Advances in Water Resources - Boston Universitypeople.bu.edu/nicleona/papers/Leonarrdi_2015_AWR.pdf · 2015-07-10 · Interplay between river discharge and tides in a delta distributary

Advances in Water Resources 80 (2015) 69–78

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Advances in Water Resources

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Interplay between river discharge and tides in a delta distributary

http://dx.doi.org/10.1016/j.advwatres.2015.03.0050309-1708/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (N. Leonardi).

Nicoletta Leonardi a,⇑, Alexander S. Kolker b, Sergio Fagherazzi a

a Department of Earth and Environment, Boston University, Boston 02215, United Statesb Louisiana Universities Marine Consortium, Chauvin, LA 70344, United States

a r t i c l e i n f o

Article history:Received 30 January 2015Received in revised form 14 March 2015Accepted 17 March 2015Available online 24 March 2015

Keywords:TidesRiver dischargeResidual currentsTidal pulsesPhase lagWavelet analysis

a b s t r a c t

The hydrodynamics of distributary channels has tremendous impact on nutrient and dissolved oxygencirculation, transport of sediments, and delta formation and evolution; yet many processes acting atthe river–marine interface of a delta are poorly understood. This paper investigates the combined effectof river hydrograph and micro-tides on the hydrodynamics of a delta distributary. As the ratio betweenriver flow to tidal flow increases, tidal flood duration at the distributary mouth decreases, up to the pointwhen flow reversal is absent. Field measurements in a distributary of the Apalachicola Delta, Florida, USA,reveal that, once the flow becomes unidirectional, high-discharge events magnify tidal velocity ampli-tudes. On the contrary, while the flow is bidirectional, increasing fluvial discharge decreases tidal velocityamplitudes down to a minimum value, reached at the limit between bidirectional and unidirectionalflow. Due to the different response of the system to tides, the transition from a bidirectional to a unidirec-tional flow triggers a change in phase lag between high water and high water slack. In the presence ofhigh riverine flow, tidal dynamics also promote seaward directed Eulerian residual currents. During dis-charge peaks, these residual currents almost double mean velocity values. Our results show that, even inmicro-tidal environments, tides strongly impact distributary hydrodynamics during both high and lowfluvial discharge regimes.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction 20,31,48,52]]. Interactions between tidal oscillations and riverine

Flow dynamics in delta distributaries determines the fate andtransport of sediments, contaminants, and nutrients with impor-tant consequences for deltaic deposits, landform evolution, waterquality, and deltaic ecosystems [e.g. [14,16,19,35,40]] . Herein weinvestigate the effect of micro-tides on the hydrodynamics of adelta distributary. We particularly focus on the interactionsbetween tides and river hydrograph, and on the system responseto tides under low and high riverine discharge conditions. Threemain aspects of the problem have been taken into account: (i)phase delay between water level and velocity fluctuations; (ii) tidalvelocity amplitudes; (iii) tidally induced Eulerian residual currents.These processes are investigated by means of a two months dura-tion instruments deployment in Apalachicola Bay, Florida, USA,where velocity and water level measurements have been collectedat the mouth of a delta distributary, and along its reaches, duringtwo consecutive flooding events.

The effect of tides has been recognized as an important factorcontrolling both the hydrodynamics of the jet exiting river mouthsand the morphology of its sediment deposits [e.g. [8,13,

runoff have been also closely related to water quality, and nutrientconcentrations [e.g. [36]]. The presence of tides has three mainconsequences: mixing is increased, and the effect of buoyancy ispartially suppressed; a bidirectional sediment transport is present;the marine-river interface moves in both the vertical and horizon-tal direction [e.g. [51]]. When tides enter the river they behave aswaves progressing upstream, they distort, and eventually dissipatedue to bottom friction and riverine flow. For a bidirectional flowand a propagating tide, the effect of river discharge on tidal ampli-tude, wave celerity, and phase lag has been explored [e.g. [6,7,48]].It has been shown that the presence of a river discharge has thesame effect than increasing friction by a factor proportional tothe riverine to the tidal discharge ratio. Thus, increasing river dis-charge enhances tidal damping, reduces tidal velocity amplitude,and wave celerity, and increases the phase lag between high waterand high water slack [e.g. [5,7]].

However, many aspects of the interaction between tidal hydro-dynamic and riverine flow deserve further investigations.

Another important mechanism, connected this time to differentriverine discharge conditions, is the backwater effect, which couldhave important geomorphological implications, when combinedwith tidal changes in water level. This backwater effect can be alsoreferred to as residual slope [e.g. [7,43]]. For a gradually varied

Nomenclature

QR river dischargeB distributary mouth widthb convergence length of the stream widthhmsl water depth at mean sea levelhFLOOD water depth during floodhFLOOD water depth during ebbH tidal range (from HW to LW)H0 slack tidal range (from HWS to LWS)HW high waterHWS High water slackLW low waterLWS low water slackO surface area of the distributary channel where the tide

propagates

P tidal prism volumeu0 velocity due to the river dischargeDu difference between tidally averaged velocities for differ-

ent discharge valuesd damping coefficiente phase lag between HW and HWSUz water level phaseUu velocity phasem tidal velocity amplitudemEBB difference between ebb velocities and velocities at

mean sea levelmFLOOD difference between flood velocities and velocities at

mean sea level

Chattahochee

70 N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78

flow, velocity varies along the channel; consequently bed slope,water surface slope, and energy line slope all differ from eachother, and a backwater profile establishes. For instance, in a sub-critical flow, the effect of a control point, such as the sea level atthe downstream boundary, propagates upstream creating a waterprofile that gradually adapts to the downstream conditions. Forlow flow conditions, the water depth at the shoreline is greaterthan the normal flow depth, and the water surface profile is con-cave up (also referred to as M1 curve, Fig. 1, green line). On thecontrary, a drawdown profile (also called M2 curve, Fig. 1 pink line)is typically established during high flow conditions, when the nor-mal flow depth is higher than the water depth at the shoreline [e.g.[11,29]].

