Creation of residual flows in a partially

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 106, NO. C8, PAGES 17,013-17,037, AUGUST 15, 2001 Creation of residual flows in a partially estuary stratified Mark T. Stacey Department of Civil and Environmental Engineering, University of California, Berkeley, California Jon R. Burau United States Geological Survey, Sacramento, California. Stephen G. Monismith Department of Civil and Environmental Engineering, Stanford University, Stanford, California Abstract. The creation of residual flows in estuaries is examined using acoustic Doppler current profiler data sets from northern San Francisco Bay. The data sets are analyzed using principal component analysis to examine the temporal variability of the flows which create the residual circulation. t is seen that in this periodically and partially stratified estuary the residual flows are created through a series of pulses with strong variability at the 24-hour timescale, through the interaction of shear, stratification and mixing. This interaction is captured through the use of a dimensionless number, the horizontal Richardso number (Ri ), which is developed to examine the local balance between the stratifying and destratifying forces at the tidal timescale. t is seen that Ri is a valuable parameter in predicting the onset of the residual-creating events, with a threshold value of 3 on ebb tides. This critical value is argued to be a threshold, above which the stratification and shear flow create a feedback effect, each further intensifying the other. This feedback results in a highly variable exchange flow which creates the estuarine residual in intermittent pulses rather than as a steady flow. Although typically attributed to baroclinic forcing, an argument is made that these pulses of residual-creating exchange flow could be created by barotropic forcing in the presence of variable stratification which is asymmetric between flood and ebb tides. This result poses a great challenge for turbulence modeling, as the timing and magnitude of stratification and shear must be correctly simulated on the tidal timescale in order to reproduce the effects seen in the data sets presented. 1. ntroduction The health of an estuarine ecosystem is strongly dependent upon its residual circulation. While tidal currents may create large fluxes of nutrients and contaminants over the short timescale, it is the net transport, which governs the net exchange of material along the axis of the estuary. As a result, the net (or residual) circulation is critical to the management of the estuarine system. The structure of residual flows involves both verti- cal variations, in the form of the familiar estuarine exchange flows, and horizontal variations, usually created by bathymetry Ll and O'Donnell, 1997]. The vertical structure of estuarine residual flows, the emphasis of Copyright 2001 by the American Geophysical Union. Paper number 2000JC /01/2000JC ,013 this paper, is well-known, with flow seaward at the surface and landward at depth [Pritchard, 1967, Hansen and Rattray, 1965]. We will examine the mechanisms which create this exchange flow and, in particular, examine the temporal variation of these flows. We begin, however, with an overview of the historical analysis of residual flows in estuaries Baroclinic Residual Circulation Many estuaries are characterized by a longitudinal salinity gradient extending from the freshwate river to the sea. The density gradient associated with this salinity gradient drives an estuarine (or baroclinic or gravitational) circulation characterized by an exchange flow, with seaward flow at the surface and landward flow at depth. Historically, analyses of estuarine circulation has focused on the baroclinically_forced residual being con-

2 17,014 STACEY ET AL.: ESTUARNE RESDUAL CREATON stant over tidal timescales. The important mechanisms where/ is the saline expansivity, vt is a turbulent difin creating this flow were thought to be riverflow (which fusivity based on the tidal velocities, and we have ascreates a longitudinal salinity gradient) and mixing, sumed that the advective terms and the horizontal viswhich was parameterized by a time-averaged eddy viscosity. On the basis of this balance, Hansen and Ratcous terms are small. Assuming an unstratified water column, we define scales for the salinity gradient tray [1966] proposed a classification system for estuaries (Os/Ox F) and the turbulent diffusivity (vt u,h based on two parameters: where u, is the friction velocity based on the tidal velocities and H is the depth), the scales of each term w, (from left to right) are the following: us/t,/ gfh, and = w u, us/h. For u,t/h >> i (i.e., considering timescales (2) long relative to the vertical mixing time, trnix H/u,), this becomes a pressure-mixing balance, and the resultwhere ul represents the freshwater flow per cross-sectional ing velocity scale for baroclinic flow is area, ut is the root-mean-square tidal current speed, Ap gh2f is the density difference between river and ocean water, us U, and H is the depth of the estuary. From these parame- (6) ters, stratification and baroclinic circulation were ana- Although this expression defines the physical scaling of lyzed using a stratification-circulation diagram [Hansen the baroclinic flow, the governing equation (equation and Rattray, 1966]. (5)) must be solved with appropriate boundary condi- The classification of estuaries was also discussed by tions to define the vertical profile. Prandle [1991] de- Fischer [1976], but he found that the use of an estuarine fined this profile shape, and found that the coefficient Richardson number Riis was superior to the classificaon the scale factor given in equation (6) reached a maxtion scheme of Hansen and Rattray: imum value of 0.03 at the bed. The factor of u, in the denominator represents turbulent mixing. As mixing p3 Ap ttf p gzt, (3) where all quantities are as above. Again, this parameter was used to characterize the level of stratification and baroclinic circulation in an estuary. n this analysis, stratification would affect the residuals through a reduction in tidal mixing. As this mixing is only a parameterization of the mean effects, stratification would only have an effect if it persisted over residual time scales. Therefore intermittent stratifica- tion at the tidal timescale would have a reduced or no effect on the residual flows as predicted by these theories. Nunes-Vaz et al. [1989] defined a dimensionless number, Ric =?RizH/1, (4) increases, the friction velocity (u,) increases and the baroclinic circulation decreases; as mixing decreases, the baroclinic flow would increase. Therefore we would expect, as argued by Nunes-Vaz et al. [1989] that the baroclinic flow should vary at timescales consistent with variations in turbulent mixing, i.e., at the tidal and spring-neap timescales. The modulation of the baroclinic circulation by turbulent mixing was also investigated by Linden and Simpson [1988]. n the laboratory they "pulsed" the turbulent mixing, using a square wave signal and concluded that baroclinic circulation increases dramatically during the nonturbulent periods. Comparison with field measurements from Liverpool Bay, Spencer Gulf, and the Columbia River, showed encouraging results and the authors suggested that the use of tidally averaged quantities may be misleading due to modulation of the baroclinic motions at timescales shorter than the tidal where 7 is a constant and Ap/l is the bulk estuarine density gradient (l characterizes the length of the period. estuary). The appearance of ut 3 in the denominator (see equation (3)) led the authors to hypothesize a baroclinic circulation that increases as tidal mixing decreases. They concluded that baroclinic flow would be most active during slack tides on the tidal timescale and during neap tides on the spring-neap timescale, 1.2. Vertical Density Stratification when turbulent mixing is at a minimum. As a result, the authors hypothesized that the residual flow would the tidal velocity (through be created through a series of pulses at the tidal and spring-neap timescales. To be explicit, we can define the scale of the baroclinically induced flow (us) by considering its governing n each of the above expressions, the level of turbulent mixing is an important parameter. n the parameterizations of Hansen and Rattray, Fischer, and Nunes-Vaz et al., the turbulent mixing was parameterized based on the friction velocity) and the depth, which is justified on scaling grounds. However, this scaling does not account for the effects of vertical density stratification, which is likely to inhibit turbulent mixing in a partially stratified estuary [Stacey et equation [Prandle, 1991]: al., 1999; Peters, 1997]. f the water column is stratified, the gravitational circulation, as scaled by equation 0 Oug (6) will increase. The balance between tidally produced at - Os + (., ), (5) turbulence and the stratifying effects of sheared advec-

