Lateral Circulation in Well-Mixed and Stratified Estuarine Flows with Curvature

Transcription

1 APRIL 2009 N I D Z I E K O E T A L. 831 Lateral Circulation in Well-Mixed and Stratified Estuarine Flows with Curvature NICHOLAS J. NIDZIEKO, JAMES L. HENCH, AND STEPHEN G. MONISMITH Stanford University, Stanford, California (Manuscript received 14 March 2008, in final form 30 July 2008) ABSTRACT A field experiment was conducted to examine stratified and unstratified curvature-generated lateral circulation and momentum balances in an estuarine tidal channel. Conductivity, temperature, depth, and current profiler data were collected vertically and laterally across the channel at a sharp bend over a fortnightly period to measure the terms of the lateral momentum budget. Well-mixed conditions allow the development of classic two-layer helical flow around a bend. Stratification strengthens curvature-induced lateral circulation, but the development of a lateral baroclinic pressure gradient opposes the resultant motions. The spatial and temporal response of this baroclinic pressure gradient is different than centrifugal acceleration, producing a three-layer profile. As the baroclinic term becomes stronger (or as centrifugal acceleration disappears as the flow exits the bend), two-layer flow with the opposite direction from curvature occurs. In both stratified and well-mixed conditions, downstream adjustment of lateral circulation (nonlinear advective acceleration) is of leading order in the lateral momentum budget; the depth-averaged term adjusts the streamline direction, while vertical deviations from the depth average account for changes in lateral circulation. The asymmetry of forcing mechanisms on flood and ebb, because of variations in stratification and strength of tidal flow, can strongly affect net lateral transport and generation of residual currents in regions of curvature. 1. Introduction Lateral circulation can redistribute momentum and scalars across an estuarine channel faster than turbulent motions alone, thus playing a major role in how estuaries function. Lateral circulation can be generated in estuarine systems by a variety of mechanisms: curvature created by channel bends and headlands (Geyer 1993; Seim and Gregg 1997; Chant 2002); interaction between barotropic tidal currents and bathymetry (Li and O Donnell 1997; Valle-Levinson et al. 2000); differential advection of longitudinal density gradients (Lacy et al. 2003; Lerczak and Geyer 2004; Ralston and Stacey 2005); boundary mixing on a sloping bottom (Chen and Sanford 2008); and Coriolis forcing (Ott et al. 2002). Because few estuaries are perfectly straight or have regular bathymetry, a combination of the above mechanisms will exist; their relative importance will further vary temporally on fortnightly to event scales, as spring Corresponding author address: Nicholas Nidzieko, Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, CA nidzieko@stanford.edu neap variations and river discharge affect the strength of stratification. The effect of stratification on these lateral circulation mechanisms is not well documented, however, particularly in the case of lateral circulation forced by curvature. Stratification affects lateral circulation through both mean and turbulent momentum terms. In the mean flow, circulatory motions forced by curvature must do work to raise a density interface, thereby opposing lateral motions (Seim and Gregg 1997; Chant 2002). Alternatively, lateral circulation may be generated by lateral straining of the axial density gradient (Nunes and Simpson 1985). In both cases, turbulence is suppressed by stratification, so the momentum budget must differ for stratified and well-mixed flows. While the role of turbulence in the lateral momentum budget as it applies to riverine systems has been studied in some detail at the laboratory scale (Blanckaert and Graf 2004; Blanckaert and de Vriend 2005a,b), it has received surprisingly little attention in the estuarine literature. Estuarine field measurements made by Lacy and Monismith (2001) suggested that lateral turbulent shear stresses were important in the absence of density gradients, though these conclusions were based on residual measurements, not DOI: /2008JPO Ó 2009 American Meteorological Society

