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The Importance of Tidal and Lateral Asymmetries in Stratification to ResidualCirculation in Partially Mixed Estuaries*

MALCOLM E. SCULLY

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

CARL T. FRIEDRICHS

Virginia Institute of Marine Science, The College of William and Mary, Gloucester Point, Virginia

(Manuscript received 2 August 2005, in final form 27 September 2006)

ABSTRACT

Measurements collected in the York River estuary, Virginia, demonstrate the important impact that tidaland lateral asymmetries in turbulent mixing have on the tidally averaged residual circulation. A reductionin turbulent mixing during the ebb phase of the tide caused by tidal straining of the axial density gradientresults in increased vertical velocity shear throughout the water column during the ebb tide. In the absenceof significant lateral differences in turbulent mixing, the enhanced ebb-directed transport caused by tidalstraining is balanced by a reduction in the net seaward-directed barotropic pressure gradient, resulting inlaterally uniform two-layer residual flow. However, the channel–shoal morphology of many drowned rivervalley estuaries often leads to lateral gradients in turbulent mixing. Tidal straining may then lead to tidalasymmetries in turbulent mixing near the deeper channel while the neighboring shoals remain relativelywell mixed. As a result, the largest lateral asymmetries in turbulent mixing occur at the end of the ebb tidewhen the channel is significantly more stratified than the shoals. The reduced friction at the end of ebbdelays the onset of the flood tide, increasing the duration of ebb in the channel. Conversely, over the shoalregions where stratification is more inhibited by tidal mixing, there is greater friction and the transition fromebb to flood occurs more rapidly. The resulting residual circulation is seaward over the channel andlandward over the shoal. The shoal–channel segregation of this barotropically induced estuarine residualflow is opposite to that typically associated with baroclinic estuarine circulation over channel–shoal bathym-etry.

1. Introduction

The understanding of tidally averaged residual circu-lation is fundamental to the study of estuarine pro-cesses. The classic representation of the two-layer es-tuarine exchange flow was represented by Pritchard(1956) as the balance between the seaward-directedbarotropic pressure gradient, the landward-directedbaroclinic pressure gradient, and the stress divergence.The internal friction associated with the exchange flowis often parameterized in terms of a vertically and tid-

ally averaged eddy viscosity acting on the tidally aver-aged vertical velocity shear. This representation of es-tuarine circulation has been the cornerstone for theclassical understanding of estuarine dynamics. How-ever, the use of a tidally averaged mixing coefficientmay often be inappropriate in characterizing the basicforcing driving tidally averaged residual circulation inestuarine systems where there are significant spatialand temporal asymmetries in turbulent mixing.

In many estuarine systems, the interaction betweenthe vertical tidal shear and the longitudinal density gra-dient can lead to tidal asymmetries in density stratifi-cation. This process, known as tidal straining (Simpsonet al. 1990), favors the development of stratificationduring ebb tides and the destruction of stratificationduring the flood and can lead to significant tidal asym-metries in vertical mixing (Stacey et al. 1999; Geyer etal. 2000; Rippeth et al. 2001). Jay and Musiak (1994)

* Virginia Institute of Marine Science, College of William andMary, Contribution Number 2744.

Corresponding author address: Malcolm Scully, Woods HoleOceanographic Institution, Mail Stop 10, Woods Hole, MA 02543.E-mail: [email protected]

1496 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 37

DOI: 10.1175/JPO3071.1

© 2007 American Meteorological Society

JPO3071

suggested that tidal asymmetries in vertical mixing as-sociated with tidal straining could play an importantrole in generating residual circulation in estuaries. Theyargued that the reduction of tidal mixing during the ebbtide resulted in an internal tidal asymmetry that couldgenerate a two-layer tidally averaged residual circula-tion qualitatively similar to a baroclinically driven ex-change flow. Consistent with Jay and Musiak’s results,Stacey et al. (2001) kinematically described the two-layer residual circulation in northern San Francisco Bayas the barotropic response to tidal asymmetries in ver-tical mixing. In a study carried out in the Hudson Riverestuary, Geyer et al. (2000) also observed tidal asym-metries in the vertical mixing coefficients associatedwith tidal straining, but concluded that the tidally av-eraged estuarine circulation was not fundamentally af-fected by this internal asymmetry. In a companion pa-per, however, Trowbridge et al. (1999) found the rela-tionship between bottom stress and the near-bedvelocity shear in the Hudson systematically departedfrom the law of the wall during ebb tides, consistentwith greater stratification due to tidal straining. Trow-bridge et al. (1999) hypothesized that the departurefrom the law of the wall on ebb was not due to localstratification, but caused by overlying stratification re-motely damping the near-bed turbulent length scale.Similar findings were presented by Scully and Fried-richs (2003), who demonstrated significant damping ofnear-bed turbulence by nonlocal, overlying stratifica-tion in the York River estuary.

These previous studies provide a somewhat contra-dictory view of the effect that tidal asymmetries in mix-ing have on estuarine residual circulation. While Jayand Musiak (1994) and Stacey et al. (2001) suggest tidalstraining of the density field enhances up-estuary re-sidual flow near the bed, Geyer et al. (2000) concludedthat asymmetries in vertical mixing had little impact onthe near-bed residual circulation. Trowbridge et al.(1999) and Scully and Friedrichs (2003) suggested thattidal straining of the density field often leads to in-creased velocity shear on ebb near the bed, but did notdocument bottom intensification of the flow duringflood due to convective mixing. Their results suggestthat given perfectly symmetric forcing, tidal strainingwould favor down-estuary residual flow if there wereno adjustment of the net barotropic pressure gradient.

