Sxnall Tidal Estuary - Earth and Planetary Scienceeps.berkeley.edu/people/lunaleopold/(082) Hydraulic... · n tidal estuary, particn1:trly at b:tnkfnll stage, iiiny help to illuiiiiiiate

Hydraulic Geometry of a Sxnall Tidal Estuary

.UNA B. LEOPOLD

PI-IYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

By ROBERT M. MYRICK and

G E 0 L 0 G I C A L S U R V E Y P R 0 I; II: S S I O N A L P A P E R 4 2 2 - B

UKITIZD STA'T12S GOVLRRMll sNI ' P R I N T I N G OFFICIC, WRSII ING'TON : 1963

UNITED STATES DEPARTMENT OF THE INTERIOR

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY

Thomas B. Nolan, Director

I;or sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C.

CONTENTS

Page I Page

FIGURIC 1 . 2. 3 . 4. 5. 6. 7. 8. 9.

10. 11. 12. 1 3 . 14.

I3ankfull discharge- - - - -. - - - - - - - - - - - - - - 9 Hydraulic characteristics a t a cross section-_ - - - -. - - - - 11 Theoretical analysis of the hydraulic gcoinctry in the

downstrean1 direction, by Waltcr 13. Langbriii-. ~ -. . - 15 hfcasiireinents of the hydraulic gcorrietry in thc down-

stream direction- - _ _ _ _ _ - -. - _ _ _ _ _ _ _ ~ - - - ___. - - - __. _ _ 17 Referpiices _____________. .____. ._________________.__ 18

ILLUSTRATIONS

TABLES --

Page

B4 4 6

TABLE 1 . 2. 3 .

F'article-size analysis of channel-bank materials _ _ _ _ ~ - -. - - -. -. - _ _ - _ _ - -. - -. -. . - - - -. . . -. . . . . -. -. . - ~ -. . - . - - - -

€'ollcn anlysis of samples of bank niatcrial at Section D - - _ _ __. ~ -~ ~ -. . __. - -. _ _ - - - - - -. __. - -. -. -. -. - ~. . ~ ~ - -

Characteristics of cross sections of tidal channel and some rivers in nearby area- - - - - ~ - ~. . - -. ~ - - - ~ -. . - . . . . - -- -

I11

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY

By R O ~ E R T M. MYRICK and LUNA I<. LEOFOLD ABYrRACT

A tidal chaniicl in a niarsli bordering the Potoiiiac River near Alwxndriii, Va , jyas iualllic~tl. aiitl cnrrc>nt-inet er lll~i~slireliit,llts o f tlisckirg \v(~r(’ rri:ide :it vnrioiis locxtions :~nd at wrioiis \ t :igw iii t h . titlal caycle. of the chaiiqe of \\ itltli, del)tli, ant1 vc4ocity will1 dih(.liiirge a I :iriou\ crokh w(.l io114 i i i id along the 1(,iigtli of the chnnnel.

‘I‘lirrc. ih i i l ~ 1)rtw~iitcd i i t litwrt~fi(*al tleveloptnent of some of 1 lies? s:inic rclnt ions I)nse.il 011 liytlr:iulic* priiiviples ilnd on the :i\\~i~nption (if :I 1111 of energy ;itid a ruiiiiiniini rat(’ of \\.orb in the I

‘l’he cliaii~,c of I\ itltli, depth, mid velocity with discharge do\vn- strc~:iiii de\ ( lolwtl froiii lhe Ecld data cliccked closely with tbe llic~oretic,ill:f drrivcvl values

Tlic wt 11: riiir c.haiint.1 differs from n terrestrial one in that tli*c.li:irge a1 :illy hecation iii an estuary I arir‘s tlepeilding on how tlw t lon sh il)cvl the entire length of tlie channel between the 1)ciint in qnt hl ion and tlie m i i i i i 1)ody of tidal \\.:iter. The result ih tlint a tillnl chiiiiiiel caliaiiges niore rapidly in width aiicl less r:ilii(lly in ( r1)t 11 a \ tliwh:ir:.e chniiges tlo\vnstream than does a t t.rrt,.;tri:il c 1i:iiitic’l.

INTRODUCTION

Thc~se Iiit’iistir(liiiCii1s ;illo\ved : i i i a l .

The shape aiid characteristics of a river chaniiel are believed to be determined mid inaiiitaiiiecl by what has been called by irrignt ioii engineers the doiriiiiniit cliscliarge. 1I’liere:is i i i :in irrigxtioii cnnal tlie i i i i ia l or tle- sign discharge is ordinarily the modal flow, the clesig- nation of L c1oniiii:iiit discliarge in a, river is iiot as easy. Tlie effect i re discharge most influentin1 in river inor- pliology and tlins eqnivnleiit to the cloniinant cliscliarge of canals is I)elie\ecl to be the flow at b:uikfull stage. This vie\\- n-as s i rengtheiietl by the findiiigs of TVolman a l ~ l T,eopolcl ( 1N+i), ant1 siibstantintecl later by Nixon (I%%), t lixt tlie I)aiikfnll discliarge 1i:ib :I ~ W I I I T ~ I I ( * ~

interval o f 1 to 2 y e w s :tiicl is siiiiilar among rivers in quite diffeiwii physiogixpliic settings.

Woliiiaii ni i t l Miller (I%iO) extended aiicl genernlizeci tliis into ;I co i i c~~p t of elfeci i v e force in geomorphology. The q e a t e s t : ~ i ~ i o i ~ t i t of geonioiyl~ic~ u orlr, t1ir.y post 1 1 -

late, clepeiids iiot only on tlie cft’wt ivcness of e:wh single event Init also on tlir freqiieiic‘y of the evriits of tliffer- eiit 1iiilg:nit ndtls. T h e conibiiintioii of effectiveness and frequency t l i a i was iiiost influeiitinl ii i ~ii:iiiit:iiiiiiig river channel foriii i i i i d jmttrim IWS tlie I)aiilcfiiIl discliarge.

A ticlal estuary is, in a seiise, iiierely one of the ex-

tremes in tlie continuum of river channels. The cIiaiine1 system of iiii estnnry bears soiiie close similarities to that of :L river systeiii. It differs, h ~ \ ~ ~ e r , in at least one est reme 1y i in 1mr t :I i i t res1 )ec t :I I I cl t 11 at is t lie frequency of t lie bnnltfull stage. If, :IS soiiie iiiveht igntors beliere, the h i i l i f i i l l s i age is tlic one that is most etfect ive in the in:iiiitenniice of a r irer cliaiiiiel and is eqiialed or exceeded once every year or tvo, one may ask ~ h y the clinniiel system of a tidal estuary so much resembles a r irer network, despite tlie fact tlint b:tnkfull stage is attained about twice :L day. Observations a ~ i d 1iie:isure- iiients of flow cliaracteristics at different sectioiis along n tidal estuary, particn1:trly at b:tnkfnll stage, iiiny help to illuiiiiiiate tlie position of estuarine channel sgsteiiis in the continuum of all natural channels.

