## 1. Introduction

Transport across continental shelves allows the exchange of heat, nutrients, larvae, sediment, and pollutants between coastal ecosystems and the open ocean. The inner shelf is a critical link in the cross-shelf transport pathway, connecting the surfzone and the continental shelf, but the mechanisms that drive cross-shelf circulation on the inner shelf are not well understood. On mid- and outer shelves, along-shelf winds typically drive cross-shelf transport following classic Ekman (1905) theory. Where the water is shallower, momentum from the wind mixes to the bottom faster than the Coriolis acceleration can turn it, so the along-shelf wind stress is ineffective at driving cross-shelf transport (Ekman 1905; Austin and Lentz 2002; Lentz and Fewings 2012). This region where the entire water column is filled by turbulent surface and bottom boundary layers is often referred to as the inner shelf.

Regional-scale studies have suggested cross-shelf winds as potentially able to complete the transport pathway between the surfzone and midshelf (Li and Weisberg 1999a,b; Cudaback et al. 2005; Estrade et al. 2008). Recent observational (Fewings et al. 2008) and modeling (Tilburg 2003) studies focused specifically on the cross-shelf wind stress have shown it to be a significant mechanism for cross-shelf transport on the inner shelf. Using a 2D idealized numerical model (no along-shelf variation), Tilburg (2003) demonstrated that when wind stress blows toward the coast, water near the surface moves onshore. As water piles up at the coast, a pressure gradient grows to balance the wind stress, and that pressure gradient drives an offshore return flow in the lower portion of the water column (Fig. 1), a result consistent with the Ekman (1905) solution for cross-shelf wind stress. For offshore winds, the circulation is reversed, with offshore flow near the surface and an onshore return flow below. Fewings et al. (2008) described observations of this circulation using wintertime measurements from the Martha’s Vineyard Coastal Observatory (MVCO). In winter, when the water column is presumably unstratified at 12-m water depth, Fewings et al. (2008) found cross-shelf wind stress to be far more effective at driving cross-shelf transport than along-shelf wind stress. Using a simple 2D unstratified numerical model, Fewings et al. (2008) also predicted that the region where cross-shelf wind is dominant should extend to around 30-m water depth for a 0.1 Pa wind stress in unstratified conditions.

A scale for the transport can be estimated as the product of the velocity and vertical length scales. The surface wind stress *τ ^{s}* provides the relevant velocity scale, the shear velocity

*ρ*is reference seawater density. In the deep-water limit and away from coastal boundaries, the turbulent boundary layer thickness provides the relevant vertical length scale. In unstratified conditions, the vertical length scale is

_{o}*κ*= 0.41 and the Coriolis parameter

*f*. Boundary layer transport then scales like the velocity scale times the vertical length scale

*h*is much less than the turbulent boundary layer thickness so the water depth is the relevant vertical length scale. The transport scales as the velocity scale times the vertical length scale

*τ*

^{sx}/

*ρ*

_{o}and

*δ*

_{Ek}as function of

*f*, we find again thatLentz and Fewings (2012) also found that velocity profiles from the North Carolina inner shelf collapsed together when velocity

*u*was normalized by

*z*by water depth

*h*, and were consistent with predictions from an unstratified 2D model (Lentz 1995).

Between the deep- and shallow-water limits, Ekman (1905) described the solution as a superposition of the two limits. The vertical length scale will be some function of both *h* and *δ*_{Ek} and the transition from deep to shallow defined by the ratio of the two vertical scales *h*/*δ*_{Ek}. For typical inner-shelf values of *f* = 10^{−4} s^{−1}, *h* = 10 m, and ^{−1}, this ratio, *h*/*δ*_{Ek} = 0.25. We proceed with our analysis by taking the inner-shelf region to behave as the shallow-water limit predicts, while recognizing that rotation plays a small but perhaps nonnegligible role in the dynamics.

Though continental shelves are typically stratified much of the year and often have buoyant river plumes flowing alongshore, the inner-shelf response to cross-shelf winds over vertical and horizontal density gradients is still unclear. In water deeper than the surface boundary layer thickness, Tilburg (2003) found that stratification decreased cross-shelf transport by cross-shelf winds in the upper portion of the surface boundary layer by limiting the depth of the surface boundary layer, causing the Ekman spiral to turn back on itself sooner. However, Tilburg’s analysis did not address the influence of the initial stratification, or the resulting density field after wind mixing, on the cross-shelf transport on the inner shelf (water shallower than the surface boundary layer thickness). South of Martha’s Vineyard, Massachusetts, Fewings et al. (2008) noticed an increased vertical shear in the cross-shelf circulation and, for offshore wind stress, an increased transport in summertime. Summer is when their field site is typically stratified, but Fewings et al. (2008) did not have density data to accompany the velocity measurements. One other recent observational study (Dzwonkowski et al. 2011) used seasonal averages of velocity and density from the Alabama shelf to conclude that cross-shelf flow was most clearly correlated with cross-shelf wind stress in the fall and winter, when the water was least stratified.

