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  • View in gallery

    In this standard schematic of tidal pumping, a jet of water exits the contraction during one-half of the tide and a potential drain of water enters during the opposite half.

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    Experiment location: (a) a broad map of the region and (b) an inset, which includes a sketch of an eddy that forms during flood tide. Both have 10-m contours. The transect path is 1–1.5 km east of the Golden Gate Bridge, which is at the narrowest part of the channel contraction.

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    Summer experiment day-1 surface salinity (psu) (a) observations compared with (b) harmonic reconstruction, and velocity (m s−1) (c) observations compared with (d) harmonic reconstruction as a function of day of year and lateral position.

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    Same as Fig. 3, but for fall experiment day 2.

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    Summer experiment cross-sectional structure of (a) time-averaged salinity observations and (b) temporal mean from harmonic reconstruction. (c) Time-averaged velocity observations also agree with (d) the temporal mean of velocity from harmonic reconstruction.

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    Same as Fig. 5, but for fall experiment.

  • View in gallery

    Flux decomposition. For each experiment (top to bottom), dispersive flux is decomposed into four components (left to right bars): tidal pumping, steady vertical exchange, steady lateral exchange, and unsteady shear dispersion. The sum of the first three flux components is presented as the right-hand bar (“1–3”). The magnitude of the advective flux is shown as a solid horizontal line for comparison with the calculated dispersive fluxes.

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    Comparison of cross-sectionally averaged salinity and velocity for (a) summer and (b) fall experiments. All data from each season are lined up by the semidiurnal zero crossings of along-channel mean velocity, and then block averaged.

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    Both tidal pumping and TER increase with the phase lag ( f, °) between mean flow [Qosin(wt)] and salinity [−Sosin(wt + π/2 + f )]. Flow and salinity amplitudes (Qo, So) are from the fall and summer experiments; w is from the M2 tide. The TER from Largier (1996) and Parker et al. (1972), estimated for fall conditions, imply much higher tidal pumping flux and much higher phase lag than those in our experiments.

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Dispersive Fluxes between the Coastal Ocean and a Semienclosed Estuarine Basin

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  • 1 Department of Civil and Environmental Engineering, University of California, Berkeley, Berkeley, California
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Abstract

Scalar exchange between San Francisco Bay and the coastal ocean is examined using shipboard observations made across the Golden Gate Channel. The study consists of experiments during each of the following three “seasons”: winter/spring runoff (March 2002), summer upwelling (July 2003), and autumn relaxation (October 2002). Within each experiment, transects across the channel were repeated approximately every 12 min for 25 h during both spring and neap tides. Velocity was measured from a boat-mounted ADCP. Scalar concentrations were measured at the surface and from a tow-yoed SeaSciences Acrobat. Net salinity exchange rates for each season are quantified with harmonic analysis. Accuracy of the net fluxes is confirmed by comparison with independently measured values. Harmonic results are then decomposed into flux mechanisms using temporal and spatial correlations. In this study, the temporal correlation of cross-sectionally averaged salinity and velocity (tidal pumping flux) is the largest part of the dispersive flux of salinity into the bay. From the tidal pumping flux portion of the dispersive flux, it is shown that there is less exchange than was found in earlier studies. Furthermore, tidal pumping flux scales strongly with freshwater flow resulting from the density-driven movement of a tidally trapped eddy and stratification-induced increases in ebb–flood frictional phasing. Complex bathymetry makes salinity exchange scale differently with flow than would be expected from simple tidal asymmetry and gravitational circulation models.

Corresponding author address: Jonathan P. Fram, Marine Science Institute, Building 520, Room 4013, University of California, Santa Barbara, Santa Barbara, CA 93106. Email: jfram@msi.ucsb.edu

Abstract

Scalar exchange between San Francisco Bay and the coastal ocean is examined using shipboard observations made across the Golden Gate Channel. The study consists of experiments during each of the following three “seasons”: winter/spring runoff (March 2002), summer upwelling (July 2003), and autumn relaxation (October 2002). Within each experiment, transects across the channel were repeated approximately every 12 min for 25 h during both spring and neap tides. Velocity was measured from a boat-mounted ADCP. Scalar concentrations were measured at the surface and from a tow-yoed SeaSciences Acrobat. Net salinity exchange rates for each season are quantified with harmonic analysis. Accuracy of the net fluxes is confirmed by comparison with independently measured values. Harmonic results are then decomposed into flux mechanisms using temporal and spatial correlations. In this study, the temporal correlation of cross-sectionally averaged salinity and velocity (tidal pumping flux) is the largest part of the dispersive flux of salinity into the bay. From the tidal pumping flux portion of the dispersive flux, it is shown that there is less exchange than was found in earlier studies. Furthermore, tidal pumping flux scales strongly with freshwater flow resulting from the density-driven movement of a tidally trapped eddy and stratification-induced increases in ebb–flood frictional phasing. Complex bathymetry makes salinity exchange scale differently with flow than would be expected from simple tidal asymmetry and gravitational circulation models.

Corresponding author address: Jonathan P. Fram, Marine Science Institute, Building 520, Room 4013, University of California, Santa Barbara, Santa Barbara, CA 93106. Email: jfram@msi.ucsb.edu

Keywords: Fluxes; Salinity; Tides

1. Introduction

Many estuaries are characterized by a bathymetric constriction at their mouth that impedes exchange with the coastal ocean. The net exchange of a scalar, such as salt, chlorophyll, or suspended solids, influences conditions along the axis of the estuarine ecosystem, as well as in the adjoining coastal ecosystem. The tidally averaged salinity field in a coastal estuary is classically described by a combination of two tidally averaged independent processes: gravitational circulation and longitudinal diffusion (Hansen and Rattray 1965). MacCready (2004) updates this model by deriving more physically realistic longitudinal diffusion coefficients, particularly near the mouth. The estuary mouth is of special importance because it is generally deeper and more bathymetrically variable than the rest of the estuary. Density-driven circulation increases with depth, and the interaction of tidal flows with bathymetric variability fosters tidal trapping and other tidal diffusive processes.