2 832 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 39 direct observations. The aforementioned laboratory work of Blanckaert and investigators demonstrated that significant feedback occurs between the momentum and turbulent kinetic energy budgets around a bend in an unstratified water column. If turbulent shear stresses are important to the mean momentum budget, how does stratification, with its concomitant reduction in turbulence, affect lateral circulation and mixing in an estuarine bend? Estuarine mixing can vary over a tidal cycle through strain-induced periodic stratification (SIPS), as the water column typically stratifies on ebb while flood tides are generally more homogenous and well-mixed (Simpson et al. 1990). Additionally, flood and ebb currents tend to be of differing strength (Friedrichs and Aubrey 1988). While this asymmetrical tidal straining and mixing is well-documented in estuarine systems of simplified geometry (Nepf and Geyer 1996; Stacey et al. 1999b; Simpson et al. 2005), it is possible that these mechanisms could be further enhanced (or negated) by how stratification interacts with channel bends and other bathymetric features. For example, a sharp channel bend may induce overturning of the water column on one phase of the tide (Seim and Gregg 1997), yet lateral gradients generated on a different phase of the tide might inhibit lateral circulation and mixing. While these asymmetries are discussed briefly by Chant and Wilson (1997), further study and observations of these interactions are warranted. Herein we present observations to clarify how stratification and tidal asymmetry influence lateral circulation in a curved tidal channel. Velocity and density measurements were made in a curving region of a shallow, narrow estuary to assess the effect of stratification on curvature-induced lateral circulation, turbulence, and mixing. In section 2 we present lateral momentum equations to provide a theoretical framework. In section 3 we describe the field measurements, which were designed to allow the calculation of all the terms in the lateral momentum budget. In section 4 we quantify the different modes of lateral circulation, describing the momentum budget responsible for those motions in section 5. Finally, in section 6, we discuss the effect of curvature on estuarine transport and mixing. Conclusions are discussed in section Lateral momentum equation The use of a curvilinear polar coordinate system aligned with the depth-averaged streamwise velocity (see Fig. 1a) simplifies investigation of the lateral momentum budget around a bend, as the coordinate rotation from a Cartesian system introduces a term for the centrifugal acceleration induced by curvature (cf. Rouse and Appel 1959). The Reynolds averaged form of the lateral momentum equation, under hydrostatic and Boussinesq approximations and the neglect of viscous stress terms, is u n t 5 u2 s u n u s R s s 1 p r 0 n fu s hu9 n u9 z i z, (1) where u i5s, n, z, is the velocity in the streamwise, lateral (streamnormal), and vertical directions, respectively, with z positive upward, referenced to a flat free surface in the streamnormal direction. Angle brackets denote an ensemble time average (the coordinate rotation is applied to each ensemble), time-average fluctuations are denoted with a prime, and overbars will denote a depth average; angle brackets have been left off terms with nonfluctuating quantities for clarity. The radius of curvature in the streamwise direction, R s, is positive for flows moving through a bend to the right. The Coriolis parameter f is s 21 at 36.88N latitude. Equation (1) is suitable for describing the main part of the flow field away from channel sidewalls for flows where the depth H is much smaller than the width B and the radius of curvature (Kalkwijk and Booij 1986). The term on the left-hand side of Eq. (1) is the time rate of change of the lateral velocity at any given vertical level z. The first term on the right-hand side is centrifugal acceleration, the leading order term in curvature-generated lateral circulation. The second term accounts for nonlocal acceleration; the other advective terms are omitted on the basis that u s u n, u z. The fourth term is Coriolis acceleration; rotation can also drive lateral circulation via vertical shear in the streamwise velocity. The last term of Eq. (1) is the turbulent shear stress in the streamnormal direction. The lateral pressure gradient p/ n decomposes into barotropic and baroclinic contributions, p n 5 p n barotropic 5 r o g h n 1 g 1 p n baroclinic ð 0 z r 0 (z 0 ) n dz0, where g is acceleration due to gravity and h is the free surface displacement from z 5 0; by separating density, r 5 r 0 1 r9(x, y, z, t), into a constant reference density, r kg m 23 ; and the resulting perturbations, p9(x, y, z, t). The integrand is from z to 0 (instead of h) as h z, which neglects the free-surface contribution to the baroclinic pressure gradient.

3 APRIL 2009 N I D Z I E K O E T A L. 833 FIG. 1. (a) Location and bathymetry of the field experiment. Labeled contours are meters below MLLW; shaded contours are meters above MLLW. Instrument locations are shown with symbols: T/CTD ADCP arrays ( d ), ADV/pressure gauges (m). The T/CTD ADCP arrays are, from left to right, west, center, and east. The streamwise coordinate system is indicated, with ebb toward the west and flood toward the east. The radii of curvature of the flood and ebb bends are shown by dotted lines. (b) Location of the experiment site (boxed) relative to Elkhorn Slough and LOBO moorings (r). (c) Cartoon representation of instrumentation setup. (d) Cross section of bathymetry across mooring array. In a water column of constant density, curvatureinduced lateral circulation is driven by a net vertical imbalance between depth-dependent centrifugal acceleration, Coriolis acceleration, friction, and the depthindependent barotropic pressure gradient (Rozovskiı 1957). In estuaries, given that streamwise velocity shear varies with stratification and tide phase (Geyer and Farmer 1989), it may be important to include depth-dependent lateral baroclinic pressure gradients in this balance. To examine this vertical imbalance in the lateral momentum equation, we subtract the depth average of Eq. (1), 0 5 u2 s u n u s R s s g h n g r 0 ð 0 z r9(z9) dz9 f u n s t b,n rh, (2) from itself. Note that, by definition, u n [ 0. The depth average of the turbulent stress is defined as the normal component of the Reynolds stress at the bed, t b,n /r 5 hu 0 n u0 z i bed, and H 5 h 1 h is the total depth of the water column. Subtracting (2) from (1) yields the driving terms responsible for lateral circulation at any z:

