## 1. Introduction

Subtidal (i.e., residual or tidally averaged) dynamics are of interest when examining transport in estuarine environments (Fischer et al. 1979). The subtidal dynamics of well- and partially mixed estuarine systems have focused on gravitational circulation (inflow at depth and outflow at the surface) resulting from a balance between tidally averaged baroclinic and barotropic forcing, and mixing associated with the barotropic tides (e.g., Pritchard 1954, 1956; Hansen and Rattray 1965, hereafter HR; MacCready and Geyer 2010). HR solved for residual velocity and salinity profiles, assuming that the tidal amplitude was small relative to the mean depth and that the eddy viscosity was constant. Since the initial analytical solutions of HR, field measurements, laboratory experiments, analytical studies, and numerical modeling have been applied to better understand subtidal estuarine dynamics (e.g., Hetland and Geyer 2004; MacCready 2004).

Unfortunately, establishing consistent models across all estuaries is difficult. In fact, recent studies have explored the importance of unsteadiness (e.g., Kranenburg 1986; Trowbridge et al. 1999; Geyer et al. 2000; Monismith et al. 2002), nonlinearities (e.g., Lacy et al. 2003; Lerczak and Geyer 2004; Scully et al. 2009), and bathymetric complexity (e.g., Valle-Levinson et al. 2000; Scully and Friedrichs 2007; Li and O'Donnell 1997) influencing residual circulation, showing that deviations from the HR balance are common. Importantly, it has become apparent that subtidal balances often reflect tidal dynamics such as asymmetries in mixing (Jay and Smith 1990; Jay and Musiak 1994, 1996). An important aspect of intratidal (within a tidal cycle) estuarine dynamics that can influence residual dynamics is straining-induced changes in the density field (e.g., Prandle 2004; Uncles 2002). Straining during ebb enhances stratification, which simultaneously suppresses turbulence and leads to enhanced shear. This strain-induced periodic stratification (SIPS; Simpson et al. 1990) generates a residual circulation in the same sense as that driven by a longitudinal density gradient. This mechanism can create shear circulation pulses on both the tidal time scale and the spring–neap time scale (Stacey et al. 2001) and can dominate residual circulation (e.g., Burchard and Baumert 1998; Burchard and Hetland 2010). Despite these extensive shortcomings, the HR solution is often revisited and is surprisingly effective in describing observed residual circulation profile shapes (MacCready and Geyer 2010).

Many estuarine theories rely on the assumption that the tidal range Δ*η* relative to the mean water depth *h*_{0} is small—that is, Δ*η*/*h*_{0} ≪ 1. However, given that many coastal plain and bar-built estuaries are characterized by Δ*η*/*h*_{0} > 0.3 (Kjerfve 1975), understanding the dynamics of estuaries that do not fit this assumption is useful. While relatively few studies of estuaries with large Δ*η*/*h*_{0} have been reported, work done thus far in such systems has shown the importance of unsteadiness and nonlinearities for residual circulation (e.g., Banas et al. 2004; West et al. 1990) as well as the importance of asymmetric mixing (Uncles and Stephens 1990).

As Uncles and Jordan (1980) discuss, the interpretation of residual velocities in a system with a large Δ*η*/*h*_{0} depends on differentiating between time averages at particular points in space (Eulerian means) and time averages following fluid particles (Lagrangian means). The difference between these two averages is defined as the Stokes drift (Longuet-Higgins 1969). While this has been recognized for several decades, many studies continue to inaccurately represent the residual circulation and salt fluxes solely with the Eulerian mean. The importance of this distinction is easily seen: for example, without freshwater input, the depth-integrated landward Stokes drift will drive an equal and opposite depth-integrated oceanward Eulerian flow (Uncles and Jordan 1980). While in some estuaries this Stokes drift component may be small, in estuaries that are dynamically shallow and/or have strong tidal forcing, Eulerian and Lagrangian means will be different (Godin 1995), complicating inferences of particle transport from observations made at fixed points in space (Uncles and Jordan 1980; Kuo et al. 1990).

A potentially useful approach to interpret residual circulation in an estuary with a large Δ*η*/*h*_{0} is to convert to a depth-normalized *σ*-coordinate system so long as one is careful to maintain a volume-conserving residual. This has been employed for residual velocities (e.g., Kuo et al. 1990) and salt fluxes (e.g., Lerczak et al. 2006; Dronkers and van de Kreeke 1986). Although some researchers have warned against this approach (e.g., Rattray and Dworski 1980), several have suggested its benefits (e.g., Kjerfve 1975; Geyer and Nepf 1996); yet, none have fully described the benefits and drawbacks of this approach.

In addition to being strongly forced, the estuary investigated in this paper is strongly stratified and exhibits a salt wedge that advects into and out of the estuary during every tidal cycle. As a result, another common assumption for partially and well-mixed estuaries—that the background horizontal stratification is constant in time and space—is not valid. Important distinguishing features of strongly forced salt-wedge systems include the tendency to be highly variable on a tidal time scale (e.g., Ralston et al. 2010; Giddings et al. 2011), different temporal and spatial patterns of turbulent vertical mixing (e.g., Geyer et al. 2008; MacDonald and Horner-Devine 2008; Tedford et al. 2009), and a decreased importance of baroclinic forcing (e.g., Ralston et al. 2010). Despite these recent investigations, the subtidal dynamics of these systems are not well understood, although it is suspected that intratidal dynamics will influence residual dynamics and a “traditional” baroclinically driven circulation will not fully explain the shear residuals.

In this paper, we examine residual circulation in the Snohomish River estuary (SRE), a strongly stratified, shallow, macrotidal estuary where the tidal amplitude is comparable to the tidal-mean water depth (Δ*η*/*h*_{0} = ~1). We investigate the proper methods with which to calculate residual circulation and examine its driving mechanisms. We start by describing the field site and instrumentation (section 2). We then investigate the kinematics of the residual circulation profiles (section 3a) including fixed vertical [section 3a(1)] and *σ* coordinates [section 3a(2)]. We show that the mass transport velocity in *σ* coordinates is a first approximation to the Lagrangian-mean velocity, as it incorporates the Stokes drift wave transport velocity [section 3a(3)]. In section 3b, we employ an empirical orthogonal function (EOF) analysis to help describe the residual profiles and split the residual into barotropic and shear parts. We investigate the terms driving these complex residual circulation profiles by analyzing the tidal momentum equations in both fixed and *σ* coordinates [section 3c(1)] and a residual momentum balance in *σ* coordinates [section 3c(2)]. Finally, the mechanisms driving the observed residual circulation are discussed in section 4. To accomplish this, we compare our results to an approximate HR-type solution in *σ* coordinates (sections 4a and 4b) and refer to our previous analyses of the intratidal system variability (section 4c; Giddings et al. 2011; 2012). Our findings are summarized in section 5.

## 2. Field site and instrumentation

### a. SRE description

The SRE is a shallow, macrotidal, strongly stratified estuary located in the second-largest drainage basin in Puget Sound (Fig. 1). The main channel curves 180° around the reinforced coastline of Everett, Washington, delineated on the west by Jetty Island before connecting to Possession Sound. Intertidal mudflats and marshes lie to the north and west of the main channel and a second, intertidal, connection to Possession Sound occurs over the mudflats north of Jetty Island, referred to as the mudflat bypass (Fig. 1c).

The *M*_{2} semidiurnal and *K*_{1} diurnal tidal constituents are dominant, resulting in a strong fortnightly cycle (which we will refer to as spring–neap, although it is technically due to the lunar declination cycle) and strong diurnal inequality. The SRE tidal range varies from 3- to 4.5-m driving tidal currents exceeding 1.2 m s^{−1}. The main channel is ~2.5 m below mean lower-low water (MLLW), which leaves only ~1.5–2 m of water in the channel at lower-low water (LLW) and 6 m at higher-high water (HHW). The value of Δ*η*/*h*_{0} ranges from approximately 0.7 (neap) to 1 (spring).

### b. Instrumentation

The observations we present were taken during the Office of Naval Research (ONR)-sponsored Coherent Structures in Rivers and Estuaries Experiment (COHSTREX) program (see, e.g., Giddings et al. 2011; Chickadel et al. 2009; Plant et al. 2009; Wang et al. 2009, 2011). The present paper focuses on results from the in situ measurements conducted 5–26 July 2006. The instrument locations during the experiment discussed in this manuscript are mapped in Fig. 1c and include the moored bottom-mounted, upward-looking acoustic Doppler current profilers (ADCPs), bottom pressure sensors at M3B and M6, and conductivity–temperature–depth (CTD) profiles along the center of the estuary. Details of the deployment and instrument configurations can be found in Giddings et al. (2011), the appendix, and section 3a(1).