MacCready (2004) notes that the challenge of characterizing exchange is that there is more involved than finding the right diffusion coefficients. These tidally averaged formulations must also capture the scaling of processes with subtidal variability (MacCready and Geyer 1999). For example, to simply model the along-channel salt field in San Francisco Bay one needs a density-driven process that scales more strongly with the longitudinal density gradient than traditional gravitational circulation (Monismith et al. 2002).

As has been done in other large estuaries, such as the Columbia (Kay et al. 1996) and the Chesapeake (Austin 2002), we focus our analysis at the mouth of San Francisco Bay, which features a narrow constriction known as the Golden Gate. Subtidal processes that create residual flow at the Golden Gate include tidal pumping, steady baroclinic flow, tidal trapping of an eddy during the flood–ebb transition, and enhanced frictional phasing by a lateral density gradient during the ebb–flood transition (Fram 2005). Scalars at the ocean–estuary interface, such as salts, are transported by these flow processes. Here, we characterize the effect of each process on salinity exchange under different oceanic conditions, tidal forcing, and freshwater input.

a. Estuarine flux decomposition

Cross-sectionally integrated salt transport can be described quantitatively using the advection–diffusion equation (Kay et al. 1996; Geyer and Nepf 1996), with rivers advecting salt seaward and dispersive processes carrying salt into the bay. For this paper, we consider net exchange of a scalar to be the mass exchange rate over longer-than-diurnal and fortnightly tidal time scales. Net exchange is, then,
i1520-0485-37-6-1645-e1
where 〈variable〉 indicates a tidal average, indicates a cross-sectional average, the x coordinate is along the main channel, U is the along-channel velocity, and S is salinity. Following Fischer’s (1972) seminal work, dispersive flux can be separated into mechanisms by decomposing flux into temporal and spatial correlations.

The traditional approach to decomposing estuarine fluxes (Fischer 1972; Prandle 1985; Jay et al. 1997) emphasizes the vertical structure by first averaging in the lateral dimension, preserving vertical variability. In this experiment, however, the flow is highly three-dimensional, and the lateral variability may make an equally important contribution to the net flux (Rattray and Dworski 1980; Dyer 1997). In this section, we will present the decomposition following the traditional approach of averaging lateral first. We note, however, that reversing the order of averaging (i.e., averaging vertically first) changes the magnitude of the individual flux components by approximately 10%, and in later sections we have averaged the two estimates. In the remainder of this section, we denote spatial averages by overbars, with superscripts denoting the dimension of averaging (h for lateral; υ for vertical); double overbars indicate a cross-sectional average. Brackets will denote temporal averages; we assume that the temporal averaging operator removes all diurnal, semidiurnal, and tidal-monthly tidal motions.

First, the cross-sectional and temporal mean 〈〉 is separated from the raw velocity [u(y, z, t)]:
i1520-0485-37-6-1645-e2
Next, we extract the cross-sectionally averaged velocity deviation from the mean (u1) and steady shear (u3), leaving unsteady shear fluctuations (u4):
i1520-0485-37-6-1645-e3
i1520-0485-37-6-1645-e4
i1520-0485-37-6-1645-e5
Combining the above, the velocity is decomposed as
i1520-0485-37-6-1645-e6
Further decomposition of u3 and u4 depends on the method of spatial averaging. Fischer (1972), Prandle (1985), and Jay et al. (1997) average laterally first to find baroclinic flow and unsteady vertical shear, with the remainders becoming laterally steady and unsteady shears:
i1520-0485-37-6-1645-e7
i1520-0485-37-6-1645-e8
i1520-0485-37-6-1645-e9
i1520-0485-37-6-1645-e10
Multiplying flow by salinity and averaging over the cross section and time gives the flux of salt through the channel. The processes associated with the scalar flux terms in (11) below are advection from river flow (a) and dispersive flux, which includes tidal pumping (b), steady vertical exchange (c), steady lateral exchange (d), and unsteady vertical (e) and lateral (f) shear exchanges:
i1520-0485-37-6-1645-e11
In the following sections we describe in more detail the flux mechanisms that are identified here.

b. Baroclinic exchange

The steady exchange resulting from vertical structure is frequently attributed to baroclinic or estuarine exchange flows created by longitudinal density gradients in estuaries (Hansen and Rattray 1965). Following Monismith et al. (2002), we can define a velocity scale for this exchange flow as
i1520-0485-37-6-1645-e12
where H is local depth and νt is a constant tidally averaged vertical mixing coefficient. This velocity structure strains the density field, converting the longitudinal salinity gradient into vertical stratification, which scales as
i1520-0485-37-6-1645-e13
where we have assumed a Prandtl number of 1, such that νt is the appropriate mixing coefficient for both momentum and salt.
Last, the along-channel flux of salt resulting from this process scales as the product of the steady velocity and stratification scales:
i1520-0485-37-6-1645-e14
where for illustration we assume that density gradients are dominated by differences in salinity [βS/∂x = (1/ρ0) ∂ρ/∂x]. The baroclinically driven flux of salt is a strong function of the density forcing and local depth, and depends inversely with the strength of turbulent mixing resulting from tidal flows.

A similar flux is created due to periodic stratification of the water column by tidal straining (SIPS; Simpson et al. 1990). In this case, the ebb tidal flows strain the density field, again converting the longitudinal gradient into stable vertical stratification. This stratification reduces mixing and increases vertical shear. When averaged with the flood tides the residual flow and corresponding salt flux that results have the same form as the baroclinic flow (Stacey et al. 2001; Bowen and Geyer 2003). In this case, however, the scaling is less well defined due to the sensitive dependence of shear on stratification through a reduction in the vertical mixing coefficient.

c. Tidal pumping and trapping

In the classic description of bathymetrically induced exchange between two subembayments, Stommel and Farmer (1952) outline a tidal asymmetry they termed “tidal pumping.” As flow enters a subembayment on the flood tide, a momentum jet is formed that penetrates into the embayment, propagating under its own momentum. Instead of reversing this flow, the ensuing ebb tide draws flow radially toward the opening as a potential drain. This jet–drain structure results in a net exchange of scalars between the two embayments, or between the coastal ocean and an estuary (Fig. 1). Dronkers and van de Kreeke (1986) refer to this salt flux as the “nonlocal” dispersive flux and find that it is the dominant mechanism for salt intrusion in regions of complex bathymetry.