3 _ STACEY ET AL' ESTUARNE RESDUAL CREATON 17,015 tion is therefore critical to determining the magnitude and timing of residual flows. 0 Os ot() OU Os _ 02 Os )' As discussed by Monismith et al. [1996] the balance Os between turbulent mixing and stratification defines an where represents the vertical stratification, K8 is additional dimensionless number, which will parame- the vertical turbulent diffusivity for salt, and U is the terize the onset of stratification. n order for turbulent (sheared) mean velocity profile. Assuming that advecmixing to be able to overcome the stabilizing effects of vertical density variation, the production of turbulence must be large enough to overcome the buoyancy flux created by the straining of the density field. As outtion and mixing come into balance to define the degree of stratification, scaling the second term on the left-hand side to be comparable to the right-hand side results in (Os/Ox F, K8 u.h)' lined by Simpson et al. [1990], this straining provides AUHF the buoyancy flux into the water column (starting from AS, U. unstratified condition) producing strain-induced periodic stratification (SPS). The discussion of this sec- where we have assumed that the vertical turbulent (10) diftion will develop the dimensionless group that defines fusivity, K scales as u.h, which is appropriate in unthis threshold for the development of stratification and stratified channel flow. t should be noted that stratthe associated reduction in mixing. ification would reduce K, resulting in an increde in Defining the friction velocity to be u. (also the scale the scale for AS and providing positive feedback to the of turbulent velocity fluctuations) and the depth to be stratification process. Substituting this expression into H, the appropriate scale for the shear production is equations (7) and (8) results in the t eshold condition a/h. Similarly, we can define the vertical buoy- for the onset of stratification: ancy flux to be B u. HN 2, where N 2 is the buoyancy frequency squared (and we have assumed w' u. and gp'/po HN '). n this discussion, we assume that g FH=AU ui shear production dominates the production of turbu- We now define the scale for the she, AU, considlence and the buoyancy flux is a sink for turbulent kiering separately the b oclinic and barotropic forcing. netic energy (i.e., the buoyancy flux is stabilizing, as on First of all, if we assume that the velocity shear (AU) ebb tides), so that the critical condition occurs when is defined by the b oclinic velocity scale, we can use they come into balance (i.e., P B). On flood tides, the expression kom equation (6) to define the she. a destabilizing buoyancy flux occurs due to the reversal Substituting equation (6) into equation (11) and rearof the SPS mechanism [Burchard and Baumert, 1998]. ranging results in the bal ce: The analysis presented here is still valid for this case, but the comparison of buoyancy flux and production will represent the importance of the production of tur- r ' ) -nil ( =) bulence by convective instabilities relative to the shear production. The role of these instabilities in the creation of residual flows will be discussed in section 1.3. where we have defined the horizontal Richardson number' Equating production and the buoyancy flux, we define the condition 3 u. u. HN. (7) H We can scale the vertical buoyancy frequency squared AS ;v (s) g FH = 2 U. which should define the t eshold between stratified and unstratified periods. Therefore, for values of Ri greater than a t eshold value (of order 1), the water column should stratify, and for subcritical values the water column should remain unstratified (i.e., tidal mixing is su cient to break-down the stratifying effects of the b oclinic flows). f we now consider the stratification to be due to ad- where AS is the vertical salinity difference and is the vection by the barotropic flows on ebb tides (flood tides saline expansivity. discussed below), we can define the velocity scale to be To determine the top-bottom salinity difference (AS), we consider the the stratifying effects of sheared horizontal advection, by either the baroclinic or, on ebb AU u., (14) tides, the sheared barotropically driven flow (i.e. a log- where u. is the friction velocity. n this case, the balarithmic boundary layer), in the presence of a longitudi- ance between stratifying and destratifying forces pronal salinity gradient. n this analysis we start with the evolution equation for the stratification (taking O/Oz of the salt transport equation and assuming that vertical advection and horizontal diffusion are negligible): duces the same dimensionless group as is expressed in equation (12), the horizontal Richardso number. t should be noted that the comparison between stratifying and destratifying forces captured in Ri is equiva-

4 17,016 STACEY ET AL.: ESTUARNE RESDUAL CREATON lent to that analyzed by Simpson et al. [1990] in developing a criteria for SPS, but here has been reformulated into a dimensionless group, Ri. For values of Ri less than the threshold value (of order 1), tidal mixing is sui cient to eliminate the stratification being produced by advection and the water column should remain well mixed. n both the barotropic and baroclinic advection cases, this stratification threshold would also indicate a point at which vertical mixing would be significantly reduced. When Ri is greater than its threshold value, stratification will increase and mixing will decrease. This decrease in mixing will allow additional shear to develop and the baroclinic flow will increase in magnitude. As a result, we would expect the residual flows to "turn on", or pulse (following Nunes-Vaz et al. [1989]) during times when the water column stratifies, or when Ri exceeds some critical value. The horizontal Richard- son number, Ri should therefore also be important in predicting the onset of residual flow creating events. As a result of this analysis, estuarine residual-creating flows would be expected to vary on the tidal and tidalmonthly timescales. Variations in turbulent mixing and stratification at those timescales allow for the develop- ment of pulses of baroclinic flow at frequencies that may not be evident with tidally filtered data Barotropic Forcing and Exchange Flows n many coastal estuaries, including San Francisco Bay, the barotropic tidal flows are an order of magnitude or more larger than the baroclinically driven exchange flows described in the previous section. f these tidal currents are symmetric between ebb and flood tides and uniform along the axis of the estuary, however, they will contribute little to the net transport of material. On the other hand, if there is a persistent asymmetry in the velocity profile induced by barotropic forcing, then this asymmetry could, in fact, result in significant net transport along the axis of the estuary. Extensive data collected on the Columbia River Es- tuary in the early 1980s has helped to examine how residual circulation appears [Jay and Smith 1990]. Jay and Smith's [1990] analysis of data from the Columbia River Estuary suggests that the baroclinic forcing creates residual flow by establishing a tidal asymmetry. The balance between the baroclinic forcing and the turbulent mixing, therefore, must be considered on a tidal and tidal-monthly (spring-neap) timescale instead of in a long-term averaged sense. n particular, Jay and Smith [1990] argued that circulation in the estuary is strongly tied to the spring-neap cycle due to variations in tidal mixing. This asymmetry is created through the superposition of a baroclinic velocity profile on the barotropic tides, and the mechanism by which the residual circulation is created is argued to be the baroclinic pressure gradient. Another likely source of an ebb-flood asymmetry is through the action of strain-induced periodic stratifica- tion (SPS, Simpson et al. 1990) and the asymmetry in turbulent mixing that is induced. During the ebb tides the tidal currents stratify the water column through the straining of the density field in the presence of a longitudinal density gradient. The flood tides reverse this process and destratify the water column. As a result, the largest stratification from this process occurs near the end of the ebb tide. The ability of this tidally periodic stratification to produce estuarine residual circulation, and the associated estuarine turbidity maximum, was first suggested by Jay and Musiak [1994, 1996]; we will outline this process briefly here. When the water column stratifies, turbulent mixing is significantly reduced [Stacey et al., 1999; Peters, 1997] and the upper portions of the water column feel a reduced effect of bottom friction. The simplified momentum balance describing the barotropic tidal flow is given by OU _ Op + o (u, ou). (15) Ot- p Ox Oz When the water column stratifies, the second term on the right-hand side is reduced through the effect of stratification on ut, the turbulent eddy viscosity. As a result, more shear develops in a stratified water column than in an unstratified one. Therefore asymmetries in turbulent mixing between ebb tides and flood tides will create asymmetries in the shear of the velocity profile, which will result in a net flow profile which is nonzero. n a partially stratified estuary, two mechanisms act to create asymmetries in mixing, thus creating a residual flow: both are tied to the presence of a longitudinal salinity gradient. First, on ebb tides the advection of the salinity distribution by the sheared tidal velocity profile provides a stabilizing buoyancy flux, increasing the stratification [Simpson et al., 1990]. Alternatively, on flood tides, advection by sheared tidal velocities creates a destabilizing buoyancy flux, which destratifies the water column and, in many cases, elevates the mixing above that which would be created in a neutrally-stable water column. Each of these mechanisms act to create a residual circulation that is similar to the profile created by baroclinic forcing, which has traditionally been credited with the creation of estuarine residual flows. To be clear, on ebb tides, the stabilizing buoyancy flux decreases the mixing of momentum and allows the surface waters to accelerate, creating a down-estuary residual flow at the surface. On the flood tides the destabilizing buoyancy flux drives convective mixing, which mixes high momentum fluid downward, intensifying the nearbed currents and creating an up-estuary residual flow at the bed. Together, the asymmetry in mixing created by advection of the longitudinal salinity gradient, and the associated buoyancy fluxes, results in a residual flow profile which has the same structure as the traditional estuarine circulation: down-estuary flow at the surface and up-estuary flow at the bed. The process of estuarine residual creation was examined by Burchard and Baumert [1998] using a numerical