### c. Meteorological, riverine, and tidal forcing

Data presented in this paper are during typical summertime through early autumn conditions with a median river discharge ~100 m^{3} s^{−1}, minimal precipitation, and light diurnal winds. The observed water level, depth-averaged along-stream velocity, and vertical stratification are presented in Fig. 2. Sampling started near the end of a neap tide and included one full spring tide and one full neap tide into the following spring. The transect survey times and averaging periods discussed in this manuscript are marked. Specifically, we incorporate three different averaging periods to examine residual circulations: one spring tide (shaded dark gray, yearday ~192.3–193.5), one neap tide (shaded dark gray, ~198.3–199.5), and a 14-day spring–neap period (shaded light gray, ~197–201).

Overall conditions during July 2006 experiment. (a) Snohomish River discharge (m^{3} s^{−1}) measured at the U.S. Geological Survey (USGS) gauge station 12150800. (b) Water depth (meters above MLLW) at mooring M3B. (c) Depth-averaged along-stream velocity ^{−1}) at mooring M3B. Positive velocity corresponds to a flood current, or into the estuary. (d) Vertical stratification at mooring M3B represented by the buoyancy frequency *N*^{2} (s^{−2}) utilizing the near-surface and near-bottom CTDs. Yearday 186 corresponds to 0000 Pacific daylight time (PDT) 6 Jul 2006. The light gray shading indicates the 14-day spring–neap cycle over which some of the tidal averaging was applied. Gray lines indicate the times of the spring and neap survey periods, respectively, while the gray bars within those lines indicate the ~24-h period over which averaging for individual spring and neap surveys was applied.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Overall conditions during July 2006 experiment. (a) Snohomish River discharge (m^{3} s^{−1}) measured at the U.S. Geological Survey (USGS) gauge station 12150800. (b) Water depth (meters above MLLW) at mooring M3B. (c) Depth-averaged along-stream velocity ^{−1}) at mooring M3B. Positive velocity corresponds to a flood current, or into the estuary. (d) Vertical stratification at mooring M3B represented by the buoyancy frequency *N*^{2} (s^{−2}) utilizing the near-surface and near-bottom CTDs. Yearday 186 corresponds to 0000 Pacific daylight time (PDT) 6 Jul 2006. The light gray shading indicates the 14-day spring–neap cycle over which some of the tidal averaging was applied. Gray lines indicate the times of the spring and neap survey periods, respectively, while the gray bars within those lines indicate the ~24-h period over which averaging for individual spring and neap surveys was applied.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Overall conditions during July 2006 experiment. (a) Snohomish River discharge (m^{3} s^{−1}) measured at the U.S. Geological Survey (USGS) gauge station 12150800. (b) Water depth (meters above MLLW) at mooring M3B. (c) Depth-averaged along-stream velocity ^{−1}) at mooring M3B. Positive velocity corresponds to a flood current, or into the estuary. (d) Vertical stratification at mooring M3B represented by the buoyancy frequency *N*^{2} (s^{−2}) utilizing the near-surface and near-bottom CTDs. Yearday 186 corresponds to 0000 Pacific daylight time (PDT) 6 Jul 2006. The light gray shading indicates the 14-day spring–neap cycle over which some of the tidal averaging was applied. Gray lines indicate the times of the spring and neap survey periods, respectively, while the gray bars within those lines indicate the ~24-h period over which averaging for individual spring and neap surveys was applied.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Under these conditions, for each tidal cycle, the system is essentially reset to initial conditions, where the channel is entirely fresh and riverine water just before LLW. On the ensuing strong flood tide, the salt wedge intrudes, creating strong vertical and horizontal density gradients. During the weak ebb, the salt wedge is strained and vertical stratification increases. Stratification is slightly decreased on the weak flood, only to be enhanced again during the strong ebb tide before the salt wedge is advected out of the estuary. See Giddings et al. (2011) for more details on the general SRE circulation and mixing dynamics.

## 3. Results

Here, we describe the observations of currents and density during the experiments in both fixed vertical and *σ* coordinates. The *σ* coordinates allow us to approximate the residual mass transport velocity, perform an EOF analysis, investigate a momentum balance in detail, and compute a tidally averaged momentum balance.

### a. Residual currents

#### 1) Fixed vertical coordinates

The data are recorded in a fixed vertical coordinate system, that is, equally spaced depth bins throughout the water column (*z* coordinates). The time-varying velocity and density are displayed in Fig. 3. The first bin of the ADCP data is at ~1 m above the bed and the top bin is ~0.25 m beneath the water surface (marked by the dashed lines in Fig. 3). The 1-Hz, 0.25-m-depth interval velocity measurements are averaged to 10-min intervals and extrapolated to the bed using a cubic spline extrapolation to zero at the bed (i.e., a no-slip condition), because it resulted in the smoothest and most reasonable near-bed shear profiles (log profiles were invalid because of strong stratification). The velocity is extrapolated to the surface, assuming no stress (i.e., that the velocity shear goes to zero at the surface), by fitting a parabola with zero slope at the surface to the two data points closest to the surface (e.g., Uncles et al. 1985). The density shown is from half-hourly CTD casts near M3B (subsampled to 10-min intervals), which reached within ~0.2 m of the bed and ~0.5 m of the surface. The 5–10 cm–resolution density profiles are bin averaged to 0.25-m bins and extrapolated to both the bed and the surface using the same parabolic fitting technique assuming no diffusive density flux (or density gradient) near the bed or the surface. These extrapolations become particularly important when converting to a *σ*-coordinate system.

Velocity and density in *z* coordinates. (a),(b) Velocity (m s^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m^{−3}) from the CTD casts between M3A and M3B are shown for the spring and neap survey periods (indicated in Fig. 2). The dotted lines near the surface and the bed indicate the regions of extrapolated data. Positive velocity corresponds to a flood current, or into the estuary. Missing density data during the neap survey was due to instrument malfunction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Velocity and density in *z* coordinates. (a),(b) Velocity (m s^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m^{−3}) from the CTD casts between M3A and M3B are shown for the spring and neap survey periods (indicated in Fig. 2). The dotted lines near the surface and the bed indicate the regions of extrapolated data. Positive velocity corresponds to a flood current, or into the estuary. Missing density data during the neap survey was due to instrument malfunction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Velocity and density in *z* coordinates. (a),(b) Velocity (m s^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m^{−3}) from the CTD casts between M3A and M3B are shown for the spring and neap survey periods (indicated in Fig. 2). The dotted lines near the surface and the bed indicate the regions of extrapolated data. Positive velocity corresponds to a flood current, or into the estuary. Missing density data during the neap survey was due to instrument malfunction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

The traditional method to compute residual circulation in an estuary is to average the velocity at given depths over a tidal cycle (or to perform a low-pass filter to remove the tides). Uncles and Stephens (1990) suggest that to conserve mass, the Eulerian residual current above LLW should be the average current during the time water is present times the fraction of time water is at that depth. Profiles of residual Eulerian velocities in this traditional *z*-coordinate system with mass conserved are complicated: exhibiting out-estuary flow at the bed, small up-estuary flow near the surface, and a gravitational circulation–like profile between LLW and just beneath HHW [similar in structure to observations by Uncles and Stephens (1990)]. This complex vertical structure is likely caused by the interaction of tidal nonlinearities (Ianniello 1979) with shear circulation (Uncles and Stephens 1990). In the SRE, water level and velocity are 77° out of phase, indicating a slightly progressive wave. Consequently, near-surface currents are always flooding at HHW, creating a near-surface up-estuary residual. Similarly, currents beneath LLW are always strongly ebbing, creating a near-bottom out-estuary residual.

#### 2) Depth-normalized coordinates

*z*coordinates cannot represent the path of a particle because at any depth above LLW, the residual includes times when there is no water at that depth and Stokes drift is missing. Therefore, as an alternative approach to compute residual circulation, we employ a

*σ*-coordinate system and compute the residual mass transport velocity. We transform our data to

*σ*coordinates, where

*σ*=

*z*/

*D*and

*D*is the time-varying total water depth. We define the tidal mean of

*D*,

*η*, such that

*D*=

*h*

_{0}+

*η*(angle brackets indicate a mean over one or several tidal cycles or similarly, a low-pass tidal filter). The resulting coordinate system has

*z*= 0 at the bed and

*z*=

*D*at the free surface, which corresponds to

*σ*= 0 and

*σ*= 1, respectively. It should be noted that the velocity in

*σ*coordinates

*u*

_{σ}does not conserve flux; thus, we must define a volume-conserving velocity. The mass transport flow between the bed and any streamline

*D*′ is

*D*′ =

*f*

_{wc}(

*D*), then we can define the residual mass transport flow between the bed and any fraction of the water column

*f*

_{wc}, which is equivalent to choosing

*σ*levels. Converting to

*σ*coordinates and differentiating with respect to

*σ*gives the mass transport flow rate as a function of

*σ*:

*u*

_{M}as a function of

*σ*(which is shown in Fig. 4):

*σ*coordinates are then computed by averaging along constant

*σ*levels. This is equivalent to averaging along streamlines if the streamlines were equally spaced throughout the water column at all time. While this is exact at the surface and bed, streamlines may not always coincide with

*σ*coordinates.