4 834 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 39 u n 5 {z} t Nonstationarity u 2 s u2 s R s fflfflffl{zfflfflffl} Centrifugal acceleration 0 g ð 0 r 0 z r9(z9) dz9 n u n u s s u u n s s fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Nonlocal adjustment ð 0 z r9(z9) dz9 n fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Baroclinic pressure gradient(bcpg) 1 C A f (u s u s ) hu9 n u9 z i t b, n. (3) fflfflfflfflfflffl{zfflfflfflfflfflffl} z rh Coriolis fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Friction Notice that the barotropic pressure gradient does not appear in (3). The terms in Eqs. (2) and (3) can be evaluated for the center of the channel using our dataset; the exceptions to this are the nonlocal adjustment (advective acceleration) terms, u s ( u n / s) in (2) and u s ( u n / s) u s ( u n / s) in (3), which we estimate as residuals of the other terms in each equation. We will use Eq. (3) to examine the effect of stratification on curvature-induced lateral circulation and how this is manifested in bends with tidal asymmetries by addressing three questions. First, how does the lateral velocity structure respond to increased vertical shear as stratification increases? Second, how do the relative sizes of the momentum budget terms change over a tidal cycle and in response to stratification? An inviscid budget assumes that stratification suppresses turbulent motions and that centrifugal (or Coriolis) accelerations are balanced by the lateral baroclinic pressure gradient in Eq. (3) (e.g., Chant and Wilson 1997). In contrast, a frictional budget assumes that the Reynolds stress plays the dominant role in balancing the driving terms (e.g., Kalkwijk and Booij 1986). We will examine how the momentum budget relates to the observed velocity structure. Third, how do mixing and transport caused by curvature vary with these different circulation patterns? 3. Data collection a. Study site Three mooring arrays were deployed in the Seal Bend region of the main channel of Elkhorn Slough, California, from 6 to 18 April 2006 (Fig. 1). Elkhorn Slough is a shallow salt marsh estuary (Caffrey et al. 2002) with steep banks in the experiment region. The channel throughout the region is on average 100 m wide and 5 m deep at mean lower-low water (MLLW); it is flanked by narrow intertidal mudflats and wide upper intertidal salt marsh tracts. Mixed semidiurnal tides with a maximum range of 2.4 m generate peak depth-average velocities of 0.75 and 0.65 m s 21 on ebb and flood, respectively, in the center measurement site. Fortnightly spring neap variations have a noticeable effect on the magnitude of the currents; spring ebb currents can be almost twice as strong as neap ebbs. The experiment site was located in a relatively straight reach of channel between two counterturning bends. The bend to the east is a 608 turn to the right, relative to ebb, and has a 375-m radius of curvature; the bend to the west is a right-hand turn on flood of 1008 and R s m. Based on these curvatures and current speeds, the leading order terms are (0.7 m s 21 ) 2 (300 m) 21 O(10 23 m s 22 ). Salinity structure in Elkhorn Slough is made complicated by freshwater sources at both the head and mouth of the estuary (Fig. 1b). Pulses of freshwater enter the main channel of the slough on the incoming tide from the Old Salinas River (OSR) channel, so large portions of the flood tend to be stratified. The ebb tides are usually well-mixed, as Elkhorn Slough is shallow, strongly ebbdominant, and has weaker freshwater inflow from Carneros Creek at its head. This tendency to stratify on floods and mix on ebbs produces a periodic stratification that can be the opposite of the traditional SIPS mechanism. It provides ideal conditions, however, for examining lateral circulation through a range of stratification conditions. b. Experimental setup Each mooring array consisted of a bottom-mounted Teledyne-RD Instruments 1.2-MHz Workhorse Monitor acoustic Doppler current profiler (ADCP) and a set of temperature and conductivity temperature depth (T/CTD) loggers set at 0.5-m increments over the top 3.5 m of the water column (Fig. 1c). The ADCPs were deployed in mode 12, sampling 16 subpings at 40-ms intervals for an effective data-recording rate of 1 Hz. They recorded beam velocities and were configured with 0.25-m vertical bins; the first bin was centered 1 m above the bed. The mode 12 configuration reduced the instrument noise floor such that lateral Reynolds stresses could be resolved (Nidzieko et al. 2006). At each mooring, T/CTD loggers were mounted on an aluminum pole suspended from a surface float, with a counterweight mounted at the end of the pole to minimize tilt caused by drag in the current. The use of the rigid

5 APRIL 2009 N I D Z I E K O E T A L. 835 TABLE 1. Depth, equipment, and sampling interval (in paretheses) of the instruments suspended in the water column at the three T/CTD ADCP arrays (d in Fig. 1). CTDs are shown in bold. Instruments were calibrated in a bath prior to, and following, the field deployment. SBE 5 Seabird Electronics, Bellevue, Washington; YSI 5 YSI Incorporated, Yellow Springs, Ohio; OS 5 Ocean Sensors, Inc., San Diego, California; RBR 5 RBR Ltd., Ottawa, Ontario, Canada. Depth West Center East 20.5 m SBE-39 (10 s) SBE-39 (10 s) SBE-39 (10 s) 21.0 m SBE-16plus (15 s) RBR TG-410 (300 s) SBE-16plus (15 s) 21.5 m SBE-39 (10 s) SBE-39 (10 s) SBE-39 (10 s) 22.0 m SBE-39 (10 s) OS200 (300 s) SBE-39 (10 s) 22.5 m YSI 600XLM (60 s) SBE-39 (10 s) YSI 6600 (60 s) 23.0 m SBE-39 (10 s) SBE-37 (15 s) SBE-39 (10 s) 23.5 m SBE-39 (10 s) SBE-39 (10 s) SBE-39 (10 s) pole maintained vertical instrument spacing throughout the tidal cycle. Sampling rates varied because of instrumentation power and memory constraints, as outlined in Table 1. The T/CTD instrumentation was concentrated in the upper half of the water column; deployment of a vertical thermistor array on the L01 (Fig. 1b) mooring prior to the experiment (6 January March 2006) showed that the strongest stratification was confined to the top 2 m (N. Nidzieko, 2006, unpublished manuscript). The upper 3.5 m corresponds to 40% 70% of the water column from high to low tide, respectively. While an extrapolation scheme was required to obtain estimates of the lower water column, this provided better resolution where the density gradient was largest and parameterization of maximum stratification is most relevant. The three main mooring arrays were complemented by Nortek 6-MHz vector acoustic Doppler velocimeters (ADVs) and Seabird Electronics SBE-26 pressure loggers located midway up each bank of the channel in ;3 m of water (Figs. 1c,d). While the location of the ADVs is included here for completeness, these data will be reported elsewhere, as will the velocity observations at the outer ADCPs. The SBE-26 tide gauges were collocated with each ADV and recorded integrated water levels every 60 s. To provide spatial and temporal context to the lateral array measurements, data were collected from the Land/ Ocean Biogeochemical Observatory (LOBO; data available online at moorings L01 and L04, located 1 km seaward and 2 km landward, respectively, of the study site. Each LOBO mooring had CTDs positioned 0.6 and 2.6 m deep, and recorded hourly. Additionally, meteorological data were collected from two nearby California Irrigation Management Information System (CIMIS) stations 19 Castroville and 129 Pajaro, 5 km to the south and 10 km to the north, respectively. c. Evaluation of terms in the momentum equation Measurements of Reynolds stresses throughout the water column were estimated with the variance method (Lohrmann et al. 1990; Stacey et al. 1999a). A 10-min ensemble average was used for the Reynolds stress averaging period. To facilitate comparison with the rest of the dataset, all other measurements were averaged to 10-min values. For each 10-min ensemble, a rotation angle f was found such that depth-mean horizontal earth coordinate velocities ( u, y) could be transformed to streamwise and streamnormal coordinates, for example, u n 5 y cos f u sin f 5 0. A 30-min low-pass filter was applied to the T/CTD measurements to reduce the effect of measurement noise and different sampling rates. Velocity, Reynolds stress, and density profiles were referenced to a normalized water column depth z/h relative to the center mooring for computation of terms in the momentum budget. Velocity was extrapolated to the bed assuming a no-slip condition and to the surface assuming a no-shear condition. For calculation of the turbulent shear stresses, smoother Reynolds stress gradients were obtained by applying a smoothing spline filter (MATLAB Spline Toolbox, smoothing parameter ) to each profile of the streamwise and lateral Reynolds stress estimates. The spline filter was also used to extrapolate the Reynolds stresses to the surface and the bed. Vertical density structure was extrapolated from the measurement locations using a combination of temperature and Reynolds stress profiles. First, measurements of salinity (S) and temperature (T) made with the CTDs on the mooring arrays were used to fit a timevarying T S relationship. Empirical orthogonal functions (EOFs; Emery and Thomson 2001) were determined to quantify the vertical structure of the ratio between temperature and salinity (T S modes); because there were only two conductivity temperature (CT) sensors on each rigid pole, only two vertical modes can be calculated for the T S relationship. The first mode is constant vertically; its amplitude changes in response to longitudinal gradients in the T S ratio created by different water mass properties in the upper and lower estuary. The second mode varies linearly with depth and

