low water on alternate days. Thefirst aspect of the inspection consists of asuperficial visual check to see if any of theinstallation has been disturbed; e.g. the staff gauge,stilling well, gauge shelter, or pressure sensormounting shifted, weakened, or damaged in anyway. If it appears that the staff gauge may havebeen shifted, its zero must be checked again bylevelling to the BMs. If an electric sight gauge(tape gauge) is in use, the elevation of its gnomonmust again be checked against the BMs if itappears the gauge shelter may have been shifted.When any alterations are required, two sets ofcomparison readings should be made and recordedas described in the tollowing paragraph, one setbefore and one set atter the alteration .

Figure 58 shows a sample of the water levelgauge Comparison Form in common use by theCanadian Hydrographic Service for recording dailygauge inspection information. The form is selfexplanatory, but the comparison procedure requiressome elaboration. Rather than completing theentries in order from column 1 to 15, the sequencebelow is recommended.1) Enter the general information requested in

columns 1 to 3 and 12 to 15, and ensure thatthe header on the sheet has been filled in.

2) Read the water level on the staff gauge andenter it in column 9; enter the true time of thereading in column 7. This reading is taken firstbecause the height readings should be asnearly as possible simultaneous, and readingthe staff may require a little time to mentallyfilter out the wave and swell fluctuations toobtain a proper reading .

3) Make a short mark with the recorder pen onthe diagram, parallel to the height axis. On apressure-activated recorder this is done bylightly moving the pen arm; on a float-operatedrecorder it is done by rotating the float pulleyback and forth, taking care not to let the floatwire slip on the pulley. The true time of thisoperation is entered in column 5.

4) If a sight gauge (tape gauge) has been

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installed, read it and enter the value in column8. The value may be either the distance of thegnomon above the water or the height of thewater above gauge zero, as long as specifiedby a note on the form.

5) Look back at the mark made with the recorderpen in step (3); read the time from the diagramat the location of the mark, and enter it incolumn 4; read the water level height from thediagram at the mark, and enter it in column 10.Step (4) is inserted before step (5) to make theheight readings on staff, automatic, and tapegauges as nearly simultaneous as possible.

6) Enter the time error in column 6; its value iscolumn 4 minus column 5 . Enter the heightdifterence in column 11; since the tape gaugesare not commonly installed at temporary gaugesites, this is taken as column 9 minus column10, and so is a measure of the height of thezero of the automatic gauge above that of thestaff gauge.

7) If it is convenient to mark on the recorderdiagram without disturbing anything, mark thetrue time (from column 5) and the date on thediagram opposite the mark made with the penin step (3 ). A soft-tipped pen should be usedto mark lightly.

The inspection is not complete untilindications of possible trouble are investigated. Ifthe record trace seems unusually smooth, if thetidal range seems unduly small on the record, or ifthe automatic gauge and the tape gauge agree witheach other, but not with the staff gauge. cloggingof the intake by silting or marine growth isindicated. Flat spots in the record may be causedby the stilling well being too short, the float wirebeing too short, the intake or pressure sensor beingexposed at low water, the water around the intakeor sensor being impounded near low water, apressure connection leaking, or by the movementof the diaphragm in a pressure sensor beingrestricted by silt. Problems peculiar to particularrecorders may also arise (clogged pen, paperjammed or off sprockets, etc . ), and these, alongwith the routine recorder maintenance (fill pen,wind clock, etc. ), must be attended to as detailedin the particular instruction manual. When anyadjustment or alteration is made that could change

the time or height readings, a gauge comparison(steps (1) to (7)) must be made immediatelybefore, a note written on the comparison formdescribing the alteration, and another comparisonmade and recorded immediately afterwards.

Small discrepancies in height or time thatcan be tolerated within the accuracy of thesounding reductions should not be removed byadjustment of the gauge or recorder. This isbecause a record with a small continuous error iseasier to treat in the final analysis than is one withfrequent adjustments. How large an error shouldbe allowed to become before it is adjusted is left tothe discretion of the field hydrographer, withinlimits that may be suggested by the Regional TidalOfficer.

7.7 Documentation of gauge records

The Temporary Gauge Data torm, theComparison Form, and the levelling sheets andsummaries are sufficiently documented andidentified when all the pertinent entries have beenmade in the spaces provided. The first step whenstarting a fresh form should always be to enter theheader intormation (station name, location, date,etc.). If it is not done at first, it may be overlookedaltogether, and the information could becomeorphaned later on. It i.s even more important toidentify each piece of water level record (sheet orstrip-chart) with the station name, date and startingtime immediately recording is commenced. Theword “Start” should be marked at the beginning ofa strip-chart record, and the word “End” marked atthe finish; station name, and the end time and dateshould also be marked at the end of the record, toavoid having to unspool a record to identity it. Ifpieces of record are cut ot-f tor use during thesurvey, each piece of the record must be identifiedat each end as described above. When recording iscompleted, all pieces of record should either betaped or pasted back together consecutively, oreach piece should be marked on the outside withits consecutive number and the total number (e.g.#l of 5), and all of them bundled together. Thisapplies only to temporary gauge records, sincerecords and parts of records may not be removedfrom permanent gauges by the field hydrographer .

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The above advice has been written withstripchart recorders mainly in mind, since they arein most common use with temporary gauges at thetime of writing. The principles, however, remainthe same regardless of the method of recording,whether it be punched paper tape, magnetic tape,solid-state memory banks, or whatever: each pieceof record and each supporting document must beed as to station, date, time (including time zone),and any other parameter that seems appropriate.

7.8 Datum notes on field sheets

Every field sheet on which soundings aremarked must have a datum note defining theelevation of the sounding datum used in thereduction of soundings shown on the sheet. Onsurveys of deep offshore regions it may be that nosounding reduction is considered necessary orpracticable, or that a constant amount is subtractedfrom each sounding just to be on the safe side. Insuch cases there is no standard format for thedatum note, but the note must still be supplied: itmight read

"Reduced to a sounding datum which is the surface(or x metres below the surface) of the water at thetime of the sounding."

There is a standard format tor the soundingnote to be used when the sounding reductions havebeen made with reference to the water level at aparticular location. The basic note in this case is toread

"Reduced to a sounding datum which at (name ofgauge location) is x metres below BM (name ofbenchmark). "

A supplement to the basic note should beadded, however, stating how the reduction at thesounding site was obtained from the water level atthe gauge site (e.g. applied directly or calculatedfrom cotidal chart).

When sounding datum has been determinedat the gauge site by levelling from a BM onInternational Great Lakes Datum (IGLD) orGeodetic Survey of Canada Datum (GD), it is

desirable to add the following sentence to the basicsounding note:

"The elevation of BM (name of benchmark) wasdetermined in (year) to be y metres above IGLD(or GD)."

This wording respects the fact that the CanadianHydrographic Service is not authorized to assignbenchmark elevations except with respect to itsown chart datums.

7.9 Submission of records and documents.

All records, documents, and explanatoryinformation pertaining to the operation of waterlevel gauges, the establishment of benchmarks, andthe method of sounding reduction are to besubmitted to the Regional Tidal Officer at the firstopportunity following completion of that phase ofthe survey. This is not, of course, an invitation toterminate operation of a gauge before sutficientrecord is obtained to permit useful tidal analysis(absolute minimum 15 days; desirable minimum of29 days). The material to be submitted includesthe originals of

1) Temporary Gauge Data form, showingsounding datum determination and relation togauge zeros and BMs, BM descriptions andsketches. and general descriptive information;

2) BM photographs;3) Water level gauge Comparison Forms;4) Complete water level gauge records (pen-on-

paper traces, punched paper tapes, magnetictapes, memory bank print-out, or whatever);

5) Levelling notes and summaries; and a copy of6) The datum notes from the field sheets.

While it is desirable to submit as much of theabove information as possible before the end of thefield season, no original records should be sent bymail or other third party carrier unless a usablecopy has been retained. It is always wise, in l:act,to make copies of records whenever possible, andto store the originals and copies separately.

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7.10 Sounding reduction - general

Sounding reduction based on surtaceelevations recorded at a gauge site is very similarin principle to the determination of sounding datumby water transfer from one gauge to another, asdescribed in sections 7.3, 4 and 5. Sounding datumis, in effect, carried by water transter from thegauge site to the sounding location. The watertranster in this case, however, is based on only onereading at each end of the line, and does not havethe benefit of averaging over several days ofrecord as in the previous case. Because of this,when accurate sounding reductions are required,the region of sounding should not be far trom thecontrol gauge. It is not possible to specity a fixeddistance from a gauge, beyond which soundingshould not be carried. That decision must take intoaccount such factors as the required accuracy ofthe survey, and the local surmay be generated bywind, seiche, river discharge, etc.