Mass transport velocity and density in *σ* coordinates. (a),(b) Mass transport velocity (*u _{σ}D*/

*h*

_{0}; m s

^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m

^{−3}) from the CTD casts between M3A and M3B are shown for the spring (left) and neap (right) survey periods (indicated in Fig. 2) in

*σ*coordinates. Compare to

*z*coordinates, Fig. 3.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Mass transport velocity and density in *σ* coordinates. (a),(b) Mass transport velocity (*u _{σ}D*/

*h*

_{0}; m s

^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m

^{−3}) from the CTD casts between M3A and M3B are shown for the spring (left) and neap (right) survey periods (indicated in Fig. 2) in

*σ*coordinates. Compare to

*z*coordinates, Fig. 3.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Mass transport velocity and density in *σ* coordinates. (a),(b) Mass transport velocity (*u _{σ}D*/

*h*

_{0}; m s

^{−1}) from the ADCP at mooring M3B and (c),(d) density anomaly (kg m

^{−3}) from the CTD casts between M3A and M3B are shown for the spring (left) and neap (right) survey periods (indicated in Fig. 2) in

*σ*coordinates. Compare to

*z*coordinates, Fig. 3.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

#### 3) Approximate Lagrangian residual currents

**u**

_{E}is used to approximately compute the net transport of particles, larvae, etc. However, the velocity-influencing mass transport should be defined as the Lagrangian residual velocity

**u**

_{L}, which is the sum of the Eulerian residual

**u**

_{E}and Stokes

**u**

_{S}velocities (Longuet-Higgins 1969):

*x*), this equation becomes

*u*,

*υ*,

*w*) are the velocity components in the along-stream (i.e.,

*x*), cross-stream (i.e.,

*y*), and vertical (

*z*) directions (Longuet-Higgins 1969). The first term on the right-hand side of Eq. (6) represents the wave transport velocity portion of the Stokes drift

*u*

_{SW}(Kuo et al. 1990).

*σ*coordinates,

*u*

_{E}(

*σ*), and the second term is the wave transport velocity portion of the Stokes drift

*u*

_{SW}(

*σ*) [i.e., the first term on the right-hand side of Eq. (6)]. Thus, we see that

Nevertheless, the residual mass transport velocity provides a truncated form of the Lagrangian residual velocity in *σ* coordinates. In addition, this approach provides a form for the residual in *σ* coordinates that conserves mass, while the *σ*-coordinate Eulerian residual does not. Residual profiles of 〈*u*_{M}(*σ*)〉, *u*_{E}(*σ*), and *u*_{SW}(*σ*) computed from Eq. (7) are shown in Fig. 5. The Stokes wave transport velocity is always upriver and is stronger during spring tides, which is to be expected because of the partially progressive nature of the tidal wave. The inclusion of this upstream transport results in a residual mass transport velocity profile with an enhanced two-layer circulation compared to the Eulerian residual [similarly observed by Kuo et al. (1990)]. This is significant: it suggests that in a tidally averaged sense, particles near the bed may be directed upriver, while particles near the surface may be directed downriver. This is certainly not evident from the *z*-coordinate residuals and suggests that even the *σ*-coordinate Eulerian residuals are misleading without inclusion of Stokes wave transport. Although, as stated above, the results here should be approached with some caution because of the exclusion of terms in the Stokes drift due to spatial variations in the velocity field.

Depth variations of residual mass transport, Stokes wave transport, and Eulerian velocities. Depth-varying residual mass transport velocity *u*_{SW}; light gray), and Eulerian residual (*u*_{E}; gray) at mooring M3B in depth-normalized coordinates [Eq. (7)] averaged over a full 14-day spring–neap cycle (solid lines). Dotted lines indicate values averaged over just the spring tide survey, while dashed lines indicated values averaged over just the neap tide survey.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Depth variations of residual mass transport, Stokes wave transport, and Eulerian velocities. Depth-varying residual mass transport velocity *u*_{SW}; light gray), and Eulerian residual (*u*_{E}; gray) at mooring M3B in depth-normalized coordinates [Eq. (7)] averaged over a full 14-day spring–neap cycle (solid lines). Dotted lines indicate values averaged over just the spring tide survey, while dashed lines indicated values averaged over just the neap tide survey.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Depth variations of residual mass transport, Stokes wave transport, and Eulerian velocities. Depth-varying residual mass transport velocity *u*_{SW}; light gray), and Eulerian residual (*u*_{E}; gray) at mooring M3B in depth-normalized coordinates [Eq. (7)] averaged over a full 14-day spring–neap cycle (solid lines). Dotted lines indicate values averaged over just the spring tide survey, while dashed lines indicated values averaged over just the neap tide survey.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

### b. EOF analysis

EOFs provide a tool to analyze spatial and temporal variability in data. The procedure finds a set of basis functions that are linearly independent (orthogonal) and whose amplitudes are uncorrelated (see Emery and Thomson 2004). EOF analysis is strictly a statistical construct that may not always be linked to physical mechanisms; however, it often correlates with physical mechanisms and can be used as a tool to examine oceanographic and estuarine processes (e.g., Stacey et al. 2001) or to act as a filter to remove specific scales of variability.

EOF analysis requires no missing data; therefore, application of EOFs to the original *z*-coordinate data could only be applied to the water column beneath LLW or with zeros in place of data above LLW. The former provides little information in a shallow, macrotidal estuary such as the SRE, while the latter results in EOFs that are difficult to interpret. Therefore, we apply an EOF analysis to the time series of volume-conserving mass transport velocity in *σ* coordinates, *u*_{σ}*D*/*h*_{0}, at individual moorings. This analysis results in a set of depth-varying basis functions (or eigenfunctions) in *σ* coordinates whose amplitudes vary in time. The first four EOFs for mooring M3B are displayed in Fig. 6. EOFs 1 and 2 account for over 98% of the velocity variance and although as noted above are statistical constructs, they correspond well to barotropic-like and two-layer shear circulation modes. Here, we use “barotropic like” in reference to velocities driven by the tides and river flow (i.e., the barotropic pressure gradient) modified by bottom friction. In light of the fact that a two-layer shear profile such as EOF 2 can be driven by various different mechanisms to be discussed shortly, we will refer to this mode as the shear exchange or two-layer shear mode, where positive shear is defined in the sense expected from gravitational effects (i.e., out estuary at the surface and in estuary at depth). The time series of the first two EOF amplitudes with the depth-mean velocity and along-stream density gradient, respectively, are presented in Fig. 7. EOF 1 resembles the depth-averaged velocity closely with a correlation of *r*^{2} = 1.00 (*N*_{eff} = 131 and *p* < 0.0001, where *N*_{eff} is the effective degrees of freedom based on the integral time scale; Emery and Thomson 2004), strongly suggesting that EOF 1 represents the barotropic (tidal plus riverine) circulation. EOF 2 is not statistically significantly correlated with the longitudinal density gradient (*r*^{2} = 0.001, *N*_{eff} = 31, and *p* = 0.44).

EOFs 1–4 at mooring M3B. Structure functions for the first four EOFs of the mass transport velocity [*u*_{M}(*σ*)] at mooring M3B in *σ* coordinates and the percent variance explained by each.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOFs 1–4 at mooring M3B. Structure functions for the first four EOFs of the mass transport velocity [*u*_{M}(*σ*)] at mooring M3B in *σ* coordinates and the percent variance explained by each.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOFs 1–4 at mooring M3B. Structure functions for the first four EOFs of the mass transport velocity [*u*_{M}(*σ*)] at mooring M3B in *σ* coordinates and the percent variance explained by each.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOF amplitudes and driving physical forces. (a),(b) First EOF amplitude [(5)^{−1}; dotted line] along with the depth-averaged along-stream mass transport velocity *u*_{σ}*D*/*h*_{0} (m s^{−1}; solid line). (c),(d) Second EOF amplitude (×2; dashed line) along with the depth-averaged horizontal density gradient (500 × kg m^{−4}; solid black line), the depth-averaged horizontal density gradient multiplied by *D*^{4} (kg; solid gray line), and the top–bottom vertical stratification (kg m^{−4}; dashed gray line). Shown are the spring (left) and the neap (right) tide surveys.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOF amplitudes and driving physical forces. (a),(b) First EOF amplitude [(5)^{−1}; dotted line] along with the depth-averaged along-stream mass transport velocity *u*_{σ}*D*/*h*_{0} (m s^{−1}; solid line). (c),(d) Second EOF amplitude (×2; dashed line) along with the depth-averaged horizontal density gradient (500 × kg m^{−4}; solid black line), the depth-averaged horizontal density gradient multiplied by *D*^{4} (kg; solid gray line), and the top–bottom vertical stratification (kg m^{−4}; dashed gray line). Shown are the spring (left) and the neap (right) tide surveys.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOF amplitudes and driving physical forces. (a),(b) First EOF amplitude [(5)^{−1}; dotted line] along with the depth-averaged along-stream mass transport velocity *u*_{σ}*D*/*h*_{0} (m s^{−1}; solid line). (c),(d) Second EOF amplitude (×2; dashed line) along with the depth-averaged horizontal density gradient (500 × kg m^{−4}; solid black line), the depth-averaged horizontal density gradient multiplied by *D*^{4} (kg; solid gray line), and the top–bottom vertical stratification (kg m^{−4}; dashed gray line). Shown are the spring (left) and the neap (right) tide surveys.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