Here, we use a numerical model of an idealized, 2D, cross-shelf transect to describe the effects of initial stratification, wind stress magnitude, surface heat flux, and cross-shelf density gradient, on the inner-shelf response to a cross-shelf wind stress. These factors are all common in field settings but difficult to isolate in observations. We demonstrate that the cross-shelf density gradient is an important control on the cross-shelf transport driven by cross-shelf winds and that the cross-shelf density gradient causes an asymmetric response to on- and offshore wind stresses. Observations from the inner shelf south of Martha’s Vineyard are used to verify the model predictions.

Section 2 describes the Regional Ocean Modeling System (ROMS) model setup; section 3 describes the field program and data processing; section 4 presents numerical experiments varying wind stress magnitude and initial stratification using the same set of forcing conditions as Tilburg (2003), while focusing on the circulation on the inner rather than midshelf; in section 5, a relationship between the cross-shelf density gradient and model inputs is derived; section 6 isolates the effect of the cross-shelf density gradient on transport; section 7 compares field observations to model results; and section 8 concludes with a discussion and summary of the results.

## 2. Numerical model

We use ROMS (Shchepetkin and McWilliams 2005) to simulate inner-shelf dynamics. This section describes the numerical model, physical parameters selected for the “base case,” and the conditions varied over subsequent suites of model runs.

### a. ROMS model

ROMS is a free-surface, terrain-following, primitive equations ocean model with a curvilinear orthogonal horizontal grid and a stretched terrain-following vertical grid based on Shchepetkin and McWilliams (2005). Time steps are 10 s for the barotropic part of the equations and 50 or 100 s for the baroclinic terms. All experiments were run for 5 model days with variables saved hourly.

#### 1) Grid

The model domain is a periodic channel with sloping boundaries to represent two along-shelf uniform coastal regions that span from just outside the surfzone to the mid continental shelf. The cross-shelf section has a shallow coastal wall (*h*_{o} = 2 m) at each boundary. Depth increases away from each wall at a constant slope (10^{−3} for the base case) until the bathymetry reaches its maximum depth (65 m for the base case) at 63 km from the wall. The two sloping boundaries are separated by a 45-km-wide uniform depth region in the center of the domain for a total domain width of 171 km. The stretched vertical grid has 32 levels with enhanced resolution near the surface and bottom. Vertical grid spacing ranges from 0.06 m near the surface and bottom in the shallowest locations to 2.81 m in the center of the domain. The periodic along-shelf direction is 8 km long and the horizontal resolution is 1 km × 1 km for the whole domain.

The model is run with the cross-shelf coordinate positive and increasing to the right as shown in Fig. 2. In all cases the model is forced with a spatially uniform, cross-channel wind stress. The opposite sides of the channel experience opposite wind stresses relative to the coast. Results from a single model run are presented as the two coasts modeled with opposite direction wind forcing. All data analysis in this paper is presented in an “East Coast,” right-handed, coordinate system. The cross-shelf coordinate *x* is zero at the coast and increases offshore. The along-shelf coordinate *y* is positive in the direction of an upwelling-favorable wind stress.

#### 2) Density

Density varies with a linear equation of state *ρ* = *ρ*(*T*, *S*, *P*) for temperature *T*, salinity *S*, and pressure *P*, and all density variation is imposed as changes in temperature. When a surface heat flux is used, it is as a temperature flux through the surface *Q*/*ρ _{o}c*

_{p}(°C m s

^{−1}) using a seawater specific heat capacity

*c*

_{p}= 3985 J kg

^{−1}°C

^{−1}.

For model runs with an initial cross-shelf density gradient, isopycnals began uniformly diagonal. The model is initialized with no velocity, but with a surface tilt that opposes the cross-shelf pressure gradient that results from the initial cross-shelf density gradient. The surface elevation is chosen so *u* = 0 at one-half of the total water depth.