In addition to the variations in turbulent mixingcaused by temporal changes in stratification, many es-tuarine systems exhibit spatial gradients in mixingcaused by lateral bathymetric variability. The impact ofbathymetry on residual estuarine circulation has beenaddressed analytically by a number of authors (Wong1994; Kasai et al. 2000; Valle-Levinson et al. 2003).

However, their analytical approach uses a spatial andtemporally constant eddy viscosity and thus cannot ac-count for asymmetries in turbulent mixing. Only theanalytical model of Friedrichs and Hamrick (1996)attempted to account for lateral variations in mixingby assuming the eddy viscosity scaled with depth.Drowned river valley estuaries often have channel–shoal morphologies, with marked lateral variations indensity stratification. Lateral mixing fronts often format breaks in topography, separating the relatively wellmixed water on the shoal from the more stratified wa-ters over the channel (Huzzey and Brubaker 1988). Incontrast to the approach suggested by Friedrichs andHamrick (1996), the absence of stratification over theshoal may enhance turbulent mixing relative to themore stratified channel. Despite the fact that lateralvariability in stratification in many systems may exceedtemporal variations arising from tidal straining, its rolein modulating residual circulation has been largely ig-nored.

In this paper, measurements collected at two laterallyadjacent locations are used to examine the forces driv-ing residual circulation in the York River estuary.The observational and analytical methods are describedin section 2 and the results are presented in section 3.The results include direct measurements that docu-ment tidal asymmetries in turbulent mixing. Lateralasymmetries in turbulent mixing are inferred fromstress estimates based on the dominant terms in theaxial momentum balance. Last, results from a numeri-cal experiment utilizing k-epsilon turbulence closureare presented to examine in more detail the physicalmechanisms that create residual flows in estuaries ex-periencing tidal and lateral asymmetries in stratifica-tion. The results are discussed in section 4, and conclu-sions are presented in section 5.

2. Methods

a. Environmental setting and observations

Observations were collected in the mesohaline por-tion of the York River estuary during the winter of2003/04 (Fig. 1). The York River is a partially mixedsubestuary of the Chesapeake Bay that forms at theconfluence of the Mattaponi and Pamunkey Rivers.Most of the York River is characterized by a mainchannel with an approximate depth of 10 m, flanked bybroad shoals with a mean depth of approximately 4 m.Although the mean tidal range is only about 0.7 m, tidalcurrents are energetic with surface currents reachingover 1 m s�1 during spring tidal conditions. Both salin-ity and temperature exhibit significant seasonal vari-ability; however, the density gradient is nearly always

JUNE 2007 S C U L L Y A N D F R I E D R I C H S 1497

dominated by gradients in salt. Top-to-bottom salinitystratification can range from well mixed to greater than12 psu.

During the winter of 2003/04, instrumentation wasmaintained at two laterally adjacent locations in theestuarine cross section near Clay Bank, located roughly13 km up-estuary from Gloucester Point. At this loca-tion in the York River, the main channel of the estuaryis located on the northeastern side of the river with amaximum depth of approximately 10 m. A broad shoalextends to the southwest of the main navigational chan-nel with a mean depth of less the 5 m and a maximumdepth of approximately 6.5 m. Instruments were de-ployed along the southwestern side of the main channelat a water depth of approximately 7 m (“channel” site)to avoid interfering with navigation, and at the deepest

location within the southwestern shoals at a waterdepth of roughly 6 m (“shoal” site). Although the in-strumented locations had a similar depth, for the pur-poses of this paper we assume that they are broadlyrepresentative of the deeper channel and shallowershoal regions found in the cross section.

During the experiment, detailed velocity measure-ments were collected by four Sontek acoustic Dopplervelocimeters (ADVs), an RDI 1200-kHz acousticDoppler current profiler (ADCP), and a Sontek 1500-kHz acoustic Doppler profiler (ADP). The four ADVswere mounted on a benthic boundary layer tripod de-ployed at the channel site to provide high-resolutionvelocity measurements at discrete elevations over thelowest 1.2 m of the water column. The ADVs weremounted 0.10, 0.43, 0.76, and 1.1 m above the bed

FIG. 1. Map of study area, York River estuary, VA, with estuarine cross section at “channel” and“shoal” deployment locations. Also shown are locations of the CBNERR CTD sites.


(mab), on an arm of the tripod that was rotated per-pendicular to the dominant orientation of the YorkRiver channel. The ADVs at 0.43 and 0.76 mab did notfunction properly and will not be discussed. The ADVscollected three-dimensional velocity measurements in5-min bursts once an hour at a sampling frequency of 5Hz. The 1200-kHz ADCP was deployed immediatelyadjacent to the tripod at the channel site and collectedvertical profiles of current velocity in 0.50-m bins. TheADCP sampled at roughly 1 Hz and collected one 10-min burst every hour. The 1500-kHz ADP was de-ployed at the shoal site and sampled at a rate of 1 Hzaveraged over 1-min bursts to provide vertical profilesof current velocity in 0.25-m bins.