7.11 Sounding reduction - cotidal charts

Cotidal charts have already been discussedin section 3.10. Their application to soundingreduction is mostly in offshore regions, where it isnot feasible to place gauges close enough to thesounding area to justify taking the correctiondirectly trom the gauge, and where a lesseraccuracy in the reduction can be tolerated. Thecotidal charts most commonly used for soundingreduction implicitly assume that the shape of thetidal curve at any point on the chart has exactly thesame shape as that at the control gauge, but that itmay be shifted in time and magnified or reduced invertical scale. This assumption may be valid overan extensive region when the tide is mostlysemidiurnal, but over only a restricted region whenthe tide is strongly diurnal. A cotidal chart isprepared and provided by the Regional TidalOfficer. atter discussion with the concerned fieldhydrographer about required accuracy, extent ofthe survey, proposed gauges, etc. The areacovered by the chart is divided into two sets ofzones, one set defined according to the timedifferences, and the other set defined according tothe amplification factors (ratios of ranges), relative

to the tide at the control gauge.Figure 59 shows a cotidal chart for a

fictitious region. Boundaries between timedifterence zones are shown as solid lines, andthose between amplification zones (range zones)as broken lines. Each time zone is labelled with theaverage number of minutes by which theoccurrence of a tidal event (e.g. HW, LW, etc.)within the zone lags behind the occurrence of thecorresponding event at the control gauge. Eachrange zone is labelled with the average ratio oftidal ranges within the zone to the tidal range at thecontrol gauge. The size of the step between timezones and between range zones must be chosen toaccord with the required accuracy of the soundingsand the average range of the tide. If, for example,the tide is semidiurnal with a 5-m range, a timeshift of 10 minutes produces an average heightchange of 0.13 m, and a maximum change of 0.21m; a shift 0.1 in range factor produces tor thesame tide an average height change of 0.32 m, anda maximum change of 0.50 m. Improved accuracyin the use of cotidal charts may be obtained ifvalues are interpolated, rather than taken asconstant over a zone. At the very least, when asounding location is closer to the boundary than it isto the centre-line of a zone, the value chosenshould be the average of those on either side of theboundary. To help illustrate the reduction method,Table 7 contains a hypothetical set of water leveldata supposed to have been read from the controlgauge for the cotidal chart of Fig. 59. The timesshown in the table are the times taken directly offthe gauge record (or diagram), corrected only for

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gauge clock error, if any.

The following two examples demonstrate how tocalculate the sounding reduction from the cotidalchart information and the water level data.

Example (a):

At 11:05 AST a sounding of 9.6 m is taken at pointA on the cotidal chart. From Fig. 59. the time lagof the tide at A relative to the tide at the controlgauge is interpolated approximately as T = 12 min.and the range factor as R = 1.03. Since the timelag is positive, it must be subtracted from thesounding time to obtain the diagram time at whichthe corresponding tidal phase occurred at thegauge. Thus, the gauge reading should be taken at11:05 - 00:12 = 10:53. By interpolation between thefirst two readings in Table 7, the gauge reading at10:53 is 4.05 - 0.35 x (13/20) = 3.82 m. This times

the range factor gives the height of the water levelat A as 3.82 x 1.03 = 3.93 m above soundingdatum. The sounding reduction, to the nearestdecimetre, is thus 3.9 m. and the correctedsounding is 9.6- 3.9 = 5.7 m below sounding datumat point A.

Example (b):

At 12:45 AST a sounding of 12.3 m is taken atpoint B on the cotidal chart. From Fig. 59, T = -7min, and R = 0.95. Since the time lag is negative,tidal events arrive at B before they arrive at thecontrol gauge, and so T must be added to thesounding time to obtain the appropriate diagramtime. The gauge reading is therefore taken at 12:45+ 00:07 = 12:52. By interpolation between theseventh and eight readings in Table 7, the gaugereading at 12:52 is 1.65 - 0.35 x (12/20) = 1.44 m.This times R gives the water level at B as 1.44 x

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0.95 = 1.37 m above sounding datum. Thesounding reduction, to the nearest decimetre, isthus 1.4 m, and the corrected sounding is 12.3 - 1.4= 10.9 m below sounding datum at point B.

As an alternative to a cotidal chart drawnout as in Fig. 59, a program might be supplied foruse in a mini-computer, to calculate R and T asfunctions of grid coordinates, and pertorm thesounding reduction from the sounding times andtabulated gauge readings. Coupling such a systemto a telemetering water level gauge can providereal-time automated sounding reduction. When it isnot feasible to have an operating gauge to controlthe sounding reduction, tidal predictions may beused in conjunction with the cotidal chart orcomputer program. Predicted water levels for thecontrol site admittedly do not include the non-tidalfluctuations that would be detected by an operatinggauge, but then, neitherdo the non-tidal oscillationsbehave in the manner prescribed by the cotidalchart.

Sounding surveys covering an extensivearea in which the tide displays a large diurnalinequality may not be well served by the type ofcotidal chart (or computer program) describedabove. because the assumptions of constant timelags and amplification factors are not justified. Inthese cases it may be necessary to supply aseparate cotidal chart for each of the major tidalharmonic constituents (usually M2, S2, 01, and K1).From the cotidal charts, the constituent amplitudesand phaselags for the immediate sounding areamay be read, and tidal predictions made in themanner described in section 3.8. No control gaugesite is directly invoved in this field procedure, butinformation from several neighbouring gauge siteswould be used in preparation of the cotidal charts.In preference to graphic cotidal charts for theconstituents, computer programs might be providedto generate the tidal constants as functions of gridcoordinates, calculate sounding datum relative toMWL, predict the water level relative to soundingdatum, and so automatically provide corrections forsoundings. In the above procedure, the distance ofsounding datum below MWL would usually betaken as the simple sum of the major tidalconstituent amplitudes.

7.12 Sounding reduction—non-tidal waters

Sounding datum on a lake is chosen as alevel surface, whose elevation is defined relative toBMs at the control gauge site. The soundingcorrection is usually taken directly as the waterelevation above sounding datum at the gauge. Thecorrection is thus accurate only insofar as thewater surface between the gauge and the soundingis level at the time. Improved accuracy may beobtained by interpolating between values from twocontrol gauges, weighting each value in inverseproportion to the distance of the sounding from theparticular gauge. Even this procedure provides nocompensation for surface slope in the offshoredirection, unless one of the control gauges is itselfoff shore. Because of this, sounding should not becarried out far from a control gauge on a lakewhen large wind set-up or seiche activity issuspected; this is particularly true for lakes that areshallow, and of large horizontal extent (see sections4.3 and 4.6).

Sounding datum on a river should be asurface that approximates closely to the actualwater surface when the river is at its lowest stageof the navigation season. If a sounding correctionis transferred directly from a single gauge on theriver, it is accurate only insofar as the river slope atthe time of sounding is parallel to the river slope atlow stage. It may be necessary, particularly onshallow rivers, to establish two control gauges andto interpolate sounding corrections between them,except when soundings can be done at or near lowstage. The method of interpolation and itsjustification are the same as those given for thedetermination of sounding datum by water transferon rivers (section 7.5).

7.13 Sounding reduction—offshore gauging

In general, offshore sounding does notrequire as high an accuracy as that near shore,because of the greater depths offshore. There areoccasions, however, when sounding must be donein shallow water far from shore, over offshorereefs, shoals, or banks. in such a case, soundingreductions taken from a distant shore gauge,whether by direct transfer or by cotidal chart, may

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not be sufficiently accurate. Some sort of offshorewater level measurement is then necessary tocontrol the soundings. If the water is very shallow,it may be possible to drive a long pole into thebottom and attach a staff gauge to it: it may not befeasible to station a vessel to read the staff everyhour, but it could be visited twice or four times aday near high and low waters to observe themaxima and minima, from which an adequate tidalcurve could be constructed. If it is not possible toset an offshore staff gauge, a vessel or launchequipped with a sounding device may be mooredover a level bottom, to take a series of soundings atthe fixed location. The vessel acts as an invertedfloat gauge, and the series of soundings is thewater level record. Alternatively, a bottom-

mounted pressure gauge of the type described insection 6.7 may be employed. Records from thesegauges require subtraction of a corresponding setof atmospheric pressure measurements beforebeing interpreted as water levels (section 4.4). Thepressure record is stored in the gauge, but mayalso be acoustically telemetered to a surface floator vessel for more immediate use in soundingreduction. While it is not possible to reference thedatums from offshore water level measurements toBMs on shore, the records may be very useful intidal studies and the preparation of future cotidalcharts. They must, therefore, be submitted to theRegional Tidal Officer, just as are the othergauging records and documents.

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CHAPTER 8

Current Measurement

8.1 Introduction

The responsibility of a Hydrographic Serviceto provide current information, mainly as an aid tonavigation, was mentioned in the Preface to thisManual . The responsibility for gathering thenecessary current information lies mainly with thefield hydrographer, and this aspect must berecognized as an integral part of any hydrographicsurvey. Current information that is obtained on asurvey will be analysed, compared with previousinformation, interpreted, and eventuallyincorporated into the charts, Tide and CurrentTables, Sailing Directions, or possibly tidalatlases. Since aiding safe and efficient navigation isthe main goal of a hydrographic survey, currentmeasurement will be called for mostly in narrow orshallow channels, harbour entrances, congestedshipping lanes, or other areas where the margin fornavigational error is small. Current informationfrom deeper and less restricted offshore areas isalso greatly appreciated by mariners, but itsmeasurement may often require the assignment ofdedicated current survey teams with specializedmooring equipment. Even in the offshore areas,however, bits of current information can begathered during the course of a sounding survey,from analysis of the drift of the ships and launches.

Various aspects of currents and havealready been described in Part I of the Manual.Some of the terminology and characteristicsassociated with tidal streams were discussed insections 1.5 and 1.11; harmonic analysis andprediction of tidal streams were dealt with insections 3.7 and 3.8; and some of the causes andcharacteristics of non-tidal currents were treated insections 4.2, 4.3, and 4.7 to 4.10. This chapter willdeal with practical aspects of gathering currentinformation as part of a hydrographic survey -preliminary investigation, methods of measurement,types of current meters, etc.