We note here that the actual form of this exchange, and, more specifically, how it will appear in flux decomposition, depends on where the analysis is applied. This was recognized by Dronkers and van de Kreeke (1986), who used a moving reference frame to evaluate this flux term. In a fixed frame, however, in the interior of the subembayment, transects across the jet–drain structure will result in a tidally averaged “steady” exchange flow, with net flow into the estuary in the center (in the jet) and net flow out of the estuary along the perimeter (influenced by the drain). This jet–drain exchange creates lateral salinity gradients, which then result in a net salt flux.

If the analysis is applied at the actual constriction instead, the velocity structure between the flood and ebb tide is relatively symmetric. In steady state, however, the net salt flux through the constriction must be equal to the flux measured just inside the embayment. In the constriction, therefore, the phasing between velocity and salinity will be shifted, with more saline flood tides and fresher ebb tides. The fact that the velocity and salinity fields are not in quadrature results in a net flux related to the phase shift between the tidal variations of the velocity and salinity.

In the region adjacent to the mouth of an estuary, the jet–drain flow structure will manifest itself in both terms of the flux decomposition (steady lateral structure, temporal correlations) resulting from partial lateral mixing of the scalar across the study location. Although both processes will appear different in the flux decomposition—one as a spatial correlation and the other a temporal correlation—they arise from essentially the same mechanism (tidal flows interacting with a bathymetric constriction), and will be considered “tidal pumping” in the discussion below.

d. Shear dispersion

In addition to the fluxes created by density-driven flows and tidal flows interacting with the local bathymetry, the combination of shear (in the vertical or lateral direction) and turbulent mixing leads to a net flux that can be loosely categorized as shear dispersion (Fischer et al. 1979). The effectiveness of this mechanism depends on the extent of cross-sectional mixing that occurs within a single tidal phase; if mixing is incomplete, the shear can actually reverse the dispersion on the ensuing tidal phase (Zimmerman 1986; Ralston and Stacey 2005).

e. Flux summary

We have introduced each estuarine flux mechanism separately, as is the traditional approach. In this presentation, tidal trapping and pumping are functions of only tidal forcing and the local bathymetry, with no influence from density forcing. Similarly, steady vertical exchange, traditionally assumed to be due to baroclinic forcing, is a function of the local depth, and does not depend on bathymetric variability in the vicinity of the measurements. As might be expected, and as documented by others (Valle-Levinson and O’Donnell 1996), however, these two processes cannot be decoupled. In this paper, we present observations from the mouth of San Francisco Bay, where we will establish that density forcing influences the magnitude of the net tidal fluxes, and bathymetric variations modify the vertical exchange.

2. Experimental setup and analysis

Velocity and scalar concentrations measurements have been made at the Golden Gate from the U.S. Geological Survey (USGS) Research Vessel (R/V) Turning Tide. The transect path was located as close to the Golden Gate Bridge as we could reliably tow our undulating profiler (Fig. 2a). The bridge spans the channel’s cross-sectional area minimum, which provides the main constraint on exchange.

Data were collected during each of the following three seasons: winter/spring runoff, summer upwelling, and fall (autumn) relaxation (Largier 1996). During the summer and fall experiments, 12-min transects across the channel were repeated continuously for 25 h for 2 days—one during spring tide and one during neap tide. Because of inadequate navigation equipment and instrument failure, the winter/spring experiment includes only daytime transects during two neap days. Combined with a diurnal tidal asymmetry, the spring experiment sampling led to a slightly biased dataset, because we happened to sample the more energetic ebb–flood tidal pair more completely than the less energetic pair.

Velocity was measured from a boat-mounted 300-kHz acoustic Doppler current profiler (ADCP; RD Instruments), which was configured to have 1-m vertical resolution. The ADCP was able to profile over the entire depth of the water column (a mean depth of 55 m), with the exception of the top and bottom few meters of the water column, because of blanking distance at the surface and sidelobe contamination at the bottom. Scalar concentrations were obtained from an undulating towed vehicle (SeaSciences Acrobat) as well as a boat-mounted CTD. The tow-yo package included instruments measuring temperature, salinity, and depth. In the energetic environment of the study site, we found that the Acrobat could reliably profile from a depth of about 10 to about 45 m. We were unable to fly over a greater vertical range because we could not turn around at the ends of our transect path without grounding the tow body if we used more than 200 m of tow cable. These depth limits provided the best compromise between vertical and lateral coverage of the channel cross section. Both the scalar and velocity fields were then interpolated and extrapolated across the cross section (Fram 2005), providing complete measurements every 12 min throughout the tidal cycles under investigation.

a. Harmonic analysis

To extract residual exchange, we have observations from a single neap and spring day. Instead of relying on a straight temporal average of the observations, which would be quite sensitive to the limits (in time) of the measurements, particularly for the temporal correlations, we rely on the integration in time of harmonic fits to the data. Specifically, we fit the observed velocity and salinity signals with harmonics at the expected tidal frequencies through adjusting each component’s amplitude and phase.