We can recreate the mass transport velocity using any number of the EOFs and their respective amplitude time series. In the present case, to understand the residual circulation, we examine EOFs 1 and 2, and their sum. Recreating the mass transport velocity field with EOFs 1 and 2, and their sum results in the residual profiles in Fig. 8. The residual mass transport velocity is a summation of a riverine-like barotropic residual and a two-layer shear residual that closely resembles the vertical profile shape of gravitational circulation despite the lack of correlation between EOF 2 and the longitudinal baroclinic pressure gradient.

EOF 1 and 2 residuals in *σ* coordinates with HR profiles. Black lines show the observed residual mass transport velocity components. Black lines show the total residual velocity in volume-conserving *σ* coordinates (thin) along with the decomposed residual mass transport velocity recreated with EOF 1 (dotted), EOF 2 (dashed), and the sum of the two (thick) for mooring M3B. Thick gray lines show the HR approximations for the barotropic residual (dotted), baroclinic residual (dashed), and total (solid). River flow *u*_{r} = −0.04 m s^{−1} was estimated from the data and the constant in front of the baroclinic term [

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOF 1 and 2 residuals in *σ* coordinates with HR profiles. Black lines show the observed residual mass transport velocity components. Black lines show the total residual velocity in volume-conserving *σ* coordinates (thin) along with the decomposed residual mass transport velocity recreated with EOF 1 (dotted), EOF 2 (dashed), and the sum of the two (thick) for mooring M3B. Thick gray lines show the HR approximations for the barotropic residual (dotted), baroclinic residual (dashed), and total (solid). River flow *u*_{r} = −0.04 m s^{−1} was estimated from the data and the constant in front of the baroclinic term [

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

EOF 1 and 2 residuals in *σ* coordinates with HR profiles. Black lines show the observed residual mass transport velocity components. Black lines show the total residual velocity in volume-conserving *σ* coordinates (thin) along with the decomposed residual mass transport velocity recreated with EOF 1 (dotted), EOF 2 (dashed), and the sum of the two (thick) for mooring M3B. Thick gray lines show the HR approximations for the barotropic residual (dotted), baroclinic residual (dashed), and total (solid). River flow *u*_{r} = −0.04 m s^{−1} was estimated from the data and the constant in front of the baroclinic term [

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

### c. Momentum budget

Next, we examine a momentum budget to investigate which terms are driving the residual. We first describe the tidally varying momentum terms in *z* and *σ* coordinates, and then present the tidally averaged momentum terms in *σ* coordinates, allowing us to proceed with the theory presented in the discussion (section 4).

#### 1) Tidal momentum equations

*z*coordinates is

*f*is the Coriolis parameter (

*f*= 1.08 × 10

^{−4}s

^{−1});

*g*is the gravitational acceleration;

*η*is the free surface elevation above the mean depth;

*D*is the total depth;

*ρ*is the density;

*ρ*

_{o}is the background density; and

*R*is the radius of curvature in the streamwise direction (e.g., Chant and Wilson 1997) and is positive when streamlines bend toward the right.

*σ*=

*z*/

*D*with

*σ*= 0 at the bed and

*σ*= 1 at the surface as described previously. This is a slight variation of the

*σ*coordinates employed in many hydrodynamic numerical models [e.g., Blumberg and Mellor (1987), where

*σ*= (

*z*−

*η*)/

*D*varies from −1 at the bed to 0 at the free surface]; however, the coordinate transformations follow similarly and result in similar equations with fewer terms:

*z*. In Eq. (9),

*ω*is a modified vertical velocity equivalent to the vertical velocity plus terms due to the water column stretching. The other major differences between Eq. (9) and the

*z*-coordinate momentum equation [Eq. (8)] include the appearance of the total water depth in several terms and the splitting of the baroclinic pressure gradient into two terms.

As discussed in the introduction, the pressure gradient and vertical stress divergence (i.e., friction) are often important terms in the momentum balance. At tidal time scales, the barotropic pressure gradient dominates and acceleration is important, whereas for subtidal flows, it is assumed that a slowly varying baroclinic pressure gradient dominates and the acceleration is nearly zero. However, the dominant balance in the SRE may differ from this classical view for several reasons. First, given the large tidal amplitude-to-depth ratio, nonlinear terms may be important. Second, the strong curvature in the system gives rise to strong lateral and vertical velocities that contribute lateral and vertical advection of momentum. Third, the baroclinic pressure gradient and friction vary strongly through the tidal cycle.

To address these issues, below we quantify all but one of the terms in the tidally varying along-stream momentum budget in curvilinear coordinates using both *z* [Eq. (8)] and *σ* [Eq. (9)] coordinates and assess their relative importance. The figures and the discussion focus on *σ* coordinates. The terms in Eqs. (8) and (9) were computed using 10-min intervals rotated to principle axes components for a position centered at mooring M3B. All of the terms except for vertical advection can be computed following a similar approach for both the *z* and *σ* coordinates, and the resulting magnitude and temporal variability of the terms are similar. To compute the terms in the *σ*-coordinate momentum equation [Eq. (9)], all variables must be expressed in *σ* coordinates. The velocity and density were extrapolated as described earlier, and the stresses were extrapolated to the surface and bed via a cubic-smoothing spline. Details on computing the individual terms and their errors can be found in the appendix.

To represent the evolving momentum balance, the terms at mid–water column (*σ* = 0.525) are shown in Fig. 9 with their 95% confidence intervals. Overall, the barotropic pressure gradient and friction dominate the tidally varying momentum balance (Figs. 9a,f in black and gray, respectively; maximum magnitudes of ~1.5 × 10^{−3} m s^{−2}). As expected, they are largest during the strong ebb and flood tides, larger during spring tides, and act in opposition to one another. The friction term also exhibits a large peak during the strong ebb as the salt wedge passes downstream, corresponding with large observed interfacial stress. There are intermittent periods during HLW and HHW when negative friction appears in the upper mixed layer (due to negative near-surface Reynolds stresses). The cause of these stresses is unknown, although we speculate that errors from the ADCP tilt sensor and/or surface wakes from a nearby obstacle may contribute. Unfortunately, the Reynolds stress extrapolations are particularly noisy, making this term suspect where extrapolations were necessary (e.g., ~day 192.5, the water column is at its shallowest and much of the *σ*-coordinate stresses are extrapolated).