2. Study site

The lower Apalachicola River is located in the Florida panhandleat the terminus of the Apalachicola-Chattahoochee-Flint (ACF)River system (Fig. 2a). Apalachicola River is the largest in Floridain terms of flow rate and it belongs to one of the largest river sys-tem in the Gulf of Mexico. The system was formed over the past6000 years as the Apalachicola River deposited sediments into ashallow shelf, the distal sands of which were reworked to formsandy barriers, spits, and islands [39]. The bay encompasses about620 km2 of open water with an average depth of 1.9 m at mean lowtide. Approximately 80% of the open water zone is composed ofsoft, muddy, unvegetated sediments and the remainder is divided

Envelope HWS

Envelope LWS

M2

M1

River mouth

x

Tidal prism

hn

hmsl

0

hn

LW

HW

hLWS

hHWS

Fig. 1. Sketch of the geometry of an idealized distributary channel. Water levels atmean sea level in case of high (pink line) and low (green line) fluvial discharge. Thelongitudinal axis, x, has origin at the river mouth and is positive in the seawarddirection. The longitudinal coordinate, x⁄, is the point up to which the tidepropagate. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

between oyster reefs and submerged aquatic vegetation [9,23].Hydrodynamics forcing observed here are likely to be present inother river mouth settings along the Gulf Coast as they are linkedto regional meteorological and oceanographic conditions [1,42].The northern Gulf Coast experiences winter cold fronts that reoc-cur with a 3–10 day periodicity. During the approaching phase ofthe front, onshore winds force water and sediment landward.After the front passes, winds shift to the north, leaving behindmudflats and other coastal deposits [1,15,42]. Discharge in theApalachicola River has ranged between 110 and 8235 m3/s from1929 to present; peak flows generally occur in late winter andearly spring, and are highly correlated to rainfall events inGeorgia, while low flows occur during late summer and early fall[9,17]. During the period of our study, river discharge ranged from200 to 3100 m3/s.

Tides in Apalachicola Bay are mixed diurnal and semi-diurnal.The two main semidiurnal components are the lunar semidiurnal(M2), and solar semidiurnal (S2), with amplitudes (half the tidalrange) equal to 0.38 and 0.12 m respectively. The two main diurnalcomponents are the two lunar constituents (K1 and O1), withamplitudes equal 0.43 and 0.37 m respectively.

AN

Sumatra

250 Km

0 50 100 1500

-1.5 z(m)

y(m)

ADCP

300 m

1 Km B

C

D

Three Brothers

Distributary mouth

Fig. 2. (A) Study area in the Apalachicola delta, Florida. Location of instrumentsdeployment: ADCP deployed at 29�45029.4400N, 84�54043.0200W; RBR deployed at29�47058.9900N, 84�59016.6600W, near the Three Brothers Creek. USGS stations nearSumatra (ID 02359170) and near Chattahochee (ID 02358000); (B) and (C)distributary mouth and location of ADCP deployment; (D) cross section of theriver mouth, the black point indicates the ADCP location, at the bottom of thedistributary.

N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78 71

In this study, we analyze processes acting at the mouth of a dis-tributary channel in the northern part of the Bay (Fig. 2b and c).The area is characterized by a gentle seaward slope and a maxi-mum channel depth of 0.5 m, with respect to adjacent, almost flatareas (Fig. 2d).

3. Methods

We deployed a Aquadopp Nortek Acoustic Doppler CurrentMeter (ADCP) at the mouth of a distributary channel(29�45024.2300N, 84�54041.8700W. Fig. 2b and c), and measured waterdepth and velocity from January 22 to March 12, 2013. The instru-ment recorded every 30 min at 2 Hz, averaging over 60 s, and withvertical cell sizes of 10 cm. Water depth was calculated by meansof ADCP’s piezometers, and pressure values were corrected fromatmospheric fluctuations using data from the NOAA station atApalachicola (NOAA station ID 8728711). Considering the shallowdepth of the channel (maximum depth of 0.5 m), we did not detectsignificant velocity variations along the vertical, and velocity mea-surements have been depth-averaged (Fig. S1). Horizontal velocitieswere then rotated to align them with the channel axis (Fig. 2c). OneRBR TWR2050 pressure transducer for water depth measurementswas deployed farther upstream, near the Three Brothers Creek(29�47058.9900N 84�59016.6600W, Fig. 2a), between the channelmouth and the USGS station at Sumatra (USGS Station ID02359170). The RBR sampled every 15 min, 512 points at 2 Hz.Surface elevations and discharge measurements from the USGS sta-tions near Sumatra and near Chattahochee (USGS Station ID02358000) were used in our analysis as well (Fig. 2a). Atmosphericdata were retrieved from the NOAA station at Apalachicola.Surface elevations were geo-referenced with respect to theNational Geodetic Vertical Datum of 1929 (NGVD29).

For signal processing, we used wavelet analysis which is a valu-able tool to analyze tidal processes deviating from the exact peri-odicity assumption, typical of classical harmonic analysis [19,27].Wavelet transforms expand time series into the time–frequencydomain, and allow finding transient periodicities as they are local-ized in both frequency (Dt), and time (Dx) [22]. Similarly to sinesand cosines for Fourier basis functions, a mother wavelet ðw0Þ isused as the basis in representing other time series. The motherwavelet is then dilated and translated in time: dilatations allowthe localization in the frequency domain, while its translationallows localization in time [27]. We used continuous waveletanalysis and the non-orthogonal Morlet wavelet. Heisenberginequality states that there is a lower limit to the product of fre-quency and time resolution. There is thus a tradeoff between local-ization in time and frequency such that if the time resolution isimproved the frequency resolution degrades. The Morlet waveletprovides a good balance between frequency and time localization,and provides realistic images of the oscillations in data with non-stationary processes such as river flow [e.g. [28]]. Continuouswavelet transform is defined as the convolution of the time serieswith the scaled and normalized wavelet and its complex argumentcan be interpreted as the local phase. We further use the cross-wavelet transform of the time series velocity and water level,whose complex argument is the relative phase lag between thetwo signals.