8.2 Preparatory investigation

The effort expended on current measurementcan be more effectively directed when something

of local conditions and navigational problems isunderstood. Before leaving for the field, thehydrographer should study the current informationgiven for the survey area in the SailingDirections, navigational charts, Tide and CurrentTables, and current atlases, if they cover that area.The Regional Tidal Officer should be consulted foradvice and for information that may not yet havebeen published, and also for copies ofcorrespondence that may have been receivedcomplaining of errors or omissions in thepublications, or requesting additional currentinformation. When in the field, the opinions ofexperienced local mariners and fishermen shouldbe sought. They should particularly be asked tocomment on the information concerning currents inthe area as shown in the Hydrographic charts andpublications. On the basis of evidence gathered asabove, and by first-hand reconnaissance, it must bedecided if and where current observations arerequired to verify or supplement the existinginformation.

8.3 Location and depth of currentmeasurement

Current measurements will normally berequired only where appreciable velocities (0.2knots or more) are encountered in shallow, narrow,or congested shipping routes, in harbour entrances,in berthing areas, etc. Where possible, theobservations should be taken where the velocity isgreatest, but if this is in the centre of heavy traffic,it may be necessary to move the observation site toone side of the channel. When this is the case, afew spot readings should be taken in the centre ofthe channel, to relate the current there to that atthe site of the more complete observations. Indeed,it is always wise to take spot readings at variouslocations and at various times (e.g. at maximumflood and ebb) during the main observations: thiscan help to define the spatial variability of thecurrents with minimal effort. The exact position ofeach main observation site will, of course, beinfluenced by the availability of a suitable bottomfor mooring current meters or anchoring a vessel,

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or of an existing platform such as a bridge footingor drilling barge.

Since the current information is intendedmainly for the use of mariners the depths ofobservation should span the depth of the deepestdraught vessels frequenting the area; if only onedepth is occupied, it should be about one-half thedeepest draught; if two depths are occupied, theyshould be about one-third and two-thirds thedeepest draught, etc. A moored current meter,however, may not be placed so near the surfacethat the subsurface float from which it issuspended could break the surface in any seaconditions or at any stage of the tide.

8.4 Time and duration of measurements

The length of record required to permitseparation of the major tidal constituents byharmonic analysis is the same for currents as forwater levels, a minimum of 29 days. This length ofrecord should, therefore, be the target for currentmeasurement in tidal waters. If, for some reason,29 days of record cannot be obtained, an honestattempt should be made to get at least 15 days.The above is intended to encourage longer seriesof current observations, but is in no way intendedto discourage short series when circumstancesabsolutely preclude the possibility of longer ones.Almost any carefully observed data are better thanno data at all. In waters whose tide shows a largediurnal inequality, observations should be obtainedover at least 25 hours, whereas if the tide showsonly a small diurnal inequality, this may be relaxedto 13 hours, if necessary. Short series are morevaluable if they can be observed when the range ofthe tide is large (usually at spring tide), since thetidal streams should then be large as well. Twoshort series taken one at spring tide and one atneap tide are much more informative than a singleseries of their combined length. In general, themore frequently the current is sampled, the morereliable is the data, because irregularities can moreeasily be smoothed out of the record. Frequentsampling is most important in short records, asampling interval of 15 minutes being desirable, butone of 30 minutes being acceptable. The samplinginterval might be extended to one hour if so doing

permitted a longer record to be obtained throughconservation of limited battery power or datastorage space. As mentioned in section 8.3, spotreadings taken at various locations and timesduring recording at the main sites are veryvaluable. While there can be no set samplinginterval prescribed for these, they are most usefulif taken near the times of maximum flood and ebb,in tidal waters. In fact, series of measurements ofthe time. rate, and direction of the current atmaximum flood and ebb (or at the times ofmaximum and minimum ebb in the case of sometidal rivers) are very worthwhile even when it maynot be feasible to observe a proper time series.

In non-tidal waters, the continuity of currentobservations is less important than in tidal waters.What is more important is to obtain measurementsover the range of conditions that influence thecurrent, these being mainly river discharge, runoff,and wind. Long continuous periods of record arecertainly satisfactory if they span a sufficient rangeof conditions, but it is frequently more convenientto schedule several shorter periods to coincide withsuch events as the spring freshet, the dry season,the stormy season, etc. The sampling interval forobservations in non-tidal waters may, in general, belonger than that recommended for tidal waters, onehour being a reasonable value unless seicheactivity is thought to contribute significantly to thecurrent (see section 4.6). The seiche contributionmay be studied from a record of duration equal toseveral seiche periods, observed with a samplinginterval about one-tenth of the seiche period:seiches are not, of course, always present, andcare should be taken that seiche activity is includedin therecord if it is to be studied.

8.5 Observation methods - general

Current observations may be categorized as“direct” or “indirect” on the basis of whether thevelocities are measured as such or are deducedfrom their relation to other measured parameters.Five examples of indirect current observation aredescribed below—the continuity, the hydraulic, thelong wave, the electromagnetic, and thegeostrophic methods. Of these, only the first three

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are likely to find even limited application onhydrographic surveys, but they may all contributeto the information in Hydrographic publicationsthrough the courtesy of other investigators. Directcurrent measurement is sub-classified aseither”Eulerian” or”Lagrangian.” Eulerianmeasurements are taken at a fixed location over aperiod of time; Lagrangian measurements aretaken by tracking the path of an object that driftswith the water over a period of time. Results fromthe two methods could be comparable only if thedrifter remained within the same current regimethroughout its path, which would usually require thepath to be very short. In general, it is easier tointerpret Eulerian than Lagrangian measurements.The methods described below that involve driftersof one sort or another may be considered to beEulerian, because it is envisaged that the drifterswould be recovered and re-set, to traverse thesame small re the series of observations.

8.6 Self-contained moored current meters

These instruments consist of sensors todetect current rate and direction, data recorder(usually magnetic tape), clock, power supply, and awatertight pressure case to house the vulnerablecomponents. In some instruments. additionalsensors may be supplied for such things aspressure, temperature, and electrical conductivity(from which salinity is determined): while theseother parameters may not be of direct concern to ahydrographer, they may be observed with no extraeffort by using an instrument that is so equipped.To date, direction sensors still rely on the magneticcompass, gyro-compasses not yet having beensuccessfully engineered into small current meters.This is a serious drawback only in the far North,where the horizontal component of the earth’smagnetic field is weak near the magnetic pole. Therate sensor is usually either a propeller mounted ona horizontal shaft, or a drum-shaped Savonius rotormounted on a vertical shaft. Some instrumentshave a tail fin and are suspended so that the wholebody of the meter turns to trace the current, in themanner of a weather vane; others, particularlyamong those using Savonius rotors, have adirection vane that rotates independently of the

instrument case. The common recording procedureis to store a pair of rate and direction readings at afixed sampling interval, typically about 15 minutes.More sophisticated instruments may beprogrammed to accumulate the vector average ofreadings taken every few seconds over a fixedinterval, and to store only the average rate anddirection after each recording interval. In this way,data storage is conserved, but wave “noise” isfiltered out of the record by high frequencysampling. Solid-state sensors are available on somemodels to replace the propeller or rotor. There aretwo acoustic types, one of which detects thechange in travel time of a sound pulse betweentwo probes, and the other of which detects theDoppler shift in frequency of a sound pulsereflected from particles in the moving water. Thereis also the EMF (electro-magnetic force) typesensor, which senses the voltage generated as themoving water (a conductor) cuts the lines of forceof a magnetic field in the sensor.

Figure 60 shows a typical configuration inwhich to moor a string of current meters inmoderately shallow coastal waters. Thesubsurface float must be deep enough not to breakthe surface at any wave or tide condition; the cablesupporting the meters is non-rotating stainless steelwire, with a swivel above the anchor, below thefloat, and above and below each meter; the groundline and the line to the surface marker are buoyantpolypropylene rope; and the anchor release istriggered by a coded acoustic signal. The groundline should be a least as long as twice the waterdepth, and be laid out in a known direction. Normalrecovery procedure is to trigger the release, pickup the subsurface float, and retrieve the metersand the rest of the mooring in order. If the releaseshould fail, the mooring may be recovered inreverse order, starting with the surface markerbuoy. If the surface buoy is lost, or was notconsidered necessary when the mooring wasplaced, the string may be recovered by grapplingfor the ground line normal to its length. In veryshallow water, a single instrument might bemounted on a tripod or frame and weighted to thebottom. Further details and alternative suggestionsfor mooring current meters may be obtained fromthe Regional Tidal or Current Officer or fromoceanographic field personnel.

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8.7 Over-the-side current meters

These instruments are basically the same asthose described above, but the internal recorder isusually replaced by a visual display and/or recorderon deck, to which the information is fed throughelectrical conductors in the suspension cable. Withthe vessel at anchor, the meter is lowered bywinch and readings taken at selected depths. Theprocedure should be repeated at intervals of about30 minutes for as long a series as is feasible. If thesystem has an automatic recorder, or if personnelare available to monitor the visual display

continuously, the meter should be left to record at aselected control depth rather than being broughtinboard between lowerings. A pressure indicatorthat can be set to zero at the surface is a desirablefeature in over-the-side current meters, to providean automatic record of the various pressures(hence depths) at which readings were taken. Ifthis is not provided, the depth must be determinedfrom the length of cable out and its angle from thevertical.

One of the self-contained meters describedin section 8.6 may be used for over-the-sideoperation instead of for mooring, if so desired. Itmust then, however, have a pressure sensor as

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well as rate and direction sensors and a clock. Arecord, independent of that in the meter, should bekept of the times at which the instrument wasrecording at particular depths. This, coupled withthe pressure record, provides additional timechecks on the record; or, if the pressure sensorfails, helps to identify the depths of the readings.Since the record is not usually accessible duringoperation, the initial estimate of the depth of theself-contained meter is determined from the lengthof wire out and its angle from the vertical. Somemeters may be monitored acoustically byhydrophone during operation, but the addedequipment, including a decoder, perhaps undulycomplicates the procedure. Again, betweenlowerings to selected depths, the instrument shouldbe left to record at a chosen control depth.