5 The residual flow created by these mixing asymmetries is qualitatively equivalent to the baroclinic circulation. The magnitude of this exchange flow, however, will not depend on the average magnitude of the turbulent mixing, as in the case of the baroclinic flow, but rather on the tidal asymmetry of the turbulent mixing. n order for this residual flow mechanism to be active, we require that the water column be characterized by significant asymmetry in the level of mixing between flood and ebb tides. The mechanism for this asymmetry that we describe here is based on the buoyancy flux created by the straining of the density field by the sheared tidal velocity profile. n order for this process to create a residual flow, we require that the buoyancy flux (sta- bilizing on ebb, destabilizing on flood) be significant as compared to the shear production of turbulence. This relative importance of the SPS-induced buoyancy flux and the tidally-driven shear production is captured by the horizontal Richardson number, Ri as discussed in section 1.2. As a result, when Ri exceeds a critical value we would expect an asymmetry to develop in the level of mixing and in the shear of the velocity profile. Note that the horizontal Richardson number does not distinguish between flood and ebb tides (i.e., the sign on the Buoyancy flux is not included in the formulation). This convention is appropriate, however, when considering the creation of residual flows, since the advective buoyancy flux (which creates SPS) acts to increase the residual flow on both ebb and flood. The discussion thus far indicates that asymmetric turbulent mixing, a reduction in mixing on the ebb tides and an increase on the flood tides, creates an exchange flow residual which is consistent with the estuarine residual circulation. t is important to note that if mixing is also reduced on the flood tides, say through persistent stratification, then a sheared velocity profile will be created during the flood tide phase which would counteract the exchange residual described previously. This "negative" estuarine circulation emphasizes the importance of the asymmetry in turbulent mixing in creating estuarine circulation through the barotropic forcing and will be evident in some of the data discussed in section 3. STACEY ET AL.: ESTUARNE RESDUAL CREATON 17,017 simulation. n their analysis, Burchard and Baumert di Summary of Exchange Flow Dynamics vided the barotropic residual creating mechanism into The two mechanisms which can create estuarine extwo parts, describing the ebb-tide mechanism (increaschange flows discussed here result from different physiing stratification) as a turbulent mixing asymmetry and cal forcing: the baroclinic flow is driven by the pressure the flood-tide mechanism (increased mixing due to congradient due to a longitudinal density gradient while the vection) as a tidal velocity asymmetry. We argue here barotropic exchange flow is driven by the tidal pressure that these two mechanisms are the result of the same gradient, but relies on asymmetries in the level of turbuprocess, the buoyancy flux created by advection along lent mixing to produce asymmetric velocity profiles on the density gradient, and we will group them into the ebb and flood tides. n the presence of a longitudinal category of "barotropic residual creation." By turning density gradient, the differential advection (or strainon and off different residual creating mechanisms, Buring) of the density field creates a stabilizing force on chard and Baumert were able to establish the fact that ebb tides, decreasing turbulent mixing and increasing barotropic residual creation, in both of its forms, could the velocity shear, and a destabilizing force on flood be an important and, at times, dominant source of estides, increasing turbulent mixing and bringing high tuarine residual flows. momentum fluid near the bed. n this scenario, both phases of the tide will reinforce the estuarine exchange flow, with the net flow being down-estuary at the surface and up-estuary at the bed. For both the baroclinic and the barotropic production mechanisms, however, the balance between the stratifying and destratifying forces is captured through the horizontal Richardson number, Ri, which should predict the onset of these exchange flows. That is, once Ri reaches a threshold value, the exchange flow mechanisms described in this section should "turn on" and pulses of residual-creating flow should result. The exact value of this threshold will be a function of the mixing efficiency of the turbulent (or the flux Richardso number, Rf = B/P), the height of the bottom boundary layer, the friction velocity and the top-bottom velocity difference, but would be expected to be of order 1. t is difficult to predict this value precisely a priori, because it would require estimates of each of these quantities, but the onset of residual events will be readily apparent in the velocity data analyzed below, allowing us to estimate the threshold value from the data. As we will see, the horizontal Richardson number exhibits significant variability at the tidal and sub-tidal timescales. Clearly, the friction velocity (u,) will vary at the tidal frequency; we will also see variability at the diurnal timescale in the longitudinal salinity gradient (F). Both of these quantities are included in Ri ; as a result, we would expect the horizontal Richardson number to vary significantly at the tidal timescale, which will permit the creation of residual flows to occur at the tidal timescale. t is important to note at this point an important limitation of the analysis presented above in the development of the horizontal Richardson number. The scaling of the turbulent mixing parameters used the entire depth as the vertical lengthscale and, as a result, neglected the effects of stratification on the mixing. Further, the analysis of section 1.2 scaled the horizontal salinity gradient as a constant, F, which was independent of depth. Once the water column stratifies, and certainly as the vertical density structure is mixed, it is likely that there will be vertical variability in the salinity gradient. n view of both of these limitations, the

6 17,018 STACEY ET AL.: ESTUARNE RESDUAL CREATON horizontal Richardson number will be predictive of the onset of stratification events, which should also repto estimate along-channel (using SCW and SCE) and across-channel (using SCW and SCS) gradients. Each resent periods of asymmetric turbulent mixing. Once station consisted of an ADCP (RD nstruments, 1200 stratification is established, however, the parameteri- khz NB-ADCP) and a pair of CTD sensors (Ocean Senzation of the turbulence, and perhaps the longitudinal sors, OS200), one deployed at the bed and one at midsalinity gradient, used in defining Riz would need to be column. The instruments were deployed on November adjusted. As such, the horizontal Richardson number is 15 and recovered on January 31, but the data returned likely to be a useful parameter in predicting the onset was reduced due to instrument failures. During the first of residual flow events, but is not expected to predict month of the study, however, all three bottom CTDs, the shut down of these events. as well as the midcolumn CTD at SCE, were functioning, and we will focus our analysis on that period. All ADCPs returned complete data sets from the duration 2. Overview of Observations of their deployment. The instruments were configured to collect data ev- The remainder of this paper will discuss the creation ery 10 min. The ADCPs recorded an ensemble every of residual flows in partially stratified estuaries using di- 10 rain which consisted of an average of 100 measurerect observations at two locations in the northern reach ments. The resulting theoretical error per ensemble was of San Francisco Bay, California. n this section, the 0.9 cm/s for the NB-ADCPs and 0.5 cm/s for the BBstudy sites and data sets are described, which will lead ADCP (RD nstruments 1993). The OS200s also colinto the analysis described in the next section. lected a data point every 10 rain, with each data point 2.1. Suisun Cutoff being an average of 99 measurements of conductivity, temperature, and pressure. n this configuration the Suisun Cutoff is a straight, narrow channel in Suisun OS200s had temperature and salinity errors of ø C Bay, in the northern reach of San Francisco Bay (Fig- and 0.01 ms/cm, respectively. These measurements are ure la). t is immediately downstream of the Sacramento- converted to salinity [Hill ½t al., 1986], which is recorded San Joaquin Delta and is influenced by significant fresh- in practical salinity units (psu) (with an error of 0.01 water forcing throughout the year, with peaks in the psu). Finally, the Data Sonde recorded a temperature late winter and spring. Suisun Cutoff is about onehalf kilometer wide and 2.5 kilometers long. Over its and conductivity measurement every 30 min, with each reading being an average of 10 readings, giving ensemwestern two thirds the Cutoff is nearly prismatic (Fig- ble errors of 0.06øC and 0.19 psu. ure lb), with the deepest part of the channel (depth n Suisun Cutoff, the direction of the channel was de- 10 m) offset slightly to the north of the channel center- termined by the bathymetry, with the SCW and SCS line. At either end of the Cutoff the bathymetry shoals stations rotated by 7 ø (channel direction 97 ø east of to 3-5 m depth, and some bathymetric effects are seen north) and the SCE station rotated by 25 ø (channel diearly in flood tides, where water enters the Cutoff from rection 115 ø east of north). The choice of along-channel two sources (Grizzly Bay on the north and the Moth- direction based on the bathymetry was necessary in orball Fleet Channel on the south). Taken as a whole, der to align the SCW station with the gradients calcuhowever, the Cutoff provides a channel with relatively lated using the three stations, although the data does ilsimple geometry and a level of freshwater forcing suffi- lustrate that this was not necessarily the primary axis of cient to produce a partially and periodically stratified the tidal ellipse. The along-channel and across-channel water column. near surface velocities for the SCW station are shown The equipment used in the Suisun Cutoff experiment in Figure 2(Figures 2a and 2b). The tidal conditions consisted of khz narrowband acoustic Doppler are characteristic of this portion of San Francisco Bay, current profilers (NB-ADCPs from RD nstruments), showing spring-neap variations and maximum surface khz broadband ADCP (BB-ADCP, also from current magnitudes of 100 cm/s. The freshwater in- RD), 5 Ocean Sensors conductivity, temperature, and flow to the bay during this period (Figure 2c) shows a depth probes (OS200) and 1 Data Sonde conductivity relatively constant, low level of flow, with the excepand temperature probe. The goal of the long-term data tion of a peak at around Julian hour 8150 (December set was to measure the relevant mean flow quantities 6, 1994) of 1000 cubic meters per second. The re- (velocity, temperature, and salinity) in a comprehen- suit of this freshwater flow event is seen in the salinity sive way. n order to examine both horizontal and ver- for this period (Figure 2d), which transitioned from a tical gradients in those quantities, we selected three range of ppt before the flow event to 5-10 ppt folstudy stations (Figure lb): one at the deepest point lowing the flow event. There is also evidence of some in the channel on the west end (SCW) in 10 m depth spring-neap variability in the salinity, with the salin- (MLLW), one in the shallower region to the south of ity increasing from hour 7900 (neap) to the hour 8050 that point (SCS) in 8 m depth, and one at the deepest (spring). Throughout the period, however, the salinpoint in the cross-section near the east end (SCE) in ity variations indicate a dynamic density field in which about 13 meters depth. These three stations allowed us shear-induced stratification could be expected.