The bnnkfiill stage of a. t i d d estuary iiiay be defined :IS the stage :it \vhich the \rater incipiently flon-s over the aclinceiit niarshlnncls. I’nliltc other streitiiis, :ti1

estuarine channel har ing 110 runoff from tlie upland^ experieiires zero velocity :it the niidir of ebb tide. Velocity increases progressively as the water deepens bnt :ig:iiii beroiiies zero :it the crest of flood tide. From tliis fact arises the quest ion concei-iiiiig how the liy- tlraiilic factors vary along the length of the channel system. There is :L collateixl qiiestioii concerning the relation of the baiiltfull stage to the effective force in cli :inn el f oriiia t ion :in t l nini i i t eii:nice.

To niidei,stnnd the hydixiilic geometry of estuaries bettei. : t i id to coin1):ire i t \I itli tliat of :in nplalitl cli:l~inel, this preliiiiin:ti*y invest igat ion was uiiclei*t:tke~i.

There are other collnt era1 c~ncstioiis coticcriiiiig tlie clilTerciices and siiiiilnrities bei v-eeii iiplaiid stre:iins and t icl:il ones. For exm~pl r , tloes tlie titlnl cliaiinel exhibit nlteriiations of slinllon s : i t i d deelis which in upl:ind c-li:iniiels we hnve i ~ i e r r e d to :IS pools aiicl rimes! Does tlie cli*aiiiage net\\ oi*k of the i ic1;il system divide into t~ilnit:ii~ies of vni-ioiis sizes i i i sncli 11 iiiiiiiiier tlint tlieir Ieiigtlis i i i i d iiiiiiibe1.s 1)e:ii- a 1og:iritlimic rcl:it ion to s t ream oi-tler :I$ i i i i i1 )I : i i id cliniiiiels

Some eridcnce oii these poiiits \ r a ~ , obtniiierl in this iiirest i p i t ioii, eiioiigli i o siiggcst tlint Eurtlier inresti- gntion ~\-oiiltl be 1ncr:itire. I 3 n t 011 :ill tlie points studied,

B1

3 2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

the present data are liiiiitecl, and any generalizations we siiggest on the basis of these clnta. should be viewed :ts working 113 potheses needing frirtlier proof.

IVe are qrateful to Estella I3. Leopold a i d Anne navis for tlieir \\-oi*k on the saniples obtained for pollen analysis, to H. I,. Svenson for notes on the tnxonoiriy of plants ii, the study area, and I O Alfred C. Redfield, for a review of the manuscript :unci perinission to include some of his nnpublislied data.

AREA OF INVESTIGATION Against 1 he west bank of the Potomac River, nearly

adjacent to the city of ,~lexanclria, Vu., there was until the winter of I%O n small area of ticlxl niarsli relatively undisturbed by construction or dredging. It can be seen in the location map of figure 1 that the southern of three estuarine channels carries away nearly all the surface rur off originating in the high ground adjacent to the marzh. Tlie central of the three channels, called by us Wrecked Recorder Creek, drains no upland area and is the ciie studied. Tlie mnrshlaiicl tributary to this channel ha:; an area of about 120 acres. The marsh area and channcls draining it receive water only as rain falling directl?r on the tributary marsh and as flow induced by tidal fluctuation. The water exchanged with the Potomac e:,tnary is here quite fresh (not salty).

The size of the channel WRS such that i t could be measured with relative ease. Where it enters the Potoinac Eiver, the channel width is about, 170 feet. The total length of cliaiinel from mouth to the most distant headwater tributary is approximately 5,000 feet. h tide gage operated by the U.S. Coast and Geodetic Survey is slightly less than 2 miles upstreani froin the mouth of the clinnnel system studied.

Cross se l ions were measured at, six locations along the channel system, as will be described. The estuarine channel was 134 feet wide a t the most downstream section and 1 3 feet wide at the uppermost section.

Tidal fluctuations affect the whole length of Chess- peake Bay, and tidewater ends at the foot of the steep reach whei-e the Potomac River descends over the Fa11 Line in tl-le vicinity of Washington, D.C. The ticlal estuary sti idied is thus about 11 river miles downstream from the 1,end of tidewater.

Fieldwork W:LS carried on cliiring late spring and suni- mer of IMI . Shortly thereafter, dredging oper a t ' 1011s for ~ ~ 0 1 i l 1 1 1 ~ 1 ~ i i 1 1 gravel px lnc t ion dest royed the channel system.

DESCRIPTION OF THE MARSH r , I h e marsh arex appears uniformly flat to the eye. In

cnrly spring the dried vegetation lias been bent over by mow, so tlint rut low tide one c:un walk with ease over nearly the full uiic1i:innclecl area. There are some small areas within the marsh on which tiinber is the primary cover. TIiougli some of these t iiiibered areas appear to have a slightly liigher elevation t Iian :uver:tge for the marsh, other timber p:tt,clles appear to he flooded regularly.

By midsuninier the inarsli vegetation is tliicli and green, staiicliiig x t least knee high. The pliotogixphs in figure 2 sliow typical cllnnnels aiicl vegetation in early suiiiiiier.

38"45'

0 H 1 MILE

FIGURE 1.-Location map of Wrecked Recorder Creek, it tributary to the Potomiic River, near Alexandrin, Va.

-.-___

EXPLANATION FOR FIGURE 2

Photographs of typical reaches of Wrecked Recorder Creek, June 10, 1960. 11, Small t r i h ~ t a r y 4 0 0 feet hc-lom arction G, right bank, stage 4.8 feet, fillling. U, Section F, viewed from 210 feet downstre;rm, stag(* 4.G fcvt. i'nlling. (,', Viom looking dowiistrmrn 260 feet upstream from S(Y' io11 I ) , sf:i&!.r 4.1; fwt , f;illiiig. I,?, Left hank nntl entr;inc(. of sni:ill tribn- tary a t Iron Hulk Gage near section A , stage 4.92 feet, falling. Zf', View- tlo\vristrenm to mouth of &'r(~ck(~l Record(.r Creek f rom position 200 feet mwnstream from section I % ; Potomac River harcly visible in cliatuncr ; atilgr 4.45 fcct, i'iilling.

n, Srction I ) . Iookiiig i ipstrwi i i , stapt, 4.6 f w t , f:illing.

HYDIIA'IJLIC GEOMETRY O F A SMALL TIDAL ESTUARY B3

A

(' I)

E F

B4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Particle size (mm.) Section designatlon (sample from right bank)

-

100 100 100 77 92 91 59 78 75 43 G4 57 40 62 48 29 48 31 22 36 28 9 20 21 3 12 11

The dominant vegetation consists of cattails, Typha la t i fo lk , and probably also T. anguatifolin. Both species art3 to be expected in localities such ;IS this, but 7’. angust ‘folia is in general dominant in areas of salty water. T ie basal par t of some stxntls of Typhn is wliit- eiied in spring, probably showing the limit of tide.