Three YSI 6000 conductivity–temperature–depthsensors (CTDs) were mounted on the tripod at thechannel site at 0.10, 0.41, and 1.4 mab and collectedpressure, salinity, and temperature every 30 min. Amooring that consisted of an YSI 6000 CTD and anInterOcean S4 current meter, outfitted with a CTD,was located immediately adjacent to the tripod at thechannel site. The CTD and S4 on the mooring werelocated 2.8 and 3.4 mab, respectively. At the shoal site,two YSI 6600 CTDs located 0.10 and 1.5 mab and anInterOcean S4 sensor with a CTD located 3.5 mab weredeployed immediately adjacent to the ADP. In additionto the field instrumentation deployed as part of thisexperiment, the Chesapeake Bay National EstuarineResearch Reserve (CBNERR) maintains four YSI 6600datasondes year-round along the York River estuary asshown in Fig. 1. Each sensor is mounted on a fixedplatform immediately below mean low water and mea-sures a variety of environmental parameters every 15min. In this paper, we use the CBNERR pressure sen-sor data to estimate the along-channel barotropic pres-sure gradient.

b. Data analysis

The primary focus of this paper is to examine howasymmetries in turbulent mixing associated with tem-poral and spatial gradients in density stratification im-pact the tidally averaged residual circulation. Directmeasurements of stress are available from the ADVdata collected at the channel site. To calculate stressfrom the ADV data, the flow measurements in eachburst were first rotated vertically such that no meanvertical velocity was present. Flow measurements werethen rotated horizontally into the predominant along-channel direction. Reynolds stress calculations weremade using the covariance method as described by Kimet al. (2000), where

�b � ��u�w��. �1�

The angled brackets indicate time averaging, and u�and w� represent the turbulently fluctuating compo-nents of the along-channel and vertical velocities, re-spectively. Although there were no direct measure-ments of stress at the shoal site, estimates of stresscan be obtained from the dominant terms in the axialmomentum balance (Geyer et al. 2000; Winant andGutierrez de Velasco 2003). Neglecting rotation andthe advective terms, the along-channel momentum bal-ance can be written approximately as

�u

�t� �g

��

�x�

g

��

�xdz �

1�

��

�z, �2�

where u is the along-estuary velocity, g is the accelera-tion of gravity, is the sea surface elevation anomaly, xis the horizontal coordinate (positive landward), is thedensity of water, � is the Reynolds stress, and z is thevertical coordinate (positive upward). We followed themethods of Geyer et al. (2000) and vertically integratedEq. (2) to solve for the vertical profile of the effectivestress (�), assuming that � /�x to be independent of z.This assumption was generally consistent with data col-lected during two longitudinal hydrographic surveysconducted during the experiment.

The acceleration term (�u/�t) was easily determinedfrom both the ADCP and ADP data. The barotropicpressure gradient was calculated from the differencebetween the pressure sensors deployed by CBNERR atthe Sweet Hall Marsh and Goodwin Island locations.Although they were separated by the greatest distanceof the four pressure sensors, the instrumented crosssection was located nearly equidistant from these twosensors. In addition, we previously compared the along-channel barotropic pressure gradient calculated fromall possible combinations of these four CBNERR sen-sors that coincided with data collected by tide gaugesduring an experiment in 2002 (Scully and Friedrichs2003; Simpson et al. 2005). The gradient measuredbetween the Sweet Hall Marsh and Goodwin IslandCBNERR stations best matched the well-constrainedlocal tide gauge data collected during 2002. When cal-culating the barotropic pressure gradient from theCBNERR pressure sensor data, there was an unknownoffset between the two sensors due to their fixedheights relative to the local geopotential surface. Con-sistent with methods of Geyer et al. (2000), this differ-ence was determined by requiring the zero crossings ofthe momentum-integral estimates of near-bed stress tooccur at the times of the zero crossings of the near-bedvelocity.

In calculating the baroclinic pressure gradient, thealong-channel salinity gradient was estimated using the


S4 data from the channel and shoal sites based on thelocal salt conservation equation, ignoring lateral advec-tion and vertical turbulent flux:

�S

�x� �

1u

�s

�t. �3�

The relatively minor contribution of the axial tempera-ture gradient was not included in the estimate of thebaroclinic pressure gradient. The estimates of �s/�x us-ing Eq. (3) were low-passed filtered to remove the high-frequency noise associated with this method. The sepa-rate estimates from the two adjacent locations werethen averaged, assuming that the along-channel densitygradient did not vary significantly in the across-channeldirection. The time series of �s/�x obtained in this man-ner has a temporal variability consistent with that de-rived from the CBNERR data. Estimates from the vari-ous CBNERR stations could not be used directly be-cause of a number of issues, including 1) the presence offreshwater at the Sweet Hall Marsh location; 2) lateraldensity differences between Clay Bank and TaskinasCreek; and 3) salinity gradient reversals near the mouthof the estuary. In calculating the baroclinic pressuregradient for both experiments, we assumed that � /�xdoes not vary with depth. Although there may be someerrors in both our estimate of the barotropic and baro-clinic pressure gradients, these errors have a similarimpact on the stress estimates at each across-channellocation because it is unlikely that there are persistentlateral differences in either � /�x or � /�x.

In integrating Eq. (2), the surface stress was set equalto the wind stress calculated using the bulk formula ofLarge and Pond (1981). Wind data were obtained fromthe Sewell’s Point meteorological station maintained bythe National Oceanic and Atmospheric Administration(NOAA), located roughly 36 km southeast of Glouces-ter Point. The coordinate system was rotated 30° westof north so that the winds conform to the main axis ofthe York River. With the relatively moderate windsobserved during the experiment, the momentum esti-mates of internal stress were relatively insensitive to theinclusion of the surface wind stress. However, the windplays an important role in modulating both the tidallyaveraged barotropic pressure gradient and the degreeof salinity stratification, as described by Scully et al.(2005).