7 STACEY ET AL.: ESTUARNE :LESDUAL C:LEATON 17,019 Figure 1. Study site, land indicated by white regions, contours at depths of 3, 6, 8, 10, and 12 m: Northern reach of San F ancisco Bay; inset shows location within San F ancisco Bay System; ADCP location for Carquinez Strait located at arrow tip; Suisun Cutoff and station locations. An important parameter in Ri:, is the local longitudinal salinity gradient F. n particular, variation in will cause variations in Ri:, and the level of stratification. As a result, we would expect that intensification of the local longitudinal salinity gradient would result in a pulse of the residual flow. The along-channel density gradient was calculated using the SCW and SCE stations. The SCE station, however, was deeper than SCW and using the densities at the two bottom CTDs resulted in erroneous density gradients. Fortunately, the midcol- urrm CTD at SCE provided a measure of the level of stratification at the SCE station. Therefore, in order to calculate the along-channel salinity (and density) gradient, the bottom (13 m depth) and mid-column (6 meters depth) salinities at SCE were linearly interpolated to the same elevation as the SCW bottom station (10 meters depth). The difference between this interpolated salinity and the SCW salinity defined the longitudinal salinity gradient, as seen in Figure 3a. The salinity gradient is clearly quite dynamic, ranging from 4 ppt/km

8 17,020 STACEY ET AL.' ESTUARNE RESDUAL CREATON loo i i ' ', ,,,,,,, OO 2OO t i i Julian Hour, 1994 Figure 2. Overview of conditions at SCW, 1994: Surface current, east-west component (positive is east or up-estuary), surface current, north-south component (positive is north), (c) freshwater inflow, and (d) bottom salinity. to less than 0.5 ppt/km, with a mean value during the deployment of m 1.9 ppt/km. A strong tidal signal is seen in the data, and is examined more closely in Figure 3b, with a strong diurnal inequality. The peaks at about hours 7980, 8005, 8030, and 8055 are tied to flood tides. The intervening flood tides (hours 7995, 8020, and 8045) also show increases in the intensity of the salinity gradient, though not as strongly. This variation could be due to a compression of the salt field during flood tides, or perhaps results from the advection of the salt field along the axis of the bay. Either way, the 24-hour variation in the strength of the salinity gradient would be expected to be important in the creation of residual flows, with stratification and exchange flows more likely (based on Rix) during the periods of strong longitudinal density gradient. Because the stratification that is produced decreases mixing, the shear is accentuated and these two mechanisms strengthen further, as described in section Carquinez Strait The second experiment to be described was performed in 1991 at the western end of Carquinez Strait (Figure la; experiment location at the head of the ar-

9 STACEY ET AL.: ESTUARNE RESDUAL CREATON 17, ' $300 Julian Hour, Julian Hour, 1994 Figure 3. Time variability of longitudinal salinity gradient, calculated from difference between SCW and interpolated value at SCE (units of ppt/km). row), a narrow channel that connects San Pablo and ties from this data set are shown in Figure 4 (Figures Suisun Bays. t is 1 km wide and is marked by severe 4a and 4b) and show a strong spring-neap variation in curvature at its eastern end. The depth at the study the tidal currents. Glear neap tides occur at around location was about 17 meters, with the channel oriented hours 1250 and 1600, with an intervening spring tide almost completely east-west (with some curvature, con- characterized by surface currents of 150 cm/s magnicave south). tude. During the neap tides a clear diurnal inequality n 1991, the USGS deployed a 1200 khz NB-ADCP is evident, with alternating strong and weak ebb tides. (RD nstruments) in Carquinez Strait and configured The freshwater inflow to the bay during this period beit to collect ensembles of velocity data every 10 min- gins with a relatively constant low level of flow, until utes with a vertical spatial resolution of 1 m for the around hour 1400, when the freshwater flow begins to period February 13, 1991, to March 19, 1991 [Gartner increase, reaching a peak at around hour From et al., 1995] (data set also discussed by Monismith et that time on, the freshwater flow is more variable, and al. [1996]). At this location the dominant channel and ranges from m 200 cubic meters per second to near 1000 flow axes were directly east-west. The surface veloci- cubic meters per second. On the basis of this overview,

10 _ 17,022 STACEY ET AL.: ESTUARNE RESDUAL CREATON o t 4O 2O -2O O (c) 1200 looo 800 6oo Julian Hour, 1991 Figure 4. Overview of conditions at Oarquinez Strait, 1991: Surface current, east-west component, surface current, north-south component, and (c) freshwater inflow. Sign conventions as in Figure 2. we would expect the two neap tides to be different in nature, with the second neap tide being more strongly buoyancy driven than the first. 3. Data Analysis The insitu ADCP data sets provide a very complete picture of the long-term evolution of the velocity fields. Because of the energetic barotropic tides, however, the small residual flows are quite difficult to distinguish and the type of analysis used becomes very important. Typ- ically, the barotropic tides have been filtered out of the data using a low-pass temporal filter, leaving only the residual information. This process, however, also filters out any temporal variability in the residual-creating ex- change flow. Liet al. [1998] used a Taylor series analysis to define a method to distinguish density-driven residuals from barotropic residuals which relies on the connection between tidal stage and barotropic flows at the spring-neap timescale (i.e., no variations on shorter timescales could be distinguished). Principal components analysis, however, provides a method to separate

11 the flow into the components of interest without losing the temporal information. n the next section we will describe the technique. Then, in the following two sections, we will apply this analysis tool to the two data sets described above Principal Component Analysis Principal components analysis (PCA) is a linear analytical tool which helps to determine the sources of variability within data sets. Preisendorfer [1988] provides an extensive definition of the analysis and discusses many related issues. Application of PCA to current measurements was performed by Koh [1977] with some success. The analysis separates the data set based on shapes, or functions, which covary. These functions are known as principal components, or, frequently, empirical orthogonal functions. n the case of a data set with spatial and temporal variability, the components could be spatial functions which covary in time or temporal functions which covary in space. From here on, we will discuss the case where the components are spatial functions which co-vary in time. n this case, each (spatial) component has an associated time series (or amplitude) which quantifies how the spatial structure varies X: in time. STACEY ET AL.: ESTUARNE RESDUAL CREATON 17,023 Data Set=Xij; i=l,...,p; j=l,... n. (16) n our application, the two dimensions are space and time, with i being the spatial dimension and j being the temporal dimension. From a complex data set a covariance matrix, C is calculated as n j=l i, k = 1,...,p, (17) where an asterisk indicates the complex conjugate and Xi is defined as n x,- (;i) j=l x,. C is a square p x p Hermitian matrix which has p realvalued eigenvalues and p complex, p x i eigenvectors. These eigenvectors are the principal components (PCs) of the data set. Because the PCs are the eigenvectors of the covariance matrix, they are mutually orthogonal and span the space of the data set. The eigenvalue associated with each eigenvector quantifies the amount of the original data set's variance which is contained in that PC. The eigenvector matrix E consists of columns which are the empirical functions used to separate the data set. As such, we can consider these vectors to be p- dimensional basis vectors which the data can be projected onto. Mathematically, projecting the data matrix, X, onto the basis defined in E produces a new matrix A: p k=l The projection of the data onto the empirical eigenbasis is known as the analysis step, and results in a p x n matrix. Each row of this new matrix is the amplitude vector of the associated PC. The final calculation is known as the synthesis step. f we vary the PC's as given by their amplitudes and sum them, the initial data set results. Mathematically, the synthesis step is as follows: Unlike filtering, there is no loss of information in the p process. All of the variability originally in the data set is still present in the principal components. The amount Xij - Ai ET j; i- 1,...,p; j - 1,...,n. (20) of variance explained by each component is also quantified by the analysis, which allows us to "rank" the com- n this representation the exact data set is returned. ponents. Thus the first principal component (PC1) is Now suppose that the first two PCs were found to conthe spatial structure which, when varied as given by its tain nearly all the variance in the data set (quantified amplitude function, explains the highest percentage of by the eigenvalues). n that case, one could create a the original data set's variance. The second component new matrix E which is only the first two columns of E is the one which explains the next highest percentage, and a new amplitude matrix A which consists of the and so on. first two rows of A. n this case, the synthesis equation The analysis requires a rectangular matrix of data, becomes 2 Xi j t tt -- Aide j; i-- 1,...,p; j-- 1,...,n, (21) and Xij is the portion of the data set which is "due" to or explained by the first two principal components, PC1 and PC2. Similarly, one could include all components except the first or any other combination of components. ADCPs record velocity data over an entire water column with a resolution which is set by the user (usually either i m or 50 cm), producing a time series of currents at each depth. When considering estuarine flows, the well-defined vertical structures in the flow (i.e., boundary layer and exchange flows) allow PCA, which relies on underlying structure to separate the data set, to separate ADCP data into these phyiscally meaningful components. To account for the vector nature of the veloc- ity data, we followed Preisendorfer [1988] and formed a single, complex data point from each velocity vector. To be specific, we defined x(z, t) = u(z, t) + iv(z, t).