#### 3) Mixing

*K*

_{m}is 10

^{−5}m

^{2}s

^{−1}, and the background value of vertical diffusivity for temperature

*K*

_{C}is 10

^{−6}m

^{2}s

^{−1}. A logarithmic bottom drag coefficient is applied at the center height of each bottom grid cell, which varies in height with the stretched grid. The resulting stress is quadratic with local velocity and scaled by the log of the distance from the apparent bottom roughness

*z*

_{0}= 1 cm:

### b. Base case

The base case run has a bottom slope of *α* = 10^{−3}, for a maximum depth of 65 m and the Coriolis parameter is a constant *f* = *f*_{0} = 10^{−4} s^{−1}. The model is initialized with a constant vertical temperature gradient of 0.25°C m^{−1}, which creates horizontal isopycnals and an initial stratification of *N*^{2} = 4.25 × 10^{−4} s^{−2}. The cross-shelf wind stress begins at zero and increases over the first two days as *τ*^{s}(*t*) = −0.1 sin(*πt*/4), where *t* is time in days. The wind stress is steady over the final three days of model time with *τ*^{sx} = −0.1 Pa and *τ*^{sy} = 0. The negative sign of the cross-shelf wind means that the stress is onshore on the left side of the channel and offshore on the right side, as shown in Fig. 2.

### c. Model runs

Other model runs start with the base case parameters, initial conditions, and wind forcing, and they typically vary only one of the model parameters. Table 1 summarizes inputs for all model runs used in this paper. Runs used for model testing but not specifically addressed here are also included for completeness. In most analyses, the base case run (top of Table 1) is included and discussed as part of the suite of model runs under investigation. In figures throughout this paper, variables are indicated as the base case (circle), varying wind stress magnitude (triangle), initial stratification (diamond), water depth (asterisk or dot), bottom slope (square), initial cross-shelf density gradient (asterisk), and surface heat flux (plus sign).

Model runs.

Parameters for the base case and model runs with wind stress magnitude, initial stratification, and bottom slope variation were chosen to match those used by Tilburg (2003). For model runs with a surface heat flux, values of *Q* (W m^{−2}) are representative of summertime daily averages of net surface heat flux over the inner shelf off of Martha’s Vineyard. For model runs with an initial cross-shelf density gradient, density gradients are representative of summertime near MVCO. Mid-Atlantic Bight climatologies of temperature and salinity (Lentz 2008) and density (Zhang et al. 2011) indicate that uniformly sloped isopycnals are representative of the seasonal mean near Martha’s Vineyard. A thermal wind shear is spun up in the first inertial period of the model run.

## 3. Field data

### a. Measurements

Field data for this analysis were generated primarily by the National Science Foundation (NSF)-funded Stratification, Wind, and Waves on the Inner shelf of Martha’s Vineyard (SWWIM) study that included mooring and tripod deployments on a cross-shelf transect south of Martha’s Vineyard. From October 2006 to February 2010, observations were made at four sites in 7-, 12-, 17.5-, and 27.5-m water depth, located 0.4, 1.5, 3.8, and 11.1 km from shore, respectively (Fig. 3). Velocity and density data from the 12-, 17-, and 27-m sites will be used in this paper.

In 17 and 27 m depth, 600-kHz Teledyne RD Instruments Workhorse Acoustic Doppler Current Profilers (ADCPs) measured velocity in 0.5-m bins from 2.25 meters above the bed (mab), up to 2.75 m below the surface. The ADCPs collected data at 1 Hz for 5 min out of every twenty and recorded only the burst averages while calculating and recording wave spectrum data at 2 Hz for 10 min every four hours.

The MVCO node at 12-m water depth provided velocity, and wave height, direction, and spectrum. The 1200-kHz ADCP there collects 2-Hz velocity data in 0.5-m bins from 2.5 to 10 mab. The MVCO data are provided as 20-min averages and are available online (at www.whoi.edu/mvco/).

Moorings located near the ADCP tripods measured temperature roughly every 2.5 m throughout the water depth with alternating SeaBird MicroCATs and Onset Temp Pros (Fig. 3). The MicroCATs also measured conductivity at their 5-m spacing. The MicroCATs sampled every 1.5 min and the Temp Pros sampled every 10 min. All temperature and conductivity data are averaged and interpolated onto the ADCP 20-min time base.

### b. Data processing

*y*axes) are oriented toward 96°, 97°, and 102° clockwise from geographic north.

To estimate the cross-shelf transport, first the ADCP velocity profiles are extrapolated to the surface and bottom. The top three measurements are used to linearly extrapolate velocity to the surface, and the bottom velocity is set to zero. The depth-average of the extrapolated velocity profile is then removed to satisfy the no net cross-shelf transport requirement of the two-dimensional model. Finally, cross-shelf transport in the upper water column is calculated by integrating the profile from the surface to the first zero crossing of the cross-shelf velocity profile.

Wind stress was estimated from the MVCO Air–Sea Interaction Tower (ASIT) wind observations using a bulk formula (Fairall et al. 2003) and rotated into the same coordinate frame as the currents, with along-shelf wind stress oriented toward 97° clockwise from geographic north. Wind stress, currents, and transports were low-pass filtered using a filter with a 24-h half amplitude to focus on subtidal variability.