3. Results

a. Observations

The 2003/04 experiment took place during a periodof elevated rainfall. Discharge from the York Rivertributaries for December 2003 was, on average, nearly 3

times the 60-yr average (Fig. 2a). The experimentspanned approximately 30 days, covering the fullspring–neap tidal cycle. The spring–neap cycle was ap-parent in the depth-averaged current magnitude mea-sured at the channel site (Fig. 2b). With the elevatedriver discharge, there was persistent vertical densitystratification throughout much of the experiment atboth the channel and shoal sites, especially during neaptide (Fig. 2c). The vertical stratification exceeded 2 kgm�4 on several occasions at both locations, and medianvalues of stratification measured between the upper-most and lowermost sensors were 0.88 and 0.58 kg m�4

at the channel and shoal sites, respectively. In addition,significant tidal asymmetries in stratification were ob-served at both locations. At the channel site, the me-dian stratification on flood and ebb was 0.81 and 0.94 kgm�4, respectively. The asymmetry had the oppositesense at the shoal site, where the flood was actuallymore stratified on average than the ebb (0.65 and 0.53kg m�4). This pattern of tidal asymmetry led to thegreatest lateral asymmetry in stratification during theebb tide, which significantly impacted the estuarine dy-namics as discussed below. At both locations, the over-all degree of stratification also was modulated by theadvective effect of wind-driven circulation (Scully et al.2005).

The subtidal near-bed currents at both sites exhibitedsignificant variability (Fig. 2d). However, a consistentpattern emerged; on average, the currents tended to bemore seaward directed at the channel site than at theshoal site, particularly during spring tidal conditions.This pattern was seen clearly in the velocity profilescollected by the ADCP and ADP. Figure 3 shows thevelocity profiles collected at each site, averaged sepa-rately over neap and spring tidal conditions. For allcases, the tidally averaged along-channel velocity pro-files demonstrated vertical shear consistent with theclassic view of estuarine circulation. However, therewas significant lateral shear during spring tidal condi-tions when the currents at the shoal site were relativelymore landward directed than at the channel site. It isimportant to note that while the depth at the two siteswas roughly the same, the deployments at the channelsite were not located at the deepest point in the mainchannel to avoid interfering with shipping traffic. As aresult, the velocity profiles at the channel site did notcapture the landward-directed residual flow necessaryto conserve mass at the subtidal time scale. We assumethat this residual inflow occurred over the deepest por-tion of the channel to the east of our “channel” loca-tion.

Previous authors have documented that the presenceof vertical density stratification causes excess shear


relative to the Prandtl–Karman law-of-the-wall rela-tionship (e.g., Smith and McLean 1977; Soulsby andDyer 1981; Friedrichs et al. 2000). For purely symmetricbarotropic forcing, it follows that the increased shearcaused by the increased stratification on the ebb tidedue to tidal straining would lead to a greater depar-ture from the law of the wall and (in the absence of anet barotropic pressure gradient) favor down-estuaryresidual flow. The shear (�u/�z) in an unstratifiedboundary layer is expected to follow the law of the wall,given as

u* � �z�u

�z, �4�

where u* is the shear velocity, is von Kármán’s con-stant (0.41), and z is the height above the bed. This

relationship can be evaluated using the ADV data col-lected at the channel location. Because our estimate ofshear is based on velocity measured at two discretelocations, Eq. (4) is rewritten to account for this finitedifference as

u* ��U�

log�z2 � z1�. �5�

The observed velocity difference (�U) was simply cal-culated by differencing the along-channel velocity fromthe two functioning ADVs at the channel site, u*(�u�w��1/2) was calculated from the stress measured bythe lowest ADV sensor (0.10 mab), and z1 and z2 arethe heights of the lowest and highest ADVs, respec-tively. Figure 4 plots �U/log(z2/z1) as a function of u*.Although the data were noisy, they exhibited a roughly

FIG. 2. Time series data December 2003–January 2004: (a) combined daily river discharge from theU.S. Geological Survey’s Mattaponi and Pamunkey gauging stations, with 60-yr daily average;(b) depth-averaged current magnitude measured by ADCP at the channel site; (c) density stratificationmeasured between the highest and lowest CTDs at the channel site (dark line) and the shoal site (lightline); (d) 35-h low-pass-filtered near-bed currents from the lowest bin of the ADCP at the channel site(dark line) and ADP at the shoal site (light line); (e) along-channel wind speed measured at NOAA’sSewell’s Point station (positive values indicate up-estuary-directed wind).


linear relationship with an intercept of approximatelyzero (regression to all data resulted in slope of 1.1 andintercept of �0.0016). In the atmospheric literature, theratio of z�u/�z to u* (defined as �m) is often used toquantify the departure from the law of the wall (e.g.,Businger et al. 1971). To estimate �m, we fit the data inFig. 4 with a least squares regression. Flood and ebbdata were fit separately, the regression was forcedthrough the origin, and all data where the sign of thenear-bed velocity and u* were opposite were excluded.As a result, 85% of the data were used in the regres-sion. The regression analysis yielded values of �m of 1.0and 1.2 for flood and ebb. This result is largely consis-tent with the results of Trowbridge et al. (1999), as wellas with the observed tidal asymmetries in stratificationat the channel site.

To more clearly demonstrate that the near-bed shearis impacted by the presence of density stratification,Fig. 5 shows the time series of �m in comparison withthe density stratification observed between the twolowest CTDs at the channel location. Because the esti-mates of �m were noisy, the time series was smoothedusing a locally weighted scatterplot smoothing (lowess)filter (Cleveland 1979) with an 8% window, after ex-cluding all negative values. As a result, roughly 20% ofthe data were excluded before smoothing, mainly dur-ing periods of low current velocity. The smoothed timeseries of �m was compared with the time series of thevertical density stratification measured between thetwo adjacent CTDs that were closest in height to themean height of the two functioning ADVs deployed atthe channel site. Because of instrument noise and pe-riodic erroneous measurements, and consistent with thetime series measurements of �m, the time series ofstratification also was smoothed using a 8% “lowess”filter after excluding all values that reported unstabledensity stratification. Consistent with previous authors,

we found a positive correlation between the departurefrom the law of the wall and the local density stratifi-cation (r � 0.76).