12 17,024 STACEY ET AL.' ESTUARNE RESDUAL CREATON A limitation of PCA is that it requires a rectangular data matrix as its input. This presents a problem when using ADCP data because the number of depth cells that have reliable velocity data varies through the tidal cycle. As the position of the water surface varies tidally, it moves through the upper depth cells, resulting in data "drop outs" for depth cells which are above the water surface (or within 10% of the surface). The use of sigma units (interpolating the ADCP profile onto a fixed number of vertical data points, with the spacing between the points varying with the water surface elevation) would result in a rectangular matrix of data but would also make temporal analysis (and the application of tidal filters) difficult, since the location of "data points" would change throughout the tidal cycle. For this reason, we decided to use the fixed grid defined by the ADCP data, but defined a rectangular matrix of data on that grid. A conservative approach on a fixed grid would be to use only depth cells up to a level which is always beneath the water surface but that results in the elimination of a significant amount of information from near the surface. Alternatively, if the maximum number of data points is used, the data set would have large gaps near the surface which would need to be filled by interpolation or extrapolation. Thus a compromise must be struck which maximizes the data used while minimizing the number of gaps in the data. n most cases, a heuristic method was sufficient, as the number of missing data points decreased rapidly away from the surface. The number chosen was selected to provide m 1% drop outs, minimizing the error due to extrapola- tion or interpolation, while still retaining most of the information in the data set. Many of the gaps in data were isolated in time. n those cases, a simple linear interpolation between the adjacent data points provided a reasonable fill. At other times, the gaps would only involve the top one or two depth cells, but would extend over several (5-10) times. n these cases, a spatial extrapolation was used assuming that the top portion of the water column (in the region of the gap) had a similar profile to the times just before and after the gap. This profile was then scaled by the data points just beneath the gap and used to fill in the gap. Using these two techniques, we were able to fill the data sets to a rectangular matrix and proceed with PCA. The spatial structures produced by PCA are chosen to maximize the variance explained by the first component (PC1). Then, the second component (PC2)is chosen to maximize the amount of the remaining variance it explains within the constraint that it be orthogonal to the first component, and so on. The components can therefore be rotated to define a new set of basis func- tions, which can be used to project the data. Using the ADCP data sets described below, we tested a variety of rotations (see Richman [1986] and Harman [1976] for 9 9 $ $ $ cm/s cm/s Figure 5. Mean flow profiles for entire data set at SCW east-west component, and north-south component. Solid line is data set, triangles are first principal component, and open circles are second principal component. Sign convenctions as in Figure 2.

13 STACEY ET AL.: ESTUARNE RESDUAL CREATON 17,025 further descriptions), but the resulting components did not improve our ability to interpret the data set, and, in for the near-bed baroclinic velocity (equation(6) multiplied by 0.03), assuming an average salinity gradient of many cases, obscured the underlying physical processes 1.9 ppt/km, the observed estuarine circulation would (see Stacey [1996] for a complete discussion). Therefore, suggest a friction velocity of 0.4 cm/s. This value in the remaining analysis, we will consider the original, is remarkably small, being more than an order of magunrotated PCs. nitude smaller than would be predicted based on the mean tidal velocities. The mean flow in Suisun Cutoff 3.2. Suisun Cutoff is, therefore, much larger than traditional theory would The mean velocity profile in Suisun Cutoff (at station predict, which could be due to stratification or due to SCW) for the entire data period has a top-bottom ve- dynamics on shorter timescales. locity difference of 13cm/s (Figure 5), the scaling of Spring-neap variability. n order to exwhich will be discussed below. Using PCA as described amine the time behavior of each component, we have in section 3.1, the ADCP data sets were separated into applied principal components analysis to the velocity their principal components. All stations produced a data from the SCW station. This station was the cenfirst component that was the barotropic boundary layer and a second component that was a sheared, exchange flow profile. Each data set was strongly dominated by the first principal component, with it consistently definter of our study and additional density data is available at certain times (discussed in section 4.1). For the purposes of displaying the results, we have reconstructed the velocity data using only individual principal coming over 98% of the variance in the data set. The second ponents (using the synthesis equation, equation (22)). component again defines almost all the remaining vari- Therefore we will examine the currents due to each in- ance. n keeping with the discussion of section 1, we will allow for the possibility that the second component is representative of a barotropically driven flow with variable stratification and mixing. Therefore we will refer to the two components as the boundary layer profile (PC1) and the shear profile or the exchange flow profile (PC2). The residual profiles due to PC1 and PC2 are shown in Figure 5 for station SCW (SCE and SCS gave simi- lar results). The first component creates an up-estuary residual with a magnitude of 3 cm/s. Clearly, this is the reverse of what would be expected in a channel in northern San Francisco Bay, where freshwater flow would be expected to create a down-estuary residual. The creation of this residual, however, could be a result of tidal pumping, due to the braided nature of the decrease, however, does not fully explain the increase channels [Fischer 1979]. t may also be an advective in energy in PC2. n fact, the change in tidal energy effect, with flood tides favoring the Northern Channel from neap tide to spring tide predicts (using the scaling through Suisun Bay and ebb tides favoring the Main of equation (6)) an increase in ug by a factor of 1.3. Channel. Either way, this component of the residual flow is a characteristic of Suisun Bay as a whole and is beyond the scope of this discussion (see J. R. Burau, manuscript in preparation, 2001). The residual profile due to PC2 is the classic picture of estuarine circulation with down-estuary flow at the surface and up-estuary flow at the bottom. The magnitude of this current is quite large, with a top-bottom dividual component and when a time series is required we will display the currents at the surface due to a given component. The energy (speed squared) of the surface currents due to PC1 and PC2 have been tidally filtered [Godin, 1972] (low-pass filter with a cutoff frequency of 50 hours) and are displayed in Figure 6. The spring-neap cycle is very clear in these time series, as the barotropic component is 40% less energetic during the neap tides than during the spring tides. At the same time, the energy of the second principal component increases by a factor of 3 during the neap tides as opposed to the spring tides. Therefore it appears that the residual flow is largely created during the neap tides, when tidal mixing is diminished. The amount that the mean tidal currents The magnitude of PC2, however, increases by nearly a factor of 2 during the neap tides. n fact, the increase in energy in PC2 is so severe, it appears that there is a threshold value which allows the shear flow to energize during the neap tides. Examining the behavior of the horizontal Richardson number (Figure 6c), the correlation between Ri and PC2 is clear. n fact, the threshold for PC2 to "turn on" appears to be at a horizontal Richardson number of 3. The fact that PC2 increases so dramatically, is most likely due to a feedback effect: as PC2 "turns on," the stratification of the water column becomes more velocity difference of 12cm/s. The time-averaged equation for the evolution of the baroclinic flow was given in equation (5). The steady state solution of this equation (creating a balance between baroclinic forcing, including the time-averaged surface pressure gradient, and pronounced (longer duration and larger magnitude) due tidal mixing) defines the velocity scale for gravitational to the advective effects of PC2. This stratification, in circulation (equation(6)). As discussed in the intro- turn, reduces vertical mixing and allows the shear in duction, the coefficient on this expression, however, is the water column to develop further [see Stacey et al., quite small, reaching a maximum near the bed of ]. As a result, once a threshold is reached in Ri, [Prandle, 1991]. On the basis of the solution of Prandle the stratifying forces exceed the destratifying forces and

14 17,026 STACEY ET AL.' ESTUARNE RESDUAL CREATON i i so i (c) Julian Hour, 1994 Figure 6. Spring-near variability in energy of PC1, PC2, and (c) Rix, SCW station. exchange flow is allowed to develop. On the basis of the low-pass filtered components of the flow, it appears that the threshold for this behavior, at the spring-neap timescale, is 3. bulence experiment [Stacey et al., 1999]), we see the expected ebb-flood variability in PC1, the barotropic component (Figure 7a). n Figure 7b, the time behavior of the second PC (exchange flow) shows a highly Tidal cycle variability. n the analysis unsteady, pulsing behavior. n particular, the large of the previous section, the dynamics of the residual pulse at hour 8197 appears late in the small ebb tide flow at the tidal and subtidal timescales are neglected. Turning to the unfiltered time series for the PCs, we now examine the relationship between tidal currents and residual flows on the tidal timescale. First focusing on a period with less energetic tides (hours were coincidental with a 24 hour tur- (ebb is defined by negative values in PC1) and extends into the ensuing flood tide (PC1 positive). The topbottom velocity difference during this period is more than 40cm/s, considerably larger than the residual flow for the station. t appears therefore that these intermittent pulses of extremely strong shear flow are what

15 STACEY ET AL' ESTUARNE RESDUAL CREATON 17,027 so t ',,,, 60 4O O Julian Hour, 1994 Figure 7. Tidal cycle variability in PC1 and PC2, neap tide, SCW station. Solid lines: along-channel component; Dashed lines: across-channel component. create the residual flow profile seen in Figure 5. This pulse of exchange flow, which is a major contributor to the estuarine residual, is likely the result of the water column becoming stratified (through the SPS mechanism), thus decreasing mixing and increasing the shear. This is the first of the barotropic mechanisms for residual creation, reduced turbulent mixing during ebb tides. The stratification which creates this shear on the ebb apparently extends into the ensuing flood tide, creating a sheared profile during that phase of the tide as well. This shear opposes the usual estuarine circulation, and is therefore represented by the short spike of PC2 into positive values at our This shear, however, is eliminated quickly around the time of the peak flood, suggesting a rapid breakdown in the stratification. Later in the same period, we see two more pulses in PC2, one at hour 8212 and one at hour 8217, which are likely to representhe second barotropic residual creating mechanism. The first appears early in the flood ebb tide, following a large ebb tide which became unstratifled late in its development (see section 4.1 for further discussion). The later pulse, at hour 8217 occurs late in the same flood tide, as the shear production of turbulent mixing by the tidal currents is reduced. Both pulses are likely to be manifestations of convective mixing events, created by the destabilizing buoyancy flux