Along-stream momentum budget at *σ* = 0.525. Terms in the *σ*-coordinate along-stream momentum budget [Eq. (9)] at *σ* = 0.525. (a),(f) Pressure terms [barotropic (black) and baroclinic (light gray)] and friction term (gray). Note that only the first baroclinic term is included, as the second is nearly indistinguishable from zero on this axis. (b),(g) Acceleration (gray), longitudinal advection (black), lateral advection (light gray), and curvature (dashed black). The curvature term error bars are less than the line width. (c),(h) Total residual (i.e., vertical advection plus errors). (d),(i) Total residual in flux form (i.e., vertical advection plus errors), as well as the depth-averaged flux form [Eq. (10), where vertical advection drops out]. Note that the units and values in these panels are different than those above because of the extra factor of *D* in the flux form of the momentum equations. Bands represent 95% confidence intervals. Spikes in the friction term during the spring (~day 192.5) are due to the shallow water column. Spikes during the large ebb tides are reduced in the depth-averaged form but do not disappear, suggesting that additional errors must exist (see the text). Missing data during the neap (~day 198.8) are due to missing CTD data. (e),(j) Water level and depth averaged velocity. Values are shown for the spring (left axis) and neap (right axis) tides.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Along-stream momentum budget at *σ* = 0.525. Terms in the *σ*-coordinate along-stream momentum budget [Eq. (9)] at *σ* = 0.525. (a),(f) Pressure terms [barotropic (black) and baroclinic (light gray)] and friction term (gray). Note that only the first baroclinic term is included, as the second is nearly indistinguishable from zero on this axis. (b),(g) Acceleration (gray), longitudinal advection (black), lateral advection (light gray), and curvature (dashed black). The curvature term error bars are less than the line width. (c),(h) Total residual (i.e., vertical advection plus errors). (d),(i) Total residual in flux form (i.e., vertical advection plus errors), as well as the depth-averaged flux form [Eq. (10), where vertical advection drops out]. Note that the units and values in these panels are different than those above because of the extra factor of *D* in the flux form of the momentum equations. Bands represent 95% confidence intervals. Spikes in the friction term during the spring (~day 192.5) are due to the shallow water column. Spikes during the large ebb tides are reduced in the depth-averaged form but do not disappear, suggesting that additional errors must exist (see the text). Missing data during the neap (~day 198.8) are due to missing CTD data. (e),(j) Water level and depth averaged velocity. Values are shown for the spring (left axis) and neap (right axis) tides.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Along-stream momentum budget at *σ* = 0.525. Terms in the *σ*-coordinate along-stream momentum budget [Eq. (9)] at *σ* = 0.525. (a),(f) Pressure terms [barotropic (black) and baroclinic (light gray)] and friction term (gray). Note that only the first baroclinic term is included, as the second is nearly indistinguishable from zero on this axis. (b),(g) Acceleration (gray), longitudinal advection (black), lateral advection (light gray), and curvature (dashed black). The curvature term error bars are less than the line width. (c),(h) Total residual (i.e., vertical advection plus errors). (d),(i) Total residual in flux form (i.e., vertical advection plus errors), as well as the depth-averaged flux form [Eq. (10), where vertical advection drops out]. Note that the units and values in these panels are different than those above because of the extra factor of *D* in the flux form of the momentum equations. Bands represent 95% confidence intervals. Spikes in the friction term during the spring (~day 192.5) are due to the shallow water column. Spikes during the large ebb tides are reduced in the depth-averaged form but do not disappear, suggesting that additional errors must exist (see the text). Missing data during the neap (~day 198.8) are due to missing CTD data. (e),(j) Water level and depth averaged velocity. Values are shown for the spring (left axis) and neap (right axis) tides.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Acceleration and longitudinal advection are the next largest contributors (maximum magnitude from ~2 × 10^{−4} to 3 × 10^{−4} m s^{−2}) with acceleration maximum as the flow direction changes and longitudinal advection maximum (and negative) during the strong ebb tide. Although not of leading order, lateral advection is important briefly during the strong flood (e.g., ~day 192.75) and strong ebb tides when secondary circulations driven by curvature and fronts create strong shear in the cross-stream velocity (Giddings et al. 2012) and lead to an input of momentum via lateral advection. Lateral advection is enhanced during spring tides because stronger along-stream currents lead to stronger cross-stream currents (see Figs. 9b,g in light gray). The curvature term is often in opposition to lateral advection but smaller in magnitude (Figs. 9b,g in dashed black). The Coriolis term is very small and does not play a significant role in the momentum balance at any time [*O*(~10^{−5} m s^{−2})] and is not included in Fig. 9.

Overall, the baroclinic terms are small and the first baroclinic term is an order of magnitude larger than the second (see Fig. 10, maximum values are ~3.5 × 10^{−4} m s^{−2} for the first baroclinic term); thus, only the first baroclinic term is included in Figs. 9a and 9f (light gray). During neap, the baroclinic gradient is slightly larger and more consistent throughout the tidal cycle. The total baroclinic pressure gradient throughout depth (sum of both baroclinic terms) is shown in Fig. 10, showing the expected increase with depth. We highlight this term separately because it behaves differently than in a partially or well-mixed estuary. The salt-wedge nature of the system creates a strong horizontal density front, leading to a baroclinic pressure gradient that varies depending on along-estuary location. At a fixed mooring, the baroclinic pressure gradient peaks rapidly as the front passes during the large ebb and large flood. This behavior differs from a well-mixed system where the horizontal density gradient is persistent over long distances and often assumed constant.

Baroclinic pressure gradient throughout depth. (a),(b) Baroclinic pressure gradient [Eq. (9), sum of the first and second baroclinic terms] throughout depth in depth-normalized coordinates computed utilizing CTD casts near mooring M3B and M6 during spring (left) and neap (right) tides. (c),(d) Near-bottom (*σ* = 0.025) baroclinic pressure gradient terms: term 1 (solid gray), term 2 (dashed gray), and total (solid black). Blank region during the neap sampling period centered at ~day 198.8 is due to missing data. (e),(f) Water level and depth-averaged velocity.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Baroclinic pressure gradient throughout depth. (a),(b) Baroclinic pressure gradient [Eq. (9), sum of the first and second baroclinic terms] throughout depth in depth-normalized coordinates computed utilizing CTD casts near mooring M3B and M6 during spring (left) and neap (right) tides. (c),(d) Near-bottom (*σ* = 0.025) baroclinic pressure gradient terms: term 1 (solid gray), term 2 (dashed gray), and total (solid black). Blank region during the neap sampling period centered at ~day 198.8 is due to missing data. (e),(f) Water level and depth-averaged velocity.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Baroclinic pressure gradient throughout depth. (a),(b) Baroclinic pressure gradient [Eq. (9), sum of the first and second baroclinic terms] throughout depth in depth-normalized coordinates computed utilizing CTD casts near mooring M3B and M6 during spring (left) and neap (right) tides. (c),(d) Near-bottom (*σ* = 0.025) baroclinic pressure gradient terms: term 1 (solid gray), term 2 (dashed gray), and total (solid black). Blank region during the neap sampling period centered at ~day 198.8 is due to missing data. (e),(f) Water level and depth-averaged velocity.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Vertical advection could not be estimated accurately because of large uncertainty in the measured *w.* We note that despite the large uncertainty, there are some consistent patterns with the largest magnitude (associated with the largest vertical velocities) occurring during the strong ebb prior to the salt wedge passing downstream and smaller peaks occurring during the peak large flood and around higher-low water (HLW) under stratified conditions. Ultimately, vertical advection may play an important role in the momentum budget as suggested below, but we do not have measurements from which to draw definitive conclusions.

Figures 9c and 9h show the residual after summing all of the terms in the momentum equation with the exception of vertical advection; therefore, this term effectively represents vertical advection plus any unaccounted for errors (black line). This residual is plotted throughout depth in Fig. 11 for both the *z*- and *σ*-coordinate analyses. There are three time periods when the residual is not zero: during the large ebb and flood when the salt wedge passes, and briefly during the neap weak ebb. The large ebb pulse (which extends throughout depth, Fig. 11) corresponds with strong vertical velocities (not shown) and, therefore, it seems reasonable that this spike may partially represent vertical advection. The imbalance during large flood occurs when the salt wedge passes upstream and there is a strong influence of the mudflat bypass, resulting in fronts and lateral circulation [described in Giddings et al. (2012)], again suggesting the importance of vertical advection as well as underestimated lateral advection. Finally, there is significant imbalance in the extrapolated regions (above and below the two gray lines in Figs. 11c,d) likely due to errors in the Reynolds stress extrapolations.

Residual momentum throughout depth. Residual momentum term throughout depth (which represents vertical advection and unaccounted for errors in the other terms). (a),(b) Results from the *z*-coordinate analysis and (c),(d) the *σ* coordinates. Values are given for spring (left) and neap (right) tides. Light gray lines indicate missing data in *z* coordinates caused by instrument limitations, while in *σ* coordinates they indicate where data were extrapolated. Blank region during the neap sampling period centered at ~day 198.8 between the two vertical gray lines is due to missing data. All other white regions indicate that the computed terms in the momentum balance fall within 95% confidence intervals of zero.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Residual momentum throughout depth. Residual momentum term throughout depth (which represents vertical advection and unaccounted for errors in the other terms). (a),(b) Results from the *z*-coordinate analysis and (c),(d) the *σ* coordinates. Values are given for spring (left) and neap (right) tides. Light gray lines indicate missing data in *z* coordinates caused by instrument limitations, while in *σ* coordinates they indicate where data were extrapolated. Blank region during the neap sampling period centered at ~day 198.8 between the two vertical gray lines is due to missing data. All other white regions indicate that the computed terms in the momentum balance fall within 95% confidence intervals of zero.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Residual momentum throughout depth. Residual momentum term throughout depth (which represents vertical advection and unaccounted for errors in the other terms). (a),(b) Results from the *z*-coordinate analysis and (c),(d) the *σ* coordinates. Values are given for spring (left) and neap (right) tides. Light gray lines indicate missing data in *z* coordinates caused by instrument limitations, while in *σ* coordinates they indicate where data were extrapolated. Blank region during the neap sampling period centered at ~day 198.8 between the two vertical gray lines is due to missing data. All other white regions indicate that the computed terms in the momentum balance fall within 95% confidence intervals of zero.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

*σ*= 0.525 (dark gray) and the depth-averaged flux form [Eq. (10); light gray] are presented in Figs. 9d and 9i. During some time periods of nonzero residual (large flood tides and the early stages of the large ebb tide), the depth average reduces to within the error bounds of zero, suggesting that indeed vertical advection played a significant role. During the mid–strong ebb tides, the vertical average is significantly reduced relative to the residual; however, it does not drop to zero. This suggests that our momentum budget and error analysis is not capturing all of the errors and that likely the 3D nature of this system (particularly during these strongly frontal periods) becomes important on scales smaller than our spatial averaging.