6 836 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 39 represents the degree of stratification. Next, a more detailed vertical structure was determined through EOF analysis of the temperature record at each mooring. The first two temperature modes (of seven) accounted for 91.9% and 6.6% of the variance in the temperature record at the center mooring (98.5% total). Though the second mode of both the T S and temperature datasets fell below the threshold of a random noise test for EOF significance (Overland and Preisendorfer 1982), retaining the variance this mode explains was critical to producing an accurate estimate. Assuming that the physics controlling the temperature modes and T S modes is the same, the higher-resolution T shapes were used to extrapolate T S vertically, allowing a reasonable extrapolation of the salinity field over the upper water column. The resultant density profiles were linearly extrapolated to the bed or the top of the bottom boundary layer, where it was assumed that r/ z 5 0; density profiles extrapolated in the bottom boundary layer are thus vertical. The depth of the bottom boundary layer was fit to the zero crossing of linear profiles of alongchannel Reynolds stress (Stacey and Ralston 2005). Following the method outlined in Geyer et al. (2000), the barotropic pressure gradient was computed by removing the baroclinic contribution from each SBE-26 pressure record prior to taking the difference of the barotropic water column heights. The undisturbed water column depth h was determined at each pressure gauge by assuming a flat water surface across the channel at slack tide. Taking the mean of the water depths collected at times of slack water (when velocity at both the east and west ADVs was near zero) referenced the two records to a flat datum. With density removed and the two records referenced to each other, the free-surface gradient was calculated by differencing the two records. A discussion of uncertainty in the measurements and calculations is presented in the appendix. 4. Observations a. Stratification conditions and circulation Stratification varied throughout the experiment due to a series of winter storms that brought significant rainfall prior to the start of the experiment (Fig. 2). Storm runoff created persistent stratification, reported as squared buoyancy frequency, N 2 5 gr 1 0 ( r/ z), for the first half of the experiment, a period of predominantly weaker neap tides. During the second half of the experiment, stratification weakened as freshwater inflow decreased, particularly from Carneros Creek; the stronger spring ebbs were well-mixed, while freshwater from OSR continued to stratify flood tides. Strong FIG. 2. Hydrologic conditions in Elkhorn Slough, April The shaded regions show the period of the field experiment. (a) Mean tide height in the main channel set in meters above MLLW. (b) Freshwater input from rainfall at CIMIS station 129 (black bars); freshwater discharge from OSR channel (broken line) and Carneros Creek (solid line). (c) Wind vectors from CIMIS station 19. Vectors indicate direction to which wind is blowing, with magnitude relative to the vertical axis. (d) Stratification (as N 2 ) at LOBO moorings L04 and L01. southerly winds between 10 and 13 April (Fig. 2c) may have contributed to the reduction in stratification. Stratification had a pronounced effect on velocity profiles (Fig. 3). Streamwise velocities were strongly vertically sheared throughout the first half of the experiment on both flood and ebb. This shear was less significant during the second half of the experiment, particularly on ebb, when water column density was more vertically uniform and streamwise profiles were roughly logarithmic. During this period, the water column restratified just prior to lower-low slack water and stratification persisted throughout the flood after

7 NIDZIEKO ET AL. FIG. 3. (a) Density anomaly, (b) streamwise velocity, (c) lateral velocity, (d) streamwise Reynolds stress, and (e) lateral Reynolds stress contours at the center mooring. The dark line represents the free surface. The dashed line in (a) indicates the bottom of the T/CTD array. Gray shading in (b) represents negative velocities generated as the tide turns. The gray line in (d) shows the bottom boundary layer inferred from the streamwise Reynolds stress. Tide is indicated with the banded bar along horizontal axis: flood is gray, ebb is black. Arrows indicate profiles in Fig. 11. APRIL