A current meter suspended from a movableplatform records the movement of the waterrelative to the platform, so it is necessary toremove the platform motion from the record or, atleast, to identify parts of the record that should beignored because of excessive platform motion. Toachieve this when the platform is a vesselpositioned by a single anchor, a record should bekept of the direction of the ship’s head and thescope of the anchor.

8.8 Suspended current cross

This is a piece of equipment that can besimply constructed to measure currents from areasonably stationary platform . Its design and useare illustrated in Fig. 61. A rigid cross, about half ametre in each of its three dimensions, is weightedat the bottom and suspended from the top by a thinwire. The drag force (D) of the water on the crossswings it in the direction of the current until thesuspension wire is hanging at an angle θ off thevertical. At this point, the horizontal component ofthe tension in the wire must balance the drag forceon the cross, and the vertical component mustbalance the weight minus the buoyancy of theweighted cross (i.e. the weight in water, W), so

(8.8.1) tanθ = D / W

The drag force of a moving fluid on a bluff object

(not streamlined) is

(8.8.2) D = ½C p Av2

where A is the cross-sectional area of the objectnormal to the flow, v is the speed of the fluidrelative to the object, p is the density of the fluid,and C is the drag coefficient, a dimensionlessconstant approximately equal to unity. Theprocedure for measuring current is as follows:

(1) lower the current cross over a fixed pulley to

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the desired depth of water (allow for wireangle),

(2) measure the angle (θ) between the downwardvertical and the wire (wire angle indicators areavailable for this purpose),

(3) estimate the azimuth toward which thesuspension wire is heading in the horizontalplane, to get current direction,

(4) record the time and date, the inclination angle,and the azimuth of the suspension wire,

(5) assure that the weight of the loaded cross inwater and the cross-sectional area of the crossplus weights are recorded for eachmeasurement, because different combinationsmay be required to give reasonable inclinationsat different speeds,

(6) to confirm the validity of the measurements, atleast some observed sets of θ, W and A shouldbe converted to current speed: if the apparatuswas supplied with calibration tables, this maybe done by table look up; if it is an ad hocpiece of equipment assembled in the field,speeds calculated from the formula

where W is the weight in water in kg, and A isthe area in m2.

The constant (0.019) in 8.8.3 was derived bysubstitution of 8.8.2 into 8.8.1, with C = 1.0, p =1020 kg m-3, g = 9.8 m s-2. Multiplication of W bythe acceleration of gravity, g, in 8.8.1 wasnecessary to convert weight units (kg weight) tofundamental force units (kg m s-2).

A current cross may be employed down todepths of 15 m if the suspension wire is thinenough (about 2 mm) to keep the drag on the wiresmall relative to that on the cross. If a cross isconstructed in the field to meet an unexpectedrequirement, it should be retained for latercalibration, because the expression in 8.8.3 is onlyapproximate, and because an error could be madein determining W/A. The area and/or the weight ofthe cross must be chosen so that a reasonableinclination angle (θ) is obtained in the currentspeeds anticipated. If the angle is too small, itcannot be read accurately; and if it is too large, thecross may plane on its side. The maximum

inclination that should be encountered duringoperation is about 40°. To assist in the design of acurrent cross for a particular application, Table 8gives the current speeds that would produce themaximum deflection of 40° for various values ofW/A, according to expression 8.8.3. If it is wishedto work in knots, a sufficiently accurate conversionis 1 m s-1 = 2 knots.

While a wooden cross is usually employed inthis application, other material and forms may beused as well. If a large value of W/A is required, itmay be convenient to use a concrete cube. Forsolid cubes, WM is proportional to the length of theside, and for a concrete cube of 30-cm side, W/A= 500 (kg wt.)/(m2). In theory, a sphere is the bestshape to use, because it presents the same cross-sectional area to the current regardless of itsattitude. A cross, however, is easy to as and itserves well.

8.9 Drift poles - general

A drift pole is a long buoyant spar of uniformcross-section, weighted at its lower end so that itfloats vertically in the water column, with only asmall portion of its upper end above the surface. Asmall flag, radar reflector, light, or even a radiotransponder may be fastened to the upper end tohelp locate it. The area exposed to the wind,however, must be kept small in relation to thatexposed to the current, to prevent the wind fromunduly affecting the drift of the pole. Forhydrographic studies, the length of the pole underthe water should approximate the deepest draughtof vessels regularly operating in the area, so that itwill experience an average current similar to that

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experienced by a ship. Drift poles may beassembled to the desired length from prefabricatedsections, or may be made up from local lumber fortemporary use. In some areas it may be difficult toacquire and to manipulate drift poles as long as thedraught of the largest vessels. How long a pole canbe accommodated will depend a great deal on thesurvey vessels, equipment, and manpower, but a l0-m pole would probably be the maximum for anyoperation.

8.10 Drift poles - tethered

This procedure is designed for use from ananchored vessel. A drift pole is tethered to thestern of the vessel (anchored at the bow and ridingthe current) at the end of a measured line. The lineshould be made of buoyant polypropylene rope,about 100 m long, and marked off in 10-m sections. Starting with the line coiled on deck, the drift poleis released and allowed to drift away from thestern of the vessel as the line is payed out fastenough to provide slack, but slowly enough toprevent fouling. The time taken for the pole to driftthe full length of the line (or a measured fraction ofit. if the current is slow) is recorded, along with therelative bearing of the pole at the end of its run andthe direction of the ship’s head. From this, thecurrent speed and direction may be calculated, if itis assumed that the vessel has not moved duringthe exercise. The ship movement should bemonitored as well as is possible, so theobservations may be adjusted accordingly. Theprocedure should not be undertaken when the tideis turning and the vessel is coming about on itsanchor, but the time of turn should be recorded assuch. The drift pole releases may be repeated at30-minute intervals to provide a meaningful seriesof current measurements.

8.11 Drift poles – free-floating

This procedure is useful when it is wished tostudy current patterns over a fairly extensive area.It requires several drift poles, at least one launchcapable of placing and recovering the poles, and afairly accurate method of position fixing (visual or

electronic). Drift poles are released one at a timein representative sections of the region, and theirpositions, along with the times, are recorded assoon as they are floating free from the launch. Asthe poles continue to drift freely, their positions arefixed about every half hour, or more frequently ifthe region is small and/or the currents are strong.A drift pole should be picked up and repositionedif:(a) it is in danger of going aground,(b) it is about to leave the region of interest,(c) the distribution of the poles no longer provides

representative coverage of the region, or(d) two poles are too close together to

independent information.

Any one of a variety of methods may be used tofix the drift pole positions: if a launch has thecapability of fixing its own position, it may comegently alongside the pole and take a fix; if thelaunch also has radar, it may fix the position of adrift pole from a fair distance, possibly fixingseveral poles from the same location; a centrallystationed vessel with proper radar may be able tomonitor all the poles, using a launch only torecover and reset them; positions may bedetermined by theodolite observations from shore;or the poles may carry radar transponders or radiobeacons and interact with an electronic positioningsystem. The number of drift poles that can bemonitored at one time will depend on the size ofthe region, the speed of the current, the numberand speed of the launches. the quality of the radaror other positioning system, etc . In a simpleregion, such as a short and narrow channel. one ortwo drift poles frequently positioned and frequentlyreset may provide better coverage than a largernumber of poles less frequently positioned andreset.

8.12 Current drogues

Current drogues are deployed andmonitored in the same manner as drift poles, butare designed to drift with the current at specificdepths, instead of with the average current over aninterval of depth. They consist of a surface markerwith floatation, from which is suspended an object

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with a large surface area to intercept the current atthe desired depth. They must be designed so thatthe area exposed to the wind and to the current atother depths is small compared to the area of thedrogue at the selected depth. Square crosses,similar to those discussed in section 8.8, may beconstructed from plywood or sheet metal andweighted at the bottom to serve as drogues. Adrogue that has a large effective surface area. butis not bulky to transport, is the ‘ parachute drogue,”which is in appearance just what its namesuggests, except that the chute is deployedhorizontally instead of vertically. Care must betaken in launching a parachute drogue to assurethat the shrouds do not get tangled or fouled. Thereis a common misconception that the chute must befully open at all times to operate properly. Thechute, however, opens only to oppose the watermovement past it, i.e. to oppose being dragged bythe surface float and marker. If the drogue ismoving freely with the current at that depth, thechute may properly appear collapsed, as long as itis free to open if needed. Figure 62 illustratescurrent drogues of the square cross and theparachute type, and also a drift pole, forcomparison .