4. Results

Fig. 3 provides an overview of the data retrieved during the per-iod of study. From January 22 to February 11, the average river dis-charge at Sumatra (FL) was 300 m3/s. After February 11, theApalachicola River was interested by two consecutive high dis-charge events, with discharge peaks of 2350 and 3100 m3/s

respectively (Fig. 3a). Fig. 3b shows surface elevations at our mea-suring stations along the distributary channel, and theApalachicola River. With increasing discharge, and consequentincrease in normal flow depth, the water surface profile shiftedfrom a concave up profile (M1 curve) to a drawdown profile (M2curve), in order to gradually adapt the upstream normal flow depthto the downstream water level (Fig. 3c). The drawdown profileforced the current within a water depth smaller than the flowdepth occurring with low river flow and high tide for a distanceof at least 3 km (Fig. 3c, point A). Freshwater discharge and incom-ing tidal waves are both time dependent, but their time scale differwidely as the tidal wave changes hourly, while the freshwater dis-charge changes in terms of days or week. The outward current u0

eventually becomes dominant as the wave progresses upstreamor for an increasing ratio of the riverine flow to the tidal flow.Once the riverine flow becomes important, the instantaneous flowvelocity of a moving particle is made of a steady component, cre-ated by the riverine discharge, and a time dependent componentcontributed by tides [e.g. [21,30]]. Eventually, if the riverine flowis high enough, the oscillating flow at the river mouth is replacedby a unidirectional flow.

Herein, we define fluvial dominated the state during whichthere is no flow reversal at the distributary mouth and the tidaldischarge is small with respect to the fluvial one. In contrast, wedefine tidally dominated the condition for which the tidal dis-charge is large enough to allow flow reversal at the distributarymouth. For the tidally dominated period, minimum velocitiesthroughout the tidal cycle (umin) are negative. On the contrary,for the fluvial dominated period, minimum velocities throughoutthe tidal cycle are positive (Fig. 4a, orange line).

Fig. 4 shows velocity and water level measurements for the per-iod of study (Fig. 4a), as well as their lower frequency constituents(Fig. 4b), diurnal (Fig. 4c), and semidiurnal (Fig. 4d) constituents.Low-frequency velocities and water levels account for frequencieslower than the diurnal one and are therefore representative forstorm surges and sea level variations different with respect tothe main tidal constituents, as well as for the riverine flow. Onthe contrary, diurnal and semidiurnal frequencies represent tidalharmonics.

Tidally averaged velocities at the distributary mouth rangedaround 0.1 m/s during low flow and reached 0.25 and 0.5 m/sduring the first and the second flood respectively (Fig. 4b).During the two high discharge events, the riverine flow was suchthat there was not flow reversal at the river mouth and mini-mum velocities throughout each tidal cycle (umin) were positive(orange line, Fig. 4a). Five main meteorological tides character-ized the period of interest. On January 30, February 7 and 24,South-West winds of 5, 8 and 4.5 m/s respectively caused anincrease in water level of around 0.6 m and consequent decreasein flow velocities (meteorological high tide, Fig. 4b). On February15 and March 3, winds of 6 and 10 m/s, coming from North,decreased water levels in the channel of 0.5 and 0.2 m respec-tively and increased mean velocity values (meteorological lowtides, Fig. 4b).

4.1. Phase lag between water level and velocity

For a tidal wave, the phase delay (e) between water level andvelocity is defined as the phase lag between high water (HW)and high water slack (HWS, i.e. when the velocity is zero). For asimple harmonic wave, e ¼ p

2 � ðUz �UuÞ, where Uz is the waterlevel phase and Uu is the velocity phase [e.g. [30,44]]. Thus, whene = 0 slack water periods occur at high and low tide. When e ¼ p

2,ebb and flood occur at low and high water respectively. The abovedefinition is maintained herein for low frequency fluctuations of

22 26 30 3 7 11 15 19 23 27 3 7 11−2

0

2

4

6

8

10

Surfa

ce E

leva

tion

NG

VD29

(m)

12ChattahocheeSumatraRBRADCP

8

16

18

20

22

Surfa

ce E

leva

tion

Cha

ttaho

chee

NG

VD29

(m)

12

10

14

B 22 26 30 3 7 11 15 19 23 27 3 7 11

January Fabruary March

2000

4000

0

River discharge at Sumatra (m /s)3

A

Days

Days

Days 0 20 40 60

SumatraBrothers creek

M2

M1

A

Q = 3100 m /ss 3

Q = 300 m /ss 3

-1

1

3

5

Wat

er le

vel (

m)

C

January February March

Fig. 3. (A) Apalachicola river discharge measurements at Sumatra (ID 02359170). (B) Water levels in the Apalachicola River and in the distributary for the period of study.Blue line: ADCP measurements at the distributary mouth; green line: RBR measurements; pink line: surface levels at the USGS station near Sumatra; red line: surface levels atthe USGS station at Chattahochee. Elevations refer to the vertical datum NAVD29. (C) Gradually varied water surface profile for high (pink line, 28 February) and low (greenline, 1 February) river discharge. Continuous and dashed lines refer to low tide and high tide respectively. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

72 N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78

the water surface (i.e. meteorologically induced fluctuations).Fig. 5b shows the phase relationship between velocity and waterlevel at the distributary mouth for different frequencies, asobtained from the cross wavelet transform of the two time series.The phase relationship is shown as arrows, with e ¼ 0 correspond-ing to an arrow pointing up and e ¼ p

2 corresponding to an arrowpointing to the left. Low-pass downstream water level variations,triggered by meteorological events, always displays a phase lage � p

2, with maximum velocities occurring when sea level is lowerthan its predicted astronomical value (Figs. 4b, 5b). This phe-nomenon is particularly evident in the first part of the study per-iod, corresponding to low flow conditions and characterized byan approximately constant, and low discharge (23 January to 1February, Fig. 4b). Mean velocity values are thus affected by stormsurges, and the percent increase in velocity is inversely propor-tional to the surge increase in mean sea level (Fig. 6). As a result,velocities in the distributary are higher during meteorologicallow tides, when wind decreases water level below its astronomicalvalue.

While surge induced water level and velocity variations arealways characterized by a phase lag e � p

2, the phase relationshipbetween tidal velocities and water levels is strongly affected by

the riverine flow, and appear related to the transition from a bidi-rectional to an unidirectional flow at the distributary mouth.During the tidally dominated period, the phase-lag between veloc-ity and water level approaches zero and slack waters occur aroundtime of high and low water (Fig. 5a and b). Once the unidirectionaloutflow condition is reached (umin > 0), the phase lag approaches p

2,and high and low water correspond to minimum and maximumvelocities respectively (Fig. 5a and b grey area). These trends areevident for diurnal and semidiurnal tidal frequencies (pink bands,Fig. 5b), and for meteorologically forced constituents. No signifi-cant trend emerges for the phase lag between velocities and waterlevels in the overtides domain.