### c. Regressions with forcing terms

*u*at each ADCP bin (with depth average removed) to a shear velocity

*τ*

^{sx}. The regression is a least squares fit toWith this definition,

*τ*

^{sx}. The regression coefficient

*a*acts as nondimensional velocity. Profiles of the regression slope represent how cross-shelf circulation responds to cross-shelf wind forcing.

*A*is the ratio of two values with units of transport, so it represents a nondimensional transport. A multiple regression analysis including cross-shelf wind stress, along-shelf wind stress, and wave forcing yields similar regression coefficients for

## 4. Inner-shelf circulation

### a. Width of inner shelf

*α*, the width of the inner shelf is (

*δ*−

*h*

_{o})/

*α*, where

*δ*is the thickness of the surface boundary layer. The expressionwhere

*N*

^{2}is the initial stratification, provides an accurate estimate of the surface boundary layer thickness (here, defined as the depth where the streamfunction of the cross-shelf circulation is zero) for both initially stratified and unstratified model runs. [This is a modification of the expression proposed by Tilburg (2003), which provided accurate estimates for stratified runs but not for unstratified runs.] Thus, when stratification is larger, the surface boundary layer is thinner and the inner shelf is narrower.

Stratification also limits the deep-water value that the cross-shelf transport must match at the offshore edge of the inner shelf. In water depths greater than *τ*^{sx}/*ρ*_{o}*f*) to 0.34 × (*τ*^{sx}/*ρ*_{o}*f*) at the center of the domain.

While *δ*_{Ek} is the relevant vertical length scale over the inner shelf.

### b. Constant density cases

Using the model initialized with a constant density (no vertical or horizontal variation), we identify how the circulation and transport vary with changes in wind stress and water depth. Once the neutral density response is understood, subsequent experiments with initial stratification will be used to isolate the effect of the density field on the inner-shelf response to wind stress.

In constant density conditions, when a cross-shelf wind stress is balanced by a cross-shelf pressure gradient, the circulation is uniform across the inner shelf when depth is scaled by the total water depth. Figure 4 shows profiles from 10-m water depth, for cross-shelf wind stresses of −0.01, −0.036, and −0.36 Pa, yielding shear velocity values ^{−1} and *h*/*δ*_{Ek} ratios of 0.13, 0.41, and 0.83, which are all within the inner shelf (*h*/*δ*_{Ek} < 1).

When the wind blows across a constant density (unstratified) shelf, the inner-shelf cross-shelf velocity profiles collapse together when nondimensionalized by shear velocity and water depth, that is, *z*/*h* for a range of wind stresses or water depths (Fig. 4). The small spread in the nondimensionalized current profiles can be explained by an increase in *h*/*δ*_{Ek} and a gradual transition from inner to midshelf following a transition from a pressure gradient/wind stress balance to a three-term balance including the Coriolis acceleration. The profile that is least sheared at middepth is from the smallest wind stress (largest *h*/*δ*_{Ek}) and the most sheared profile is from the largest wind stress (smallest *h*/*δ*_{Ek}). A higher wind stress thickens the boundary layer, moving the transition to midshelf dynamics farther offshore. In Fig. 4, an onshore wind stress is used, but the response is essentially symmetric in unstratified experiments, so the results for an offshore wind stress are nearly identical.

Cross-shelf transport is proportional to *h*/*δ*_{Ek} < 1 or *z*/*h* scalings that seem appropriate for the modeled velocity profiles suggest that the coefficient between *U* and *h*/*δ*_{Ek} (or *U* approaches a constant. The symbols showing varying water depth can be seen falling away from the 1.3 slope line as *h* increases. This is in contrast to the symbols showing increasing wind stress, which represent decreasing *h*/*δ*_{Ek} values as *τ*^{sx}/*ρ*_{o}*f*) to 0.38 × (*τ*^{sx}/*ρ*_{o}*f*) at the center of the domain when *κ**f* is less than the water depth of 65 m.

### c. Initially stratified “base case”

When wind blows across an initially stratified shelf, a surface mixed layer develops. For water depths less than the thickness of the surface mixed layer (the inner shelf), a circulation develops that is similar to the unstratified case (cf. Figs. 1 and 6). Both on- (Fig. 6, top) and offshore (Fig. 6, bottom) wind stresses drive near-surface circulation in the direction of the wind stress with a return flow in the lower portion of the water column (Fig. 6b, and indicated by arrows in Fig. 6a). Vertical mixing of the initial stratification over the sloping bottom creates a cross-shelf density gradient on the inner shelf because the bathymetry intersects increasingly dense isopycnals with depth. For all cases with initially horizontal isopycnals, the resulting cross-shelf density gradient is positive regardless of wind direction.