8.13 Continuity method

It follows from the continuity principle,which is basically the principle of the conservationof mass, that the rate at which the total mass ofliquid inside a container increases must equal thenet rate of transport through the entrance to thecontainer. Neglecting the small changes in densitythat may occur, we may equally well say that therate at which the volume of water inside anembayment increases must equal the net inwardvolume transport through its entrances. Thesimplest application of this principle to currentdetermination is to estimate the average tidalstreams flowing through the single entrance to anembayment. It is required to know the surfacearea of the embayment (A), the crossectional areaof the entrance (a), and the harmonic constants forthe main tidal constituents in the mean vertical tideover the surface of the embayment. Suppose Hand g are the average amplitude and phaselag of a

constituent of the vertical tide in the embayment.Let h(t) be the contribution of the constituent to thetidal height in the embayment at time t, and V(t) beits contribution to the volume of water in theembayment. Assuming that the surface area isroughly the same at all stages of the tide,

(8.13.1) V(t) = Ah(t) = AH cos(ωt – g)

where ω is the angular frequency of theconstituent. Differentiation of 8.13.1 with respectto time, and use of some simple trigonometricrelations, gives the rate of change of V(t) as

(8.13.2) d/dt V(t) = -ωAH sin (ωt - g) = -ωAH cos (ωt – g + 90)

By continuity, the rate of change of volume mustequal the rate at which water is being transportedin through the entrance, and this, divided by thecrossectional area of the entrance, is the meantidal stream speed through the entrance, u(t),whence, from 8.13.2, divided by a,

(8.13 .3) u(t) = (ωAH/a) cos (ωt – g + 90)

The amplitude and phaselag of the tidal streamconstituent are thus ωH(A/a) and (g – 90) in theappropriate units. Consider the following numericalexample, using the tidal constituent M2, for whichω = 0.00014 radians per second. Let A = 20 km2, a= 0.007 km2, H = 1.0 m, and g = 025°. By 8.13.3,the amplitude of the average M2 tidal stream in theentrance is 0.00014(1.0)(20/ 0.007) = 0.4 m s-1,and its phaselag is 025°-90° = -065°, or, adding360° to conform to the convention that phaselagsare positive angles, 295°.

The field hydrographer would not beexpected to perform the continuity calculationsnecessary to convert water level observations intocurrent information, but could be expected toprovide sufficient water level data from inside anembayment to support the calculations.Observations from one gauge site are sufficient formany small embayments, but observations fromseveral gauge sites may be required to representthe average tidal surface of a long and shallowinlet. The usefulness of the method has beendemonstrated by the calculation of tidal stream

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constants for various crossections of the St.Lawrence estuary and river between Pte. desMonts and Lake St. Peter. A steady riverdischarge through the system is not reflected in thewater levels, and its effect on the current at theoutlet must be added as a constant current, withspeed equal to the volume discharge rate dividedby the cross-sectional area of the outlet.

8.14 Hydraulic method

The cross-sections of some narrow andshallow passages are too small to accommodate

the large volume transports of water associatedwith the propagation of very long waves (tides,seiches, etc.). The result is that the water levelrises or falls at one end of the passage, creating ahydraulic head between the two ends. The flow inthe passage is said to be “hydraulic” if waterenters the passage at very nearly zero velocity andis accelerated down the pressure gradient createdby the hydraulic head. Neglecting frictional losses,the law of the conservation of energy tells us thatthe gain in kinetic energy per unit mass (v2/2)along a streamline must equal the loss in potentialenergy (gh), where v is the current speed, g is theacceleration of gravity, and h is the hydraulic head.

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Thus, we might expect to find the relation(8.14.1) v2 = 2gh

In practice, however the gauges at the ends of apassage might not be far enough apart to detectthe full hydraulic head, and their zeros might not beset to exactly the same datum. The practical formof 8.14.1 is, therefore

(8.14.2) v2 = ah + b

where a and b are constants to be determined bycalibration against direct current measurements.The constant b may also include allowance for theinitial kinetic energy possessed by water enteringthe system at a non-zero velocity.

Self-contained pressure gauges of the typedescribed in section 6.7 are convenient to place ateach end of the passage to measure the fluctuatinghydraulic head, since they detect the sum ofhydrostatic and atmospheric pressure, and it is tothis combined pressure gradient that the waterresponds. These gauges can be left to operateunattended for several months, but during part ofthe period direct current measurements should betaken in the passage to permit determination ofappropriate values for the constants a and b.Whether or not the method is applicable in aparticular passage is revealed by how well theresults of the calibration conform to equation8.14.2, with a and b constant. It may be reasonableto permit choice of two sets of constants, one forflow in one direction and one for flow in the other,but if the results cannot even then be fittedsatisfactorily to expression 8.14.2, it must beassumed that the flow is not sufficiently hydraulicfor the method to apply.

8.15 Long wave method

The relation between the particle motion andthe wave form was discussed for long waves insection 1.5, both for standing and progressivewaves. If water level measurements in a regionare available from enough locations, it may bepossible to identify the propagation characteristicsof long waves (e.g. tides and seiches), and so todeduce a great deal about the streams associated

with them. This is another reason why water levelmeasurements during a hydrographic survey shouldnot be limited to the absolute minimum needed forsounding reduction.

8.16 Electromagnetic method

This method works on the same principle asthe electric dynamo: if an electrical conductor ismoved through a magnetic field, a voltage isdeveloped along the conductor in proportion to therate at which the conductor cuts through lines ofmagnetic force. Although pure water is a very poorconductor of electricity, most naturally occurringwater (especially seawater) is a reasonably goodconductor, because of the dissolved salts in it. Thevertical component of the earth’s magnetic fieldtherefore causes an electric voltage to begenerated in water that flows through it, and thevoltage is proportional to the speed of flow as wellas to the strength of the magnetic field. In theory,therefore the flow through a channel may bemeasured by placing the probes of a sensitivevoltmeter in the water, one at each side of thechannel, to detect the voltage generated by theflow. The voltage that is measured can be shownto be proportional to the total transport through thechannel, rather than to the flow at a particulardepth. This is because higher voltages generated atdepths where the flow is greater are partiallyshort-circuited by the water at depths where theflow (and hence the voltage) is less, so that anaverage voltage is detected. A further complicationarises from the fact that the material on the bed ofthe channel, and beneath it, is not a perfectinsulator, so that part of the signal is also short-circuited by this path. Because the conductivity ofthe material in and below the bed is never wellenough known to permit calculation of its effect onthe measurements, direct observations of the flowmust be made during part of the installation period,to permit calibration of the system.

There are many practical difficulties to facein implementing such a measuring system. Aninsulated electrical cable must be led from thevoltage recorder to the electrode at the far side ofthe channel; this is usually done by laying it alongthe bottom, unless there is a bridge along which it

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can be strung. The electrodes at the two sides ofthe channel must be carefully matched, since thevoltage generated by their “battery effect” mayotherwise be as great as the signal beingmeasured. ln channels through which the flow issmall, it is difficult to separate the signal fromextraneous “noise” generated in the system. lnchannels through which the flow is large enough togenerate a strong signal, it is often difficult to layand maintain the electrical cable intact to the farside. This is not a method of current measurementthat is recommended for routine use onhydrographic surveys.

8.17 Geostrophic method

The Coriolis force, that results from theearth’s rotation and deflects currents to the right inthe Northern Hemisphere, was discussed in section1.8. The forces acting on a current thatexperiences no acceleration must be in balance;the balance in the direction of flow being betweenthe pressure gradient and the frictional forces, andthe balance in the direction normal to the flowbeing between the pressure gradient and theCoriolis forces. If, in addition to zero acceleration,the assumption is made that friction is negligible,there would be nothing to balance a component ofthe pressure gradient in the direction of flow. Thecurrent would then have to flow in a directionnormal to the pressure gradient, and at a speed justsufficient to produce a Coriolis force equal andopposite to the horizontal pressure gradient force.Such a current is said to be in “geostrophic”equilibrium. The hypothetical current whoseCoriolis force just balances the horizontal pressuregradient force is called the “geostrophic current,”and if the pressure field in the ocean is known, therate and direction of the geostrophic current mayeasily be calculated. The geostrophic current will,however, resemble the actual current only insofaras the assumptions of zero acceleration and zerofriction are true, and as the horizontal pressuregradient is accurately determined.

The horizontal pressure gradient in theocean depends upon the horizontal atmosphericpressure gradient, the slope of the sea surface, andthe distribution of water density within the body ofthe ocean. Oceanographers can calculate the

density distribution from measurements of thewater temperature and salinity, and so determinehow the horizontal pressure gradient changes withdepth. To convert these relative values to absolutevalues of the pressure gradient requires aknowledge of the actual pressure gradient for atleast one depth. Since it is rarely possible to knowthe slope of the sea surface, the assumption isusually made that the horizontal pressure gradientis zero at some large depth (e.g. 2000 m). Thisdepth is called the “depth of no motion,” becausethe horizontal pressure gradient can be zero only ifthe Coriolis force, and therefore the current, is alsozero. The relative geostrophic currents calculatedfrom the density distribution may then be referredto zero at the depth of no motion, to obtainestimates of the absolute values of geostrophiccurrent. In the open ocean, where friction andaccelerations are small, the geostrophic currentsresemble the actual currents reasonably well, andmuch has been learned about ocean circulation bythis method.

In coastal waters, where friction andacceleration may not properly be neglected, greatcare must be taken if geostrophic currents are tobe interpreted in terms of actual currents. If thecurrent is known to have a fairly uniform directionof flow, we may consider only the balance offorces normal to the flow, and so avoid friction,which acts parallel to the flow. Also, ifobservations are taken over a long enough period,the effects of acceleration must average to nearzero. Therefore, if the density distribution isdetermined as an average over a period sufficientlylong to remove acceleration effects, and if only thecomponent of the horizontal pressure gradientperpendicular to the main flow is used, the profileof the geostrophic currents so calculated shouldresemble reasonably well the profile of the actualaverage current over the same period. Selection ofa reference for a geostrophic current profile incoastal water is difficult, since a depth of nomotion cannot be assumed reasonably to exist inshallow water. This difficulty may be partlyovercome by adjusting the current profile to satisfythe estimated transports of water and salt throughthe channel, or by measuring the average currentdirectly at a particular depth, and fitting the profileto that value at that depth.

This discussion of geostrophic currents is

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given here not with the expectation that fieldhydrographers would be required to execute theassociated observations or calculations, but withthe realization that a great deal of the existing bodyof ocean current information has come from thissource, so that the method should be understoodand appreciated.