A phase lag close to zero is what is often found in stronglytapering estuaries, while a phase lag close to p

2 corresponds to aprogressive wave, frequently observed in river channels with con-stant cross section. This change in phase lag is thus in agreementwith the transition from estuarine to riverine conditions. Thedrawdown backwater curve further creates a situation where thecross-sectional area is more constant. In fact, the width conver-gence that we observe during low flow is now compensated byan upstream increase in flow depth, and this adds to the riverinebehavior.

26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 27 1 3 5 7−0.5

0

0.5

1

9

z (m)u (m/s)

26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 27 1 3 5 7 9−0.5

0

0.5

26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 27 1 3 5 7 9−0.5

0.5

z (m)u (m/s)

Days

Days

A

B

C

TIDAL DOMINATED FLUVIAL DOMINATED

0

0

26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 27 1 3 5 7 9

0.5

D

Diurnal frequency

Semi-diurnal frequency

Storm surge

−0.5Eb

bFl

ood

Ebb

Floo

dEb

bFl

ood

Ebb

Floo

d

Days

Days

umin (m/s) umax (m/s)

z (m)u (m/s)

z (m)u (m/s)

January Fabruary March

January February March

January February March

January February March

Fig. 4. (A) Water levels (red line) and Velocity (blue line) measurements for the period of study; envelopes of maximum (yellow line), and minimum (orange line) velocitiesthroughout a tidal cycle. The fluvial dominated period is indicated with a solid line while the tidally dominated period is indicated with a dotted line. (B) Low frequencyvelocity and water level constituents. (C) Diurnal tidal constituents. (D) Semidiurnal tidal constituents. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78 73

4.2. Tidal velocities

To qualitatively assess the system behavior under low and highdischarge conditions, it is convenient to look at velocity fluctua-tions for one tidal cycle at low flow and during the highest dis-charge event (Fig. 7a). Around the peak of the second flood, thelow tide further enhances spatial accelerations created by thedrawdown profile. Due to the change in phase lag between velocityand water level fluctuations, low water levels also correspond tomaximum velocities. Therefore, the combined effect of a draw-down profile, and the presence of tides promotes a dramaticincrease in peak velocity (points A and A0, Fig. 7a). Difference inmaximum tidal velocities between high flow and low flow are thenhigher than differences in minimum tidal velocity (the distancebetween points A–A0 is 60% higher than the distance betweenpoints B–B0, Fig. 7a). It is then worth noticing that, going from

low to high discharge conditions, there is a strong increase in tidalvelocity amplitude (tidal velocity amplitude, mA0B0 , for high dis-charge conditions almost double the tidal velocity amplitude, mAB,corresponding to low river discharge conditions).

Fig. 7b and c represent tidal velocity amplitudes for the diurnaland semidiurnal tidal components as a function of the minimumvelocity throughout the tidal cycle, umin. umin is positive in the pres-ence of unidirectional flow, and negative in the case of flow rever-sal. During the tidally dominated period, tidal velocity amplitudesdecrease with umin, for both diurnal and semidiurnal tidal compo-nents (left side Fig. 7b and c). Vice versa, during the river domi-nated period, the amplitude of velocity fluctuations increase withflow velocity (right side of Fig. 7b and c).

To explain this behavior we use a simplified tidal prism model.Considering the river discharge, QR, the instantaneous flowbetween the estuary and the ocean, Q , and the instantaneous

26 30 3 7 11 15 19 23 27 3 7 128

32

64

16

8

4

2

P

erio

d (H

ours

)

First high discharge event=

=0Second high

discharge event

HW

HWS

Water levelVelocity

Days

A

B

2

Fig. 5. (A) Sketch of water level, velocity fluctuations, and phase lag between HW and HWS. (B) Phase relationship between water level and velocity fluctuations as a functionof time, for different frequencies. Phase lag e ¼ 0 corresponds to a pointing straight up arrow, and e ¼ p

2 corresponds to an arrow pointing to the left. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

−0.15 0 0.15 0.3−100

−50

0

50

Surge (m)

% M

ean

velo

city

cha

nge

R =0.82

Fig. 6. Mean velocity changes due to changes in water level induced by surges formJanuary 23 to February 1 2013. On the vertical axis the percentage of velocitychanges with respect to the mean velocity value. On the horizontal axis, surgeinduced water level fluctuations.

74 N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78

volume between the free surface and the low tidal level, Pi (seeFig. 1), from the continuity equation, it follows that:

dPi

dt¼ Q R � Q ð1Þ

For a simple tidal harmonic with period T , the Volume Pi can beapproximated as:

Pi ¼P2

sin2pt

T

� �þ 1

� �ð2Þ

where P is the tidal prism. Substituting equation (2) into equation(1), it follows that:

Q ¼ Q R �pPT

cos2pt

T

� �ð3Þ

From which it is possible to notice that, as the ratio QRT=pPincreases, the flood duration decreases up to the point when flowreversal is negligible and seawater does not enter the estuary, ifgravitational circulation is not present [e.g. [34]]. The tidal velocityamplitude, m is then proportional to:

m¼ Q EBB

B0hEBB� Q FLOOD

B0hFLOOD¼Q R

B0

HsinðeÞh2

msl�H2

4 sin2ðeÞ

!þpP

T2hmsl

h2msl�H2

4 sin2ðeÞ

!

ð4Þ

where hmsl, hEBB and hFLOOD are water depths at mean sea level, ebb,and flood respectively, H is the tidal range, B0 is the width of the dis-tributary mouth, hEBB ¼ hmsl � H

2 sinðeÞ, hFLOOD ¼ hmsl þ H2 sinðeÞ. When

the river discharge QR is much larger with respect to the tidal dis-charge, and flow reversal is negligible (umin > 0), the first term onthe right-end side of Eq. (4) is dominant and the tidal velocityamplitude increases with increasing discharge. Moreover, eapproaches p

2, and low water level to tidal range ratios promote hightidal velocity amplitudes. Thus, when meteorological tides are pre-sent, tidal velocity fluctuations can be decreased (in case ofmeteorological high tide), or further increased (in case ofmeteorological low tide), in agreement with a more pronounceddrawdown profile.