Anothe- plant common to the area is the arrow arum, Peltandra virginica. Some of the plants are unusually luxuriant and might be taken for the yellow water lily, Nuphlrr adwena, especially where partly submerged.

The trees growing in some areas slightly higher in elevation ‘han average mxrsh surface include green ash, Frasinus lanceolata, which is a common tree in such places in the tidewater region. The willows are probably Bnlir: nigra, conimon along the Potomac in the Wasliingt on area.

TJnderl~yiug most of tlie n i : t r d i iireii is :L thick layer of gravel, presumably Pleist o w i e in ~ g e , wliich is being dredged for use in Iiiglinny fills a i r c l other coiistruction. The operitors of the dredge stated tha t gravel layers oc(’ur 10 o 40 feet h l o u - the sI1rfnc.e of the marsh.

few jxmples were txken for pollen analysis at section D ai, cleptlis ranging froni t lie ni:trsli s u h c e to 14 inches below i t . ‘I’liese samples were brown silty

clny which in the field appeared to be undisturbed by m y recent lateral movement of the channel. A sum- mary of the pollen analysis is presented in table 2. The following interpretation of these samples is quoted from an unpublished report by Dr. Estella R. Leopold (written communication to authors concerning Paleon- tology and Stratigraphy I h n c l i shipment WR-60- 11D).

The samples show a decrease of pine and a corresponding increase of hickory from the base to the top of the sampled section. These features, plus the presence of warm-temperate tree types and a small amount of spruce pollen represent a n as- semblage which is similar to that of the late pine zone or the postglacial pollen Zone B of Deevey (1939). Assunling the sam- lrles a r e postglacial rather than interglacial in age, an interpolation from a marsh section talien near Blackbird, Delaware. \ ~ o u l d suggrst an age of a t least 3000 to 7000 years B. E‘.

Tlic small amount of oak, the large amount of pine and the presence of spruce make the samples quite unlike deciduous tree Zon~?s C-1 through C-3 a s represented a t Blackbird, Dela- ware, as well as in several postglacial sections from Delaware, Virginia, and New Jersey.

TABLE 2.-Pollen analysis of samples taken along channel bank at section D

Samples assigned USQS Paleobotanical location Nos

D1599-C 1 D1599-B 1 D1599-A

-

Depth (inches) below surface

0-2 I 7-9 I 12-14 I 0.7

35.5 . 7

__.____..___ 12.8

. 7

2.6 2. 5 41.8 50.0

. 7 1.5 ..___.._..__ . 5

. 7 __._____.___ 5.2 8 . 4

__.______.__ 2. 5 . 7 1.0

1.9 1. 9 1.3 6.9

1.0 . 5

Polygonum. . . - -. - - - - -. - - - -. . - -. . . - - -. - - - - Umhelliferae ... . - - -. . - - - -. . - -. - -. - -. - - _.__

Undetermined conifers. --..-._...in percent.. .5 Undetrrmined dicots ... . - _ _ - - .___ .- _ _ _ _ do ___. Undetermined monocots.-. do ._-.

This preliniinary study of pollen from a few samples indicates that tlie niarsh has existed for a considerable tinie, a t least 5,000 years. This Irnowledge is usafiil in tlint, it iniplies sufficient time for establishment of a ciii:~si-equilibl.iiiiii between forces of erosion and deposi- t ion in the construction and inxinteiiance of the channel system.

HYDRAULIC GEOMETRY O F

METHOD OF STUDY

The priiicipal observational da te consisted of the gage height and discharge a t selected locations along the length of the estuarine channel. The location of the six measuring sections is shown in figure 3. This figure shows in detail only one tributary, the one on which section F is located. Recording gages were installed near the upper and lower sections, and a t each there was established a Semipermanent staff gage. A level survey established the elevations of the staff gages above an arbitrarily chosen datum. A t each of the six cross sections, several complete discharge measurements were made by means of a current meter operated from a canoe. Later a continuously recording current meter and, concurrently, a recording water-stage apparatus were installed and operated, first at one section and then at another, over a period of several weeks at each. *4 longitudinal profile of the bed extending from

section A to the headwater tributary was obtained by

A SMALL TIDAL ESTUARY B5

sounding and is shown in figure 4. Some samples of bed and bank material for particle-size analysis were collected.

CHANNEL SHAPE AND DRAINAGE NETWORK

Before embarking on a discussion of the hydraulics of the tidal channels, a description of the drainage network in conjunction with the planimetric maps presented in figures 3 and 5 will provide the reader with some picture of the channels studied. Cross sections up to the level of the marsh-that is, to bankfull stage- are presented in figure 6 for each of the six principal measurement locations. I n table 3 the principal characteristics of these channel cross sections are compared with river cross sections considered typical for river channels of similar size in the vicinity of Washington, D.C. The table compares width-depth ratios, banlrfull discharge, and some other channel parameters.

N

FIGURE 3.-Map of Wrecked Recorder Creek at Potomac River, south of Alexandria, VL, showing location of cross sections and recording instruments.

675126 0-63-2


drh. Vn.: Section A ............................

H.. .............................. c ............................... D ................................ E - . .............................. F . . ..............................

Watts Branch ncar Rockville, Md ....... Seneca Creek near Dawsonville, Md ..... Brandywine Basin, Pa ...................

I I I I I 1 I 0 1000 2000 3000 4000

DISTANCE, IN FEET

I

FIGURn 4.-Longitudinal profile of bed of Wrecked Recorder Creek ; vertical scale is depth in f ee t below average level of marsh, which is also equal to 7.2 feet on the arbitrary gage datum of the present study

134 4.13 32.4 I566 82 4.77 17.2 I461 84 3.83 21.9 I338 45 3.38 13.3 1216 32 3.00 10.7 1136 18 2.38 7.6 I 47 15 3.4 4.4 170 XB 4.2 21.6 1,320

100 5.35 18.5 2,700 65 3.4 19.1 1,OOo 38 2.0 19.0 310

Table 3 shows that the width of estuarine channels iiicreases with respect to discharge faster than in up- laiicl channels. The table does not include enough examples to demonstrate conclusively any geiiersliza- tion about width-depth ratio. Tidal cliaiiiiels are often sliwllow near the mouth owing to deposition where an estnarine cliaiinel joins a h g e body of water.

TABLE 3.--Clbaracteristics of cross sections of tidal channel and some nearbu rivers f o r comparison

Width Depth Width/ Bankfull 1 (It) I (ft) 1 depth Idischarge (CIS)

Wrecked Recorder Creek near Alexan- I I I I

I

I For flood tide having a maximum stage of 8.6 It and a range in stage of 3.5 It; this is om! comhlnation of range of stagc and maximum ktage at which maximum velocity during tho tidal cycle occurs whcn thr chnnncl is hankfull.