Other than these few exceptions, the residual is often near zero, indicating that the primary balance in this system is a barotropic pressure–friction balance. A regression analysis of the barotropic pressure and friction terms in the mid–water column (*σ* = 0.525, 3-h low-pass filtered) suggests a strong overall balance between these terms (slope = 1.1 ± 0.1, *r*^{2} = 0.80, *N*_{eff} = 16, and *p* < 0.0001, where *N*_{eff} is an effective degrees of freedom to account for the filtering). A correlation of the total pressure gradient (barotropic plus baroclinic) with the friction term yields statistically similar results (slope = 1.1 ± 0.1, *r*^{2} = 0.81, *N*_{eff} = 16, and *p* < 0.0001). Importantly, however, correlations of the pressure and friction terms are significantly improved in *σ* coordinates relative to *z* coordinates. In *z* coordinates, the correlations decrease rapidly with increasing height above the bed and the balance breaks down around ~0.5 m above MLLW when the correlation coefficient and the relationship between friction and pressure becomes statistically insignificant at the 95% level (*r*^{2} < 0.2, *N*_{eff} < 16, and *p* > 0.06). This may reflect the importance of additional terms higher in the water column, but also reflects the decreased number of data points with increasing height. Conversely, the *σ*-coordinate momentum equations, despite noisy stress extrapolations, exhibit a barotropic pressure–friction balance throughout depth: pressure and friction are correlated at the 95% confidence interval at all depths. The correlation decreases near the bed and near the surface; however, *r*^{2} > 0.35 (*N*_{eff} = 16 and *p* = 0.008) throughout the water column and exceeds 0.8 in the middle of the water column. The slope of the linear fit is ≈1 except near the bed, and the intercept is ≈0 except near the surface. Overall, the statistically significant pressure–friction balance throughout depth demonstrates that *σ* coordinates improve the ability to examine momentum balances above LLW.

Despite complex interactions among shear, stratification, and mixing observed in this system, strong lateral circulations, and the large tidal amplitude-to-depth ratio, as expected from traditional estuarine theory, the barotropic pressure gradient and friction dominate on the tidal time scale. This simple pressure–friction balance breaks down briefly during the large flood and ebb when acceleration and advection are important. The budget also does not close during the large flood and ebb when the salt wedge passes and vertical advection, as well as unaccounted for errors due to the 3D nature of the flow may become significant.

#### 2) Tidally averaged momentum equations, *σ* coordinates

It is important to note that tidally, acceleration and advection are often on the same order (or larger) of magnitude than the baroclinic term. To understand the role of baroclinic pressure gradients on subtidal time scales, we examine the tidally averaged momentum budget in *σ* coordinates. This is done for a single spring and a single neap tide in Fig. 12. The drawback of this approach is that errors in the stress extrapolations are emphasized as spurious values dominate the residual; therefore, we must focus on the other terms. On the subtidal time scale, the barotropic term remains dominant, yet the baroclinic pressure gradient becomes relatively more important and the first of the two baroclinic terms dominates. The unsteady term averages to near zero and the lateral advective and curvature terms, although nonzero, approximately balance each other. The longitudinal advection term becomes important and contributes to balancing the barotropic pressure near the surface. The residual (representing vertical advection plus friction plus unaccounted for errors) approximately matches what might be expected. In particular, the residual is larger during the spring survey than during the neap survey. Overall, we see that in a tidally averaged sense, the dominant terms in the *σ* momentum equation are the pressure and friction terms. Additionally, however, we note the importance of longitudinal advection near the surface. Stacey et al. (2010) similarly found advection (in their case, lateral advection) contributed to the near-surface tidally averaged momentum balance in a partially mixed estuary.

Tidally averaged terms in the *σ*-coordinate momentum equation. Tidally averaged terms in the *σ*-coordinate momentum equation, Eq. (9), for the (a) spring and (b) neap surveys. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some of the terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Note that the friction terms are excluded from these figures because of noisy extrapolations as discussed in this text. Residual is a sum of the computed terms, thus representing vertical advection plus friction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Tidally averaged terms in the *σ*-coordinate momentum equation. Tidally averaged terms in the *σ*-coordinate momentum equation, Eq. (9), for the (a) spring and (b) neap surveys. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some of the terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Note that the friction terms are excluded from these figures because of noisy extrapolations as discussed in this text. Residual is a sum of the computed terms, thus representing vertical advection plus friction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Tidally averaged terms in the *σ*-coordinate momentum equation. Tidally averaged terms in the *σ*-coordinate momentum equation, Eq. (9), for the (a) spring and (b) neap surveys. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some of the terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Note that the friction terms are excluded from these figures because of noisy extrapolations as discussed in this text. Residual is a sum of the computed terms, thus representing vertical advection plus friction.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

## 4. Discussion

The dominance of the pressure–friction terms in the tidal and mean momentum analysis and the resemblance of the residuals in *σ* coordinates to traditional theory lead to the expectation that the shear residual is driven by the mean longitudinal density gradient as in HR theory; however, EOF analysis showed insignificant correlation between the shear residual and the longitudinal density gradient (Fig. 7). Here, we investigate the mechanisms driving the residual shear by testing the HR theory extended to *σ* coordinates.

### a. Approximation of an HR-type balance in σ coordinates

*z*-coordinate momentum equation and assuming a Fickian eddy viscosity model to represent the stress. If we follow a similar approach with the

*σ*momentum equations, we find that the equation becomes slightly more complicated. Applying a Fickian eddy viscosity model to represent the Reynolds stresses,

*v*

_{t}is the eddy viscosity, incorporates yet another factor of

*D*

^{−1}into the friction term. To write the equations in a form similar to HR, we must multiply each term by

*D*

^{3}

*h*

_{0}

^{−1}

*v*

_{t}

^{−1}before tidally averaging, such that the residual mass transport velocity will come out of the friction term. Assuming that the eddy viscosity is a constant (i.e., an effective eddy viscosity), we can write this tidally averaged

*σ*-coordinate momentum equation as

*η*/

*h*

_{0}, these equations will collapse to those formulated by HR. The terms in Eq. (11) (without the constant

*h*

_{0}

^{−1}

*ν*

_{t}

^{−1}) are plotted in Fig. 13. The results are markedly different than Fig. 12 because each term now contains nonzero correlations with the total water column depth. For example, the unsteady term becomes more important in this formulation because it correlates more strongly with the water depth than compared, for example, to the first baroclinic term.

Tidally averaged terms in the HR formulation of the *σ*-coordinate momentum equation. Tidally averaged terms in the HR form of the *σ*-coordinate momentum equation [Eq. (11)] for the (a) spring and (b) neap surveys. Note that the constant *h*_{0}^{−1}*v*_{t}^{−1} shown in Eq. (11) is not included. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Friction is not included here except as part of the residual. Thus, the residual represents vertical advection plus friction plus unaccounted for errors. Note that the differences from Fig. 12 are due to the correlation of each term with *D*.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Tidally averaged terms in the HR formulation of the *σ*-coordinate momentum equation. Tidally averaged terms in the HR form of the *σ*-coordinate momentum equation [Eq. (11)] for the (a) spring and (b) neap surveys. Note that the constant *h*_{0}^{−1}*v*_{t}^{−1} shown in Eq. (11) is not included. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Friction is not included here except as part of the residual. Thus, the residual represents vertical advection plus friction plus unaccounted for errors. Note that the differences from Fig. 12 are due to the correlation of each term with *D*.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Tidally averaged terms in the HR formulation of the *σ*-coordinate momentum equation. Tidally averaged terms in the HR form of the *σ*-coordinate momentum equation [Eq. (11)] for the (a) spring and (b) neap surveys. Note that the constant *h*_{0}^{−1}*v*_{t}^{−1} shown in Eq. (11) is not included. Legend is on the right-hand side of (b). Light gray bands indicate 95% confidence intervals. For some terms, for example the two baroclinic terms, the confidence intervals are smaller than the width of the line, such that they are barely visible. Friction is not included here except as part of the residual. Thus, the residual represents vertical advection plus friction plus unaccounted for errors. Note that the differences from Fig. 12 are due to the correlation of each term with *D*.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