The main velocity tidal harmonics measured at a Golden Gate current station over 2 yr are M2 (12.42-h period, 106.0 cm s−1), S2 (12-h period, 28.9 cm s−1), N2 (12.66-h period, 24.8 cm s−1), K1 (24-h period, 22.2 cm s−1), 01 (25.8-h period, 18.4 cm s−1), and M4 (6.21-h period, 10.0 cm s−l) (Walters et al. 1985). To reduce the number of parameters, we chose to omit N2 from our harmonic calculations because we found virtually the same quality of fit for our data (measured by R2) by removing either it or S2. Even though it is a small part of the cross-sectional mean signal, we keep the M4 harmonic because of its role in mixing and stratification. Typical R2s for velocity and salinity are 0.97 and 0.90, respectively.

Harmonics are fit to the center 2500 m of the 3300 m of our cross section, on which a grid spanning the cross-sectional arcs follow the mean boat track. It is broken up into 17-m-deep by 250-m-wide (jxk) blocks, dimensions close to the velocity correlation lengths. As applied to this analysis, U(j, k, t) is velocity perpendicular to the grid. For each block, we minimize;
i1520-0485-37-6-1645-e15
where Ai and θi represent the amplitude and phase of the ith tidal harmonic, which has tidal frequency ωi and tidally averaged velocity A0 (i.e., frequency zero).

The multiple-parameter optimization results in estimates of the amplitude and phase for each of the tidal constituents identified at the site. To ensure convergence of the optimization, some constraints on the relative magnitude of the amplitudes were applied (requiring that the semidiurnal components be 50% greater than the other components, e.g.); the results of the optimization were not, however, pegged at these constraints, and they simply helped the gradient search converge.

To verify this harmonic approach, we used historical time series of sea surface height and salinity (J. Cuetara 2004, personal communication), which had 15-min resolution for periods of several months. These time series were sampled similarly to our observational method (two separate days—one during spring, one during neap), and the resulting harmonics compare favorably to those estimated from a continuous 2-week record (Table 1).

b. Overview of observations

The six experiments (both spring and neap tides, for three different seasons) each result in a time series of the cross-sectional velocity and salinity structure. To provide an overview of the observations, and to facilitate later discussion of some of the detailed dynamics, we present here an overview of the surface velocity and salinity for two of these six observational periods. During the summer (Fig. 3), the observations begin during a flood tide, and then encompass a weak ebb tide, a strong flood, and a strong ebb tide (Fig. 3c). This diurnal asymmetry between consecutive ebb–flood tide pairs is characteristic of the mixed tides at the site, and is well captured by the harmonic fit to the observations (Fig. 3d). From the fall experiment (Fig. 4), we present a tidal day with a more limited diurnal asymmetry, and the observations extend from a flood tide through two flood–ebb pairs (Figs. 4c and 4d).

The presence of a significant diurnal asymmetry creates a challenge for capturing the net fluxes at the study site. It is critical that we capture a representative tidal period, which requires a 24-h sampling strategy (or at least a number of days that span different portions of the tidal period). In the absence of such a sampling strategy, we would be faced with the same challenge encountered by Dronkers and van de Kreeke (1986), who had to correct their flux estimate for the nonperiodic nature of their observations. To address the representativeness of our sampling protocol, we compare the temporal average of the observations to that estimated by the harmonics (Figs. 5 and 6). If we did not, in fact, capture a representative period for the system, the straight temporal average of the data would diverge from the harmonic estimate of the mean resulting from the “extra” (or “missing”) measurements in the tidal cycle. For both salinity and velocity, the agreement between observational and harmonic means is remarkable. In the summer experiment (Fig. 5), the average salinity structure is characterized by primarily vertical gradients, but with slightly fresher conditions on the north side of the channel, and stratification of approximately 1.5 psu. In the fall experiment (Fig. 6), the structure is similar, but with greatly reduced gradients and a representative stratification scale of about 0.3 psu. In both seasons, the residual velocity field (Figs. 5c, 5d, 6c, 6d) is characterized by a double jet in the center, more inflow at depth, and strong outflow along the northern edge of the transect (see Fram 2005 for details on this structure).

3. Net fluxes

Accurately describing the net scalar flux at a location like the Golden Gate is confounded by the fact that the instantaneous flux is two orders of magnitude larger than the net flux. In this section, we compare the advective flux out of the estuary with the dispersive flux into the estuary based on a volume integration of Eq. (1):
i1520-0485-37-6-1645-e16
where the dispersive flux terms are all of the components described in Eq. (11) with area incorporated into them. The time variation of area, which is up to 4% per tidal cycle, is included in Eqs. (2)(11) by replacing each velocity with flow.

a. Advective flux

Advective flux is the mean flow times the difference in mean salinity at the upper and lower ends of the estuary. Mean salinity at the upper end is near zero, which is 20–34 psu less than that at the Golden Gate. Our mean salinity from harmonics is within 0.1 psu or 0.3% of the data mean for each season. Mean flow from our data is known to a much lower percent accuracy because it is only ∼1% of the rms instantaneous flow (∼60 000 m3 s−1). Furthermore, there is a net seaward flow out the side of our transect path, so the harmonic results systematically underpredict net advective flux. Instead of using harmonics, we estimate mean flow from a mass balance based on river and tide gauges.

The main flow contribution comes from the delta (see information online at http://cdec.water.ca.gov). Streams downstream of the delta contributed less than 2% of the inflow during each of our field campaigns. The local municipal utility district (MUD) outflow, which acts as a freshwater source, was ∼10 m3 s−1, which is 9% of the fall experiment flow, and less than 3% of the spring and summer experiment flows. The volume of the bay varies during each 30-h study because of meteorological effects (e.g., storm surges) and fortnightly or longer tidal changes. Bay volume change and its resulting effect on mean flow out of the bay is found from a low-pass filter on the bay’s five tide gauge signals. Volume change is the surface area times the elevation change, and constitutes a large contribution in the spring experiment (23%), but not the fall or summer experiments (<2%). Evaporation is a big portion of advective flux (20%) only during the fall experiment (Conomos 1979). Summing these sources gives the advective flux to within a few percent (Table 2),
i1520-0485-37-6-1645-e17
where each Q term represents a freshwater source, and the last two terms account for bay volume loss.