The principal parts of the c1i:unnel network were mapped by planetnble, :tiid the entire headwater network of one sni:~ll tribut:rry ~ : L S sketched by pace-and- coinpass methods. The maps of figures 3 :Ind 5 :we sufficient to make :I pre1iiiiiii:xry IIortoii :tn:ilysis (Hor- ton, 1945), relating stream order to tlie number aiicl

average length of channels. The smallest channel without, a tributary is considered to be of first order a d is labeled 1 in figure 5. Channels having only first-order tributaries are labeled 2, meaning that they are of second order. The sm:dlest channel where cross-section and discharge measurements were made is a channel of fourth order (fig. 5 ) . The main charinel in the network, 011 which are located sections A to G , is of fifth order.

I t is recognized that a map of only one example does not represent the variety of conditions that may occur in the field. Yet it, is informative to note the rei* '1 t ' 1911s of strenni order to average length of stream shown in figure 7. As can be seen there, first-order cliannels average about 20 feet in length, and the fifth-order main channel 11:s :L length of 5,000 feet. The length-order relations for the estuary are compared with tlie same relation for epliemernl streams (I~eopold ancl Miller, 1956,p. 18) and for 1)erennial channels in Peiinsylvanin ( BI~LISII, 1961, p. 1%). This estuary increases in length with stre:im order faster than do either ephemeral 01'

perennial river chnnels . The ratio of the number of streams of a given order

to the nninber of tlie nest higher order (called by Hor- ton the hifurration ratio) ranges for river channel net- works from :% to &, often with a n average of 3.5. Be- muse oiily oiie f i.ibutary of the study estuary was iri:ippecl in det:tiI, the est iiiiate of tlie number of streams of various orders is crude. From the limited chta for Wrecked Recorder estuary, this ratio averages 3.4. This

HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B7

figure may not be typical of other estuaries, and other STAGE-DISCHARGE RELATIONS similar studies would be informative; yet, as can be seen in figure 8, this one s:iiiipl(> is very similnr to two other types of channels in very different environments.

Tlie relation of water stnge to discharge has been deterniinecl for tidal estuaries in many localities, and the cliaracteristic difference between the form of the curve on ebb and flood tide is well known. Neitrly all previous observations, however, have been niade on relatively large estuaries. With the exception of the studies made by Gilbert (1917, 1’. 108) in a tidal marsh near Sail Francisco, no publislied records are known to lis for smd1 tidal channels having no drainage :\rea in the upland. The present, investigation adds to Gil- bert’s work only tlie comparison of the relations between different cross sections located along the length of the estuarine channel.

Velocity, and thus discharge, at a. given stage was dependent both on tlie maximum stage attained by tlie p:trticdilr tidal cycle ancl on the range of stnge in the tidal cycle. Thns, every tidal cycle requires a slightly different stage-discharge ciirve.

The dependence of velocity on range of tide is known from other investigations. I3radley (1957, p. O M ) , also working in :L relatively small estuary, found that the near-bed velocity varied directly with tlie range of tide. His excellent work does not help solve tlie particular questions here investigated. All his flow measurements were made close to the estuary bed. Further, lie was concerned with a small bay rather than with rirerlike estuarine c1i:innels.

It was impossible for a party of two to make a complete set of discharge measurements a t all sections along the stream simultaneously through a single tidal cycle. We chose, therefore, to make tis maiiy discharge measurements as possible through a single cycle at each section, ancl on different clays, making similar measurements on different cross sections. Now tlie cross-sec- tional area of each section for any given stage is linomn from the surveys of the section. The problem is to determine the mean velocity for the whole section a t selected stages in order to construct ;t relation of discharge to stage. Fo r two tidal cycles the relation of velocity to stage is shoi\-n in figure 0, using section D as tin esaniple. Such graphs :ilone, however, do not give sitfficient information to determine the iiifluence of either tidal range 01’ maxiniuni stage.

T o incorporate these :idditional variables, two sectioiis were clioseii for 11 more intensive series of velocity nie:isurenieiits covering different tidal cycles. A t sections 11 and If’ ;i recor(1ing current meter was installed, :tlong wi th a device tliat recorded water stage simultaneously. Thus, :kt :L particulnr distance above the bot-

froin spritig to neap tides.

toll), velovity W:\S nie;isnred ~ie:\r the center of t lie c h - l l r l ~ol~tillllorl.;]y for t I,.() \\-eel<s to s;\lllple tictnl cycles

3

Iris Bank staff gage

50 100 FEET

F I G ~ R ~ 5.-Detailud m:lI) of one t r i I ~ ~ ~ t u r y of Wrecked Recorder Creek near Alexnndrin, Va., including :ill nlinor channels. Order number of each channel is shown, some of which are estimated ( e s t ) .


Section A Section D

VERTICAL SCALE

4 0 FEE1

Section E

10 0 10 20 30 FEET l , a , , l I I I

Section F v- - - - - - - - -______-_ HORIZONTAL SCALE

Cross sections viewed Section C looking downstream 7

FIGURE 6.--Cross sections viewed looking downstream of each of the principal mensoring places along the channel of Wrecked Recorder Creek. Vertical scale is feet below average level of marsh.

The location and depth of the recording current. meter were selected on the basis of velocity reddings from previous measurements to provide a position where point velocity best correlated with mean velocity. Owing to the variation in water depth throughout the tidal cycle, the depth below the water surface where the meter was installed changed. The position of the recording meter varied from three-tenths to seven- tenths of the full depth below water surface at section D, and from two-tenths to eight-tenths at section F. The velocities recorded :it the single point during a given cycle correlated reasonxbly well with measured mean velocities for the whole section. From this cor- relation, the relation of inean velocity to stage and to range of stage could be constructed.

For each cross section, curves of the type shown in figure 10 were developed to show the stage-velocity relation for excli of several values of range in stage. The curves shown by full lines in figure 10 are merely selected examples wl-hich comprise three values of maxi-

mum stage, each with a, different range in stage. Fo r one of these examples, an additional graph is shown (dashed line) representing the condition of another value of range in stage but identical niaximum stage. It will be visualized that there may be a very large number of curves making up such a family as that shown in figure 10.

The graphs drawn in the interpolation procedure are considered reasonably reliable except for very low velocities, but even there the error in velocity does not exceed 10 percent. Minor variations in individual cycles were neglected in the construction of the graphs.

Most data in tidal studies have been presented as graphs of stage o r velocity as 'a function of time in a tidal cycle ( for example, Gilbert, 1917, p. 124; Ahnert. 1960, p. 397). Our data plotted in that usual way exhibit the nsnnl fe:xtures (fig. 11). Minor variations occur in id iv idua l tidal cycles ; velocity decreases toward zero with time in a somewhat irregular manner in the early part of the ebb tide.