On the other hand, when the tidal discharge is larger than thefluvial one, the second term on the right hand side of equation(4), becomes relevant and tidal velocity amplitudes mainlydepends on the volume of the tidal prism. The tidal prism, P, isthe water volume enclosed between the envelopes of high waterslack (HWS) and low water slack (LWS) (Fig. 1, and [43]:

P ¼Z 0

�1H0Bds ð5Þ

Semidiurnal component

-0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tidal dominated

Diurnal component

River dominated Tidal dominated River dominated

Δu

0 12 24

Time (hours) u

A’

AB’

B

Q =3100 m /ss3

Q =300 m /ss3

-0.25

0

0.25

0.5

0.75

1Ve

loci

ty (m

/s)

A’B

’

AB

min umin

A B C

Fig. 7. (A) Tidal velocity fluctuations during one tidal cycle at the distributary mouth (ADCP location) corresponding to low, and high river discharge conditions, and to thewater surface profiles indicated in 3C. (B) Tidal velocity amplitudes of the semidiurnal tidal constituents as a function of the minimum velocity throughout the tidal cycle. (C)Tidal velocity amplitudes of the diurnal tidal constituents as a function of the minimum velocity throughout the tidal cycle.

N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78 75

where H0ðxÞ is the range between HWS and LWS as a function of dis-tance (difference between envelopes of the water levels occurring atHWS and LWS along the estuary). BðxÞ is the estuary width and x isthe longitudinal coordinate, starting from the river mouth and con-sidered positive when moving seaward (Fig. 1). Assuming that thewavelength is large compared to the length of the river and that dif-ferent water levels are reached instantaneously along the distribu-tary channel, the tidal range, H, exponentially changes along theestuary axis with damping coefficient d (d < 0 when the tidal waveis damped; d > 0 when the tidal wave is amplified). The tidal rangebetween water slacks, H0, is then related to the tidal range, H, by:

H0 ¼ HcosðeÞ ð6Þ

where e is the phase lag between HW and HWS, assumed constantalong the estuary.

Considering the tidal damping and substituting equation (6)into (5) it is possible to obtain the following expression to estimatethe tidal prism volume [43,45]:

P � HO1� db

cosðeÞ ð7Þ

where b is the convergence length of the stream width, and O is thesurface of the river interested by tidal oscillations. The tidal prism isthus a function of the tidal range, H, of the surface area where thetide propagates, O, of tidal damping, d, and of the phase lag, e.While flow reversal at the river mouth is still allowed, increasingriver discharge decreases the upstream area where tides propagate(Fig. 1, the longitudinal coordinate x⁄ moves downstream).Increasing river discharge also decreases tidal amplitudes far fromthe river mouth (higher absolute values of the damping coefficient),and increases the phase lag (e) (Fig. 3b, green line; Fig. 5). All thesemechanisms contribute to a reduction in flood volume storedwithin the river, and a consequential decrease in tidal velocityamplitudes. Moreover, by substituting equation (7) into Eq. (4), itis possible to notice that as e increases from 0 to, the first term inequation 4 gains precedence over the second term. In fact, the firstterm contains HsinðeÞ, while the second term contains HcosðeÞ. Thechange in phase lag can be thus considered a clear indicator of the

changed hydraulic behavior, and it is one of the reasons behind thenon-linear behavior illustrated in Fig. 7.

4.3. Eulerian residual currents

Eulerian residual currents are defined as the averaged velocitiesat a fixed location, and over a tidal cycle. These second-order cur-rents, driven by the nonlinear tidal dynamics, have been recog-nized as a significant component of the flow field in shallowareas, and can be relevant to investigate river deltas, and estuarieshydrodynamics [e.g. [2,12,24–26,37,38,49,53]]. It is important tonotice that the net mass transport of water is not only dependentby the mean velocity, but it is rather controlled by Lagrangean tra-jectories, obtained as the sum of Eulerian velocity and Stokes drift,with the Stokes drift being a mathematical artifact of the Eulerianframework [e.g. [45]].

Several mechanisms contribute to the formation of tidal resid-ual currents, which can be decomposed into three main con-tributions [e.g. [3,10]]: (1) the density driven flow whichdepends on buoyancy gradient; (2) asymmetric tidal mixing, whichis connected to the correlation between eddy viscosity and verticalshear, and thus to tidal straining, relevant to tidal asymmetry. (3)Vertically averaged tidal mean velocity, connected to the residualriverine flow and to non-linear flow mechanisms, which can beexplained by using the various terms in the De Saint–Venant equa-tions including the non-linear frictional term, the non-linearadvective term in the momentum equation, and the non-linearterm in the continuity equation [e.g. [41,47]]. Residual currentscalculated here are depth averaged, and are thus only representa-tive of the third component described above. Fig. 8 shows semi-diurnal (blue arrows) and diurnal (black arrows) residual currents.Residual currents exit from the river mouth for most of the recordperiod, and maintain a direction parallel to the mean flow.Moreover, while low discharge residual currents are of the orderof magnitude of the mean flow (compare arrows to the dashed line,Fig. 8), as the river discharge increases, they almost double. This isbecause once the unidirectional outflow condition is reached,acceleration of the flow at low tide are higher and not

Days

Semi-Diurnal Residual Diurnal Residual

Res

idua

l Ve

loci

ty (m

/s)

x y

Low pass filtered velocity

22 27 1 5 10 15 20 25 2 7 120.6

0.5

0.4

0.3

0.2

0.1

0

-0.1 Landward

Seaward

Fig. 8. Eulerian residual currents. Blue arrows correspond to diurnal averaged residual currents. Black arrows correspond to semidiurnal residual currents. The dotted linerepresents the low pass filtered velocity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

76 N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78

compensated by the corresponding flow deceleration at high tide.In fact, for unidirectional flow, and the increase in velocity duringebb with respect to the velocity at mean sea level can be writtenas:

mEBB ¼Q R

B0hEBB� QR

B0hmsl¼ Q R

B0

H2 sinðeÞ

hmslðhmsl � H2 sinðeÞÞ

!ð8Þ

However, the corresponding decrease in velocity during flood islower, being equal to:

mFLOOD ¼Q R

B0hmsl� Q R

B0hFLOOD¼ QR

B0

H2 sinðeÞ

hmslðhmsl þ H2 sinðeÞÞ

!ð9Þ

The average over a tidal cycle will thus result in a net seawarddirected velocity component. Moreover, the value mFEB � mFLOOD

increases with increasing discharge, and with the tidal range towater depth ratio. The formation of seaward directed, depth-aver-aged Eulerian residual currents after the transition from tidal to flu-vial dominated conditions is also in agreement with previousnumerical model results [32]. On the contrary, high meteorologicaltides enhance the possibility of landward-directed residual currents(Fig. 8).