8 838 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 39 lower-low water, the (weak) ebb after lower-high water, and the first part of the flood after higher-low water; thus, the profiles of the floods and weak ebb were generally more strongly vertically sheared than the strong ebb. Lateral circulation varied with stratification as well. Helical flow profiles are readily apparent in the lateral velocity record (Fig. 3c); the surface layer flowed toward the outer bend and the bottom layer toward the inner bend. Two-layer flows with this circulation sense only occurred when there was no stratification (i.e., during the latter half of the experiment). In contrast, different vertical profiles of lateral circulation were observed on flood and ebb when the water column was stratified. Stratified ebbs were three-layered, with an outwarddirected surface layer that was thinner and had a return flow in the middle of the water column; the flow at the bed was often directed toward the outer bend. These three-layer profiles adjusted with the varying water column depth throughout the tidal cycle. Stratified floods produced two-layer flow with a rotational sense opposite of that expected from curvature (i.e., outward at the bed and inward at the surface), consistent with a baroclinic flow driven by the observed lateral density gradient. Some of these density-driven flows had a thin surface layer directed toward the outer bend. b. Reynolds stress Reynolds stresses (Figs. 3d,e) were small throughout the first half of the experiment, when the water column was stratified. The bottom boundary layer did not extend significantly into the water column, as turbulent motions were suppressed by strong stratification. As stratification weakened throughout the experiment, the magnitude of the lateral Reynolds stresses increased at both the bed and in the midwater column in response to stronger bottom shear stress. The lateral Reynolds stresses act to rectify the momentum imbalance between the depth-variable centrifugal acceleration and the depthindependent barotropic pressure gradient, and are the manifestation of lateral shearing of the bottom boundary layer along a curving streamline (cf. Schlichting and Gersten 2000), so curvature-induced lateral stresses are directed toward the inside of the bend at the bottom and toward the outside in the upper water column (Fig. 4). The lateral stresses in the upper water column were much stronger and more clearly defined on ebb. On flood, the bottom boundary layer height (as a fraction of the total depth) was more variable (see Fig. 3d) and so the flood hodograph in Fig. 4a does not have the same shear as the ebb observations, despite the similarity in the bed stress ratios. Consider a steady ( u n / t, u n / s 5 0) frictional momentum balance; integration from the bed to any level z: FIG. 4. Scatterplot of streamwise and lateral Reynolds stresses at the center mooring for (a) flood and (b) ebb. Dashed line and equation are result of linear regression of the bed stress measurement (gray points). A hodograph of mean observations for Fr9. 1 is shown as a heavy line; vertical increments of z/h along the hodograph are shown by thin lines. yields ð z ð z h h u 2 s u 2 s R s dz 1 ð z h g h ð z dz 5 n h z hu9 n u9 z i dz, (4a) dz 1 g (z 1 h) h R s n hu9 n u9 z i z5 h 5 hu9 n u9 z i z. (4b) Equation (4b) shows how midwater column Reynolds stresses arise from centrifugally induced shear, as any vertical variations in streamwise velocity cannot be accounted for by the barotropic pressure gradient (which is depth independent) and therefore must be balanced by friction. If the integration in (4a) is performed over the full depth, frictional loss is due to lateral bed stress alone: u2 s H R s 1 gh h n 5 hu9 n u9 z i z5 h 5 u 2 n, (4c) where u n is the lateral analog to shear velocity u *. The lateral bed stress is an order of magnitude smaller than

9 APRIL 2009 N I D Z I E K O E T A L. 839 either centrifugal acceleration or the barotropic pressure gradient, which provides the basis for a zero-order (steady, fully developed) scaling of the free-surface gradient given by Rozovskiı (1957): h n a u2 s gr s. (5) The coefficient a nc d k 2 accounts for vertical variation in the streamwise velocity; C d is the drag coefficient, k is von Kármán s constant, and n is an O(1) coefficient determined from laboratory studies to be ;0.8. Substituting (5) into (4c) and using C d u 2 s u2 suggests how lateral and streamwise shear velocities should be related to the bend geometry u 2 n 5 n u2 H R s k 2. (6) Regression between observations of u 2 H/(R sk 2 ) and u 2 n gives n (not shown, R ). Though it neglects any stratification dynamics, the implications of (6) are that deeper water and smaller R s increase shear of the boundary layer and the potential for strong lateral circulation. We observed lateral bed stress to be roughly 30% of the streamwise bed stress (Fig. 4), with the ratio nearly the same for floods (20.33, R ) and ebbs (20.29, R ). Based on geometry, u 2 n /u2 should be (1/225 m)/(1/375 m) times larger for the flood bend, though the observations do not support this. While scaling of the ensemble temporal resolution [ u n / t ; 0.1 m s 21 (600 s) 21 ; m s 22 )] suggests the assumption of ensemble steadiness is valid, the assumption of a fully developed flow is invalid, as the location of the mooring array was not equidistant from both bends. Consequently, the discrepancy between observations and predictions for fully developed flows highlights the significant spatial adjustment that must occur in a short distance. The importance of this downstream adjustment to the momentum balance will be discussed further in the following section. c. Modes of circulation The lateral velocity structure can be quantified by determining what percentage of the vertical profile is captured by each of a series of empirical orthogonal functions. The first three modes (Fig. 5) account for 92% of the variance in u n throughout the experiment. Positive mode 1 amplitudes can be loosely interpreted as curvature-driven helical two-layer flows and negative mode 1 amplitudes as baroclinic two-layer circulation with the opposite rotation from curvature. Increased amplitudes of the second and third modes during the first half of the experiment reflect weaker lateral velocities, but also stronger shear. We binned first and second mode amplitudes according to tide (ebb or flood) and stratification (based on the bulk internal Froude number, described below) in Fig. 5e. The influence of stratification is clearest in the first mode, as well-mixed conditions tended to have larger positive values, reflecting the helical flow profiles. In contrast, stratified floods had negative mode 1 amplitudes, reflecting a baroclinic influence. The strength of the mean flow relative to stratification can be assessed via the bulk internal Froude number Fr9 5 u s N max H, (7) which compares the inertia force to the buoyancy force in the water column. In general, increased Fr9 (Fig. 6) reflected Ð greater lateral velocity magnitude ju n j H h j u njdz, decreased shear magnitude u n z 5 u n dz, and increased vertical eddy viscosity 1 H Ð 0 h z e V 5 1 H ð 0 h hu9 su9 z i dz, (8) u s / z though not as a rule. However, the use of the maximum water column value of N provides a criterion for assessing when the water column can be mixed completely. A uniform water column permits full, unimpeded vertical excursion of any fluid element, and thus there is no buoyancy restriction to the development of a lateral circulation cell for Fr9. 1. As a result, strong helical flow was only observed when Fr9. 1 (Fig. 7), evidence that well-mixed conditions are a requirement for full water column two-layer lateral circulation generated by curvature. Seim and Gregg (1997) suggest a scaling of Fr9 as an indication of when curvature-generated lateral currents should be strong enough to overturn the density field. Based on an inviscid lateral balance, they argue gfr9 2 B R s # 1 (9) as a condition for baroclinic stability against lateral overturning; B is the channel width and the parameter g 5 (u 2 s u2 s )/u2 s is a measure of the vertical shear, which we have evaluated as a depth-averaged RMS value for each 10-min ensemble. Their scaling indicates that there were few instances when the lateral circulation cell should have been strong enough to overturn the density interface (Fig. 6a). This discrepancy with the scaling described by Eq. (7) suggests that turbulent motions were responsible for vertically mixing the water