The interaction of the cross-shelf circulation with the cross-shelf density gradient is the primary cause of the difference between the response to on- and offshore wind stresses. An offshore wind stress blows from the less dense nearshore region toward the denser, deeper region of the inner shelf (Fig. 6a, bottom) creating a small positive stratification as a result of buoyancy advection. The stratification reduces vertical mixing and in turn allows greater vertical shear and greater cross-shelf transport. The increased cross-shelf transport across the cross-shelf density gradient feeds back positively on the stratification. The transport does not run away, unbounded, with increasing stratification because the process is still limited by shear instability. For offshore wind (red lines in Fig. 7), both *N*^{2} and Richardson number (Ri) peak where there is near zero shear in the velocity at the height of the peak onshore return flow. The peak Ri is about 0.4 and limits the eddy viscosity to a constant 10^{−4} m^{2} s^{−1} in the bottom few meters of the water column while eddy viscosity increases approximately linearly away from the bottom boundary in the neutral case (black line). The weak positive stratification throughout the water column creates an Ri of 0.05–0.01 throughout most of the water column, causing a decreased eddy viscosity relative to the neutral density case. The resulting slight increase in shear throughout the water column allows a higher cross-shelf transport, which will be quantified in the next several sections.

An onshore wind stress blows from the denser offshore water toward the less dense nearshore water (Fig. 6a, bottom) creating a small negative stratification (Fig. 7b, blue line), which allows greater mixing and so reduces shear and transport. The reduced transport decreases the buoyancy advection that creates the negative stratification and the increased mixing also weakens the negative stratification. Both of these processes bring the density profile closer to neutral. This negative feedback causes the buoyancy advection by onshore winds to have a smaller effect on transport than the opposite direction buoyancy advection by offshore winds. The resulting *N*^{2} and Ri are negative (Figs. 7b,c; blue lines). The negative stratification has the largest magnitude near the surface, but the profile is nearly linear, with no pronounced maximum as for the positive stratification in offshore wind stress case. The negative Richardson number has the largest amplitude at the height of zero shear of the velocity profile, but it is the negative value throughout the water column that causes a higher eddy viscosity over the neutral case. The increased mixing causes lower shear throughout the lower two-thirds of the water column, which causes reduced transport compared to the neutral density and stratified offshore wind stress cases.

Our examination of this base case model run has suggested that differential advection of the cross-shelf density gradient creates a small final vertical density gradient that enhances or reduces the mixing relative to the unstratified value. As a result, an initial stratification increases *U*/(*U*/(

### d. Varying cross-shelf wind stress magnitude

For an initially stratified shelf, varying the wind stress magnitude or water depth affects the cross-shelf transport in the same way as for the unstratified cases (Fig. 5), where *U* increases with *h*. In Fig. 5, the bumps up in the blue line of asterisks and down in the red line are near the offshore edge of the inner shelf, where *τ*/*ρ*_{o}*f* occurs where *h*/*δ*_{Ek} value given by

To isolate the effect of the density field on the transport from the effects of water depth and wind stress, we compute a nondimensional transport fraction *U*/(1.3*h*/*δ*_{Ek} for on- and offshore wind stresses of varying magnitudes. For both directions of wind stress, the inner- to midshelf transition occurs near

For onshore wind stress (Fig. 8, top), the cross-shelf density gradient is constant across the inner shelf and the gradient is well predicted by depth averaging the initial stratification over sloping bathymetry *h*/*δ*_{Ek}, then decrease across the inner shelf, with a widening gap between the stratified runs and the unstratified case (included for comparison). At the offshore edge of the inner shelf, where the surface boundary layer has separated from the bottom, the depth continues to increase while the surface density and transport are constant, so the density gradients fall to zero and the transport fractions are proportional to 1/*h*.

For offshore wind stress (Fig. 8, bottom), upwelled water makes the outer edge of the inner shelf denser than for onshore wind stress (as can be seen in Fig. 6a), creating a higher cross-shelf density gradient near the coast. The density gradient decreases with distance from the coast over the inner shelf, in contrast to the runs for onshore wind stress. The transport fractions all fall together, increasing across the inner shelf, with a widening gap between the stratified runs and the unstratified case. As for the onshore wind, where water depth is greater than the surface boundary layer thickness, the cross-shelf density gradients fall to zero and the transport fractions are proportional to 1/*h*.