5. Discussion

Despite of Apalachicola Bay being a micro-tidal environment,our results suggest that tides can significantly affect the hydrody-namic of distributary mouths, for both low and high river dischargeregimes. When the tidal discharge is large compared to the fluvialone, the presence of a riverine flow increases tidal damping, anddecreases the amplitude of tidal velocities. However, once the riverdischarge becomes sufficiently high, the tidal flow becomes neg-ligible, and a unidirectional outflow establishes, with the systemtransitioning from tidally dominated to river dominated condi-tions. Once this fluvial dominated state is reached, the effect oftides at the distributary mouth is surprisingly amplified. Theunidirectional flow promotes a change in the phase relationshipbetween velocity and water level fluctuations, with ebb and floodoccurring at low and high tides respectively. Moreover, the ampli-tude of tidal velocities increases with increasing discharge. Thepresence of tides in the river dominated case (i.e. during high flowregimes) also promotes the establishment of seaward directedEulerian residual currents at the distributary mouth. During thetwo high discharge events, residual currents almost double meanvelocity values providing evidence of the possible contribution oftides to the outward mass transport.

All of the above mentioned mechanisms could significantlyaffect biological and ecological processes at distributary mouths.As an example, tidal velocity amplitudes contribute to nutrientflushing, and have been found to rapidly dilute both phytoplanktonbiomass and nutrient concentration in the lower reaches of dis-tributary channels [4,50]. Our results may have implications forsediment transport processes as well. For instance, bathymetricsurveys of the Wax Lake Delta have shown that tides may promotethe erosion of deltaic channels tips, and play a major role in chan-nel kinematics [18,46]. Moreover, even if rivers are often assumedto decelerate at their mouth, the region connecting the upstreamreaches to the offshore plume can be an erosional area when adrawdown profile establishes [29]. Under high discharge regimes,the presence of tides enhances the drawdown profile and increasestidal velocity amplitudes, which in turn may promote channelsscour. On the other hand, tides may also affect those areas wheredeposition is expected. For instance, when the fluvial discharge islarge, higher tidal velocity amplitudes have been connected towider deposits at distributary mouths, as well as to the most likelyoccurrence of tidal bedding features. Specifically, laminationextent and difference in mud content between successive layershave been found to increase with tidal velocity amplitudes [e.g.[33]]. In the presence of bidirectional flow, low tidal velocityamplitudes are associated to mouth deposits with a compactshape, while higher tidal velocity amplitudes generally correspondto deposits dissected into multiple channels [e.g. [32,51]]. Theeffect of tides on distributary hydrodynamics could also be exacer-bated by the shallow depths typical of distributary mouths, whichare of the order of 1–2 m. For these depths, even an oscillation inwater level of few tens of centimeters can have dramatic conse-quences for the velocity field.

6. Conclusions

We conclude that even very small tides can strongly impact thevelocity field at distributary mouths, during both low and highflow regimes. Specifically, while during low discharge conditionsthe presence of a river discharge increases tidal damping anddecreases tidal velocity amplitudes, during very high flow regimesthe effect of tides at the distributary mouths is magnified with anoteworthy increase in tidal velocity amplitudes and seawarddirected Eulerian residual currents. High discharge regimes leadto a phase change between water level and velocity fluctuations,with minimum and maximum velocities occurring at low and hightides respectively. This change in phase lag appears a key determi-nant for the amplification of tidal velocities. The effect of tides atthe mouth of distributaries appears thus intensified during very

N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78 77

high discharge conditions when even small water level fluctua-tions strongly impact the velocity field.

Acknowledgments

This research was supported by the ACS-PRF program award51128-ND8, by the ONR award N00014-14-1-0114, by the ExxonMobil Upstream Research Company award EM01830, and throughthe NSF VCR-LTER program award DEB 0621014. We would like tothank Jennifer Harper and Lauren Levi at the Apalachicola NERR forthe wonderful hospitality and technical support, CyndhiaRamatchandirane and Annalise Muscietta at LUMCON for helpingduring the field work. Data for this paper are available at USGSNational Water Information system, stations ID 02359170(Sumatra), ID 02358000 (Chattahoochee); at NOAA dataset number8728690. For more information about data contact [email protected]. We would like to thank the Editor, and the anonymousReviewer whose thoughts strengthened this manuscript.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.advwatres.2015.03.005.

References

[1] Allison MA, Kineke GC, Gordon ES, Goñi MA. Development and reworking of aseasonal flood deposit on the inner continental shelf off the Atchafalaya River.Cont Shelf Res 2000;20:2267–94. http://dx.doi.org/10.1016/S0278-4343(00)00070-4.

[2] Burchard H, Hetland RD. Quantifying the contributions of tidal straining andgravitational circulation to residual circulation in periodically stratified tidalestuaries. J Phys Oceanogr 2010;40(6):1243–62. http://dx.doi.org/10.1175/2010JPO4270.1.

[3] Burchard H, Hetland RD, Schulz E, Schuttelaars HM. Drivers of residualestuarine circulation in tidally energetic estuaries: straight and irrotationalchannels with parabolic cross section. J Phys Oceanogr 2010;41(3):548–70.http://dx.doi.org/10.1175/2010JPO4453.1.