10 840 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 39 FIG. 5. Principal components analysis of lateral velocity profiles at the center mooring, (a) (c) amplitudes of the first three modes and the percentage of variance accounted for by each mode. Tide is indicated with the banded bar along horizontal axes: flood is gray, ebb is black. (d) First three modes, (e) median and interquartile range (gray bars) of first and second mode amplitudes, binned according to tide and stratification; ebb, left-pointing triangles; flood, right-pointing triangles; Fr9. 1, filled triangles; Fr9 # 1, open triangles. column, rather than an overturning produced by lateral circulation. We surmise that in deeper channels, where the effects of friction are diminished, Eq. (9) would more accurately describe the limitation to helical flow. An alternative scaling to Eq. (7) is the horizontal Richardson number (Monismith et al. 2002), Ri x 5 gbgh2, (10) u 2 which expresses the ratio of buoyancy flux to turbulent production, scaled according to the geometry of the flow. Though these two terms are similar (Ri x 1/Fr9 2 ), the buoyancy flux in Ri x is scaled as the vertical straining of a longitudinal salinity gradient G and so Ri x. 1 is an effective predictor for determining when tidal straining can stratify the water column. The turbulence parameterization is based on well-mixed open channel flow, however, and thus its predictive capability for the transition from stratified to unstratified flow is diminished [i.e., Ri x, 1 does not guarantee the breakdown of stratification (Stacey et al. 2001)]. Consequently, (10) was not as good an indicator of helical flow as (7), as the condition Fr9. 1 is roughly equivalent to Ri x, 1. The variations in lateral velocity structure as a function of tide and stratification (Fig. 8) can be summarized as three general patterns along a baroclinic barotropic continuum: density-dominated circulation with a circulation sense predominantly opposite of curvatureinduced circulation that occurred primarily on stratified floods, three-layer circulation with strong vertical shear that occurred on stratified ebbs, and two-layer helical flow that occurred when stratification became negligible. In the following section we focus on how the lateral momentum budget varies between these different circulation modes. 5. Lateral momentum budget a. Depth-independent terms The depth-independent terms in the lateral momentum Eq. (2) are shown in Fig. 9. Centrifugal acceleration

11 APRIL 2009 N I D Z I E K O E T A L. 841 FIG. 7. Scatterplot of mode 1 amplitude and Fr9. Small gray points are observations. Large black circles are mean mode 1 amplitudes binned by Fr9 (50 observations per bin). Range bars indicate 6 one standard deviation. FIG. 6. (a) Bulk internal Froude number [Eq. (7), black line] and internal Froude number scaled by bend geometry [Eq. (9), gray line], (b) depth-averaged lateral velocity magnitude, (c) depthaveraged lateral shear magnitude, and (d) depth-averaged eddy viscosity. Tide is indicated with the banded bar along horizontal axis: flood is gray, ebb is black. was always positive, as both the flood and ebb bends had the same streamwise direction of curvature. The barotropic pressure gradient increased with the transition from neap to spring tides, but appeared to balance centrifugal acceleration only on ebb. On flood, the consistent direction of the barotropic pressure gradient suggests that the channel had begun to turn left, into the next curve, at the location of the mooring array; there is experimental evidence that the free surface set-down along the inner bend precedes the region of curvature (Kalkwijk and de Vriend 1980). If this were the case, the curvature term at this location would be negative, with R s m; using the flood bend radius of curvature (R s m) misapplies local centrifugal acceleration in our calculations. The observed velocity and salinity redistribution, however, were consistent with centrifugal acceleration driven by the flood bend. The depth-averaged baroclinic pressure gradient (BCPG) was much larger during the first half of the experiment, because of freshwater runoff, and had the same order of magnitude as centrifugal acceleration at times. The depth-averaged BCPG was the same sign as centrifugal acceleration, reflecting baroclinic compensation of the barotropic pressure gradient. The BCPG is not necessarily opposite the barotropic pressure gradient, however, because the time required for baroclinic adjustment is long compared to the free-surface setup (i.e., surface waves travel across the channel much faster than internal waves). The instances when the BCPG opposed centrifugal acceleration were likely due to differential streamwise advection, rather than lateral velocity redistribution of the density field, as this only occurred early in the experiment when longitudinal salinity gradients were strongest. The depth-averaged lateral friction term was an order of magnitude smaller than the other terms, even in this energetic, shallow system. The lateral bed stresses were at most O( ); division by H makes this term even smaller. Additionally, the depth-averaged Coriolis term was negligible relative to the sharp bends [Rossby number, Ro 5 u s /f R s O(10 2 ); i.e., dominated by curvature], though in many channels of more gentle (or no) curvature this term can be of leading order (Ott et al. 2002; Lerczak and Geyer 2004). It is clear from Fig. 9 that the pressure gradients, centrifugal acceleration, and bottom friction were not in balance, leading us to conclude that advective accelerations due to the downstream adjustment of the lateral velocity must be important. In defining a streamwise coordinate system, we have necessarily prescribed a radius of curvature for both bends via geometric estimates; the actual radius of curvature at the mooring