8.18 Current surveys - general remarks

Shipping and marine interests should benotified in a Notice to Mariners of any plannedcurrent survey that might in any way interfere withtheir operations. The information should appear ina Notice about a month before the commencementof the survey, early enough to be acted upon, butnot so early as to be forgotten. It should tell thepurpose of the survey, the general area and timeperiod involved, the nature of the operation (e.g.current meter moorings, anchored vessels, trackingof drift poles, etc. ), and should describe theappearance of any surface markers and surfacedrifters that are to be used. The locations ofmoorings and anchor stations that are plannedshould be given as closely as possible. An attemptshould also be made to have the same basicinformation broadcast on the marine radio duringthe survey, particularly if the operation is in or nearshipping lanes or fishing grounds. Another effectiveoutlet for the information is sometimes the fishingand marine broadcast over the local commercialradio station. Such notices not only reduce the riskof lost or damaged equipment and of lost data, butthey foster better public relations, by satisfyingnatural local curiosity. The issuing of notices,however, does not relieve the surveyor of the needto choose mooring locations and conductoperations in a manner that will cause the leastdisruption of other interests, while still providing thedesired data. From the standpoint of equipmentsafety, the mooring of current meters in an areafrequented by fishing draggers is particularlyhazardous.

The convention for quoting the direction ofcurrents is the exact opposite of that for winds: thecurrent direction is the direction toward which it isflowing, whereas the wind direction is that fromwhich it is blowing. The compasses in currentmeters are designed to record in agreement with

this convention, and all manual records mustaccord with it as well.

As it is for water level observations,accurate time keeping is also important for currentobservations, particularly in tidal waters, where it iswished to relate the phases of the tidal streams tothose of the constituents in the equilibrium tide. Inthe mooring and recovery logs, the time of everyevent that could be reflected in the current recordshould be recorded and described; e.g. rotorspinning in the wind, meter in water, anchor onbottom, anchor release tripped, etc. Thisinformation provides supplementary time checks onthe records, which can frequently resolveuncertainties caused by an error in either the initialor final time check. In all records involving time,the zone time being used must be clearly indicatedon every sheet by the appropriate abbreviation(GMT, AST, PDST, etc.).

It may be that the same current meter is tobe moored more than once during the season, oreven during a single survey. Even though the datastorage capacity of the meter may be large enoughto accommodate the combined records fromseveral moorings, the data record should always beremoved, and replaced by fresh magnetic tape (orfilm, etc.), before the meter is moored again .Keeping separate records for each installationreduces the possibility of confusion later on, but themain reason for removing the record beforeresetting is to protect if from loss or damage. Thetime, effort, and expense invested in a successfulmooring are worth a great deal more than thematerial on which the data are recorded, and thereis no economy in risking the data simply toconserve magnetic tape. There is, of course, noobjection to cutting off the used portion andcontinuing with the unused portion of a tape or film,if this can be done with no risk to the record and ifsufficient storage space remains on the unusedportion. Magnetic tape records should be stored inferrous metal containers and kept out of strongmagnetic fields and excessive heat until the datahas been extracted from them. All current recordsand supporting documentation should be submittedto the Regional Tidal Officer at the end of theseason, or earlier if an opportunity is afforded.Where possible, copies should be made of recordsand documents, and stored separately from theoriginals.

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BIBLIOGRAPHY1. Canadian Hydrographic Service publications

CANADIAN TIDE & CURRENT TABLES(published annually)

Vol. 1 : Atlantic Coast and Bay of FundyVol. 2: Gulf of St. LawrenceVol. 3: St. Lawrence and Saguenay RiversVol. 4: Arctic and Hudson BayVol. 5: Juan de Fuca and Georgia StraitsVol. 6: Barkley Sound and Discovery Pas-

sage to Dixon Entrance

SAILING DIRECTIONSNewfoundlandNova Scotia and Bay of Fundy Gulf and River St.LawrenceGreat Lakes (two volumes)British Columbia (two volumes)Great Slave Lake and Mackenzie RiverLabrador and Hudson BayArctic Canada (three volumes)

SMALL CRAFT GUIDESSaint John RiverTrent-Severn WaterwayBritish Columbia (two volumes)

CURRENT ATLASESTidal Current Charts, St. Lawrence Estuary, OrleansIsland to Father Point, along the main shippingroute. (I 939). Tidal Publication No. 21 (out ofprint),

British Columbia, Tidal Current Charts, VancouverHarbour. Tidal Publication No. 22 (out of print).

Atlas of Tidal Current Charts, Yuculta Rapids,Cordera Channel, British Columbia. (1970). TidalCurrent Publication No. 23.

Tidal Currents, Quatsino Narrows, BritishColumbia. (I 97 1). Tidal Current Publication No.24 (out of print).

Atlas of Tidal Current Charts, Vancouver Harbour,British Columbia(1973). Tidal Current PublicationNo. 30.Atlas of Tidal Currents - Bay of Fundy and Gulf ofMaine. (1981).

Current Atlas - Juan de Fuca Strait to Strait ofGeorgia. (1983).

CHART NUMBER ONE(Defines and illustrates chart symbols andabbreviations, including those for tides and currents)

NAVIGATIONAL CHARTS(Approximately 1 600 in number, most containsome current and tide or water level information)

NOTICES TO MARINERS(Issued weekly to update information on charts;they are published jointly by the CHS and theCanadian Coast Guard)

TIDES IN CANADIAN WATERS (G.C. Dohler)(Descriptive pamphlet on the origin and nature oftides, with examples from Canadian waters; 14pages plus 6 fold-outs)

TIDAL & METEOROLOGICAL INFLUENCES ONTHE CURRENT in LITTLE CURRENTCHANNEL. (W.D. Forrester, 196i) (Manuscriptreport, demonstrating the part played by wind set-up, seiche, and tide to produce the reversing currentin Little Current Channel between Georgian Bay andthe North channel of Lake Huron; 55 pages)

THE CANADIAN HYDROGRAPHIC SERVICE (I979) (Illustrated pamphet on the organization andoperations of the Canadian Hydrographic Service;17 pages)

2. Introductory reading

CARSON, R.L. The Sea Around Us. Oxford UniversityPress, New York, NY, 1961 (revised edition), 237 p.

DEFANT, A. Ebb and Flow. Ambassador Books, Ltd.,Toronto, Ont. 1958 (translation), 121 p.

DARWIN, G.H. The Tides and kindred Phenomena of theSolar System. 1898, republished 1962, Greenmanand Cooper, San Francisco, CA. 378 p.

3. Supplementary reading

LEBLOND, P.H., AND L.A. MYSAK. Waves in theOcean. Elsevier Scientific Publishing Co., NewYork, NY, 1978, 602 p. POND, S., AND G.L.PICKARD. Introducing Dynamic Oceanography.

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Pergamon Press, Oxford, 1978, 241 p.THOMSON, R.E. Oceanography of the British Columbia

Coast. Can. Spec. Pubi. Fish. Aquat. Sci. 56;1981, 291 p.

WHEELER, W.W. A practical Manual of Tides andWaves. Longmans and Green, London, 1960, 201 p.(includes a good history of the developments in tidaltheory)

Rouch”, J. Les Marées. Payot, Paris, 196 1,230 p. (traiteI’histoire et la théorie des marées et des phénomènessemblables d’une manière intelligible)

LISITZIN, E. Sea-level changes. Elsevier ScientificPublishing Co., New York, NY, 1974, 286 p.

PROUDMAN, J. Dynamical oceanography. Methuen,London, 1954, 490 p.

NEUMANN, G., AND W.J. PIERSON. Principles ofPhysical Oceanography. Prentice-Hall, EnglewoodCliffs, NJ, 1966, 545 p.

DOODSON, A.T., AND H.D. WARBURG. AdmiraltyManual of Tides. H.M. Stationery Office, London,1941, reprinted 1952, 270 p.

ADMIRALTY MANUAL OF HYDROGRAPHICSURVEYING, volume two, chapter 2. Tides andTidal Streams. H.M. Stationery Office, London,1969, 119 p.

ADMIRALTY TIDAL HANDBOOK No. 2 Datums forHydrographic Surveys. H.M. Stationery Office,London, 1975, 38 p.

NATIONAL OCEAN SURVEY, NOAA TechnicalReport NOS 64. Variability of Tidal Datums andAccuracy in Determining Datums from Short Seriesof Observations. U.S. Government Printing Office,Washington, 1974, 41 p.

NATIONAL OCEAN SURVEY, Publication 30-1.Manual of Tide Observations. U.S. GovernmentPrinting Office, Washington, 1965, 72 p.

NATIONAL OCEAN SURVEY, Special Publication No.215, revised. Manual of Current Observations.U.S. Government Printing Office, Washington,1950, 87 p.

NATIONAL OCEAN SURVEY (Bowditch). AmericanPractical Navigator (1958), Part 6, Oceanography,p. 691 to p. 762. Part 8, Hydrography, p. 837 to p.899. U.S. Government Printing Office, Washington,1966.

NATIONAL OCEAN SURVEY, Special Publication No.135, revised. Tidal Datum Planes. (H.A. Marmer).U.S. Government Printing Office, Washington,1951, 142 p.

SERVICE CENTRAL HYDROGRAPHIQUE. Marées.École d’application. Cours de M. Villain. Paris,1943-44.

SERVICE CENTRAL HYDROGRAPHIQUE. 14exercices de marées. Par M.A. Courtier. Paris,1937.