It follows from the analysis of the boundary layershear at the channel site that the increased stratificationon ebb due to tidal straining resulted in an intensifica-tion of ebb velocities throughout the water column rela-tive to flood, given the same bottom stress (u*). At thechannel site, the depth-averaged current magnitude isgreater on ebb (�0.41 m s�1) than on flood (0.33 ms�1). However, there was no strong lateral asymmetryin current magnitudes at peak flood or peak ebb, evenduring spring tides, when strong lateral shear was ob-served. This indicates that the preferential down-estuary flow at the channel site was driven by lateralasymmetries in the duration of flood and ebb tides. Thedepth-averaged currents were ebb directed over 57% ofthe time at the channel site, as compared with roughly

FIG. 4. Test of law of the wall using ADV data at channel site.Error estimates for �m correspond to one standard error.

FIG. 3. Depth-averaged along-channel velocity profiles for (a) neap and (b) spring tidal conditionsmeasured by the ADCP at the channel site (solid line) and ADP at the shoal site (dashed line).


52% of the time at the shoal site. This pattern was evenmore pronounced during spring tidal conditions. Thislateral asymmetry in duration can be seen clearly byexamining an average tidal cycle representation of thedepth-averaged currents at both sites. The average tidalcycle values were calculated by dividing the entire timeseries into sections of length equal to the M2 tidal pe-riod and then averaging all values with the same rela-tive tidal phase. Figure 6a shows the average tidal cyclevalues for the depth-average currents at both sites forthe entire experiment. From this view, the lateral asym-metry in duration is clear. At the channel site the ebbtide was consistently longer than the ebb observed atthe shoal site. This occurs because the transition fromebb to flood occurred much later at the channel sitethan at the shoal site, while the transition from flood toebb occurred more simultaneously at both locations.

The lateral phase difference at the end of ebb ap-pears to be related to both tidal and lateral asymmetriesin turbulent mixing caused by changes in stratification.The average tidal cycle values of stratification at thechannel and shoal sites are plotted in Fig. 6b. Consis-tent with the classic view of tidal straining, stratificationat the channel site generally increased throughout theebb tide, and decreased during the flood. The tidal pat-tern is less clear at the shoal site, which is generally lessstratified than the channel consistent with the shallower

bathymetry adjacent to the shoal site. Figure 6c plotsthe lateral difference in stratification between the twosites as a function of tidal phase. The data for springand neap tidal conditions are plotted separately. Thelateral difference in stratification was greater duringspring tides than it was during neaps, even though theoverall level of stratification was greater at both loca-tions during neap conditions. This difference is particu-larly pronounced between the two locations at the endof ebb. This occurs because tidal straining generallycreates stratification throughout the ebb at the channelsite, while the mixing was sufficient to break down thestratification during peak ebb at the shoal location dur-ing spring tidal conditions. Given this lateral differencein stratification, the greatest lateral asymmetry in fric-tion is assumed to have occurred during the later half ofthe ebb tide during spring tidal conditions. As a result,at the end of ebb the barotropic pressure gradient(which is assumed to be the same at both locations) isbalanced to a greater extent by friction over the shoalthan at the channel. With acceleration contributingmore significantly to the dynamic balance at the chan-nel location than the shoal, the tide begins floodingearlier over the shoals than at the channel. This lateralphase difference is not observed at the end of floodbecause the opposite sense of tidal straining reduces thelateral asymmetry in turbulent mixing. The net result of

FIG. 5. Smoothed time series of �m estimated between the upper and lower ADVs (solid line) anddensity stratification measured between the two lowermost CTDs (dashed line), both at the channel site.


this pattern is to favor down-estuary residual flow overthe deeper channel regions with residual inflow overthe shoals.

A more direct examination of the impact that thetidal variations in stratification have on friction at thechannel site can be examined using the ADV data tocalculate a drag coefficient given as

CD ��b ��

u |u | , �6�

where u is the velocity measured by the ADV 1.1 mab,and �b is calculated from the ADV data following Eq.(1). Figure 6d shows the average tidal cycle values ofCD for the entire deployment at the channel site. Therewas a decrease in CD during the ebb tide, reaching aminimum value at the end of ebb. During the flood, thedrag coefficient generally increased, reaching a maxi-mum value near the end of flood. From this view, the

drag coefficient was only slightly greater on flood thanon ebb in the average sense. However, CD was overtwice as large at the end of flood (0.002) than the end ofebb (0.001). This pattern is consistent with the tidalvariations in stratification at the channel site and thegeneral pattern of stratification expected as a result oftidal straining. Although no direct estimates of stresswere available from the shoal site, we infer that thegreatest lateral difference in internal mixing occurred atthe end of ebb during spring tidal conditions, when thelateral difference in stratification was at its maximumvalue.