b. Flux comparison

The advective and dispersive fluxes should differ by the bay’s salt content rate of change (Table 3, Fig. 7). The dispersive fluxes are all oriented into the bay (positive flux in Table 3). Dispersive flux totals include cross-sectionally averaged unsteady correlations and steady lateral and vertical correlations [Eq. (11): b, c, and d]. The advective fluxes shown in Fig. 7 are directed out of the bay at all times (negative flux in Table 3). The net salt flux is into the bay during the spring and summer experiments (dispersive flux exceeds advective flux), but the reverse is true during the fall experiment. This is to be expected because both spring and summer experiments were during a period of decreasing freshwater flow into the bay, such that the bay was becoming more saline with time during those two experiments. Salinity time series from the USGS (J. Cuetara 2004, personal communication) from three stations in central San Francisco Bay have been averaged and low-pass filtered (to remove tidal variability) to define a rough estimate of the rate of change of the central bay salinity (Table 3, second to last row). During the fall experiment, freshwater flow was increasing slightly, and the bay was freshening, which is also consistent with a negative net salt flux.

Net flux values are checked quantitatively with a simple box model [Eq. (16)]. Net salt flux is equal to the change in salinity of the central bay in time. The central bay provides a reasonable volume for consideration here, resulting from constrictions between the central and north and south bay. Here, ∂〈S〉/∂t is the trend of the diurnally filtered mean of the three signals, which taken from the box model is within 30% for the spring and summer experiments (Table 2). Smaller salinity changes and instrument fouling make the ∂〈S〉/∂t uncertainty 50% for the fall experiment, while it is 5% during the wet seasons. Nonetheless, calculated net salt fluxes match the rise of the central bay salinity well, considering the short dataset length and the dynamically complex field site.

4. Dispersion mechanisms

As described in the introduction, we can associate physical processes to one or more of the terms in the flux decomposition [Eq. (11)]. In this section, we quantify each mechanism separately by integrating the harmonics associated with the various spatial structures.

a. Tidal pumping

Tidal pumping at a contraction is due to a temporal correlation of cross-sectional mean salinity and velocity. The net flux created by this mechanism is from subtle shifts in the phasing of velocity and salinity.

To be specific, we define the cross-sectional average of salinity as
i1520-0485-37-6-1645-e18
and of velocity as
i1520-0485-37-6-1645-e19
where the summation is over the five tidal harmonics described above (with frequencies ωi). Here and represent the amplitudes of the tidal variability (for harmonic i) of the cross-sectionally averaged salinity and velocity, respectively. Last, and represent the phase associated with each harmonic component. Integrating these harmonics over time gives the net tidal pumping flux, where it is seen that net flux is defined in terms of the relative phase of salinity and velocity:
i1520-0485-37-6-1645-e20

As shown in Table 3, and as predicted by Walters et al. (1985) and Walters and Gartner (1985), tidal pumping salt flux is the largest contributor to dispersive flux during all three seasons. This flux, however, scales more strongly with freshwater flow than is predicted by standard theory. In Stommel and Farmer’s (1952) classic geometric explanation of jet–drain tidal pumping, it scales linearly with the along-channel salinity gradient [Flux = ]. Tidal pumping increases with tidal excursion, and thus with rms velocity, by a proportion dependent on a channel’s geometry (Fig. 1). Spring experiment tidal pumping is greater than that of summer, consistent with it having a higher rms velocity at a nearly identical salinity gradient. Spring experiment rms velocity is higher than that of summer and fall because of a sampling bias, so our detailed look at the effect of changing freshwater flow on tidal pumping will be between just the summer and fall experiments. Summer experiment tidal pumping flux is 6.8 times that of fall despite only a 3.5 times as large , where [] is change in salinity over the tidal excursion []. Because the summer and fall experiment velocities are of the same magnitude, the tidal pumping discrepancy implies a change in the relative phase of mean salinity and velocity signals from fall to summer. We will first show how tidal pumping depends on the timing of the salinity and velocity signals, and then we will discuss two sources for a seasonal change in this timing.

We emphasize the velocity-to-salinity phase relationship by writing tidal pumping flux as a function of one tidal harmonic,
i1520-0485-37-6-1645-e21
where ϕ refers to a shift in the phase of salinity away from quadrature. To tidally pump salinity into the bay, salinity must follow velocity by less than 90°. To find this mean salinity-to-velocity phase lag, we use the harmonics generated from fits to all of each season’s mean salinity and velocity data. From a long time series recreated from the five component harmonics, we define the time lag as the time from zero crossings of [] to zero crossings of (t). Fall and summer experiment lags are 15 and 29 min, respectively, which correspond to 4.5° and 9° phase lags in a system with only M2 tides. Doubling the phase lag between seasons nearly doubles tidal pumping flux, so the phase lag change is a large portion of the total dispersive flux.
The effect of seasonal phase change on tidal pumping flux is larger than the calculated uncertainty (Table 3). Uncertainty of the tidal pumping flux is estimated as the standard deviation of the difference between measured and harmonic cross-sectional mean tidal flux,
i1520-0485-37-6-1645-e22

The fundamental process that creates the tidal pumping flux is the asymmetry between the ebb and flood salinities, as would be predicted in rough terms by most basic analysis of tidal pumping. In the remainder of this section, we discuss two processes that modify this dynamic and create seasonal variability in the tidal pumping exchange.

1) Flood–ebb tidally trapped eddy

The seasonal variation in the tidal pumping coefficient, which would normally be expected to depend only on tidal forcing, is due to a subtle shift of the relative phase between velocity and salinity between seasons. The harmonic reconstructions, however, tell us only the effective relative phasing of these variables, and do not inform an understanding of the mechanism that is creating the phase shift. To examine the underlying dynamics, we turn to the data themselves, rather than the harmonic reconstructions. During both summer and fall seasons, the surface salinity fields (Figs. 3a and 4a) display nonmonotonic variations during the ebb tide. To be specific, during the summer (Fig. 3a), at the beginning of the ebb tide at day 161.8, the salinity on the north side of the cross section decreases initially, as would be expected; but it then increases at around day 161.9. This reversal in the temporal variability at a short time scale cannot be completely captured by the harmonic reconstruction (Fig. 3b), where the effects of this detailed structure are manifested in a phase shift of the harmonics. A similar feature develops during the fall experiment (Fig. 4a) at around day 302.7, but the magnitude of the feature and the extent of its influence are greatly reduced relative to the summer experiment.