B9

occurs twice in every tidal cycle, and thus there are also two occurrences of high discharge in each cycle. There remain different frequencies of the different values of the peak discharge from cycle to cycle, but the range of values possible is much smaller than the range of possible values of flood flows in an upland river of the same size.

Because of this difference, i t may be fruitful to ex- amine the frequency of occurrence of the factors which do vary.

Figure 12 shows the percentage of time the high and low tides equal or exceed a given stage at the Alexandria, Va., tide gage. The ordinate shows the variation in terms of the datum uesd in Wrecked Recorder Creek.

H Y D R A U L I C G E O M E T R Y O F A S M A L L TIDAL E S T U A R Y

* 3 5

STREAM ORDER 1°1

w h

6 FIGURE ?‘.-Relation of stream length t o stream order for Wrecked 2

Recorder Creek a n d foi some o ther drainage areas. T h e da ta a f rom Brush (1961) were extended t o acc’ount for differences in 8 map scale, a n d his order 1 w a s comparable to order 5 in t h e present study. Dots represent measurement da t a from this study.

3 ti a

U BANKFULL DISCHARGE

0

The evaluation of the banks relative to gage datum was determined by visual observations at several sections in the reach during the flood tides. The stage a t which the flow started to inundate the marsh rniiged from 7.1

\ t rary datum) for the reach. The difference between i

the elevation of the marsh at the headwaters and a t

I n upland river systems, the higher the discharge the less frequently it is experienced. Except for very low

quency of occurrence is a general characteristic of

: 10 5 I - - - 4

to 7.3 feet, and averaged 7.2 feet (relative to our arbi-

the mouth is very small.

flows, this inverse relation of discharge rate and fre-

- X

\ \ \ \

\ - \

\ \

1 \I \j )

\

1 2 STREAM

EXPLANATION Relation between indicz order streams in estua

o----o 1st and 2d

x- - - -x 2d and 3d -

+---- + 3d and 4th

~

1st through 5th

4 )RDER

5

rivers. FIGURE 8.-Relatlon of number of streams to stream order : d a t a for Wrecked Recorder Creek a r e compared wi th average relations for a n a rea of perennial streams a n d a n a rea of ephemeral streams. I n tidal channels, on the other hand, zero discharge

B 10

8.0

7.0

7 l- a n

a e L

W c3

6.0 W Y

z J 0 a bi

-

5.0

4.0


VELOCITY. IN FI

I I I I I I I I I I

T PER SECOND

T

/ I /

/

EX PLAN AT ION

Discharge measurements

Mean velocity measurements

A+----

0.

Ebb tide

I I I I I I I I I I -100 - 50 0 50 100

DISCHARGE, IN CUBIC FEET PER SECOND

&WX7RE Q.-stage-discharge relation ( l ight line) and st: ge-velocity relation (llpal-y l inr) ; a nearly complete tidal cyclr llaring a maximum Data are for

(The velocity shown is mean velocity in the cross section. stage of 6.8 feet i s shown and the ebbing limb only is shown for a tidal cycle in which maximum stage was 7.6 feet. section D .

A comparison of our observations with the concurrent values of stage of the Potomac estuary a t Alexandria showed that the stage at the two locations varies in :I comparable manner, and this indicates that tJhe frequency of stage a t the two locations is comparable. To eliminate the effect of the spring and neap tides, a period of 2 lunar months, 58 days, was used in the preparation of the frequency data. The extremes shown by dashed lines were extended on the basis of a straight-line projection.

From the graph we find the median high tide for this area is 7.27 feet relative to our local datum. Bank- full stage, 7.2 feet, is, interestingly, approximately the median high tide and is therefore attained on the average every other tidal cycle or once a day. A terrestrial river reaches its bankfull stage but once every 1 or 2 years.

This, though true, leaves an incomplete and perhaps false impression in the mind of the student of channel morphology. W e are interested, after all, in tlie frequency of the effective discharge.

I n an estuarine channel, at some value of stage near high tide, velocity must be zero, for the direction of flow must reverse. In Wrecked Recorder Creek, the velocity is highest a t a stage mucli lower than high tide.

The velocity and the st age corresponding to maximum discharge both increase wit,h the maximum stage of of any given tidal cycle. The stage a t ~7hich the maximum velocity OCCLII’C: is not well defined by the available da ta ; but as a rongh approximation, a. tide reaching a rnaximum stage of 8 to 8% feet is required for the mix--

irrium velocity to occur at a. stage of 7.8 feet, which is the bankfull stage. Thus, because of the dependence of dis-

HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B11

8.0

7.0 s I-

d Y a G L W LL

z $ 6.0 2 v)

5.0

- 1 .oo -0.5 0 0.5 VELOCITY, IN FEET PER SECOND

1 .oo

FICUKE lO.-Sample relations of stage to velocity a t section D showing effects of maximum stage and of range i n xtage. The samples include Two samples are for the same maximum stage (8.0 feet) h u t illustrate different ranges of various maximum stages, 7.0, 7.3, 8.0 feet.

stage, 3.5 feet and 2.0 feet, respectively

charge on velocity at a give11 stage, tlie top of banks is also the level of iiiaxiiiiuiii discharge when the high tide occurs x t the stage between 8 xnd 8!/2 feet.

From an xiialysis of records of the ,~lexniidria tide gage, a. high tide of 8.5 feet is equaled or exceeded about 0.0 percent of the time, and 8.0 feet about 9 percent of the time. Nixon (1‘3.59) foimd that bxiilrfull stage is equnled or exceeded 0.6 percent of tlie time in Englisli rivers. These results :ire of the s:me order of magni- tucle. Wlietlier this is coincitle1itnl or morpliologically iniportant caiiiiot be stated i i i the 1)resent state of knoml- edge. It is a probleni worth further work. I t is hoped that further investig:!.atioiis on estnariiie stre:iiiis mill consider the problem of the dominant discharge.

HYDRAULIC CHARACTERISTICS AT A CROSS SECTION

In terrestrial rivers. c.liniiges in width, depth, and yelocity at any cross section accompany a change iii clischrirge. These \-ari;tl)les :ire related to discharge i l s simple power fuiict ions expressed by I m p o l d and Maddock ( I D M ) in the following form :

ic;=aQb d=cQf

w=h-Q?II

where Q is discliarge, w. d, xiid represent tlie water- surface width, Liean depth, and me:m velocity, respec- tirely. The coefficients, I / , c, and IC, are constants, ancl 0 , f, aiid I H :ire exponents representative of the section.