The impact of this inferred lateral difference in fric-tion can be seen more clearly in Fig. 7, where the depth-integrated barotropic pressure gradient scaled by thetidal frequency (�) is plotted against the depth-inte-grated transport for both sites. If the pressure gradientwere balanced solely by friction, the data in Fig. 7would fall roughly on a straight line (not exactly linear

FIG. 6. Average tidal cycle values of (a) depth-averaged current velocities (solid line the channel siteand dashed line the shoal site); (b) density stratification at the channel site (circles) and shoal site(triangles); (c) the lateral difference in stratification obtained by subtracting the stratification measuredat the shoal site from the channel site for spring (solid line) and neap (dashed line) conditions; (d) dragcoefficient calculated from ADV data at the channel site (vertical lines represent one standard error).


because of the quadratic frictional dependence), whilea frictionless balance between the pressure gradientand acceleration would appear as a circle. The mostnoticeable difference between the two sites occurredduring the second half of the ebb tide, when the trans-port values at the channel site fall to the left of those forthe shoal site. These data coincided with spring tidalconditions and mostly occurred during the period of21–28 December, when the strongest lateral shear wasobserved. The data in Fig. 7 suggest that there was lessfriction to balance the pressure gradient during thelater half of ebb at the channel site as compared withthe shoal site.

b. Momentum balance

Although there were no direct measurements ofstress at the shoal site, the stress at this location wasestimated from the dominant terms in the momentumbalance, as discussed above. Integration of Eq. (2) gavean estimate of the bed stress that could be comparedwith a quadratic drag representation of bed stress asexpressed by Eq. (6). In general, the momentum esti-mate of near-bed stress was well correlated with thequadratic bottom velocity (Figs. 8a and 8b). The corre-lation coefficient, r, was 0.70 and 0.82 for the channeland shoal sites, respectively. However, there appearedto be both lateral and tidal asymmetries in the relation-ship between bottom velocity and bottom stress. Theseasymmetries can be seen clearly in the estimates of thedrag coefficient. The drag coefficient was calculated us-ing a least squares regression between the quadraticvelocity and the momentum estimate of the bed stress.Flood and ebb data were treated separately and theregression was forced through the origin.

Consistent with ADV measurement, this analysissuggests that CD was consistently greater during the

flood than the ebb tide at the channel site (0.0019 and0.0011, respectively). At the shoal site, the tidal asym-metry was less pronounced and had the opposite sense,with a slightly lower CD on flood as compared with ebb(0.0023 and 0.0026, respectively). This asymmetry isconsistent with the overall greater degree of stratifica-tion observed at the channel site, as well as the patternsof tidal asymmetry in stratification that were observed.In addition to the tidal asymmetry in CD, there wereimportant lateral asymmetries as well. This was particu-larly evident during the ebb phase of the tide. While theestimate of CD during ebb was significantly greater atthe shoal site than the channel site, the difference wasmuch less pronounced during the flood phase of thetide. This pattern of lateral asymmetry is consistentwith the observations that showed the tide reversingfrom ebb to flood significantly later at the channel sitethan the shoal site, but currents reversing from flood toebb more simultaneously.

By using the momentum estimate of stress and theobserved vertical velocity shear, profiles of the verticaleddy viscosity were estimated following:

Az ��u�w��u��z

. �7�

Figures 8c–f show the estimates of the median verticalprofiles of eddy viscosity for the two sites. The datawere segregated by the phase of the tide as well asspring and neap tidal conditions. Depth-averaged val-ues of eddy viscosity derived from the momentum bal-ance are consistent with tidal and lateral asymmetries instratification. In general, values of eddy viscosity werehigher during spring tidal conditions and greater at theshoal site than at the channel site. At the channel site,where stratification was greater on average on the ebbphase of the tide, the depth-averaged eddy viscositywas consistently greater on flood as compared with ebb.

FIG. 7. Depth-integrated transport multiplied by the M2 tidal frequency (�) measured by the (a)ADCP at the channel site and (b) ADP at the shoal site plotted against the estimated depth-integratedbarotropic pressure gradient.


The shoal site generally demonstrated the opposite pat-tern, consistent with the observed stratification at thatlocation. Although tidal and lateral asymmetries wererelatively weak during neap conditions, large asymme-tries were present during spring tidal conditions. At thechannel site, the depth-averaged eddy viscosity wasnearly 2 times as large during the flood as during theebb, despite strong down-estuary flows during this pe-riod. No significant tidal asymmetry was present at theshoal site, which led to a strong lateral asymmetry dur-ing the ebb phase of the tide.

The estimates of eddy viscosity from the momentumbalance also were compared with the measured valuesfrom the ADVs at the channel site. Figure 9 shows acomparison between the time series of eddy viscositymeasured by the ADV 0.6 mab and the momentumestimate of eddy viscosity that corresponds to the low-est bin of the ADCP (1.4 mab). Because there wasnoise in both the direct measurement and momentumestimate of eddy viscosity, especially near periods oflow current magnitude, both time series were smoothedusing a lowess filter with an 8% data window. Before

FIG. 8. Results from the along-channel momentum balance: scatterplot of estimated bed stress fromthe momentum integral and the quadratic velocity measured 1.4 mab for the (a) channel and (b) shoalsites; median profiles of the eddy viscosity estimated for both the (c), (e) channel and (d), (f) shoal sitesduring (c), (d) neap and (e), (f) spring tidal conditions (solid line denotes flood and dashed line denotesebb).


the smoothing was performed, all negative values andall values that exceed the maximum expected value foran unstratified boundary layer (i.e., u*z) by more than20% were removed. As a result, roughly 80% of thedata were used for both the direct measurement andthe momentum estimate of eddy viscosity. The goodagreement between the magnitude and relatively highcorrelation between the time series of these two esti-mates of eddy viscosity provides confidence that theestimates of stress from the momentum integral aremeaningful.

c. 1D modeling results

To illustrate more clearly the creation of residualflows due to internal asymmetries in mixing, a numeri-cal experiment was conducted using the one-dimen-sional General Ocean Turbulence Model (GOTM).GOTM was run employing a k–epsilon turbulence clo-sure scheme as described by Burchard and Bolding(2001) with the stability parameters of Canuto et al.(2001). The model was used to simulate a simple tidalflow, forced by a tidally varying barotropic pressuregradient and a depth-invariant along-channel salinitygradient (�S/�x � 3 � 10�4 m�1). The model was runfor a depth of 10 m, roughly consistent with the mainchannel of the York River, as well as a depth of 5 m,generally consistent with the shoal region of the YorkRiver. A net seaward-directed barotropic pressure gra-dient was added so that the sum of the tidally averagedtransport for the two depths resulted in no net volumetransport. The net barotropic pressure gradient was im-posed assuming that the shoal area (5 m deep) and thechannel area (10 m deep) had equal cross-sectional ar-eas, a pattern generally consistent with the bathymetryof the York River.