### e. Varying initial stratification

The strength of the initial stratification affects the inner shelf in two ways: it increases the cross-shelf density gradient generated by vertical mixing over sloping bathymetry and it limits the deep-water boundary layer thickness *h*/*δ*_{Ek} value, which ranges from 0.32 to 1.15. The inner- to midshelf transition occurs near *h* is less than *h*/*δ*_{Ek} that sets the place in parameter space that tells us how important the Coriolis acceleration is relative to vertical mixing. If these data were plotted as a function of *h*/*δ*_{Ek}, the too-small vertical length scale would cause transport for both on- and offshore wind stresses to appear larger for the stratified cases than the constant density case, hiding the result that a negative final stratification should decrease transport.

For onshore wind stress (Fig. 9a), higher initial stratification increases the cross-shelf density gradient uniformly across the width of the inner shelf. The higher cross-shelf density gradient allows a larger (more negative) unstable stratification to be produced over the inner shelf by the wind-driven advection (Fig. 10a, blue symbols). Consequently, increasing initial stratification results in reduced transport fraction over the inner shelf (Fig. 9b and Fig. 10b, blue symbols). Note that the magnitude of the unstable stratification over the inner shelf is 100 times smaller than the initial stratification.

For offshore wind stress, the cross-shelf density gradient (Fig. 9c) increases with increasing initial stratification, but the density gradient is not uniform across the inner shelf. The maximum ∂*ρ*/∂*x*, located just a few kilometers from shore, is larger than that predicted purely by vertical mixing of initial stratification. (See values generated by onshore wind stress in Fig. 9a for comparison.) The wind-driven circulation acting on the larger cross-shelf density gradient creates a stronger inner-shelf stratification. Thus, for offshore wind stress, stronger initial stratification results in stronger inner-shelf stratification (Fig. 10a, red symbols). As noted for onshore wind stresses, the inner-shelf stratification is two orders of magnitude smaller than the initial stratification. The stronger stratification over the inner shelf allows a greater shear and transport to be maintained before the positive feedback process is limited by mixing. Consequently, the inner-shelf transport fraction (Fig. 9d) increases with stronger initial stratification (red symbols in Fig. 10b).

## 5. Predicting the cross-shelf density gradient

Results from the previous section indicate that the cross-shelf density gradient is a key component in setting the balance between circulation and mixing. Here, we show that a simple one-dimensional model provides accurate estimates of the cross-shelf density gradient over the inner shelf for uniform initial stratification (Table 1, runs 1 and 10–20), as well as for model experiments adding a constant surface heat flux (runs 35–40) or an initial positive or negative cross-shelf density gradient (runs 21–31).

*α*

_{T}

*Qt*/

*hρ*

_{o}c_{p}, where

*α*

_{T}is the thermal expansion coefficient in the linear equation of state. The cross-shelf density gradient generated by the surface heat flux

Inner-shelf cross-shelf density gradients (Fig. 11a) match the prediction from Eq. (10) for most of the model runs, regardless of whether the density gradient was initially imposed or created by mixing of initial stratification or surface heat flux. However, there are small differences between model and prediction for a few of the model runs (Fig. 11b) when buoyancy advection over the inner shelf is significant. The cross-shelf density gradients plotted in Figs. 8a, 8c, 9a, and 9c showed that offshore wind stress creates variation in the cross-shelf density gradient across the inner shelf, while the onshore wind stress creates a nearly constant density gradient. Correspondingly, the offshore wind (red) data in this subset (Fig. 11b) show a greater spread than the onshore wind (blue) data. In the model runs, advection is most important in modifying the horizontal density structure for offshore wind stress, particularly near the outer edge of the inner shelf. Overall, the close match of ∂*ρ*/∂*x* with Eq. (10) indicates that the cross-shelf density gradient is set primarily by one-dimensional processes (vertical mixing) on the inner shelf in this 2D model.

## 6. Effect of the cross-shelf density gradient on cross-shelf transport

As discussed in section 4, a cross-shelf density gradient modifies the wind-driven transport by changing the values of shear and stratification that control vertical mixing and close the governing equations. To quantify the relationship between transport fraction and the horizontal density gradient, we look for the relevant scales to characterize the influence of the cross-shelf density gradient on vertical mixing.

*h*times the stress scale

_{x}is a signed value:The Ri

_{x}is positive when wind-driven advection will create positive stratification that limits mixing. Momentum from the wind will be confined nearer to the surface, leading to increased transport in the upper portion of the water column. The Ri

_{x}is negative when the advection will create negative stratification that enhances mixing. Mixing moves high-momentum surface water downward, where it combines with water moving the opposite direction from the lower portion of the water column. Velocities in both the upper and lower portions of the water columns are reduced, so the resulting transport is smaller.