4. Advanced harmonic analysis references

FOREMAN, M.G.G. Manual for Tidal Heights Analysisand Prediction. Pacific Marine Science Report 77-10, Institute of Ocean Sciences, Sidney, B.C., 1977,97 p. GODIN, G. The Analysis of Tides.University of Toronto Press, Toronto, Ont., 1972,264 p. (A philosophical and highly mathematicaltreatment - not recommended as a practical guide toharmonic analysis).

ADMIRALTY TIDAL HANDBOOK No. 1. TheAdmiralty Semi-graphic Method of Harmonic TidalAnalysis. H. M. Stationery Office, London, 1958.

ADMIRALTY TIDAL HANDBOOK No. 3. HarmonicTidal Analysis (short periods). H.M. StationeryOffice, London, 1964.

NATIONAL OCEAN SURVEY, Special Publication No.98, revised. Manual of Harmonic Analysis andPrediction of Tides. U.S. Government PrintingOffice, Washington, 1940, 317 p.

NATIONAL OCEAN SURVEY, Special Publication No. 260.Manual of Harmonic Constant Reductions. U.S.Government Printing Office, Washington, 1962,74 p.

5. Glossaries of terms

INTERNATIONAL HYDROGRAPHICORGANIZATION, Special Publication No. 28.Vocabulary concerning Tides - Vocabulaireconcernant les Marées. International HydrographicBureau, Monaco, 1932, 22 p.(Terms defined in both English and French).

INTERNATIONAL HYDROGRAPHICORCANIZATION, Special Publication No. 32.

Hydrographic Dictionary, Part I - DictionnaireHydrographique, Partie 1, InternationalHydrographic Bureau, Monaco, 1974.

NATIONAL OCEAN SURVEY. Tide and CurrentGlossary, revised. U.S. Government PrintingOffice, Washington, 1975, 25 p.

119

APPENDIX A

MAJOR TIDAL HARMONIC CONSTITUENTS

This is by no means a complete list of all the possible tidal harmonic constituents, but it does containall the larger ones.

The “ratio” in column 2 is the amplitude of the constituent in the equilibrium tide divided by theamplitude of the M2 constituent in the equilibrium tide. The “speed” in column 3 is the angular speed ofthe constituent in degrees per solar hour.

Description Symbol Ratio Speed °/h

Mean value Zo - 0.0000

Annual constituent Sa 0.013 0.0411(see section 2.6)

Semi-annual constituent Ssa 0.080 0.0821(see section 2.6)

Monthly constituent Mm 0.091 0.5444(see section 2.6)

Fortnightly constituents Mf 0.172 1.0980(see section 2.6) MSf 0.009 1.0159

Diurnal constituents K1 0.584 15.0411(section 2.5) 01 0.415 13.9430

P1 0.193 14.9589

Semidiurnal constituents M2 1.000 28.9841(section 2.5) S2 0.465 30.0000

N2 0.194 28.4397K2 0.127 30.0821L2 0.028 29.5285T2 0.027 29.9589

Quarter-diumal shallow- M4 - 57.9682water constituents MS4 58.9841(section 3.5)

120

APPENDIX B

121

122

123

Temporary Gauge Data

This form combines a computation sheet for datums and Bench Mark elevations, a gauge history for past andpresent data and a Bench Mark sketch with descriptions along with detailed instructions for standardizing its use.The gauge history will be supplied along with Bench Mark data when required.

Section 1. This section is for computing the elevation of all Bench Marks and the zero of the automatic gaugerelative to chart or sounding datum. The Bench Marks should follow the pattern indicated by the letters a and b, andthe convention + above and - below should be used throughout having careful regard to the exact context.

Section 1A. is used by successive parties when the elevation of Chart Datum below a Bench Mark is known. Theelevation of all Bench Marks in the net and the zero of the staff gauge are related to Chart Datum. The zero of theautomatic gauge relative to the zero of the staff gauge is found by comparing simultaneous readings recorded on thecomparison form and hence the zero of the automatic gauge relative to Chart Datum.

Section 1B. This section is used by the initial party to transfer Chart Datum from a place (Z) where it has alreadybeen established. The table should be completed using the days with maximum tidal range available. The meanvalues are used as indicated to compute r, R, m, M, and hence “d” the amount the zero of the automatic gauge isabove (+) or below (-) Sounding Datum. The zero of the staff gauge relative to the zero of the automatic gauge isfound by comparing simultaneous readings recorded on the comparison form and hence the zero of the staff gauge isfound relative to Sounding Datum. The elevation of all Bench Marks in the net are then computed above SoundingDatum. The Higher High Water Large Tides (HHWLT) datum for heights can be calculated using the HHWLT at (Z)and r/R as indicated. The HHWLT at (Z) is found by applying the HHWLT height difference at Z to the HHWLTheight at the appropriate reference port. If this data is not published, it will be supplied by the Tides and WaterLevels Section.

Section 2. All parties insert appropriate data (In sections 3, 4 and 5 all parties complete one section of each).

Section 3. Indicate type of record and insert a brief description of the site.

Section 4. Insert a brief description of the method used for establishing the Bench Mark elevations e.g. water leveltransfer from “Z” or levelling run from Geodetic Bench Mark No. “CCX”. Similarly for successive years e.g., levellingrun from controlling Bench Mark No. 1.

Section 5. The elevations of Bench Marks are accepted as those computed by the party which establishes them.Successive parties tabulate the results of levelling lines and identify the controlling Bench Mark with an asterisk e.g.12.23*.

Section 6. The initial party inserts Datum for Heights. The calculation of this Datum in section 1 B is a water leveltransfer and it is therefore referred to Sounding Datum. Its elevation above Chart Datum will be entered by the Tidesand Water Levels Section.

Section 7. The initial party inserts Bench Mark data and concise descriptions using the printed format. Theelevations of Bench Marks are always above Sounding Datum unless a Bench Mark elevation above Chart Datumhas been used in the levelling run. Otherwise, elevations above Chart Datum will be entered by the Tides and WaterLevels Section. Photographs of each Bench Mark with their number and location marked, should be submitted alongwith other gauge data. Successive parties insert the condition of each Bench Mark e.g., good, unreliable, destroyedor not located, and check accuracy of descriptions making amendments if necessary.

Section 8. The initial party draws a sketch showing location of Bench Mark, automatic gauge and staff gauge,showing distances from conspicuous fixed points. Successive parties check for amendments if necessary.

F.TWL-502/83

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Aliasing, 3.6Amphidrome, 1.10Amphidromic point, 1.10Amphidromic system, 1.10Amplification, by funneling, 1. 12Amplification, by reflection, 1.6Amplification, by resonance, 1.6Amplification, by shoaling, 1.12Amplitude, wave, 1.2Amplitude, harmonic constituent, 2.1, 3.3Amplitude modulation, 2.5Analysis, tidal, 3.1Analysis, harmonic, 3.3Aneroid, 6.5Anti-cyclonic, 4.8Anti-node, 1.3Aphelion, 2.5Apogee, 2.5Arctic, 2.9Ascending node, 3.4Atlantic Ocean, 1.6, 1.10, 2.9, 4.8Atmospheric pressure, 4,4

Backsight, 6.17Bay of Fundy, 1.6, 1.12, 3.11, 4.6Belle Isle, Strait of, 1.3Bellows, 6.5Benchmark, 5.9, 6.12Benchmark, descriptions, 6.15Benchmark, special types, 6.14Benchmark, standard CHS, 6.13Bore, 1.12Bourdon tube, 6.5Bubbler gauge, 6.6Capillary wave, 1.2Celestial sphere, 3.4Centrifugal force (acceleration), 1.8Centripetal acceleration, 1.8, 2.1, 2.2Chart datum, 5.6Circulation, estuarine, 4.9Classification of tides, 2.9Constants, harmonic, 2.1Constituents, diurnal, 2.5, 2.7Constituents, harmonic, 2.1, 3.3Constituents, long-period, 2.6, 2.7Constituents, semidiurnal, 2.5, 2.7Constituents, shallow-water, 3.5Continuity method, 8.13Co-phase, 3. 1 0Co-range, 3. 1 0Coriolis force (acceleration), 1.8, 4.8Coriolis parameter, 1.8

Cotidal charts, 3.10, 7.11 Crest, wave, 1.2Crustal tilting (or movement), 5.5Current cross, 8.8Current drogue, 8.12Current measurement, general, 8.1, 8.18Current measurement, location and depth, 8.3Current measurement, methods, 8.5Current measurement, preparation, 8.2Current measurement, time and duration, 8.4Current meters, moored, 8.6Current meters, over-the-side, 8.7Currents, geostrophic, 8.17Currents, inertial, 1.9, 4.8Currents, ocean, 4.8Currents, residual, 1. IICurrents, wind-driven, 4.2Cyclonic, 4.8

Data recorders, digital, 6. 1 0Data recorders, drum, 6.10Data recorders, strip-chart, 6. 10Data recorders, tele-announcing, 6. 1 0Datum, chart, 5.6Datum, for elevations, 5.6Datum, Geodetic, 5.4Datum, International Great Lakes (1955), 5.5Datum, sounding, 5.6, 7.1, 7.2, 7.3, 7.4, 7.5Datum notes, 7.8Datum of vertical reference. 5.1Datum surface, 5. 1Deep-sea pressure gauge, 4.4, 6.7Deep-water wave, 1.4Diaphragm-type pressure gauge, 6.5Dispersion, 1.4Distortion, shallow-water, 1.12Diurnal constituents, 2.5, 2.7Diurnal inequality, 2.5Diurnal tides, 2.5, 2.9Documentation of records, 7.7Drift poles, free-floating, 8.11Drift poles, general, 8.9Drift poles, tethered, 8. 10Drogues, current, 8.12Dynamic elevation, 5.3