*σ*coordinates:

*σ*:

*D*

^{4}, we must assume something about the variation of the entire term with

*σ*. Assuming that the tidal-mean baroclinic term in Eq. (12) varies linearly with depth, which appears to be a reasonable assumption (Fig. 13), implies that the term on the right-hand side of Eq. (13) must be independent of

*σ*. Employing this assumption, Eq. (13) can be integrated three times and solved with appropriate boundary conditions. Using the following boundary conditions:

*σ*= 0 (i.e., no slip);

*σ*= 1 (i.e., no stress at the surface); and that the depth integral of the residual mass transport equals the river flow,

*σ*coordinates:

Resulting profiles approximated from the HR-type solution are shown in Fig. 8 split into the barotropic contribution, baroclinic contribution, and the total along with EOFs 1 and 2. The constant driving the baroclinic residual is calculated from the measurements of the time-varying total depth and the depth-averaged horizontal density gradient. The eddy viscosity can be calculated from the data assuming an eddy viscosity model—that is, *v*_{t} = 0.0007 that is used in Eq. (14). This is an upper bound on the baroclinic circulation, as filtering the observed *v*_{t} before averaging by removing extreme outliers produces a tidal mean closer to 0.002 that further reduces the HR baroclinic shear.

The two-layer shear residual is significantly underpredicted, suggesting that it is not established by a tidally averaged density gradient and friction, as the HR model assumes [similar to Burchard and Hetland (2010)]. Rather, additional factors must contribute to the subtidal shear, including shear pulses, because of the changing horizontal density gradient and vertical stratification.

### b. Validity of the HR assumptions

As already discussed, a pressure–friction balance is a decent approximation for the lower half of the water column (Fig. 13); however, near the surface, advection and particularly acceleration become important. This approximation is better even higher into the water column during neap tides, when the effects of acceleration are reduced and baroclinic forcing is enhanced. Although the assumption of the dominance of the first baroclinic term is strong (Fig. 13), a close inspection of this first baroclinic term shows that the assumption that this term varies linearly with depth is certainly not exact. The right-hand side of Eq. (13) is not depth independent and deviations from the depth average of this term are smaller than the depth average but only by a factor of ~3 (Fig. 14).

Depth variations of baroclinic term. Depth variations of the constant term in the HR-type residual circulation formulation, the term on the right-hand side of Eq. (13). Thick line shows the full term, thin line shows the depth average of this term (i.e., the approximation used to carry through with the HR-type formulation), and dashed line is the depth deviation of this term. Black lines are averaged over the spring tidal cycle, and gray lines are averaged over the neap tidal cycle.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Depth variations of baroclinic term. Depth variations of the constant term in the HR-type residual circulation formulation, the term on the right-hand side of Eq. (13). Thick line shows the full term, thin line shows the depth average of this term (i.e., the approximation used to carry through with the HR-type formulation), and dashed line is the depth deviation of this term. Black lines are averaged over the spring tidal cycle, and gray lines are averaged over the neap tidal cycle.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

Depth variations of baroclinic term. Depth variations of the constant term in the HR-type residual circulation formulation, the term on the right-hand side of Eq. (13). Thick line shows the full term, thin line shows the depth average of this term (i.e., the approximation used to carry through with the HR-type formulation), and dashed line is the depth deviation of this term. Black lines are averaged over the spring tidal cycle, and gray lines are averaged over the neap tidal cycle.

Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-12-0201.1

However, probably the most critical assumption that breaks down is the application of a tidally and vertically averaged eddy viscosity. Our data suggest that the eddy viscosity varies significantly over the tidal cycle and with depth, due to the strong, temporally varying vertical stratification. In fact, as illuminated by Stacey et al. (2010), an effective eddy viscosity following a Fickian model may be an improper assumption, particularly in estuaries where the baroclinic forcing is small relative to the barotropic forcing (i.e., small Simpson number).

### c. Intratidal pulses of residual shear

In the HR theory, baroclinic shear is correlated with the longitudinal density gradient. The strongly forced, salt-wedge nature of the SRE leads to a strongly temporally and spatially varying horizontal salinity gradient. The maximum horizontal salinity gradient occurs during the strong flood tide when the salt wedge enters the estuary, yet there is minimal shear during this time. In fact, during parts of the flood tide, the shear is destabilizing (i.e., reverse shear). As shown in an EOF analysis (Fig. 7), the shear circulation is enhanced particularly during weak ebbs and again briefly during early strong ebbs prior to the salt wedge exiting the estuary. The two-layer shear EOF is not significantly correlated with Δ*ρ*/Δ*x* or *D*^{4}Δ*ρ*/Δ*x* (*r*^{2} = 0.09, *N*_{eff} = 9, and *p* = 0.22). It is, however, significantly correlated with the top/bottom stratification (*r*^{2} = 0.40, *N*_{eff} = 56, and *p* < 0.0001), thus confirming the importance of stratification in driving residual shear (see Fig. 7).

The changes in vertical stratification are driven by a combination of along-stream straining (similar to SIPS; Simpson et al. 1990), lateral straining during peak currents, and along-stream advection of stratification (Giddings et al. 2011). Periods of enhanced stratification coincide with periods of enhanced shear as vertical mixing is suppressed. Therefore, we hypothesize that residual circulation is strongly driven by pulses on the tidal time scale that are linked to a combination of straining, advection, and vertical mixing.

Overall, we find that the HR-type solution explains less than half of the residual shear circulation and that many of the assumptions critical to HR theory break down. Residual circulation pulses driven on a tidal time scale are such that the overall residual retains a similar profile to traditional theory but of a larger magnitude than that driven by the tidally averaged horizontal density gradient and mixing alone. Effectively, as discussed in Monismith et al. (1996), the eddy viscosity represents the complex effects of tidally varying stratification as well as other tidal nonlinearities, and so cannot be specified a priori. It is a common paradox that the classical theory appears to work well in many systems despite the neglect of important driving factors (MacCready and Geyer 2010).

## 5. Conclusions

This analysis of residual circulation in a shallow, macrotidal estuary in both *z* and *σ* coordinates highlights the advantages and disadvantages of each of these methods. The transformation to *σ* coordinates allowed us to approximate a Lagrangian residual—that is, a residual mass transport velocity—that incorporates the Stokes wave transport velocity, and it appears that it may be a promising method to extend the understanding of residuals in estuaries with a large Δ*η*/*h*_{0} to those with small Δ*η*/*h*_{0}. The transformation to *σ* coordinates facilitated an EOF analysis that split the residual into barotropic-like and two-layer shear components. Examination of the tidal and mean momentum balance throughout depth and the comparison of residual circulation to traditional estuarine theory were also made clearer through the use of *σ* coordinates. Despite the failure of the momentum budget to close at all locations and all times, *σ*-coordinate analysis appears to be robust, providing insight into the momentum balance over depth and averaged over a tidal cycle. Similarly, the derivation of the HR approximations in *σ* coordinates enabled examination of the mechanisms driving residuals.

We find that the mass transport velocity has an enhanced two-layer flow relative to an Eulerian mean due to the Stokes wave transport velocity directed upstream at all depths. Thus, as a first-order correction, the volume-conserving mass transport velocity should be used when examining residuals in *σ* coordinates. This has significant implications on transport in this system, suggesting that despite the large river flow and strong tidal nonlinearities, a near-bed particle may travel upriver, as suggested by traditional estuarine theory, contrary to Eulerian residuals in either *z* or *σ* coordinates. We caution, however, that additional terms that contribute to the Stokes drift due to spatial variability in the flow field were not included in our analysis and may influence the actual particle pathways. Additionally, deviations of streamlines from the *σ* surfaces affect computed residual calculations. Indeed, it is possible that this approach may break down in a Fjord-like estuary or other strongly hydraulically controlled systems where there may be significant differences between streamlines and *σ* surfaces.

EOF analysis reveals the residual and the intratidal variability of the barotropic-like and two-layer shear components of the flow. Residual profiles were a sum of a barotropic riverine-like residual and a two-layer shear residual of similar strength. Although these residuals resemble HR theory, the two-layer shear component does not correlate strongly with the along-stream density gradient, suggesting that other mechanisms must play a role in driving the residual shear.