The model predicted a time-varying pattern of strati-fication where stratification was generally created dur-ing ebb and destroyed during flood at both the channel

and shoal locations (Fig. 10a). The greatest stratifica-tion was predicted when velocity near the bed began toflood and the upper water column was still ebbing. Thisprocesses created significant stratification at the end ofebb that persisted into the flood, and as a result theflood tide was slightly more stratified on average thanthe ebb for the 10-m simulation. However, the time-varying pattern of stratification predicted by the modelexhibited the key feature that the end of ebb was sig-nificantly more stratified than the end of flood. Addi-tionally, the stratification predicted over the lower halfof the water column was over twice as large during theebb, on average, as compared with the flood for the10-m simulation. The overall degree of stratificationwas lower for the 5-m location relative to the deeper10-m location because, unlike at the deeper location,boundary layer mixing at the shallow location was ca-pable of reaching all the way to the surface. As a resultof the tidal asymmetry in stratification and internalmixing, the eddy viscosity was greater during the floodthan during ebb (Fig. 10b). While this asymmetry waspredicted at both the channel and shoal sites, it wasmore pronounced at the deeper location. Large lateralasymmetries in eddy viscosity in the upper portion ofthe water column were predicted, largely due to theinviscid conditions predicted in the upper water columnat the channel site.

The profiles of residual circulation predicted by themodel (Fig. 10c) are largely consistent with the obser-vations from the York River. Relatively strong down-estuary residual circulation was predicted over the up-per portion of the water column for the channel loca-tion, relative to the shoal site. The depth-averagedresidual circulation was down-estuary over the channeland up-estuary over the shoal despite the stronger in-tegrated baroclinic forcing associated with the deeperlocation. A net up-estuary residual was predicted overthe deepest portion of the 10-m location. Again, we

FIG. 9. Smoothed time series of the eddy viscosity at the channel site estimated from the momentumintegral 1.4 mab (solid line) and measured by ADVs 0.6 mab (dashed line).


assume a similar residual inflow occurred over thedeepest part of the channel in the 2003/04 experiment,but was not resolved because of the instrument loca-tions. Although the predicted ebb velocities at thechannel location are larger than both the flood veloci-ties at the channel and the ebb velocities on the shoal,much of the down-estuary residual at the channel site iscaused by longer relative duration of the ebb tide. Con-sistent with the observations, both the lateral and ver-tical phase differences were significantly larger duringthe transition from ebb to flood than during the tran-sition from flood to ebb. This is largely a consequenceof the reduced friction at the end of ebb caused by tidalstraining of the density field, which delays the onset ofthe ebb tide in the channel.

It is important to note that in order to have zero netflow through the estuarine cross sections over the tidalcycle it was necessary to impose a net ebb-directedbarotropic pressure gradient. This is consistent with theclassic analytical solution for density-driven estuarine

circulation (e.g., Officer 1976). However, the patternof tidal asymmetries in the vertical density stratifica-tion, which lead to increased vertical velocity shearduring the ebb tide, reduced the net barotropic pres-sure gradient significantly from the value predictedanalytically. In fact, the net ebb-directed barotropicpressure gradient necessary to ensure no net volumetransport through the cross section was nearly 70%smaller than that predicted based on the Officer (1976)solution (i.e., � /�x � �3h/8� /�x for case with negli-gible river flow).

The role that tidal asymmetries in mixing play in themodification of the net barotropic pressure gradientcan be clearly demonstrated with another simple nu-merical experiment. The model was next run imposingthe time-varying salinity field from the first model run,but artificially setting the baroclinic pressure gradientto zero. To highlight the role of asymmetries in mixing,no net barotropic pressure gradient was used. The re-sulting residual circulation profiles are presented in Fig.

FIG. 10. Results from 1D turbulence model runs: (a) time series of density stratification predicted bythe model for depths of 5 and 10 m (circles represent flood tide and crosses represent ebb tide); (b)median eddy viscosity profiles from model run with baroclinic and net ebb-directed barotropic pressuregradient (solid lines represent flood tide and dashed lines represent ebb tide); (c) along-channel residualvelocity profile for model run with baroclinic and net ebb-directed barotropic pressure gradient; (d)along-channel residual velocity profile for model run with no baroclinic or net barotropic pressuregradient, but forced with tidal variations in salinity field. In (c) and (d), the dark line represents 5-msimulation and the light line represents 10-m simulation.


10d. Clearly the time-varying pattern of stratificationleads to strong down-estuary residual flows, as no baro-clinic or net barotropic pressure gradient forcing wasused in this simulation. Of course in a real estuary, thenet barotropic pressure gradient would adjust to ac-count for this net volume transport in order to conservemass. The important role that tidal asymmetries in mix-ing have on the residual velocity is demonstrated in theprofiles shown in Fig. 10d, which have nearly the samevertical form as those presented in Fig. 10c despite nobaroclinic forcing. This result suggests that the two-layer structure of the residual estuarine circulation isdriven largely by the interaction of tidal asymmetry andthe barotropic tide. The along-channel density gradientis necessary to create the asymmetries in stratificationthat drive this process; however, the process itself islargely barotropic in nature.