16 _.. 17,028 STACEY ET AL.: ESTUARNE RESDUAL CREATON 8O 6O 4O 2O -2O -4O -6O Julian Hour, 1994 Figure 8. Tidal cycle variability in PC1 and PC2, spring tide, SCW statiofi. Solid lines, along-channel component; and dashed lines, across-channel component. on the flood tide. The instability created by the advection of dense waters over light waters has carried high momentum fluid downwards, which has created a pulse in PC2 that reinforces the estuarine circulation. Next, we can compare this period to the spring tide just before this neap tide (hours 8025 to 8055, Figure 8). During this time of high tidal energy the behavior of PC2 is much less active than during the neap tide, and consists of pulses of shorter duration and smaller magnitude. During the ebb tides such as at hour 8032, brief pulses of PC2 are evident, again due to the stratification of the water column. t is interesting to note, however, that the largest pulse of PC2 occurs at hour 8053, which is late in the flood tide. Further, this flood tide is of larger magnitude than the preceding ebb tide, suggesting that the SPS mechanism would almost certainly produce an unstable density structure during the flood tide. This pulse of PC2 therefore is likely to be a result of the mixing of high momentum fluid downwards by the convective instability established by the straining of the density field on the flood tide. Taken as a whole, the Suisun Cutoff data set demonstrates quite clearly the dynamic nature of residual creation in a partially and periodically stratified estuary. The dynamics are readily explained by considering the effects of SPS acting on the barotropic flow, and the resulting asymmetries in turbulent mixing. Measurements of turbulent Reynolds stresses during this same

17 STACEY ET AL.' ESTUARNE RESDUAL CREATON 17, i i i i 16,,, i i cm/s cm/s Figure 9. Mean flow profiles for entire data set at Carquinez Strait east-west component, and north-south component. Solid line is data set, triangles are first principal component, and open circles are second principal component. period [Stacey et al., 1999] indicate that there is, in fact, a persistent asymmetry in the level of turbulent mixing between flood and ebb. The analysis of the ADCP data presented here indicates that this asymmetry has a pronounced impact on the creation of residual flow in this environment Carquinez Strait From the Carquinez data set, principal components analysis produced a set of 14 eigenfunctions, and associated amplitudes. The eigenvalues quantify the variance explained by the first EOF as 97.5% and the second as 2.1%, (over 80% of the remaining variance). Figure 9 displays the mean flow profile for the entire data set and the mean flow due to PC1 and PC2. The profile shape is consistent with an estuarine exchange flow (down-estuary at the surface, up-estuary at the bed). As would be expected, the second principal component dominates the production of this exchange flow. The mean flow due to the first PC is largely across-channel; examining the station location in Figure la, we can see that the Carquinez station lies in a portion of the bay which is strongly curved. Such curvature would cause the barotropic flows to produce a cross-channel residual [Geyer, 1993], as is evident in the residual flow of the first PC Spring-neap variability. Just as with the Suisun Cutoff data, the low-pass filtered (50-hour cutoff period) component amplitudes are used to examine the spring-neap variations in the principal components (Figure 10). The spring-neap cycle is quite evident in the time series for the first PC (PC1), with a period of 350 hours or 14.6 days. The average energy in the top depth cell is seen to vary from a minimum of 6000 cm2/s 2 to a maximum of nearly cm2/s 2. Just as in the Suisun Cutoff data set, the second PC (PC2) also displays a strong spring-neap dependence. The most pronounced peak in PC2 appears during the second neap tide in PC1, around hours The energy in PC2 increases to 4-5 times its background value while the energy in PC1 is decreasing. Freshwater input into the system (Figure 4c) also increasedramatically just before this "pulse" of PC2. Thus the large peak in exchange flow at hours is seen to be due to the combined forces of an increase in buoyancy (freshwater flow) and a decrease in mixing energy (barotropic tides). During the first neap (around hours ) the energy in the exchange flow (PC2) increases by nearly a factor of 2 as the barotropic flow (PC1) diminishes in amplitude. t is clear from the time series of freshwater input into the system that this increase in exchange flow

18 17,050 STACEY ET AL' ESTUARNE RESDUAL CREATON oooo 9000 t , E ,,,, ulian Hour, 1991 Figure 10. Spring-neap variability in energy of PC1 and PC2, Carquinez station. is not due to an increase in buoyancy but rather only due to a decrease in tidal mixing. (up-estuary) flow near the bottom. Once again, the residual flow profile is created through a series of intense Thus there are two distinct pulses of exchange flow pulses of shear flow, rather than a constant, background evident in this data set, each falling during a neap tide in the barotropic flow. The largest pulse is due to the combined effects of a buoyancy influx and a decrease in circulation. Many of the longest lasting pulses of exchange flow appear during the weak ebb tides in the barotropic flow mixing, but a pulse in the exchange flow also appears (negative values of PC1). The pulse seem to extend during a time of low buoyancy input, due only to a de- from near the maximum ebb tide to just after the start crease in mixing. On the basis of the above discussion, of the flood tide. The timing of these pulses is consiswe know that the residual flow in the system is almost tent with a water column undergoing SPS and the asentirely contained in PC2, but, just as in Suisun Cutoff, sociated variations in mixing. The reduction in mixing there is an unsteady subtidal process that creates that late in the ebb due to the strain-induced stratification mean profile. producesheared exchange flow (PC2) and the strong Tidal Cycle Variability. The behavior pulses of exchange flow which begin during the weak of the first two principal components during the first ebbs (at hours 1220, 1245, and 1270) are most likely neap tide are displayed in Figure 11. The first principal created through this mechanisms. During the strong component displays a strong diurnal inequality, with flood tides consistently reaching a maximum speed of ebb tides (at hours 1210, 1235, 1260, and 1285) the pulses are shortened and reduced due to greater turbu- 100cm/s and ebb tides alternating between 130cm/s lent mixing during the ebb tide (particularly clear at and 30cm/s, characteristic of the neap barotropic tides hour 1265). The increase in energy from weak ebb to in this part of San Francisco Bay. As was the case in Suisun Cutoff, we see that PC2's time series consists of a series of pulses of exchange flows in Carquinez Strait. n particular, the second principal component tends to turn on for periods ranging from 1 or 2 hours to as many as 5 hours. During these pulses, the surface current strong ebb would manifest itself in a 24-hour variation in Riz, and the reduction in the magnitude of the pulses of PC2 during the strong ebbs is also consistent with the SPS mechanism. During this period we do not see evidence of the second barotropic production mechanism, which would require an unstable density structure to is strongly westward (down-estuary), and the exchange develop during the flood tides. Although there is no flow has a magnitude of as much as 35 c_m. Based on the structure of the second PC, this implies an eastward stratification data available from this site, it would appear that the water column remains stratified, so that

19 STACEY ET AL.' ESTUARNE RESDUAL CREATON 17,051 loo 50,t \ / -5o -loo -15o 121o 122o 123o 124o 125o ' '. ' 1: t ' tli; '":J,, Julian Hour, 1991 ] Figure 11. Tidal cycle variability in PC1 and PC2, first neap tide, Carquinez station. Solid lines, along-channel component; and dashed lines, across-channel component. the advection of the density field by the flood tides does pear and disappear in less than an hour. Further, the not produce an unstable distribution. spikes tend to occur as the barotropic tide turns from During the spring tide (hours ) the first two ebb to flood, rather than just after the maximum ebb. principal components have the time behavior shown in Once again, the time variability of PC2 is consistent Figure 12. Now the first principal component is of uni- with the SPS mechanism: as the ebb tide develops, form strength on ebb and flood ( 130cm/s) and no stratification also develops, reducing turbulent mixing significant diurnal inequality is evident. The increase and allowing additional shear to develop in the water in mixing energy, and its uniform temporal distribu- column. During this period, the negative spikes of PC2 tion, alters the behavior of the second principal com- are frequently followed by small positive spikes (reversed ponent. nstead of sustained pulses of exchange flow, estuarine circulation), which would indicate a sheared the time behavior of the exchange flow consists of a se- flood tide produced when the water column remains ries of "spikes." While the exchange flow would pulse stratified following the end of the ebb tide. for several hours during the neap tide during the spring During the second neap tide the barotropic forcing is tide it only appears in short, sharp spikes which ap- similar to the first neap tide (Figure 13a), but a large