B12

7 .O

z I- 4

W

W 4

6.0

“, I- W W Y

z W-

a W

I- v)

5.0

4.0

- \ \ J I

L I I 1 I I 1 I I I I I I I -

PHYSIOGRAPHIC AND HYDRAULIC STUDIES O F RIVERS

0 7 5

1 0 0

I \ y, / /

I

\ I \ I \

/ \ Stage versus time. Apr. 20.1960 / v

\ \ \ \ \ \

I (fl; versus time, Apr. 21,1960 I I

I I I I \

\

I I

--.-..-I I

I

‘ I Apr. 20,1960

I \ I \ I \ I I I

/

I

Median height of tide ADr. 19 to June 17,1960

- -1.00

- -0.75

0 z 0

- - o s 0 g B +

-0.25

HYDRAULIC GEOMETRY O F A SMALL TIDAL ESTUARY B13 c 9

FIGURE 12.-Duration curves (cumulative frequency) of values of Datum stage at high and low tide, Alexandria, Va., tide gage.

j s the arbitrary datum used at Wrecked Recorder Creek.

Figure 13, thus presents for Wrecked Recorder Creek two sets of curves ; each set, represents a different tidal cycle and is indicated separately by crosses and tri- angles. The results of three current meter measure-.

nearly identical on flood and ebb limbs of a tide. Thus, it is possible to discuss the relation of these slopes as exponents in power function equations, and to compare the exponents with those applicable to terrestrial rivers.

For Wrecked Recorder Creek, values at given cross sections during flood tide of the exponent constants, h, f , and m, are :

Decreasiiig v e l o d t ~

Change in width . . . . . . . . . . . . . . . . . . . . b=O.@4 b= -0.01 Change in depth . . . . . . . . . . . . . . . . . . . . f= f=- .@4

Change in velocity _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ m= .785 m= 1.05

For an upland stream from Leopold and Middock (1953, p. 26) the average values are :

Increasing valocitii

.175

b=O. 26 f= .40

m= .34

I n the values for the tidal channel, the left-hand col- umn is uncloubtedly the more important, from the stand- point of channel morphology. The increasing velocity is related to flow in the channel, whereas when the velocity decreases on R flood tide a t least part of the discharge is usually governed by conditions of overflow rather than chilnnel hydraulics.

The small value for the exponent h in the estuarine channel mas expected because of the relatirely vertical banks prominent throughout the reach except a t the bends.

The median range in tide in the vicinity of the proj- represellted by dots, are s~ lo~v l l for ' ect areik is 2.8 feet, although under extreme conditions

Widths for stages above 7.2 feet (bankfull) were determined by a straight-line extension of i i width-stage curve, :ind discharges were l)ilsed on these widths. The set of curves indicated by a dnshecl line represent a flow below bankfull stiige during the complete tidal cycle.

The two tides giro siich siiiiilar results that the slopes for each set of curves are nearly identical. Each tidal cycle will give a separate set of curves, but the slopes are consistent.

I t can be visualized that in :I flood limb of a tidal cycle, width and depth increase progressively from zero discharge through iiiaximuni to zero discharge. Ve- locity, on the other liancl, begins with zero, reaches a maximuin value a t or IleiLr tlie stage of inaximum discharge, and decreases to zero at the highest value of depth when flow reverses. Because figure 13 presents a flood tide only, the arrows show only one direction of progression aroiind the hysteresis loop.

The respective slopes of the lines representing relations of width, depth, and velocity to discharge were

a range of more than 4 feet may be expected. Iiicreas- ing velocity occurs principally in the lower se,ment of the range in stage. As an example, on n high tide of 8.6 feet with a range in stage of 3.6 feet the maximum velocity occurs at, a stage of 7.2 feet, or 1.4 feet below the maximum stage.

I t is not known whether the value for the exponent f for Wrecked Recorder Creek is also representative of tlie numerous :ireas where a large range in stage occurs during each tidal cycle. Some data are available for comparison as discussed below, but more examples are needed.

Old Mill Creek differed froiii Wrecked Recorder Creek by being longer and deeper. The former also c h i n s some areit of upland whereas tlie latter does not.

Gilbert (1917, pl. 33) presented a graph which included the velocity, discharge, nnd stage for Ravens- wood Slough on the southwest shore of San Francisco Ray. The range in stage for one of the flood tides illustrated was over 7 feet. From his published data, width at nny given stage could be measured, and depth

B14

n z 0

In LL W a

+ W W LL

Y

z - 0

W > s 0.1

0.01


a'

I I I 1 I I l l

EX PLAN AT1 ON

R a v e n s w d Slough; southwest shore of San Francisco Bay, Calif. (Gilbert, 1917)

e I3 Flood flow

e Ebb flow

Moxmum 'loge I? 6 feel. range ~n rloge 7 O+ fed

Old Mill Creek near Lewes, Del. (Author's data:

Flwd flow

Wrecked Recorder Creek, Section D

Flood flow M~~~~~~ $loge a o feet rOnge tn I~oge 3 5 feet

+ Flood flow

a

Moximum m g e 7 0 feel r m g e ~n stage 2 0 feel

Flood flow Acluol rneowremenli of rnoximum rloge 6 85 feel

1 1 I I I I I I I I I I I l l 1

z 1-

D 3

I-

1 I I I 1 I l l I I I I I l l

1 10 I 1 I 1 I l l I 1 I I I l l

1000 10,009 100,000


FIGURE 13.-Chnnges at-a-station of width, depth, nud velocity a8 funrtions of dischnrge. Data for Wrecked Recorder Creek were collected Also included are observations a t Old Mill Creek estuary, near Lewes, Del., and Rnvexiswood Slough, San Francisco Bay. a t section D .

Calif.

width depth

/ I J ______- See. D, Wreckcd Rccordcr Crcek .__.. ~ . . 0. 04 0. 18 0.78 Old Mill Crcck _____._..______.._______ 1 . 08 I . 14 1 . 78 Raveliswood Slough. -. __. - - -. ~ ~ . 14 . 08 . 7 8

veloclty

m

__-- Avrragc- - -. -. - - - - -. - - - - - - . . . 7 8

Averagc for terrestrial rivcrs __________. I . ig I : ii 1 . 34

As can be seen in figure 13, the curves for width alid depth versus discharge have slightly negative slopes as discharge decreases from its maximum value to zero at high tide. Thus the values of tlie exponents h and f given xb,ove would be smaller rather tlmn larger if averagecl witla those applicable to decreasing disclinrge on tlie Aooclticle. I t must be conc.ludect then that the principal difference in the at-a-station hydraulic characteristics of estuarine and terrestrial streailis is that the foiiner have a niucli more rapidly changing velocity with discliarge than do terrestrinl rivers. This is ~0111- pensated by less rapidly clianging depth and width with discharge.