12 842 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 39 FIG. 8. Average profiles of stratification (filled gray), streamwise velocity (solid black line on lhs), and lateral velocity (solid black line on rhs), sorted by stratification [(a),(b) well mixed, Fr9. 1 vs (c),(d) stratified, Fr9 # 1] and tide [(a),(c) ebb vs (b),(d) flood]. Only observations where u s. 0.2 are shown. Dashed lines show logarithmic streamwise velocity based on averaged observations and corresponding analytic solution for curvature-generated lateral circulation given by Kalkwijk and Booij (1986). location is more difficult to discern. Expansion of u n in the nonlocal adjustment term from (2) (substituting a/ s 5 21/R s, cosa 5 s/ x, and sina 5 s/ y), u n u s s 5 u s s (y cos f) (u sin f) s 5 u s (u cos f1y sin f) f s s y 1 u s R s x s u s u s y s 5 u2 s 1 cos a y sin f u s s, (11) cancels the centrifugal acceleration term, leaving Cartesian advective terms. While the second and third terms on the rhs of (11) are not necessarily negligible, since we cannot estimate their magnitude, we are forced to interpret the depth-averaged advective term as a downstream adjustment of the streamline direction. When u s ( u n / s) is positive, the streamline adjusts to the left downstream (canceling curvature); negative u s ( u n / s) to the right (increasing curvature). Since our residual estimate relies on our choice of R s, we have attempted to bound uncertainty in interpreting the dynamics by calculating the downstream adjustment term with and without u 2 s /R s; the former represents the maximum centrifugal acceleration in the curve, whereas u 2 s /R s 5 0 describes the condition exiting the bend where the radius of curvature becomes infinite. With centrifugal acceleration included, advective acceleration was positive, indicating that observed pressure gradients were insufficient to balance the prescribed radius of curvature and that downstream adjustment of the streamline should be to the left. In contrast, with no centrifugal acceleration term, the advective term was negative on ebb and positive on flood. The positive flood value is consistent with the streamline straightening out following the flood bend and beginning to curve to the left entering the subsequent bend; on ebb, the negative value suggests that the flow is still in the curve and centrifugal acceleration is part of the momentum balance. The true local advective term is somewhere between these two extremes; the potential for the advective term to be leading order, however, suggests that this term should not be neglected in regions with abrupt changes in curvature. b. Driving terms The depth-variable centrifugal acceleration, BCPG, and friction terms from Eq. (3) are shown in Fig. 10.

13 APRIL 2009 N I D Z I E K O E T A L. 843 FIG. 9. Depth-averaged lateral momentum Eq. (2) terms (dark gray). (a) Centrifugal acceleration, (b) barotropic pressure gradient, and (c) baroclinic pressure gradient. The nonlocal adjustment term calculated without centrifugal acceleration is shown (f) as a black line. Light gray shading indicates standard error calculations from the appendix. Tide is indicated with the banded bar along horizontal axis in each panel: flood is gray, ebb is black. Note that the vertical axis scale on the (d) friction and (e) Coriolis terms is an order of magnitude smaller than the other terms. While the terms described in this section are most properly vertical deviations from the depth average, we will refer to the terms generally and omit this distinction to avoid repetition of the phrase. Representative profiles of measurements and the resulting terms are shown in Fig. 11. The profiles in Fig. 11 are marked with arrows in Fig. 10. The first row in Fig. 11 (1255 PST 12 April) is a stratified ebb with weak curvature forcing and a dominant BCPG that produced lateral circulation predominantly with the opposite sense of rotation from curvature. The second row (1915 PST 12 April) is a strongly sheared, stratified flood with three-layer lateral circulation. The third row in Fig. 11 (0045 PST 13 April) is a (relatively) well-mixed ebb and has a classic twolayer helical flow. The time derivative and Coriolis terms in Eq. (3) are not shown, as they were both an order of magnitude smaller than the other terms. Curvature always accelerated fluid at the bed toward the inside of the bend, as bottom drag requires slower moving fluid there. When stratification enhanced streamwise velocity shear such that the velocity maximum was midwater column, centrifugal acceleration at the

14 JOURNAL OF PHYSICAL OCEANOGRAPHY FIG. 10. (a) (e) Contours of the vertical deviation from the depth-average terms in the lateral momentum Eq. (3). (f) Lateral velocity contours. Tide is indicated with the banded bar along horizontal axis in each panel: flood is gray, ebb is black. Arrows indicate profiles in Fig. 11. The baroclinic pressure gradient does not extend all the way to the bed as the center mooring is deeper than the side moorings. 844 VOLUME 39