Transport fractions at 10-m water depth plotted as a function of the horizontal Richardson number for all the model runs tend to collapse onto a single curve (Fig. 12). The collapse of all the model runs onto a single curve supports the idea that Ri_{x} is the key nondimensional number characterizing the inner-shelf response to stratification, surface heat flux, and cross-shelf density gradients. For density values increasing in the direction of the wind stress (Ri_{x} > 0), an increased density gradient increases transport by increasing stratification and reducing mixing. For a wind stress opposing the density gradient (Ri_{x} < 0), a larger cross-shelf density gradient decreases the transport fraction by creating negative stratification and more mixing for a particular wind stress magnitude. The steeper slope of the transport for positive Ri_{x} values than negative ones indicates the transport’s stronger response to wind blowing with a given cross-shelf density gradient (from lighter toward denser water) than against it. The larger change in transport for off- than onshore wind stresses shown in section 4e was not just an effect of the higher nearshore density gradient created by offshore wind stresses, but an effect of the two directions of wind stress acting in different ways on the cross-shelf density gradient. By using both positive and negative density gradients with both positive and negative wind stresses, we ensure the asymmetry observed is a true response to Ri_{x} and not an artifact of each sign of Ri_{x} only being generated by transport toward shallower or deeper regions. Comparison of transport fraction to horizontal Richardson number at a fixed *h*/*δ*_{Ek} value, to ensure the effect of Ri_{x} on transport fraction is fully isolated, yields a result very similar to the one shown in Fig. 12.

The dependence in Fig. 12, with a larger slope for positive Ri_{x}, and smaller slope for negative Ri_{x}, is similar to the response of wind shear to vertical flux Richardson number *R _{f}* as described by Monin and Obukhov (1954) similarity theory and demonstrated with atmospheric field measurements by Businger et al. (1971). Monin–Obukhov similarity theory predicts that the nondimensional wind shear in the atmospheric boundary layer will be a function of

*R*=

_{f}*z*/

*L*, where

*B*

_{z}the vertical buoyancy flux from surface heating (e.g., Kundu and Cohen 2004, pp. 566–567; Foken 2006).

*c*= 1/6 to fit the observed relationship between Ri

_{x}and normalized transport in Fig. 12,where we have taken

*γ*= 15 and

*β*= 5 following Garrat (1992) (dashed line, Fig. 12). This relationship analogous to Monin–Obukhov similarity theory provides an accurate representation of the model result dependence for Ri

_{x}< 1.

## 7. Comparison to field observations

Our modeling results indicate the importance of the cross-shelf density gradient in controlling the efficiency of the cross-shelf wind stress at driving cross-shelf transport. We use measurements from the inner shelf south of Martha’s Vineyard to test if the same relationship seen in Fig. 12 and summarized in Eq. (15) is demonstrated by field observations. To ensure that we only consider times when the moored array is on the inner shelf, we limit the analysis to times when density is well mixed at the 27-m site (near-surface to near-bottom density difference less than 0.05 kg m^{−3}) so that _{x} is estimated using the cross-shelf density gradient, shear velocity from cross-shelf wind stress at the MVCO ASIT in 15-m water depth, and the water depth of each observation site. Cross-shelf velocity and transport estimates from the 12-, 17-, and 27-m sites are regressed (as described in section 3c) against shear velocity and then examined for different range bins of Ri_{x}.

### a. Velocity profiles

Cross-shelf current profiles driven by cross-shelf wind stress from the model and observations exhibit a similar vertical structure and dependence on Ri_{x} (Fig. 13). Velocity profiles from the model (solid) and regression slope profiles from observations (circles) both show larger magnitude and middepth shear for positive Ri_{x} and a lower magnitude and middepth shear for a negative Ri_{x} relative to the zero Ri_{x} profiles.

The normalized currents ^{−1}) and the resulting mixing will reduce shear in the observed velocity profiles. Any along-shelf component of the wind stress would also contribute to mixing. By comparison, Lentz and Fewings (2012) found a closer match between modeled and measured profile magnitudes on the North Carolina inner shelf, where tides are weak. This effect of tides reducing wind-driven circulation on the inner shelf was demonstrated numerically by Castelao et al. (2010) for upwelling wind stresses.