[4] Caffrey JM, Chapin TP, Jannasch HW, Haskins JC. High nutrient pulses, tidalmixing and biological response in a small California estuary: variability innutrient concentrations from decadal to hourly time scales. Estuarine CoastShelf Sci 2007;71(3–4):368–80. http://dx.doi.org/10.1016/j.ecss.2006.08.015.

[5] Cai H, Savenije H, Yang Q, Ou S, Lei Y. Influence of river discharge and dredgingon tidal wave propagation: Modaomen estuary case. J Hydraulic Eng2012;138(10):885–96. http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000594.

[6] Cai H, Savenije HHG, Toffolon M. Linking the river to the estuary: influence ofriver discharge on tidal damping. Hydrol Earth Syst Sci 2014;18(1):287–304.http://dx.doi.org/10.5194/hess-18-287-2014.

[7] Cai H, Savenije HHG, Jiang C. Analytical approach for predicting fresh waterdischarge in an estuary based on tidal water level observations. Hydrol EarthSyst Sci Discuss 2014;11(6):7053–87. http://dx.doi.org/10.5194/hessd-11-7053-2014.

[8] Canestrelli A, Fagherazzi S, Defina A, Lanzoni S. Tidal hydrodynamics anderosional power in the Fly River delta, Papua New Guinea. J Geophys Res: EarthSurface 2010;115. http://dx.doi.org/10.1029/2009JF001355. F04033.

[9] Chanton J, Lewis FG. Examination of coupling between primary and secondaryproduction in a river-dominated estuary: Apalachicola Bay, Florida, USA.Limnol Oceanogr 2002;47:683–97. http://dx.doi.org/10.4319/lo.2002.47.3.0683.

[10] Cheng P, Valle-Levinson A, De Swart HE. A numerical study of residualcirculation induced by asymmetric tidal mixing in tidally dominated estuaries.J Geophys Res: Oceans 2011;116(C1):C01017. http://dx.doi.org/10.1029/2010JC006137.

[11] Chow VT. Open-channel hydraulics. New York: McGraw-Hill; 1959. 680 pp.[12] Zhou Z, Coco G, Jiménez M, Olabarrieta M, van der Wegen M, Townend I.

Morphodynamics of river-influenced back-barrier tidal basins: the role oflandscape and hydrodynamic settings. Water Resour Res2014;50(12):9514–35. http://dx.doi.org/10.1002/2014WR015891.

[13] D’Alpaos A, Lanzoni S, Marani M, Rinaldo A. On the tidal prism–channel arearelations. J Geophys Res: Earth Surface 2010;115(F1):F01003. http://dx.doi.org/10.1029/2008JF001243.

[14] Dalrymple RW, Choi K. Morphologic and facies trends through the fluvial–marine transition in tide-dominated depositional systems: a schematicframework for environmental and sequence-stratigraphic interpretation.Earth-Sci Rev 2007;81(3–4):135–74. http://dx.doi.org/10.1016/j.earscirev.2006.10.002.

[15] Draut AE, Kineke GC, Huh OK, Grymes JM, Westphal KA, Moeller CC. Coastalmudflat accretion under energetic conditions, Louisiana Chenier-plain coast,USA. Mar Geol 2005;214:27–47. http://dx.doi.org/10.1016/j.margeo.2004.10.033.

[16] Edmonds DA. Stability of backwater influenced bifurcations: a study of theMississippi-Atchafalaya bifurcation. Geophys Res Lett 2012;39:L08402. http://dx.doi.org/10.1029/2012GL051125.

[17] Elder JF, Mattraw HC. Riverine transport of nutrients and detritus to theApalachicola Bay estuary, Florida1. JAWRA J. Am. Water Resour. Assoc.1982;18(5):849–56. http://dx.doi.org/10.1111/j.1752-1688.1982.tb00081.x.

[18] Esposito CR, Georgiou I, Kolker AS. Efficient delivery of sediment through anactive crevasse splay. Geophys Res Lett 2013;40:1540–5. doi: 1510.1002/grl.50333..

[19] Ganju NK, Schoellhamer DH, Warner JC, Barad MF, Schladow SG. Tidaloscillation of sediment between a river and a bay: a conceptual model.Estuarine Coast Shelf Sci 2004;60(1):81–90. http://dx.doi.org/10.1016/j.ecss.2003.11.020.

[20] Geyer WR, Woodruff J, Traykovski P. Sediment transport and trapping in theHudson River estuary. Estuaries 2001;24(5):670–9. http://dx.doi.org/10.2307/1352875.

[21] Godin G. The propagation of tides up rivers with special considerations on theupper Saint Lawrence river. Estuarine Coastal Shelf Sci 1999;48(3):307–24.http://dx.doi.org/10.1006/ecss.1998.0422.

[22] Grinsted A, Moore JC, Jevrejeva S. Application of the cross wavelet transformand wavelet coherence to geophysical time series. Nonlinear Process Geophys2004;11(5/6):561–6. http://dx.doi.org/10.5194/npg-11-561-2004.

[23] Huang W, Sun H, Nnaji S, Jones WK. Tidal hydrodynamics in a multiple-inletestuary: Apalachicola Bay, Florida. J Coastal Res 2002;18(4):674–84. http://dx.doi.org/10.2307/4299119.

[24] Hunt JN, Johns B. Currents induced by tides and gravity waves. Tellus1963;15(4):343–51. http://dx.doi.org/10.1111/j.2153-3490.1963.tb01397.x.

[25] Huthnance JM. Tidal current asymmetries over the Norfolk Sandbanks.Estuarine Coastal Mar Sci 1973;1(1):89–99. http://dx.doi.org/10.1016/0302-3524(73)90061-3.

[26] Ianniello JP. Tidally induced residual currents in estuaries of variable breadthand depth. J Phys Oceanogr 1979;9(5):962–74. http://dx.doi.org/10.1175/1520-0485(1979) 009<0962:TIRCIE>2.0.CO;2.

[27] Jay DA, Flinchem EP. Interaction of fluctuating river flow with a barotropictide: a demonstration of wavelet tidal analysis methods. J Geophys Res:Oceans 1997;102(C3):5705–20. http://dx.doi.org/10.1029/96JC00496.

[28] Labat D. Recent advances in wavelet analyses: Part 1. A review of concepts. JHydrol 2005;314(1–4):275–88. http://dx.doi.org/10.1016/j.jhydrol.2005.04.003.