15 APRIL 2009 N I D Z I E K O E T A L. 845 FIG. 11. (rows) Vertical profiles of measurements (dark gray line, upper axis) and their respective driving terms (black line, lower axis, with light gray shading indicating error bounds) at three selected time points (see arrows in Fig. 10). (columns) (1) streamwise velocity, normalized by depth-averaged streamwise velocity, and centrifugal acceleration term; (2) s t at inner (solid) and outer (dashed) mooring, relative to streamwise direction, and lateral baroclinic pressure gradient term; (3) hu9 n u9 z i and friction term; and (4) lateral velocity and nonlocal adjustment term calculated with (solid) and without (dashed) centrifugal acceleration. surface was negative, toward the inside of the bend, while acceleration at midwater column was positive toward the outer bend, driving a three-layer profile that occurred primarily on stratified flood tides. When the water column was well-mixed, the upper water column was faster than the depth average and the resulting acceleration was positive and toward the outer bank. Some of the strongest two-layered lateral circulation profiles during the later ebbs (13 17 April) had relatively weak acceleration toward the outer bend. In these instances, lateral redistribution of streamwise momentum throughout the bend had affected the streamwise velocity profile, shifting it toward the outer bend and lower into the water column (de Vriend 1981; Blanckaert and Graf 2004). The depth-dependent BCPG almost always opposed centrifugal acceleration, consistent with an inviscid balance, except when centrifugal acceleration reversed sign near the surface in strongly sheared flows (e.g., 1915 PST 12 April). In several cases the adjustment of the BCPG following slack water accelerated fluid in the same direction as curvature until the density field was reorganized in opposition to centrifugal acceleration. In the absence of lateral differences in stratification, the vertical profile of the BCPG term would be linear, dependent solely on z; the deviations are due to lateral variations in the depth and magnitude of stratification. Although stratification increases streamwise velocity shear, this shear is squared in the centrifugal acceleration term, and so vertical profiles of the BCPG and centrifugal

16 846 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 39 acceleration were rarely mirrors of one another; the result was a more complex lateral velocity structure that was generally three-layered. When the BCPG was stronger than the curvature term (e.g., 1255 PST 12 April), the lateral velocity profile had the opposite sense from that driven by curvature, consistent with a densitydriven flow, though in many cases the influence of curvature was retained near the surface. The friction term became important as stratification weakened. The friction term at the bed was always positive (toward the outer bend), in opposition to centrifugal acceleration and lateral velocity. Although the inner bend was stratified in the 0045 PST 13 April example (Fig. 11), the outer bend was not. The stratification could have been generated by the mudflats and marsh to the northeast (there was precipitation on 12 April); this stratification broke down as it was advected across the channel, however, and mixed out by the strong turbulent flow. Similar turbulent controls on lateral advection of salinity were observed by Lacy et al. (2003). As in the depth-independent discussion, we calculated nonlocal adjustment as the residual of the other terms in Eq. (3) with and without centrifugal acceleration. In both cases the nonlocal adjustment is leading order. While the depth-averaged advective term is responsible for steering the streamline in the axial direction, the deviating advective term represents spindown or spinup of lateral circulation. Scaling u s ( u n / s) u s ( u n / s) as u s (u n /L) ; O(10 3 ), where L represents the spindown length required for curvature-induced lateral velocity to go to zero, yields L ;100 m. This agrees well with analytic formulas for spindown length (Kalkwijk and Booij 1986), and supports our assumption that the downstream adjustment is the largest advective term. Our conceptual view of these dynamics is summarized in Fig. 12. Lateral circulation is driven by centrifugal acceleration even in the presence of stratification. When the water column is stratified, ebbs are strongly sheared, leading to intensified lateral circulation by stronger centrifugal acceleration in Eq. (3). This circulation is simultaneously modified by the BCPG, however, which becomes dominant and drives a baroclinic flow with the opposite sense of rotation relative to that expected by curvature as the flow exits the bend and centrifugal acceleration disappears. The lag required for the BCPG to spin up can lead to a seiching downstream of the region of curvature (Chant and Wilson 1997; Lacy and Monismith 2001). Because stratification can produce a vertical BCPG that does not necessarily mirror centrifugal acceleration, three-layer circulation can be generated as the relative magnitudes of the terms vary over the depth. The three-layer profile is particularly evident on stratified floods, as the streamwise velocity is reduced at the surface. When stratification is sufficiently weak such that turbulence dominates, the streamwise velocity profile is nearly logarithmic; the result is classic helical flow around a bend. In all cases, the downstream adjustment of lateral velocity and the streamline direction plays a significant role in the lateral momentum budget. 6. Discussion We have shown in the previous sections how stratification and tide phase generate different lateral circulation profiles as the flow interacts with the bend. We discuss here the impact of these circulation patterns on dispersion and residual transport in the estuary. The efficiency of longitudinal dispersion depends on the homogenization of the cross section (Fischer et al. 1979); we are interested here in how curvature affects T nz, the time required for complete lateral and vertical mixing, and how T nz compares to the advective travel time through the bend, as a shorter bend would be less effective at augmenting dispersion than a longer bend. These time scales provide qualitative insight to of the impact of curvature. The augmentation of longitudinal dispersion due to channel curvature can be considered by comparing the ratio of time scales for lateral and vertical processes in a curve T nz to a canonical estimate for transverse dispersivity in straight open-channel flow T straight nz 5 B 2 (0.6Hu ) 1 (Fischer et al. 1979). For T nz /T straight nz, 1, cross-sectional dispersion is faster (provided that vertical mixing is faster than the lateral processes), and curvature increases longitudinal dispersion. A second time-scale ratio relates cross-sectional processes to the streamwise advective time T s 5 L/u s through a bend of length L. For T nz /T s, 1, complete cross-sectional transport or mixing occurs in the bend and curvature increases longitudinal dispersion. For T nz /T s. 1, the bend may still augment cross-sectional mixing, but the travel time through the bend is too short to allow complete mixing. Combining these time-scale ratios, we can assess the impact of different cross-sectional mixing mechanisms on longitudinal dispersion as T 5 T nz T nz. (12) T straight T nz s Note that because of the complexities of longitudinal dispersion processes, there is not necessarily a one-toone relationship between T and changes in longitudinal dispersion. Cross-sectional mixing involves lateral and vertical mechanisms. The relevant lateral time scales are set