THEORETICAL ANALYSIS OF THE HYDRAULIC GEOM- ETRY IN THE DOWNSTREAM DIRECTION

By WALTER B. LANGnEIN

Pillsbury (1939, p. 228-230) defined an ideal estuary as one in which tlae tidal range, depth, :tnd current are uniform through the length of tho channel. 13y analysis of the motion of water in tlie c~linnnel RS indncect by tlie tides at the moutli aiicl as retnrded by friction, Pillsbury showed f1i:it in sncli :in estrinry the width decreases exponent inlly with tlie dist:tnce up the estuary ; thus

7,j =w(,e-".' c o t 9

where w=widtli at distance a npstream from the month where width is wo, e is l m e for natnral logxritlinis, 'p is the lag ( in degrees) between tlie time of ni:ciiniim~ slope and the time of maximurn velocity, and n is the quantity a/ d D where a is the mean angular speed of the semi- diurnal lunar tide= 0.00014 radians per second, D is

mean depth a t a given cross section, and g is the accel- eration of gravity. This equation describes an estuary decreasing in width expoiientially with distance.

Pillsbury showed that tidal estuaries tend to approxi- mat e this expoiieiitial form, generally with a sinuous aliiienieiit . The analysis is based on an tmuiript ion of a single c1i:Liiiiel estuary. Presuniably, if applied to a branched estuary the width, w, must be tlie total width at distance 2. There is, however, a well known tend- ency for natural clianiiels to become sliallower :is they beconie narrower. It will be instruotive to deduce the hydraulic geometry of an estuary in the terms defined by Leopold and Mnddock (1953), and to introduce concurrently some of the principles of hydraulic probability described by Leopold and Langbein (1962). That report treats the distribution of energy in natural systems, in wliich known physical relatioris or constraints are insufficient to yield unique solutions-in other words, a. system with remaining degrees of freedom. From analogy to entropy production in thermodynamics, it was sliown that a river system, in adjusting to the remaining degrees of f reedom, would tend toward uni- forin clistribution of energy, and a minimum rate of work in the system as a whole. Pillsbury, for example in postulating equal depth and velocity in an estuary iii- trocluced only the first of these conditions. This is incomplete. We shall explore the effect of niinimizing total work as well.

The amount of work clone by the tide in filling and emptying the estuary is :I function of the volume of tlie tidal prism and of the hydraulic friction. Both these quantities are related to tlae width and depth of the estuary.

If change in depth is described by the functioii / )=D,e-~L where D, is mean depth at the month of the estuary, an approximate equation describing the width of an estuary is as follows :

w=woe - (n cot p f k ) z

According to this equation, the estuary will narrow and thus the tidal prism lessen more rapidly, tlie greater the rate of decrease i n depth. Other hydraulic relations that govern flow in :in estuary are as follows :

Continuit?y--Since Q = vdzu (at each cross section, tlie relation m+f+b= 1.0 innst be satisfied.

Nope-depth wlation.--Slope in tliis case i s the surface slope of the water surface created by the progress of tlie tide in the estuary. Tlie slope varies sinusoid- ;illy during tlie tidnl cycle. As I'illsbuiy shows, the value of s (amplitude of the slope) is :L Rinction only of the depth, thus :

s = A a / c D where A is the tidal amplitude (half the range be-

Bl6 pHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

tween mean high water and mean low water), and a and D are as described previously.

The average value during one phase of the tidal cycle will be 0.63 S and -

S=O.63Aa/GD

Since

Hence z= -+.f

Velocity-depfh rr7ution .-Tlie amplitnde of the tidal current is as follows :

v=A J$D sin 4

where 4 is the :uigiiI:Lr lag of the priniar-y cnrrent he- hind the hydraulic slope. The value of + is itself a function of depth mid chnnnel friction, and is evaluated by tables given by Pillsbury (1939, p. 123). Using a Chezy coefficient of 80 and mean dept I i of cross sections ranging from 2 to 10 feet, the valne of the amplitude of the current is c;ilcnlated to he

V=0.35AD1/'

Velocities during the t,idal cycle will vary sinusoidally in proportion to the ;Lmplitiide V . Me:ln velocities at different cross sections c i ry as tlie fifth root of the mean depths, therefore, rn= f / 5 .

Thus we have the three following hydraulic condi- t ions :

m +f+ b = 1 .o m=f/5 z=-f/2

There are fonr iinknowns ; the remaining statement will be supplied by the necessity that the estuary appro:icli a. state in which energy is as uniformly distrib- u ted as is consistent with tlie necessity tliat total \vork in the estii:wy as :I 1vlioIe be :I niininiiim.

The first conditioil is sntisfietl if velocity and cleptli are uniform ; hence, m+f must n p p r o : ~ ~ h zero. Since n?, and f are of the s;ime sign, each of the p:wameters, m and f , also would approach zero. Since m + f + h=1.0, ( 1 - 6 ) approaclies zero.

The second condition, that of minimlull total work, is iiiet as the intcgrnl ram &sd.r approacli~~s zero. 1 1 1 t lie 11pln11d i-ivei.. 0, tlte tlisc*linrge, is tlie 1:incl drnin:ige and is iiiclepeiiclriit of tlie c~iniiiiel gronirti*y, and s l o p is thc olily vari:ible. Tlic cliscliarge in an estiiary, however, is iuniqiiely :L fwicf ion of its size :tiid tliiis of its wicltli, depth, md length, : ~ n c l so the problem is more complex.

The relative work performed during n tidal cycle in an estuary is as follows :

(1) n cot 'p

1 . 7 f n c o t 'p

where P is tlie ratio of the integral xa Qsdx for a n estuary in w1iic.h the depth varies as to the same integral for an estiiary in which the depth does not ra~y, that is, k=O.

This equation sliows that the relative work decreases as tlie depths decrease iipstretunl from the mouth. As was shown preriously, the decrease in depth affects tlic estuary width, and so affects the value of the exponent h in the relation between wi.idtl1 and tlischnrpe, tr cc Q", where 20 is the clinnnel wicltli a t a given cross section and Q is the haiikfull clischnrge. For :m estiiary iii wliicli depth does not dem-e:ise, the vnlne of b is unity. For a n est ii:wy ill wliicli depths decrease iipsti.e:irii from the iiioutli, tlie value of 7, decrenses. The relation is :Is folloltis :

P=

kfncotcp 2.2kfn cot 'p

b=

Formulas 1 and 2 can be solved to determine the relation between the relative work P and the width exponent 6 , thus :

1.36 - 0.59 or as a more convenient 0.36 + 0.41

P=

approximation : I Pz1.76 - 0.7

As stated previously, the condition of uniform energy distribiition requires (1 - 6 ) to appro:xcli zero. Thus b should be high. On the otlier Iiand, minininni work requires I .7b - 0.7 to approach zero-in other words, b slionld be low. Following tlie logic developed in the p:iper by Leopold aiicl Tmigbein (1062), the most prolxhle ~.:ilue o f h is that for which the probability of tlir de\ iation of (I - h ) from 0 eqiials tlie prob:il)ility of a deviation of 1.76-0.7 from 6. These two proba- bilit ies :ire eqiial when