[29] Lamb MP, Nittrouer JA, Mohrig D, Shaw J. Backwater and river plume controlson scour upstream of river mouths: implications for fluvio-deltaicmorphodynamics. J Geophys Res: Earth Surface 2012;117(F1):F01002. http://dx.doi.org/10.1029/2011JF002079.

[30] Lanzoni S, Seminara G. On tide propagation in convergent estuaries. J GeophysRes-Oceans 1998;103(C13):30793–812. http://dx.doi.org/10.1029/1998jc900015.

[31] Lanzoni S, Seminara G. Long-term evolution and morphodynamic equilibriumof tidal channels. J Geophys Res 2002;107(C1):1–13. http://dx.doi.org/10.1029/2000JC000468.

[32] Leonardi N, Canestrelli A, Sun T, Fagherazzi S. Effect of tides on mouth barmorphology and hydrodynamics. J Geophys Res: Oceans2013;118(9):4169–83. http://dx.doi.org/10.1002/jgrc.20302.

[33] Leonardi N, Sun T, Fagherazzi S. Modeling tidal bedding in distributary-mouthbars. J Sediment Res 2014;84(6):499–512. http://dx.doi.org/10.2110/jsr.2014.42.

[34] Luketina D. Simple Tidal Prism Models Revisited. Estuarine Coastal Shelf Sci1998;46:77–84. http://dx.doi.org/10.1006/ecss.1997.0235.

[35] Mariotti G, Falcini F, Geleynse N, Guala M, Sun T, Fagherazzi S. Sediment eddydiffusivity in meandering turbulent jets: Implications for levee formation atriver mouths. J Geophys Res: Earth Surface 2013;118(3):1908–20. http://dx.doi.org/10.1002/jgrf.20134.

[36] Montani S, Magni P, Shimamoto M, Abe N, Okutani K. The effect of a tidal cycleon the dynamics of nutrients in a tidal estuary in the Seto Inland Sea, Japan. JOceanogr 1998;54:65–76. http://dx.doi.org/10.1007/BF02744382.

[37] Nihoul JCJ, Ronday FC. The influence of the ‘‘tidal stress’’ on the residualcirculation. Tellus 1975;27(5):484–90. http://dx.doi.org/10.1111/j.2153-3490.1975.tb01701.

[38] Olabarrieta M, Geyer WR, Kumar N. The role of morphology and wave-currentinteraction at tidal inlets: an idealized modeling analysis. J Geophys Res:Oceans 2014;119(12):8818–37. http://dx.doi.org/10.1002/2014JC010191.

[39] Osterman L, Twichell D, Poore R. Holocene evolution of Apalachicola Bay,Florida. Geo-Mar Lett 2009;29(6):395–404. http://dx.doi.org/10.1007/s00367-009-0159-1.

[40] Paola C, Twilley R, Edmonds DA, Kim W, Mohrig D, Parker G, Viparelli E, VollerV. Natural processes in delta restoration: application to the Mississippi Delta.Ann Rev Mar Sci 2010;3:67–91. http://dx.doi.org/10.1146/annurev-marine-120709-142856.

[41] Parker BB. Tidal hydrodynamics. New York: John Wiley; 1991, ISBN9780471514985. 883 pp. http://books.google.com/books?id=zd0XbSrQFI0C.

[42] Walker ND, Hammack AB. Impacts of winter storms on circulation andsediment transport: Atchafalaya-Vermilion Bay Region, Louisiana, U.S.A.. JCoastal Res 2000;16(4):996–1010. http://dx.doi.org/10.2307/4300118.

78 N. Leonardi et al. / Advances in Water Resources 80 (2015) 69–78

[43] Savenije HHG. Salinity and tides in alluvial estuaries. Amsterdam: Elsevier;2006, ISBN 9780444521071. 197 pp.

[44] Savenije HHG, Toffolon M, Haas J, Veling EJM. Analytical description of tidaldynamics in convergent estuaries. J Geophys Res 2008;113(C10). http://dx.doi.org/10.1029/2007jc004408.

[45] Savenije HHG. Salinity and tides in Alluvial Estuaries, 2nd completely revisededition: salinityandtides.com; 2012.

[46] Shaw JB, Mohrig D. The importance of erosion in distributary channel networkgrowth, Wax Lake Delta, Louisiana, USA. Geology; 2013. http://dx.doi.org/10.1130/G34751.1

[47] Tee KT. Tide-induced residual current—verification of a numerical model. JPhys Oceanogr 1977;7(3):396–402. http://dx.doi.org/10.1175/1520-0485(1977) 007<0396:TIRCOA>2.0.CO;2.

[48] Toffolon M, Savenije HHG. Revisiting linearized one-dimensional tidalpropagation. J Geophys Res: Oceans 2011;116(C7):C07007. http://dx.doi.org/10.1029/2010JC006616.

[49] Van der Vegt M et al. Modeling the equilibrium of tide dominated ebb-tidaldeltas. J Geophys Res 2006;111:F02013. http://dx.doi.org/10.1029/2005JF000312.

[50] Valiela I, McClelland J, Hauxwell J, Behr BJ, Hersh D, Foreman K. Macroalgalblooms in shallow estuaries: controls and ecophysicological and ecosystemconsequences. Limnol Oceanogr 1997;42(5part2):1105e1118. http://dx.doi.org/10.4319/lo.1997.42.5_part_2.1105.

[51] Wright LD. Sediment transport and deposition at river mouths: a synthesis.Geol Soc Am Bull 1977;88(6):857–68. http://dx.doi.org/10.1130/0016-7606(1977) 88<857:STADAR>2.0.CO;2.

[52] Yang Z, De Swart HE, Cheng H, Jiang C, Valle-Levinson A. Modelling lateralentrapment of suspended sediment in estuaries: the role of spatial lags insettling and M4 tidal flow. Cont Shelf Res 2014;85:126–42. http://dx.doi.org/10.1016/j.csr.2014.06.005.

[53] Zimmerman JTF. Dynamics, diffusion and geomorphological significance oftidal residual eddies. Nature 1981;290(5807):549–55. http://dx.doi.org/10.1038/290549a0.