East Coast, 3.10, 4.8Ebb, 1.11Ekman, 4.2Electromagnetic method (currents), 8.16Elevation, dynamic, 5.3Elevation, geopotential, 5.3Elevation, orthometric, 5.3

INDEX

125

Ellipse, tidal, 1.11, 3.7Equi-geopotential surface, 2.4Equilibrium tide, 2.2, 2.4, 2.7Equipotential surface, 2.4Erie, Lake, 4.3, 4.6Estuarine circulation, 4.9Estuary, 4.9Evaporation, 4.7Extrema, tide and tidal stream, 3.9

Fall (of tide), 1.11Float gauge, 6.4Flood, 1.11Flow, 1.11Forced oscillation, 1.6Forces, tide-raising, 2.1Foresight, 6.17Foucault pendulum, 1.9Fourier theorem, 3.2Free oscillation, 1.6Freezing, 4. 1 0Frequency, 1.2Frequency, natural, 1.6Fundamental frequency, 3.2Fundy, Bay of, 1.6, 1.12Funneling, 1. 12

Gaspé Current, 4.9Gauge, bubbler (gas purging) type, 6.6Gauge, deep-sea, 6.7Gauge, diaphragm type, 6.5Gauge, float type, 6.4Gauge, sight (tape), 6.9Gauge, staff, 6.8Gauge inspection, 7.6Gauge operation, 7.1Gauge record documentation, 7.7Gauge shelter, 6.3Gauge site selection, 6.11Gauge zero, 6.23Geoid, 5.2Geopotential, 2.4Geopotential elevation, 5.3Geostrophic current, 8.17Gore Bay, 4.6Gravitational attraction, 1.8, 2.1, 2.2Gravity, 1.8, 2.1Gravity wave, 1.3Gulf of Maine, 1.6, 3.11, 4.6Gulf of St. Lawrence, 1. 10, 2.9, 3.10, 4.8, 4.9

Half pendulum day, 1.9

Harmonic analysis, tidal, 3.3Harmonic analysis, tidal stream, 3.7Harmonic constants, 2.1Harmonic constituents, 2.1, 3.3Harmonic frequency, 3.2High water (HW), 1.2Higher high water (HHW), 1. I IHigher high water, large tide (HHWLT), 5.7Higher high water, mean tide (HHWMT), 5.7Higher low water (HLW), 1.11Hudson Bay, 3.10, 3.11Huron, Lake, 4.6Hydraulic method, 8.14

Inequality, diurnal, 2.5Inertial circle, 1.9Inertial current, 1.9, 4.8Inertial period, 1.9Internal tides, 1.7Internal waves, 1.7International Great Lakes Datum (1955), (IGLD),5.5Inverted barometer effect, 4.4, 6.7

James Bay, 3.10Juan de Fuca Strait, 3.11

K1 constituent, 2.5K2 constituent 2.7

L2 constituent, 2.5Lake Erie, 4.3, 4.6Lake Huron, 4.6Lake Ontario, 4.3, 4.6Langara Island, 4.11Length of record, 3.6, 6.1Level surface, 2.4, 5.2Levelling, 5.8, 6.16Levelling, accuracy, 6.22Levelling, closure (closing error), 6.17Levelling, differential, 6.17Levelling, equipment, 6.18Levelling, instrument adjustments, 6.20Levelling, instruments, 6.19Levelling, procedures, 6.21Levelling, recording, 6.21Levelling, terminology, 6.17Long-period constituents, 2.6, 2.7Long wave (shallow-water), 1.4Long wave method, 8.15Loop, 1. 3Low water (LW), 1.2

126

Lower high water (LHW), 1.11Lower low water (LLW), 1.11Lower low water, large tide, (LLWLT), 5.6, 5.7Lower low water, mean tide, (LLWMT), 5.7Lowest astronomic tide, (LAT), 5.7Lowest normal tide, (LNT), 5.6

M2, 2.6M4, 3.5MS4, 3.5Mf, 2.6Mm, 2.6Mean lower low water (MLLW), 5.6Mean low water, springs, (MLWS), 5.7Mean sea level (MSL), 5.2Mean water level (MWL), 5.2, 5.7Mediterranean Sea, 4.9Melting, 4.10Merian’s formulae, 1.6Meters, current, 8.6, 8.7Mixed tides, 2.9Modelling, numerical, 3.11Modulation, amplitude, 2.5 11 ,Modulation, phase, 2.5Moncton, 1. 12Moon’s tide-raising force, 2.3

N2, 2.5Natural frequency (or period), 1.6Neap tide, 2.8Neuville, 1. 12Newton’s law of gravitation, 2.2Newton’s law of motion, 2.2Nineteen-year modulation, 3.4Nodal correction (u), 3.4Nodal factor (t), 3.4Node, 1. 3Node, ascending, 3.4Notes, datum, 7.8Numerical modelling, 3.11

01, 2.5Ocean, Atlantic, 1.6Ocean currents, 4.2, 4.8 Ocean, Pacific, 1.6Ontario, Lake, 4.3, 4.6Orthometric elevation, 5.3Oscillation, forced, 1.6Oscillation, free, 1.6Overfall, 1.12Over-tides, 2.1

P1, 2.5

Pacific Ocean, 1.6, 4.8Particle motion, 1.2, 1.4, 1.5, 1.7Pendulum day, 1.9Pendulum, Foucault, 1.9Pefigean tide, 2.8Perigee, 2.5Perihelion, 2.5Period, 1. 2Period, inertial, 1.9Period, natural, 1.6Petitcodiac River, 1.12Phase, 1. 2Phase modulation, 2.5Phaselag, 1.2, 2.1, 3.3Port Alfred, 1.6Potential, tidal, 2.4Precipitation, 4.7Prediction, extrema, 3.9Prediction, tidal, 3.1, 3.8Pressure, atmospheric, 4.4, 6.5, 6.7Pressure, hydrostatic, 4.4, 6.5, 6.7Pressure, wave, 1.4, 6.5Pressure gauge, bubbler-type, 6.6Pressure gauge, diaphragm-type, 6.5Progessive tide, 1.11Progressive wave, 1.3

R2, 2.5Race, tide, 1.12Range, 1.2Rayleigh criterion, 3.6Record length, 3.6, 6.1Recorders, data, 6.10Rectilinear flow, 1.11Reduction of soundings, 7.10, 7.11, 7.12, 7.13Refraction, 1.4Regional relations, 3.6Residual current, 1.11Resonance, 1.6Reversing falls, 1.12Rip, tide, 1.12Ripples, 1.2Rise (of tide), 1.11Rotary flow, 1.11Runoff, 4.7

S2, 2.5Sa, 2.6Ssa, 2.6Saguenay fjord, 1.6Saint John, 1.12St. John River, 1.12

127

St. Lawrence Estuary, 1.6, 1.7, 1.8, 4.9St. Lawrence River, 1. 12Sampling interval, 3.6Seiche, 1.6, 4.6Seismic sea wave, 4.11Semidiumal constituents, 2.5, 2.7Semidiumal tide, 2.5, 2.9Set-up, wind, 4.3Shallow-water constituents, 3.5Shallow water effects, 1.12Shallow water wave, 1.4Shelter, gauge, 6.3Shoaling, 1.12Short wave, 1.4Shubenacadie River, 1.12Sight gauge, 6.9Slack water, 1.11Sounding datum, 5.6, 7. 1, 7.2, 7.3, 7.4, 7.5Sounding reduction, 7.10, 7.11, 7.12, 7.13Speed, angular, 1.2Speed, wave, 1.2, 1.4, 1.5, 1.7Spring tide, 2.8Staff gauge, 6.8Stand of tide, 1.11Standing tide, 1.11Standing wave, 1.3Strait of Belle Isle, 1.3Strait of Georgia, 3.11, 4.8Storm surge, 4.5Stream, tidal, 1.11, 3.7Submission of records and documents, 7.9, 8.18Sun’s tide raising force, 2.2Surface wave, 1.2Surge, storm, 4.5Sydney Harbour, 4.6

T2, 2.5Tadoussac, 1.6Tape gauge, 6.9Tidal analysis, 3.1Tidal bore, 1.12Tidal ellipse, 1.11, 3.7Tidal potential, 2.4Tidal prediction, 3.1, 3.8, 3.9Tidal streams, 1.11, 3.7Tide, definition, 1.1Tide, diurnal, 2.5Tide, earth, 2.2Tide, equilibrium, 2.2, 2.4, 2.7Tide, internal, 1.7, 1.11Tide, neap, 2.8Tide, semidiumal, 2.5

Tide, spring, 2.8Tide race, 1.12Tide rip, 1.12Tide waves, 1.11Tide-raising forces, 2.1, 2.2, 2.3Time of turn, 3.9Tofino, 4. 1 I-Tractive forces, 2.2Transfer of datum, 7.1, 7.2, 7.3, 7.4, 7.5Trois Rivières, 1.12Trough, wave, 1.2Truro, 1.12Tsunami, 4.10Turning point, 6.17

Upwelling, 4.2

Velocity, particle, 1.4, 1.5, 1.7Vertical datum, 5.1

Water transfer, 5.5, 5.8, 7.3, 7.4, 7.5Wave, definition, 1.2Wave, capillary, 1.2Wave, gravity, 1.2Wave, internal, 1.7Wave, long (shallow water), 1.4Wave, progressive, 1.3Wave, short (deep water), 1.4Wave, standing, 1.3Wave, surface, 1.3Wave number, 1.2Wave speed, 1.2, 1.4, 1.5, 1.7Wavelength, 1.2Well, stilling, 6.2West Coast, 3.10Wind set-up, 4.3Wind-driven currents, 4.2

Zo, 3.3