F , ( b ) =F.(O) OF,

where P, (0 ) = 1.7b - 0,7 ant1 F 2 ( 0) = ( 1 - b ) . t r F , autl trl.', are (lie stniitlartl clcviat ions of those quantities. TI10 nbsoliite valiies of the staiiclarci deviations need itot be k1101vii. E a c h can be expressed in terms of the st:iiid:ird deviation of h, thus

oF2=a6

~~

crF',=1.7rrb X I I ~

HYDRAULIC GEOMETRY O F A SMALL TIDAL ESTUARY B17

5.0

L i - W V -

f z a x 1 .o

Hence

-

.,X)* /*/ ,.&- - ex-*'

I I 1 I 1 I I I I I 1 1 l 1 1 1

1.7b-0.7 1 - b

Tidal estuaries -

Theorctical Wrecked (Lanphein) Recorder Barnstable

Creek Marsh, (this Mass.1

report)

__ Exponent of width, h - . ~ ~ 0. 71 0. 77 0. 74 Exponrnt of depth, f- -. . . 24 . 2 3 . 1 7

E x p o n e n t o f s l o y ) c _ _ _ ~ ~ _ . - . I 2 ~ _.__._. ..__._. ~

Exponent of velocity, vi- . 05 too .09

1.7ab ab or b=0.71, which leads to the following set of values of the hydraulic geonict ry exponents :

In,= 0.05

b= .71 z= -. 12

These results derived without reference to field meas- nrements will be coinpared in the next section with data obtained in the tidal creek studied.

f = .24

Rivers

Average for rivers in mid- western United States a

______

0. 50 . 40 . 10

- . 4 9

MEASUREMENTS OF THE HYDRAULIC GEOMETRY IN THE DOWNSTREAM DIRECTION

I n the discussion of channel characteristics, the changes in width, depth, and velocity in the downstream direction are also important. A representative discharge for all cross sections is required for this analysis. The dominnnt or bankfull discharge is probably the most meaningful for terrestrial rivers. The estuarine cliaiinel presents the problem of what value of discharge is pertinent. It is reasonable to assiinie that the discharge occurring a t the time of maximum velocity a t a section is a pertinent discharge. This occurs nearly simultaneously a t all sections in a channel as short as Wrecked Recorder Creek because the time lag along the length of the channel is small. It is this discharge that is used in the computation of the downstream curves in figure 14.

The data used in construction of the curves represent flood and ebb flow during a tidal cycle having a high tide of 8.6 feet, for which some measurements are sum- marized for floodtide in table 3.

From the slopes of the lines in figure 14 the values of the exponents in the hydraulic geometry may be determined. The curves represent the average of both flood aiicl ebb flows.

I h l u e s of exponents tn the hydraulic geoinetry, with respect to increasing dischauqe dou)nstrcam

1 Preliminary estimates hy Alfrrd C. Redfield and Lincoln Ilollister communicated

* Leopold and Middock (1653) and Lcopold (1953). by letter to Leopold Mar. 30 1YG2.

Langbein's theoretkdly derived values of these exponents are compared below with the values determined from the present field investigation of Wreclred Re- corder tidal stream. Also included are values from field data collected and analyzed by Redfield and Hol- lister a t Barnstable Marsh, Mass.

100

L W LL

z I I- :

i n I I I I I I I l l 1 I I I I I I l l -- 10 100 1000


F I G U E ~ 14.-Downstream or along-the-channel changes in width, depth, and velocity a8 functions of discharge, at bankfull stage in Wrecked Recorder Creek.

The data from Redfield and Hollister are particularly valuable because the channel studied ranged in width from 10 feet near the headwaters to 3,800 feet near the mouth. This is a much greater range in channel size than in our study and yet there is remarkable apeenlent of the results.

The agreement between the field results of this investigation and the theory is very satisfnctory. The field .data, however, are few and the scatter is large. More field data are needed.

Both theory and field data show that in tidal estuaries, depth tends to be more conservative (a low value of f )


than in upland rivers, so that the width-depth ratio varies rapidly downstream. nt the mouth, a tidal estuary is wide and relatively sliallow ; at its head, it is narrow and relatively deep. The reason for this is that, in contrast, to rivers, the discharge at any section is itself a dependent variable depending 011 how the flow shaped the channel in :dl the channel length between the point in question and the main h y or body of tidal water. I n a terrestrial river, discharge is independent in that it is produced by the watershed and the channel is accommodated to it, rather than by such accommodn- tion modifying the discharge itself.

It, is this influence of the channel on discharge that makes the rates of change of width, depth, and velocity long the channel different than in terrestrial rivers.

REFERENCES

Ahnert, Frank, 1960, Estuarine meanders in the Chesapeake Bay area: Geog. Rev., v. 50, no. 3, p. 39&401.

Bradley, W. H., 1057, Physical and ecologic features of the Sagadahoc Bay Tidal Flat, Georgetown, Maine : Geol. Soc. America, Mem. G7, p. 641-682.

Brush, L. M., Jr., 1961, Drainage basins, channels, and flow characteristics of selected streams in central Pennsylvania : U.S. Geol. Survey Prof. Paper 282-F, p. 145180.

Deevey, E. S., Jr., 1939, Studies on Connecticut lake sediments : Am. Jour. Sci., v. 237, no. 10, p. 691-724.

Gilbert, G. K., 1917, Hydraulic mining debris in the Sierra Nevada : U.S. Geol. Survey Prof. Paper 105, 154 p.

Horton, R. E., 1945, Erosional development of streanis and their drainage basins-hydrophysical approach to quantitative morphology : Geol. SOC. America Bull., v. 56, p. 275-370.

Leopold, L. B., 1953, Downstream changes of velocity in rivers : Am. Jour. Sci., v. 251, no. 8, p. GOG624.

Leopold, I;. B., and Langbein, W. B., 1962, The concept of entropy in landscape evolution : U.S. Geol. Survey Prof. Paper 5 W A , p. 1-20.

Leopold, L. B., and Maddock, Thomas, .Jr., 1953. Hydraulic geometry of stream channels and some physiographic impli- cations: U.S. Geol. Survey Prof. Paper 252, 57 p.

Leopold, L. B., and Miller, J. P., 1956, Ephemeral streams- hydraulic factors and their relation to drainage net : U.S. Geol. Survey Prof. Paper 282-A, p. 1-37.

Nixon, AIarshall. 1939, A study of hankfull discharge of rivers in England and Wales: Inst. Civil Engineers Proc., v. 12,

Pillsbiiry, G. R., 1939, Tidal hydraulics : Washington, U.8. Army Corps of Engineers, Prof. Paper 34, 281 p.

Wolnian, 11. G., and Leopold, L. B., 1957, River flood plains; some observations on their formation : U.S. Geol. Survey Prof. Paper 282-C, p. 87-109.

Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency of forces in geomorpliic processes: Jour. Geology, v. 68, no.

p. 167-174.

1, p. 54-74.

U.5. GOVERNMENT PRINTING OFFICE; 1963 0 4 7 5 1 2 6