The process that creates this particular salinity variation during the ebb tides involves the creation and propagation of a tidally trapped eddy that forms during the second half of the flood tide between Point Cavallo, Angel Island, and Tiburon (Fig. 2b) (Fram 2005). Here we focus on the salinity signature of the eddy. As is evident in the surface salinity fields (Figs. 3a and 4a), the eddy is characterized by a lower salinity than that of the surrounding waters in the channel. That is, as the eddy moves into the channel early in the ebb tide (actually around slack water), the observed salinity drops abruptly (see, e.g., day 161.8 in Fig. 3a), but recovers after the eddy passes (around 161.9 in Fig. 3a). This structure and variability is also present in the fall (day 302.7, Fig. 4a), but with a much smaller difference between the salinity of the eddy and the surrounding waters.

The distinct salinity signature of the eddy creates a density difference between the eddy and the surrounding water that is stronger during the wet (Fig. 3a) than dry (Fig. 4a) seasons. This salinity difference creates a baroclinic pressure gradient that accelerates the eddy into the channel, and should provide a mechanism for seasonal variation in the phasing. A scaling of the seasonal change in barotropic acceleration shows that this mechanism is sufficient for causing the phase lag increase. Cross-channel acceleration of the eddy by the gravitational circulation is slowed by friction,
i1520-0485-37-6-1645-e23
Assuming that acceleration is constant and the eddy edges are accelerated for an hour, they will move seaward into the channel 475 m ahead of where they would be in the dry season,
i1520-0485-37-6-1645-e24
This puts the eddy 18 min ahead in the tidal cycle,
i1520-0485-37-6-1645-e25

In this analysis, friction has been neglected, because it is an order of magnitude smaller than the baroclinic forcing. In summary, density forcing moves a tidally trapped eddy 15–20 min ahead in the tidal cycle. This eddy is a major feature in exchange for more than 3 h of every M2 tidal cycle.

2) Ebb–flood frictional phasing

Tidal pumping is also strengthened during wet seasons because of increased frictional phasing around low slack tide. Mean salinity starts increasing about 40 min earlier in the tidal cycle during the summer experiment than during the fall. In Fig. 8a, the cross-sectionally averaged salinity reaches its minimum at around hour 5 (relative to a high slack tide) during the summer experiment. In the fall, the minimum in salinity is shifted later, with a minimum just after hour 6 (Fig. 8b). This shift is not accounted for by changes in the cross-sectionally averaged velocity, but rather coincides with a reversal of the near-bed velocities. The strong lateral density gradient during the wet season drives a lateral exchange flow [(dV/dz)], which increases stratification (Fram 2005). Thus, stratification at low slack is proportionally higher in the wet season by more than the along-channel density gradient [d is 3.5 times as large in the wet as in the dry season, while (dρ〉/dz) is 4.7 times as large overall and 6.8 times as large just after low slack]. This additional stratification strengthens along-channel shear and increases the vertical frictional phasing at low slack during wet seasons. As a result, most of the seasonal phase lag change in salinity is in the bottom half of the water column.

3) Tidal exchange ratio

The tidal exchange ratio (TER) is an aggregate measure of tidal exchange applied to estuary mouths where tidal pumping is the main exchange mechanism. TER through the Golden Gate has been estimated by Largier (1996) and found experimentally by Parker et al. (1972). Parker et al.’s experiment was done in early fall to maximize the proportion of tidal pumping to baroclinic exchange. Both estimates are a few times as high as our findings (Table 4 and Fig. 9), which we will explain by relating TER to the tidal pumping flux.

The tidal exchange ratio is the ratio of a flood and ebb scalar concentration difference and an oceanic and ebb concentration difference. Specifically, we follow Parker et al. (1972) and Chadwick and Largier (1999):
i1520-0485-37-6-1645-e26
We use salinity as the tracer for calculating our flood and ebb concentration averages (Table 4). For offshore salinities, we use typical depth-averaged values for non–El Niño years (Collins et al. 2002; Parker et al. 1972). Parker et al. calculated TER from repeated vertical profiles at four stations across the ocean side of the Golden Gate Channel [Point Bonita (Pt. B.)], as well as oceanic salinity values from a concurrent survey of the Gulf of the Farallones. The present fall experiment’s tidal salinity range is similar to that of Parker et al.’s, yet our difference between the mean ebb and flood salinities is much less. Parker et al.’s experiment, which predates ADCPs, understandably did not accurately measure when the tide turned. By assuming a large []-to-(t) lag, TER was overestimated at Point Bonita, which Largier (1996) calculates should have a TER roughly double that of the TER at the Golden Gate Bridge. Parker et al.’s TER result implies a tidal pumping flux 5–9 times the tidal pumping flux from our fall experiment; however, their advective flux was only 2 times that of our fall experiment advective flux. As shown in section 4, tidal pumping flux is roughly proportional to advective flux for low-to-moderate flows.