20 17,052 STACEY ET AL.' ESTUARNE RESDUAL CREATON loo 50 o " w, -ZO Julian Hour, 1991 Figure 12. Tidal cycle variability in PC1 and PC2, spring tide, Carquinez station. Solid lines, along-channel component; and dashed lines, across-channel component. pulse of freshwater flow has altered the buoyancy forcing (Figure 4c). During this period the second principal component is again characterized by a series of pulses, but the pulsing is now much stronger than during the first neap. n fact, the strongest pulse (just after ½m hour 1585) results in a surface velocity of the basis of the shear of PC2, a surface velocity of this magnitude would correspond to a bottom velocity of m 41cm/s, or a top-bottom velocity difference of 91cm/s. Clearly, the effect of freshwater flow has been to intensify and prolong the pulses of exchange flow. The mechanism that has created this change is most likely an increase in the salinity gradient (either horizontal or vertical), induced by the large freshwater flow at hour t appears that the intensification of buoyancy forcing now allows even the strong ebb tides (e.g., hour 1595) to exhibit persistent shear, with sustained pulses of exchange flow extending throughout each ebb tide. The stratification at this site has clearly increased from the earlier period (Figure 11), and appears to no longer be periodic, but rather persistent, during the period shown in Figure 13. The effect of persistent stratification is seen in the positive values for PC2 during the flood tides (such as at hours 1565, 1593, and 1603). As was seen during the spring tide, and even at the Suisun Cutoff location, these instances of reversed estuarine circulation are readily explained by considering

21 _ STACEY ET AL.: ESTUARNE RESDUAL CREATON 17, / ' \ /, 50 / '" -, , f\ '-,...,,',,.,d i.1¾ r, \,.,, f-- -2O -4O Julian Hour, 1991 Figure 13. Tidal cycle variability in PC1 and PC2, second neap tide, Carquinez station. Solid lines, along-channel component; and dashed lines, across-channel component. the effects of a stratified water column on a flood tide, which would produce a positive pulse in PC2 (i.e., a reversal of the estuarine exchange flow). The strength of these reversals in PC2 has increased during this second neap tide as compared to either of the earlier periods, which is consistent with the influx of buoyancy due to freshwater flow preceding this period. 4. Discussion 4.1. Residual Flow Dynamics n all of the data sets discussed above, the estuarine residual exchange flow is evident, but with a time behavior which is highly unsteady. The pulses appear to be tied to changes in the level of turbulent mixing (and hence to the magnitude and duration of stratification) and the strength of the longitudinal density gradient. These parameters are combined into a single dimensionless group, the horizontal Richardson number Ri=. As discussed above, the horizontal Richardson number should predict the onset of the pulses of shear flow seen in the second principal component of the ADCP data sets. For the Suisun Cutoff data set, we are able to calculate the horizontal Richardson number at the tidal timescale using the fixed CTDs to define the salinity gradient. n section 3.2, we focused on a 24-hour pe- riod and observed a strong pulse in exchange flow occurring late in the weak ebb tide (Figure 7b, hour 8197). This pulse had a magnitude of 40 cm/s (top-bottom velocity difference of 70 cm/s) and lasted for several hours. n order to examine the 24-hour variability, we now expand our view to include additional tidal periods before and after the window discussed above. n Figure 14 (Figures 14a and 14b), the first two prin- cipal components in Suisun Cutoff are displayed for a 50-hour window during a neap tide (actually just preceding the neap, see Figure 6). The pulse that was evident at hour 8197 is seen to be repeated late in each weak ebb tide, at hours 8173, 8197, and n each case, the pulse appears midway through the weak ebb tide and extends into the ensuing flood tide, consistent with the stratification of the ebb tide by the SPS mechanism. The fact that the exchange flow itself varies at the diurnal frequency is not surprising, given the fact that during neap tides the tidal mixing has a significant diurnal inequality, as does the longitudinal salinity gradient (Figure 3b). Two pulses (at hours 8212 and 8217) during this period appear to be created by a different mechanism. As was discussed above, these pulses of estuarine circulation are likely the result of a convectively driven mixing event created by the flood phase of the SPS mechanism (which is destablizing). n each case, however, it is the balance between the advective buoy-

22 17,034 STACEY ET AL.: ESTUARNE RESDUAL CREATON 5O -5O ]_o -2O (c) ' 0 xlo 2 1.: ' 1-0.:5-0 81' Julian Hour, 1994 Figure 14. Overview of residual creation in Suisun Cutoff: tidal currents (at surface), exchange flow, (c) horizontal Richardsonumber, and (d) top-bottom density difference. ancy flux and vertical turbulent mixing that sets the flood tide mechanism also correlated with increases magnitude and timing of these pulses. the horizontal Richardson number (at hours 8212 and To capture the balance between the longitudinal salin- 8217), but the connection is not as clear in this case and ity gradient (the buoyancy flux) and vertical mixing, requires further examination. Ri is displayed for this period in Figure 14c. The 24- hour variation in the horizontal Richardson number is also clearly evident, with maximum values attained at the same time as the pulses were evident in the ex- change flow. t appears that there is a critical value of Riz above which the largest pulses of exchange flow develop, with the critical value appearing to be 3. The The effects of the exchange flow on the level of stratification is demonstrated in Figure 14d, where the topbottom density difference is calculated from density profiles collected at the location (see Stacey et al. [1999] for details on the 24-hour turbulenc experiment). The pulse of exchange flow which begins at hour 8197 is also the onset of a stratification event, with the stratification

23 STACEY ET AL.: ESTUARNE RESDUAL CREATON 17,055 increasing to a maximum at hour The increasing stratification would reduce turbulent mixing and allow the exchange flow to be sustained, resulting in the extended pulse of exchange flow during each weak ebb tide in Figure 14. The event is triggered by the horizontal Richardson number moving to a value greater than a threshold ( 3), which signals the onset of exchange flow and stratification. The stratification that is produced reduces turbulent mixing, allowing the exchange flow to develop further. ncreased shear in the flow then further stratifies the water column, providing the feedback mechanism that creates the pulse-like nature of the exchange flow. The second barotropic mechanism is represented at hours 8212 and 8217, where convective instabilities appear on the flood tides. Notice that the pulse of exchange flow which occurs early in the flood tide at hour 8212 develops following an unstratified period, as would be expected, although the water column does restratify during the flood tide through a process that is not clear from this data set (and, unfortunately, the density data does not extend to hour 8217 when the second flood 4.2. Barotropic and Baroclinic Residual Creation n both data sets discussed here, residual flows were created through pulses of exchange flow which were strongly linked to the tidal frequency. Although traditional theory would attribute these exchange flows to baroclinic forcing, because many of the significant pulses were seen during ebb tides, they could have equally (or perhaps more likely) been the result of barotropic forcing acting on a stratified water column. That is, during the weak ebb tides seen in Figure 14a, the water column may stratify through SPS on the weak ebb tide, and destratify on the ensuing flood tide (as is evident in Figure 14d). The stratification that develops during the weak ebb tides would then allow shear to develop in the water column independent of the baroclinic forcing. During the strong ebb tides in Figure 14, pulses of exchange flow do not develop, which is evidence that the turbulent mixing is s,fficient to prevent the development of SPS. The horizontal Richardson number predicts the development of SPS, and a critical value of 3 seems to explain the behavior seen in the exchange flow. Even the pulses of residual flow created during the flood tide appear to be correlated to elevated values of Rix, although the threshold appears to be lower during these periods. Determining whether the flows observed were due to tide pulse occurs). The horizontal Richardso number also increases during these pulses, which reflects the fact that the destabilizing buoyancy flux (due to SPS on the flood tide) is becoming more important in the production of turbulent energy. Although the horizontal Pdchardson number was not barotropic or baroclinic forcing unequivocally is difficult available for the Carquinez Strait data set, the interdue to the fact that both processes axe defined by the pretation of the Suisun Cutoff data set, as illustrated same dimensionless group (Rix) for onset conditions. in Figure 14, is completely consistent with the discus- Considering the magnitude of the exchange flow, howsion of the Carquinez data set in section The inever, we can examine what assumptions would be reteraction between turbulent mixing and stratification, quired in order for the baroclinic forcing to produce the specifically that created through the advection of a lonobserved flows. n both Suisun Cutoff and Carquinez gitudinal salinity gradient by a sheared velocity profile, Strait, the peak shear flow velocity scales were 40 is ubiquitous in estuarine flows. The feedback mechacm/s. f we consider a balance between turbulent mixnism that has been described here would therefore be ing and the baroclinic pressure gradient, equation (6) expected to be seen more broadly, and pulses of exdefines the physical scaling of the baroclinically driven change flow are expected be seen whenever the horizonflow, but the actual magnitude of the baroclinic flow is tal Richardson number exceeds some threshold value. defined by this quantity times a coefficient of about 0.03 This feedback mechanism may also provide an impor- (for the near-bed maximum [Prandle, 1991]). Using reptant control on the salt distribution in the estuary, as resentative depths of 10 m for Suisun Cutoff and 17 m is discussed by S. G. Monismith, et al. (Structure and for Carquinez Strait and a longitudinal salinity gradient flow-induced variability of the subtidal salinity field in of 1.5 ppt/km in Suisun Cutoff (from data, Figure 3) northern San Francisco Bay, submitted to Journal of and 0.5 ppt/km in Carquinez Strait [see Monismith et Physical Oceanography, 2001). al., 1996], the observed exchange flow (40 cm/s) would The fact that the residual flow is created through a require a turbulent velocity scale of cm/s for series of pulses has important implications in the transeach location. This value is exceptionally small, even port of scalars in the estuary. f the substance of interest for stratified conditions. f stratification is sufficient to has a time-variability in its concentration which is also reduce mixing to this level, then the appropriate baltied to the tidal cycle (e.g., suspended sediment), then ance in equation (5) is likely to be between inertia and the relative phasing between its concentration and the the baroclinic pressure gradient. This balance would residual-creating flow will be critical in determining its define a baroclinic velocity scale to be net transport. For example, if residual flow is created only during weak ebb tides, when suspended sediment u gfht, (22) concentrations are reduced, then the net transport of sediment will be significantly less than if the residual where T is the timescale over which the baroclinic flow flow were a constant, background circulation. is allowed to develop in the absence of turbulent mix-