Similarly, while the momentum analysis indicates a pressure–friction balance in the tidally varying momentum equations and the dominance of pressure gradients in the tidally averaged momentum budget, traditional theory cannot explain the strength of the two-layer shear residual. The extension of the HR theory to *σ* coordinates incorporates the time-varying water depth; however, several approximations required to create a simple HR-type balance in *σ* coordinates are broken because of the importance of acceleration and advection near the water surface, depth variations in the horizontal density gradient, and time and depth variations in vertical mixing. Thus, the HR balance underestimates the observed two-layer shear residual as is to be expected in such a tidal, salt-wedge type of estuary. The EOF analysis and detailed investigation of intratidal variability (Giddings et al. 2011, 2012) suggest that additional two-layer shear residual is driven by pulses in shear due to the interactions of straining and advection of the salinity field with vertical mixing, as seen in other systems (e.g., Stacey et al. 2010; Burchard and Hetland 2010; MacCready and Geyer 2010; among many others).

Thus, given that a volume-conserving *σ*-coordinate system works well for systems with large Δ*η*/*h*_{0} as well a small Δ*η*/*h*_{0}, it may be the best general approach for describing estuarine flows. Ideally, one would have velocity and salinity information throughout depth and a cross section to estimate what is termed the “total exchange flow” in isohaline coordinates (MacCready 2011). Chen et al. (2012) examined the differences between an isohaline and an approximate *σ*-coordinate Eulerian framework, thus demonstrating the value of a combined approach to understanding subtidal and tidal fluxes. Additional comparison among various methods for computing residuals is desirable, particularly if done to assess errors associated with undersampling (i.e., one or two vertical profiles versus a cross section). Finally, a comparison with the full Lagrangian velocity and drifters and/or particle tracking results from numerical simulations would aid interpretation of these exchange flow methods to approximating Lagrangian exchanges.

## Acknowledgments

Thanks to those at the Stanford EFML; APL-UW; and other members of the Coherent Structures in Rivers and Estuaries Experiment (COHSTREX) team who provided help in the field, particularly N. Nidzieko, J. Hench, K. Davis, L. Walter, B. Hayworth, P. J. Rusello, T. Litchendorf, E. Boget, C. Craig, and F. Karig. Special thanks to Parker MacCready and Rocky Geyer for the helpful discussions, as well as three anonymous reviewers of this paper for their constructive criticisms. This research was supported by the Office of Naval Research through Grants N00014-05-1-0485 and N00014-10-1-0236. Additional support for SNG was provided by the National Science Foundation, a Wells Family Stanford Graduate Fellowship, and the Achievement Rewards for College Scientists Foundation.

## APPENDIX

### Momentum Budget Calculations and Error Analysis

The terms in the momentum balance [Eqs. (8) and (9)] were computed for a position centered at mooring M3B (or as closely as possible). Of the moorings with a full dataset of velocity, density, and stress measurements (M2A, M2B, M3A, and M3B), M3B is the most representative of dynamics in the main channel, exhibiting minimal effects of the sill and the mudflats. The 1200-kHz Teledyne RD Instruments ADCPs at these moorings were operated in mode 12, averaging 10 subpings per 1-Hz sample with data recorded in beam coordinates in 0.25-m-depth bins throughout the water column with the bottom bin centered at ~1 m above the bottom. This manuscript only utilizes the along-stream CTD transects conducted with a Sea-Bird Electronics (SBE) 19 CTD at five cast stations along the estuary every half hour (Fig. 1c) during the representative 30-h spring and neap surveys (12–13 and 18–19 July 2006, respectively). The CTD sampled at 2 Hz and was lowered at 10–20 cm s^{−1}, resulting in density profiles with 5–10-cm resolution in the vertical. The velocity and density were extrapolated as described earlier, and the stresses were extrapolated to the surface and bed via a cubic-smoothing spline described below. Coordinates were determined via principal axes analysis and the quantities are averaged (or subsampled) in 10-min intervals.

*t*distribution and using a calculation of the standard error. For any function

*q*(

*x*

_{1},

*x*

_{2}, … ,

*x*

_{n}), the standard error

*σ*

_{q}can be found as (e.g., Emery and Thomson 2004)

*σ*

_{u}= 7.85 mm s

^{−1}), which includes long-term instrument error (5 mm s

^{−1}reported by Teledyne instruments for a mean flow of ~1 m s

^{−1}), statistical uncertainty of each 10-min ensemble (182 mm s

^{−1}per ping reported by Teledyne instruments equates to 2.35 mm s

^{−1}per 10-min ensemble), and errors due to pitch and roll (0.5 mm s

^{−1}estimated assuming the reported tilt and roll accuracy). The uncertainty in the time stamp is approximately one-hundredth of a second (

*σ*

_{t}= 0.01 s). As an example (not shown for the other terms), applying Eq. (A1) to the acceleration term Δ

*u*/Δ

*t*yields the following equation for the total error:

Longitudinal advection was estimated utilizing ADCP velocity data from mooring M3B and the upstream mooring M6. This estimate of longitudinal advection may be an underestimate because of the distance between the moorings smoothing out larger local gradients. The error estimate includes the ADCP velocity error (above) and an error in the along-stream distance based on the GPS accuracy (*σ*_{x} = 10 m).

Lateral advection was estimated two ways: using the across-stream ADCP velocities from moorings M3A and M3B, and using the ADCP transects. It was found that the latter method gave similar results but of larger magnitude and noisier. The lateral advection computed with the across-stream mooring is presented in the manuscript; we note, however, that it is likely an underestimate due to the spatial smoothing. The error estimate was computed similarly to longitudinal advection.

Vertical advection could not be estimated accurately. The standard error associated with the computed vertical advection (which includes errors in *w* due to the long-term instrument error, statistical uncertainty, and errors from the pitch and roll sensors, as well as errors in the calculation of Δ*u*/Δ*z*) is on the order of 5 × 10^{−3} m s^{−2}, similar in magnitude to the term itself and larger than the other terms. The error estimate was computed similarly to longitudinal advection; however, it is important to note that *σ*_{w} > *σ*_{u} because errors in pitch and roll are exacerbated in the vertical velocity component.

The curvature term was estimated using the along- and cross-stream velocities at mooring M3B and a radius of curvature, *R* = 860 m; *R* was calculated at a midchannel location utilizing the change in the principle axes of flow direction between the mean of the upstream moorings M3A and M3B and the downstream mooring M2A (e.g., Chant and Wilson 1997). The error estimate was computed similarly to longitudinal advection.

The Coriolis term was estimated using the cross-stream velocities at mooring M3B. The error estimate included the ADCP velocity error and an estimated value for *σ*_{f} based on the latitude range for the study site, which is negligible.

The total pressure gradient was estimated using bottom pressure sensors at M6 [a Richard Brancker Research (RBR) pressure sensor sampling once per minutre] and M3B (an SBE16 CTD sampling once per minute): the pressure data were averaged to 10-min intervals, detrended to remove any long-term instrument drift, and the hydrostatic pressure difference between the depth of the two instruments was removed. In addition, an offset was estimated assuming that when the near-bed velocity was zero, the pressure gradient and acceleration should balance (Geyer et al. 2000). The baroclinic contribution to this total pressure gradient was removed to isolate the barotropic pressure. We found that the baroclinic pressure gradient was best estimated using along-stream CTD casts rather than the mooring data that had less depth information. To estimate the errors associate with the total pressure gradient, the reported instrument pressure error was used (2.5 mm for M6 and 2 mm for M3B per 1-min sample). To estimate the errors associated with the baroclinic pressure gradient, the instrument reported uncertainty in temperature, pressure, and salinity that were put into the United Nations Educational, Scientific and Cultural Organization (UNESCO) equation of state to determine a worst-case scenario for the density uncertainty (*σ*_{ρ} = 0.0015 kg m^{−3}). The errors associated with the barotropic gradient were then estimated from the errors in both the total pressure gradient and the baroclinic pressure gradient.

Finally, friction was estimated using along-stream Reynolds stresses computed at M3B from the ADCP variance technique (Lu and Lueck 1999; Stacey et al. 1999; Nidzieko et al. 2006). The Reynolds stresses were smoothed with a cubic spline before taking a first-order finite difference. The smoothing spline parameter was chosen (MATLAB csaps function *P* = 0.99), such that the difference between the raw and smoothed stresses retained noise characteristics similar to the original data [the stress noise floor was 2 × 10^{−5} m^{2} s^{−2} calculated per Nidzieko et al. (2006), Stacey et al. (1999), and Williams and Simpson (2004)]. The error estimate for the stress term is based on the error at each depth bin [see Stacey et al. (1999) and Williams and Simpson (2004) for formulas] and an adjusted value of the vertical bin size resulting from the smoothing spline. Unfortunately, the extrapolated stresses seem unreasonable: they are noisy and exhibit a near-surface negative bias, suggesting unaccounted for errors associated with the extrapolation, thus making this term suspect. Using a lower smoothing parameter decreases this noise and the near-surface bias; nevertheless, we use a smoothing parameter consistent with the measured noise characteristics.

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