Largier (1996) suggests larger TERs than those of Parker et al. (1972) by attributing tidal exchange to ebb jet–flood drain geometric exchange (Fig. 1). In this model, the jet does not spread laterally or entrain water through its sides. Despite its hourglass surface appearance, jet–drain exchange is not as large as suggested by this idealization because the channel cross-sectional area contracts only by a factor of 2 at the Golden Gate. Similarly, the ocean side of the channel is much deeper than the horseshoe-shaped ebb bar, which bounds half the volume of the tidal prism. The jet spreads laterally after going through both of these contractions because the channel shallows. Lateral spreading of the jet reduces the jet–drain geometric exchange.

b. Steady fluxes

Steady vertical and lateral fluxes make up 18%–26% of the total flux for each season (Fig. 7). Steady vertical fluxes are 10 times as large in the summer as in the fall, and our estimates of spring steady vertical fluxes are another factor of 2 larger than those of the summer. We note, however, that the spring values are biased by sampling primarily during energetic tidal phases. The lateral steady fluxes show a similar, but weaker, seasonal variation, with summer values exceeding fall values by a factor of about 5.

We estimate the effects of experimental error on the flux components by their sensitivity to changes in harmonic constraints and changes in the number of harmonic parameters, and from a Monte Carlo simulation based on the variances of the residuals between the velocity and salinity data and the harmonic fits. Uncertainty values from sensitivity and Monte Carlo analyses are less than or equal to those in Fig. 7, which are calculated by propagating the uncertainty of the same residuals (Taylor 1997). For steady vertical flux,
i1520-0485-37-6-1645-e27
where “υ” indexes 17-m vertical blocks and σ2Rsυ is the variance of the residual of the horizontally averaged salinity at a vertical bin “υ.” Uncertainty from pumping is smaller because spatial averaging over a larger area reduces noise, and because the mean tidal velocity and salinity signals are stronger than the residual flow signals.

Uncertainties are small enough for us to glean some physical results from the steady fluxes. The proportion of steady vertical exchange to steady lateral exchange increases with outflow, as it should, because freshwater outflow increases baroclinic exchange more than it increases tidal pumping exchange. Tidal pumping creates steady lateral exchange from the net seaward flow of fresher water through the transect sides and net landward flow of saline water through the transect center. In terms of total steady flux to unsteady (tidal pumping) flux, summer and fall proportions of tidal pumping flux are indistinguishable within uncertainty, while spring has proportionally more steady shear. Spring and summer experiment density forcing is the same, so one might expect them to have the same proportion of tidal pumping flux; however, higher spring experiment velocities increase steady shear (Fram 2005).

c. Unsteady shear flux

Unsteady lateral and vertical shear fluxes are shown in Fig. 7, but are not significantly different from zero (based on their uncertainties) because the high-frequency variability causing these fluxes is not well resolved by the harmonic analysis. Despite these uncertainties, there appears to be a positive unsteady lateral shear flux and no significant unsteady vertical shear flux.

5. Dispersion coefficients

Seasonal scaling of the flux mechanisms can be estimated with bulk along-channel dispersion coefficients [Flux = ]. During the fall and summer experiments, the dispersion coefficients are 380 and 720 m2 s−1, respectively (Table 5). We put our Kx values in context by comparing them to earlier estimates and to standard scalings of Kx. Monismith et al. (2002) calculated the bulk dispersion coefficient [Kx(x, Q)] along the axis of the San Francisco estuary over a range of delta inflow rates from 30-yr datasets of monthly along-axis salinity (USGS 2004) and daily freshwater flow (information online at http://cdec.water.ca.gov). And, although Monismith et al. (2002) did not suggest that their approach would extend to the mouth of the bay, where they fixed the salinity as their downstream boundary condition, Kx from the model at our transect path is within 33% of this experiment’s values for the same seasonal freshwater flow rates (Table 5).

Monismith et al. (2002) described [Kx(x)] as a function of several exchange mechanisms: gravitational circulation, bathymetrically induced exchange (e.g., tidal pumping), and SIPS. To separate Kx by mechanism, we focus on its scaling with the along-channel salinity gradient [],
i1520-0485-37-6-1645-e28
where n is 0 for purely tidal processes, such as standard tidal pumping and trapping. Here, n will be positive for those processes that are a function of density forcing. For example, n is 2 for classical baroclinic circulation. Monismith et al. (2002) found that salt flux along the estuary scales more strongly with flow than could be achieved by either purely tidal or classical baroclinic circulation mechanisms. In other words, mean n is greater than 2. While the result that is greater than 2 for the entire bay is valid, at the Golden Gate we find a weaker dependency of Kx on dS/dx.

From the model of Monismith et al. (2002), we solve for n in our range of flows (100–1000 m3 s−1) 1.5 km from the Golden Gate; n is 6.2 in this model, which is considerably more than the n of 0.51 found from combining our fall and summer data (the spring estimate was too uncertain for inclusion). Model results for Kx are consistent with our data, so it is divergent dS/dx values that cause the disagreement in dispersion scaling (Table 5). By fixing Golden Gate salinity to 32, the model overestimates dS/dx. As shown in this experiment, mean salinity 1.5 km from the Golden Gate drops significantly (by 2) even under moderate flows (406 m3 s−1). Thus, the Golden Gate Bridge is not strictly the downstream boundary of the salinity field.

We now examine how each flux mechanism contributes to our scaling of the bulk dispersion coefficient. As described in section 4a, tidal pumping increases with flow more than a purely tidal process because of density-driven effects. Thus, n for tidal pumping is greater than 0 (0.54). For steady lateral exchange n is also greater than 0 (0.32), but for a different reason; increases proportionally more than between our fall and summer datasets because of a north-to-south bay salinity gradient created by the south bay’s longer salinity recovery time after spring runoff (Fram 2005). The steady lateral velocity gradient is essentially the same across the seasons, so the rise from dry to wet seasons in the steady lateral dispersion coefficient comes entirely from a rise in lateral salinity gradients.

In the vertical direction, n for steady shear flux (0.50) is much less than n from standard baroclinic scaling (2) because most of the net vertical shear and stratification derive from asymmetric flow over a sill, which is forced more by tides than by density gradients (Fram 2005).