24 17,036 STACEY ET AL.: ESTUARNE RESDUAL CREATON ing. Using the same values for H and r as above, this expression defines the timescale for the baroclinic flow to develop to a magnitude of 40 cm/s to be i hour in Suisun Cutoff and 1.7 hours in Carquinez Strait. Although these values are not unrealistic, it does appear that the exchange flow develops extremely quickly (e.g., Figures 7b and 12b), perhaps more quickly than can be accounted for with baroclinic forcing. The barotropic forcing, however, is of much greater magnitude than the baroclinic forcing. As a result, shear in the water column can develop to a larger magnitude more quickly when barotropic forcing is in the presence of stratification than can the baroclinic forcing. The velocity scale for the shear created under these conditions is dependent upon the details of the stratification distribution and can not be calculated from the observations presented here. t does appear, however, that the magnitude of exchange flows observed are more likely a result of barotropic forcing in a periodically stratified water column than the typically assumed baroclinic forcing. This conclusion is consistent with numerical models of tidally-driven stratified flow. As described in the introduction, numerical modeling work by Burchard and Baumert [1998] indicated that when SPS is present, the barotropic forcing may dominate the production of the estuarine exchange flow, not the baroclinic forcing. The dynamics of the bottom-mixed layer and the associated stratification and shear were captured effectively by Monismith and Fong [1996], and significant shear was created at the interface due to the barotropic forcing acting on the stratified water column. A more complete model of the dynamics being examined was developed by Bowen [2000], who found that tidal asymmetry is the primary contribution to the tidally averaged shear. The implications of this possibility on turbulence modeling are profound. f residual flows are created through the action of a baroclinic force, then the magnitude of the residual flow can be accurately predicted using a bulk turbulent viscosity to capture the effects of turbulent mixing. That value could be a constant over depth and time and the magnitude of the residual current could still be calculated correctly (although tem- /poral variations will be lost). f, however, the residual flows are being created through asymmetries in the tidal currents due to SPS, then the turbulence model must correctly predict the level of stratification and shear on the tidal timescale. Without accurate representation of the subtle asymmetries between ebb and flood tides, which will depend critically on the turbulence model employed, the residual current, with regards to either magnitude or timing, will not be accurately represented. Acknowledgments. The authors gratefully acknowledge the support of NSF (OCE ) during the field data collection and analysis. The field experiments were supported, in part, by the nteragency Ecological Program and equipment used in this study were purchased through the U.S. Department of the nterior's Placed-Based Program. The assistance of Jeff Gartner (USGS, Menlo Park) during the field experiments was greatly appreciated. References Bowen, M. M., Mechanisms and variability of salt transport in partially-stratified estuaries, Ph.D. thesis, Mass. nst. of Technol., Cambridge, Burchard, H., and H. Baumert, The formation of estuarine turbidity maxima due to density effects in the salt wedge. A hydrodynamic process study, J. Phys. Oceanogr., 28, , Fischer, H. B., Mixing and dispersion in estuaries, Annu. Rev. Fluid Mech., 8, , Fischer, H. G., E. J. List, R. C. Y. Koh, J. mberger, and N.H. Brooks, Mixing in nland and Coastal Waters, Academic, San Diego, Calif., Gartner, J. W., M. R. Simpson, and R. N. Oltmann, Velocity measurements by acoustic Doppler current profiler and conventional current meters in Suisun Bay, San Pablo Bay, and Carquinez Strait, California, , U.S. Geol. $urv. Water Resour. nvest. Rep., Geyer, W. R., Three-dimensional tidal flow around headlands, J. Geophys. Res., 98, , Godin, G., The Analysis of Tides, Univ. of Toronto Press, Toronto, Canada, Hansen, D. V., and M. Rattray, Gravitational circulation in straits and estuaries, J. Mar. Res., 23, , Hansen, D. V., and M. Rattray, New dimensions in estuarine classification, Limnol. Oceanogr., 11, , Harman, H. H., Modern Factor Analysis, Univ. of Chicago Press, Chicago, ll., Hill, K. D., T. M. Dauphinee, and D. J. Woods, The extension of the 1978 practical salinity scale to low salinities, EEE J. Oceanic Eng., 0E-11(1), , Jay, D. A., and J. M. Musiak, Particle trapping in estuarine tidal flows, J. Geophys. Res., 99, , Jay, D. A., and J. M. Musiak, nternal tidal asymmetry in channel flows: Origins and consequences, in Mixing Processesin Esturies and Coastal Seas, edited by C. Pattiaratchi, , AGU, Washington, D.C., Jay, D. A., and J. D. Smith, Residual circulation in shallow estuaries, 2, Weakly stratified and partially mixed estuaries, J. Geophys. Res., 95, , Koh, R. C. Y., Analysis of multiple time series by principal components with application to ocean currents off San Francisco, Tech. Memo. 77-, Cal. nst. Technol., Pasadena, Li, H., and J. O'Donnell, Tidally induced residual circulation in shallow estuaries with lateral depth variation, J. Geophys. Res., 102, 27,915-27,929, Li, H., A. Valle-Levinson, K. C. Wong, and K. M. M. Lwiza, Separating baroclinic flow from tidally induced flow in estuaries, J. Geophys. Res., 103, 10,405-10,417, Linden, P. F., and J. E. Simpson, Modulated mixing and frontogenesis in shallow seas and estuaries, Continental Shelf Res., 8(10), , Monismith, S.G., and D. A. Fong, A simple model of mixing in stratified tidal flows, J. Geophys. Res., 101, 28,583-28,597, ].996. M6n';smith, S.G., J. R. Burau, and M. T. Stacey, Stratification dynamics and gravitational circulation in northern San Francisco Bay, in San Francisco Bay: The Ecosystem, edited by J. T. Hollibaugh, Am. Assoc. for the Adv. of Sci., San Francisco, Calif., Nunes-Vaz, R. A., G. W. Lennon, and J. R. de Silva Samarasinge, The negative role of turbulence in estuarine mass transport, Estuarine Coast. Shelf Sci., 28, , 1989.

25 STACEY ET AL.: ESTUARNE RESDUAL CREATON 17,037 Peters, H., Observations oœ stratified turbulent mixing in an estuary: Neap-to-spring variations during high river flow, Estuarine Coast. Shelf $ci., 5, 69-88, Prandle, D., Tides in estuaries and embayments (review), in Tidal Hydrodynamics, edited by B. B. Parker, pp , Wiley nterscience, New York, Preisendorœer, R. W., Principal Components Analysis in Meterology and Oceanography, Elsevier, New York, Pritchard, D. W., Observations of circulation in coastal plan estuaries, in Estuaries, edited by G. Ward and W. Espey, pp , AAAS, Washington, D.C., RD nstruments, Broadband Acoustic Doppler Current Profiler Technical Manual, San Diego, Calif., Richman, M. B., Rotation oœ principal components, Y. Climatok, 6, , Simpson, J. H., J. Brown, J. Matthew, and G. Allen, Tidal straining, density currents, and stirring in the control of estuarine stratification, Estuaries, 13, , Simpson, J. H., and J. Sharples, Dynamically-active models in the prediction of estuarine stratification, in Dynamics and Exchanges in Estuaries and the Coastal Zone, edited by D. Prandle, pp , Springer-Verlag, New York, Stacey, M. T., Turbulent mixing and residual circulation in a partially stratified estaury, Ph.D. thesis, Stanford Univ., Stanœord, Calif., Stacey, M. T., S. G. Monismith, J. R. Burau, Observations of turbulence in a partially stratified estuary, J. Phys. Oceanogr., 29, , J. R. Burau, U.S. Geological Survey, Placer Hall, 6000 J Street, Sacramento, CA (jrburau@usgs.gov) S. G. Monismith, Civil Engineering Department, Stanford University, Stanford, CA e. st anford. ed u) M. T. Stacey, Department of ntegrarive Biology, VLSB 30b0, University of California, Berkeley, Berkeley, CA (mstacey@socrates.berkeley. edu) (Received July 31, 2000; revised February 9, 2001; accepted March 7, 2001.)