4. Discussion

The results from the 1D model simulations, whichsuggest that the majority of the vertical shear in theresidual velocity profile is generated by tidal asymme-tries in mixing and not directly by the baroclinic pres-sure forcing, are generally consistent with the mecha-nism proposed by Jay and Musiak (1994). However, it isimportant to note that in both the observations andmodeling results, the near-bed intensification of thelandward velocity is not the result of convective mixingduring the flood, but is caused by the modification ofthe net barotropic pressure gradient. This occurs be-cause the ebb velocity profile is more sheared than theflood throughout the water column, which leads to anasymmetry in transport that must be balanced barotro-pically. While the barotropic pressure gradient re-sponds to the integrated transport through the entireestuarine cross section, the vertical velocity shear re-sponds locally to the degree of turbulent mixing. As aresult, distinct laterally sheared residual flows can de-velop in systems that exhibit both tidal and lateralasymmetries in mixing due to changes in the transversebathymetry. In systems with more uniform lateralbathymetry, it is likely that tidal asymmetries in strati-fication will act more or less uniformly across the estu-ary, leading to a vertically segregated two-layer re-sidual. The data from the York River suggest that theresidual lateral shear is modulated over the spring–neap transition. Significant lateral shear is observedduring spring conditions because this is when the great-est lateral difference in stratification was observed (Fig.6c).

While this mechanism is related to asymmetries instratification caused by tidal straining of the along-

channel density gradient, it is a barotropic processnonetheless. The baroclinic forcing will act to offset thelateral pattern of residual circulation described aboveby favoring inflow over the channel areas with outflowover the shoals (Wong 1994; Kasai et al. 2000; Valle-Levinson et al. 2003). The large impact of lateral asym-metries in mixing in this experiment was highlightedbecause the relatively similar depths at the two instru-mented locations negated significant difference in baro-clinic forcing. Had our instrumentation been deployedin the deepest portion of the channel, the impact of thelateral asymmetries in mixing may have been more dif-ficult to isolate. In fact, the large lateral phase differ-ence we observed at the end of ebb may have beenreduced or even reversed had we measured the flow inthe deepest portion of the channel where the impact ofthe baroclinic pressure gradient may have caused thetide to turn from ebb to flood sooner because of theincreased magnitude of the baroclinic pressure gradi-ent.

Separating the barotropic and baroclinic contribu-tions to residual circulation is difficult. Li et al. (1998)attempted to separate the barotropic and barocliniccomponents of the residual circulation in the JamesRiver and found the barotropic residual to exhibit out-flow over the channel with inflow on the shoals, withthe baroclinic residual displaying the opposite pattern.Although their analysis was kinematic and they did notidentify the physical mechanism driving the flow, theJames River is dynamically similar in many ways to theYork River, and the barotropic residual described by Liet al. (1998) is consistent with the mechanism describedin this paper. However, there are a number of otherbarotropic mechanisms that can generate a similar lat-eral distribution of residual flow. In systems with later-ally varying bathymetry, tidal rectification of stokesdrift (Li and O’Donnell 1997) or the nonlinear advec-tive terms in the momentum equation (Winant andGutierrez de Velasco 2003) can result in a laterallysheared barotropic residual. It is important to note thatthe mechanism detailed in this paper is fundamentallydifferent from either of these processes because it arisesfrom tidal asymmetries in internal mixing. However, itis analogous in some respects because it involves therectification of asymmetric transport driven by tidalnonlinearities. It is unlikely that these other nonlinearprocesses play an important role in the data discussedabove. The contribution of stokes drift in the YorkRiver can be estimated from the data and was only onthe order of 1 cm s�1, consistent with the results ofFriedrichs and Hamrick (1996) for the James River.Additionally, estimates of the nonlinear advectiveterms from the data are not consistent with pattern of


residual circulation described by Winant and Gutierrezde Velasco (2003) or Lerczak and Geyer (2004).

5. Conclusions

This study demonstrates the importance of asymme-tries in mixing associated with tidal straining to residualestuarine circulation. The presence of stratificationcauses the shear in the boundary layer to deviate fromthe expected law of the wall relationship. This deviationis strongly correlated with the strength of the observeddensity stratification. As a result, tidal asymmetries instratification lead to increased shear during the morestratified phase of the tide. When the expected patternof tidal straining is observed, the increased shear duringthe ebb tide favors down-estuary transport that must bebalanced barotropically in order to conserve mass overthe tidal cycle. When tidal asymmetries act uniformlyacross an estuary and there are no significant lateralgradients in turbulent mixing, this will result in a verti-cally segregated two-layered exchange flow. However,many drowned river valley estuaries have channel–shoal bathymetries that lead to lateral variability indensity stratification and turbulent mixing. As a result,tidal straining can significantly increase density stratifi-cation during the ebb tide over deeper areas, whileshallower areas remain relatively well mixed. This re-sults in large lateral variations in friction, causing thetide to turn from ebb to flood considerably earlier onthe shoals than in the channel. This increases the dura-tion of the ebb tide over the channel and favors down-estuary residual flow that is balanced by residual inflowover the shoals.

Acknowledgments. The authors thank Grace Cart-wright, Todd Nelson, Bob Gammisch, Frank Farmer,Sam Wilson, Wayne Reisner, and Don Wright for theirassistance with the preparation, deployment, and re-trieval of the instrumentation used in this experiment.Willy Reay kindly provided the YSI 6000s used in theexperiment. This manuscript benefited from the helpfulcomments of two anonymous reviewers. Support forthis research was provided by the National ScienceFoundation Division of Ocean Sciences Grant OCE-9984941.

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