Density forcing plays a role in generating the tidal pumping flux in addition to the vertical steady shear flux, so the vertical steady shear flux is not equal to the density-driven flux. Instead, we estimate the overall seasonal baroclinic flux by assuming that the density-driven fluxes scale with -like baroclinic exchange (n = 2), and that the remaining flux is strictly tidal (n = 0). From this assumption, the dispersion coefficient scaling of n = 0.51, and for the seasonal values we estimate that approximately 10% of the fall flux and nearly 50% of the summer experiment flux are density driven.

6. Summary

The net flux of salt between a coastal estuary and the ocean has been quantified using cross-sectional observations of velocity and salinity. The tidally averaged net dispersive salt flux for each season was validated with a box model using independently collected data. Decomposing the dispersive flux by temporal and spatial correlations showed that the temporal correlation of cross-sectionally averaged salinity and velocity (tidal pumping flux) is the largest part of the dispersive flux of salinity into the bay. From the tidal pumping flux portion of the dispersive flux, it was shown that there is less exchange through the Golden Gate than was found in earlier studies.

Ocean–estuary exchange is often described as a combination of two independent mechanisms: density-driven exchange, such as gravitational circulation, and tidal processes, such as shear dispersion, tidal pumping, and tidal trapping. In this study we found that exchange is also governed by an interaction between these mechanisms. Tidally trapped eddies created in shallow shoals are mixed into the main channel earlier in the tidal cycle during the wet season because the eddies are pushed seaward by gravitational circulation. The resulting increase in tidally averaged dispersive salt flux into the bay indicates that a dispersion coefficient scaling for tidal pumping flux based only on tidal forcing will be inadequate. Instead, an appropriate dispersion coefficient to describe tidal pumping will require some dependence on density forcing. In contrast, the vertical steady shear exchange, which is usually assumed to be baroclinic, was found to also be a result of bathymetric forcing from a sill adjacent to the study site, as well as density-driven increases in ebb–flood frictional phasing. This results in a weaker scaling of this flux component with density forcing than would be expected with a baroclinic approach.

Taken together, we have established that density forcing and bathymetry both influence ocean–estuary exchange, as would be expected. The two forcing mechanisms cannot be decoupled, however, and both enter into quantification of the tidal pumping and vertically sheared steady flux.

Acknowledgments

The data described in this paper were taken from the USGS R/V Turning Tide, operated by Jay Cuetara, Jon Yokomizo, Chris Smith, and Curt Battenfield. Field support was provided by David Ralston, James Gray, Stefan Talke, Deanna Sereno, and Sarah Giddings. This research was funded by California SeaGrant R/CZ-170 to Mark Stacey and Thomas Powell and by NSF OCE-0094317 to Mark Stacey.

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Fig. 1.
Fig. 1.

In this standard schematic of tidal pumping, a jet of water exits the contraction during one-half of the tide and a potential drain of water enters during the opposite half.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 2.
Fig. 2.

Experiment location: (a) a broad map of the region and (b) an inset, which includes a sketch of an eddy that forms during flood tide. Both have 10-m contours. The transect path is 1–1.5 km east of the Golden Gate Bridge, which is at the narrowest part of the channel contraction.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 3.
Fig. 3.

Summer experiment day-1 surface salinity (psu) (a) observations compared with (b) harmonic reconstruction, and velocity (m s−1) (c) observations compared with (d) harmonic reconstruction as a function of day of year and lateral position.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 4.
Fig. 4.

Same as Fig. 3, but for fall experiment day 2.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 5.
Fig. 5.

Summer experiment cross-sectional structure of (a) time-averaged salinity observations and (b) temporal mean from harmonic reconstruction. (c) Time-averaged velocity observations also agree with (d) the temporal mean of velocity from harmonic reconstruction.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 6.
Fig. 6.

Same as Fig. 5, but for fall experiment.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 7.
Fig. 7.

Flux decomposition. For each experiment (top to bottom), dispersive flux is decomposed into four components (left to right bars): tidal pumping, steady vertical exchange, steady lateral exchange, and unsteady shear dispersion. The sum of the first three flux components is presented as the right-hand bar (“1–3”). The magnitude of the advective flux is shown as a solid horizontal line for comparison with the calculated dispersive fluxes.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 8.
Fig. 8.

Comparison of cross-sectionally averaged salinity and velocity for (a) summer and (b) fall experiments. All data from each season are lined up by the semidiurnal zero crossings of along-channel mean velocity, and then block averaged.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Fig. 9.
Fig. 9.

Both tidal pumping and TER increase with the phase lag ( f, °) between mean flow [Qosin(wt)] and salinity [−Sosin(wt + π/2 + f )]. Flow and salinity amplitudes (Qo, So) are from the fall and summer experiments; w is from the M2 tide. The TER from Largier (1996) and Parker et al. (1972), estimated for fall conditions, imply much higher tidal pumping flux and much higher phase lag than those in our experiments.

Citation: Journal of Physical Oceanography 37, 6; 10.1175/JPO3078.1

Table 1.

Harmonic verification. Harmonic fits to 2 days of data spaced 1 week apart produces similar results as harmonic fits to two continuous weeks of data. The second row of each section is the R2 value for 2 weeks of data against a 2-week time series generated from the 2-day harmonic fit. Spring R2 values are lower because data were taken on consecutive days instead of days 1 week apart.

Table 1.
Table 2.

Freshwater flow calculations (m3 s−1).

Table 2.
Table 3.

Comparison of net advective flux with aggregate dispersive flux. Temporal salinity derivative calculated from USGS data (second to last row) and based on simple box model with observed fluxes (last row). Fluxes: thousands of psu m3 s−1; salinity derivatives: psu s−1.

Table 3.
Table 4.

Tidal exchange ratios based on Eq. (26). “FOGG” refers to the experiments analyzed here (“24hr*2” indicates two 24-h experiments; “12hr*2” indicates two 12-h experiments). Salinity: psu; tidal range: m. “Parker” refers to Parker et al. (1972).

Table 4.
Table 5.

Dispersion coefficients inferred at the mouth of San Francisco Bay. “FOGG” refers to this analysis. The exponent n is defined in Eq. (28).

Table 5.
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