### b. Cross-shelf transport

The dependence of transport fraction estimates from the 12- (blue symbols, Fig. 14), 17- (red), and 27-m (cyan) sites on Ri_{x} is similar to the model dependence (Fig. 12) and both follow the curve predicted by Monin–Obukhov similarity theory (black dashed line). As with the model output, the observations are normalized to make the transport fraction one when Ri_{x} is zero. The regression slope *A*_{0} between transport *U* and transport scale

At all three sites, the observations show a reduced transport fraction for Ri_{x} < 0 and enhanced transport fraction for Ri_{x} > 0 relative to the transport fraction for Ri_{x} = 0. For |Ri_{x}| less than one, the data show a higher slope for positive Ri_{x} than for negative Ri_{x}, consistent with model results. Transport fraction at the 12-m site shows the strongest response (larger change from *U*/*A*_{0}*ρ*/∂*x* for the 27-m site would place points at too large a value of Ri_{x}, making the effect of Ri_{x} on transport fraction appear weaker. Conversely, the response may appear too strong for the 12-m site if the cross-shelf density gradient has been underestimated.

For Ri_{x} greater than one, the data (Fig. 14) and limited model runs (Fig. 12) show that the transport fraction appears to change less rapidly in response to further increases in Ri_{x} than for Ri_{x} less than one. When Ri_{x} is small, the wind stress term is much larger than the baroclinic pressure gradient term in the cross-shelf momentum balance. In these cases, the cross-shelf density gradient affects the circulation by how it alters the vertical mixing, but does not directly drive the flow. For larger Ri_{x}, the baroclinic pressure gradient term is similar in magnitude to the wind stress and the two terms are equal in magnitude when Ri_{x} equals two. When the baroclinic pressure gradient is a significant term in the momentum equation, the cross-shelf density gradient will directly drive circulation in addition to its effect on the vertical mixing; in this case the system is in a different dynamical regime from the one Monin–Obukhov theory explains and it is not surprising our model and observations diverge from the empirical relationship [Eq. (15)].

## 8. Conclusions

This paper was motivated by field observations and previous modeling work that left open the question: how does cross-shelf wind stress drive circulation on an initially stratified inner shelf? How does the density field affect the circulation and why is the response to on- and offshore wind stresses asymmetric? We use an idealized, 2D, cross-shelf transect in ROMS to describe the effects of initial stratification, wind stress magnitude, surface heat flux, and cross-shelf density gradient on the inner-shelf response to cross-shelf wind stress. To first order, the cross-shelf transport *U* scales like *U*/*h*/*δ*_{Ek}.

Over sloping bathymetry, vertical mixing of initial stratification or a surface heat flux creates a cross-shelf density gradient. Under cross-shelf wind stresses, the resulting cross-shelf gradient for water depths less than the deep-water boundary layer thickness is well represented by a prediction based on complete vertical mixing of all contributions to the density field [Eq. (10)]. Straining of the resulting cross-shelf density gradient increases transport by offshore winds and decreases transport by onshore winds, but does not generate a strongly stratified inner shelf for either wind direction. For a positive cross-shelf density gradient (lighter water near the coast), an offshore wind stress moves lighter over denser water, creating a small positive stratification, which allows increased vertical shear in the cross-shelf velocity profile and increased transport. The increased transport feeds back positively on the increased stratification until the process is limited by shear instability. An onshore wind stress moves denser over lighter water, creating a small negative stratification, which increases mixing and decreases transport. The decreased transport reduces the buoyancy advection that creates the negative stratification and the increased mixing also destroys the negative stratification, so the process that alters transport by onshore wind stress is more self-limiting than that for offshore wind stress.

The horizontal Richardson number is the scale for the cross-shelf density gradient. Transport fraction is a function of Ri_{x} and the relationship collapses for all model runs. Figure 12 demonstrates the empirical relationship [Eq. (15)] between the cross-shelf density gradient and the efficiency of the cross-shelf wind at driving transport.

This result is particularly important for planning field investigations and interpreting data because the cross-shelf density gradients analyzed here, along with wind stresses and water depths, are measurable with common observational techniques. In contrast, the small stratification resulting from horizontal buoyancy advection across the inner shelf, as described in sections 4c–e and shown in Fig. 10a are not readily measurable from the SWWIM dataset or from other similar moored observations. Observations from the inner shelf south of Martha’s Vineyard support the modeling results. The similar results for model and field observations in Figs. 12 and 14 provide compelling evidence that it is the cross-shelf rather than vertical density gradient that is critical to predicting transport driven by a cross-shelf wind stress.

## Acknowledgments

This work was funded by Ocean Sciences Division of the National Science Foundation Grant OCE-0548961 and by the Woods Hole Oceanographic Institution through the Academic Programs Office and the Coastal Ocean Institute. Data central to this study were provided by the Martha’s Vineyard Coastal Observatory, which is funded by WHOI and the Jewett/EDUC/Harrison Foundation. Thanks to Craig Marquette, Captain Ken Houtler, and crew member Ian Hanley of R/V *Tioga* for instrument assembly, deployment, and recovery and thanks to Rocky Geyer for suggesting investigating the Monin–Obukhov comparison.

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