The Effect of Stokes Drift and Transient Rip Currents on the Inner Shelf. Part II: With Stratification

Nirnimesh Kumar University of Washington, Seattle, Washington

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Falk Feddersen Scripps Institution of Oceanography, La Jolla, California

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Abstract

This is Part II of a two-part study focused on Stokes drift and transient rip current (TRC) effects on the unstratified (Part I) and stratified (this paper) inner shelf. Part I focuses on funwaveC–Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) coupling and TRC effects on mixing and exchange on an unstratified inner shelf. Here, two simulations (R3 and R4) are performed on a stratified inner shelf and surfzone with typical bathymetry, stratification, and wave conditions. R3 is a COAWST-only simulation (no TRCs), while R4 has funwaveC–COAWST coupling (with TRCs). In R4, TRCs lead to patchy, near-surface cooling, vertical isotherm displacement, and increased water column mixing. For both R3 and R4, the mean Lagrangian circulation has two nearly isolated surfzones and inner-shelf overturning circulation cells, with a stronger, R4, inner-shelf circulation cell. The R4, inner-shelf, vertical velocity variability is 2–3 times stronger than a simulation with TRCs and no stratification. Relative to R3, R4 eddy diffusivity is strongly elevated out to three surfzone widths offshore due to TRCs and TRC-induced density overturns. The R4 inner-shelf stratification is reduced nearshore, and mean isotherms slope more strongly than R3 because of the TRC-enhanced irreversible mixing. At six surfzone widths offshore, both R3 and R4 are in geostrophic balance, explaining the stratified (summertime) observed deviation from Stokes–Coriolis balance. In this region, baroclinic pressure gradients induced by sloping isotherms induce an alongshore geostrophic jet offshore, strongest in R4. In R4, TRCs result in an enhanced (2–10 times) cross-shore exchange velocity across the entire inner shelf, relative to R3. Accurate, stratified, inner-shelf simulations of pollution, larval, or sediment transport must include transient rip currents.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: N. Kumar, nirni@uw.edu

Abstract

This is Part II of a two-part study focused on Stokes drift and transient rip current (TRC) effects on the unstratified (Part I) and stratified (this paper) inner shelf. Part I focuses on funwaveC–Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) coupling and TRC effects on mixing and exchange on an unstratified inner shelf. Here, two simulations (R3 and R4) are performed on a stratified inner shelf and surfzone with typical bathymetry, stratification, and wave conditions. R3 is a COAWST-only simulation (no TRCs), while R4 has funwaveC–COAWST coupling (with TRCs). In R4, TRCs lead to patchy, near-surface cooling, vertical isotherm displacement, and increased water column mixing. For both R3 and R4, the mean Lagrangian circulation has two nearly isolated surfzones and inner-shelf overturning circulation cells, with a stronger, R4, inner-shelf circulation cell. The R4, inner-shelf, vertical velocity variability is 2–3 times stronger than a simulation with TRCs and no stratification. Relative to R3, R4 eddy diffusivity is strongly elevated out to three surfzone widths offshore due to TRCs and TRC-induced density overturns. The R4 inner-shelf stratification is reduced nearshore, and mean isotherms slope more strongly than R3 because of the TRC-enhanced irreversible mixing. At six surfzone widths offshore, both R3 and R4 are in geostrophic balance, explaining the stratified (summertime) observed deviation from Stokes–Coriolis balance. In this region, baroclinic pressure gradients induced by sloping isotherms induce an alongshore geostrophic jet offshore, strongest in R4. In R4, TRCs result in an enhanced (2–10 times) cross-shore exchange velocity across the entire inner shelf, relative to R3. Accurate, stratified, inner-shelf simulations of pollution, larval, or sediment transport must include transient rip currents.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: N. Kumar, nirni@uw.edu

1. Introduction

This is Part II of a two-part study that focuses on exploring Stokes drift–driven and transient rip current effects on an unstratified (Kumar and Feddersen 2016, hereinafter Part I) and a stratified (Part II, this manuscript) inner shelf. Background and motivation is provided in Part I and is revisited here as relevant to the stratified inner shelf. The nearshore region consists of the surfzone [from the shoreline to the seaward extent of depth-limited breaking (LSZ)] to the inner shelf (from 5 to ≈15 m). Surfzone and inner-shelf cross-shelf exchange processes, important for tracer evolution (e.g., larvae, pollutants), are three-dimensional, complex, and forced by a variety of mechanisms. However, the relative role of surface gravity wave–driven processes, such as Stokes drift and transient rip currents, in driving cross-shore exchange on a stratified inner shelf is not understood.

On an alongshore uniform bathymetry, cross-shelf exchange from the surfzone across the inner shelf is often due to a nonzero, mean, cross-shore Lagrangian circulation in the vertical z and cross-shore x plane, where the mean, cross-shore Lagrangian velocity is the sum of the mean Eulerian velocity and the onshore, wave-driven Stokes drift (e.g., Lentz and Fewings 2012). In particular, for inner-shelf weak vertical mixing (e.g., weak winds) and weak lateral variations, a Stokes–Coriolis balance (e.g., Xu and Bowen 1994; Lentz et al. 2008) develops:
e1
where f is the Coriolis parameter, resulting in zero-mean Lagrangian flow = 0 and no cross-shelf exchange. For subtidally (>33-h time scale) averaged, Eulerian, cross-shelf velocities and unstratified conditions, a Stokes–Coriolis balance (1) was observed in h = 12-m depth for weak winds (Lentz et al. 2008). However, for summer (stratified) conditions, near the bed was onshore, and in the upper–midpart of the water column was offshore, inconsistent with a Stokes–Coriolis balance (Lentz et al. 2008). This result was unexpected, as weak vertical mixing due to stratification should allow Stokes–Coriolis balance to dominate. However, spatial variation in stratification, wave forcing, or other processes may have been important (Lentz et al. 2008).

On an alongshore, uniform bathymetry, exchange from the surfzone across the inner shelf is also induced by horizontal eddies (vertical vorticity). Finite-crest-length wave breaking generates surfzone eddies (Peregrine 1998; Johnson and Pattiaratchi 2006; Spydell and Feddersen 2009; Clark et al. 2012; Feddersen 2014) that coalesce, inducing episodic, 10–50-m, alongshore, length-scale transient rip currents (TRCs) that dominate (over nonzero ) observed and modeled surfzone to inner-shelf exchange (Hally-Rosendahl et al. 2014, 2015; Hally-Rosendahl and Feddersen 2016). Even in the inner-shelf TRC-generated, eddy-induced exchange is larger than an estimated Stokes drift–driven exchange for two to five surfzone widths LSZ, as indicated through a depth-integrated, wave-resolving model study (Suanda and Feddersen 2015).

Inner-shelf stratification can be strong even within 80 m of the surfzone inhibiting vertical tracer mixing (Hally-Rosendahl et al. 2014). A stratified inner shelf enhances cross-shelf exchange due to cross-shelf winds in both observations (Fewings et al. 2008) and models (Tilburg 2003; Horwitz and Lentz 2014), relative to an unstratified inner shelf. On a stratified inner shelf, TRCs also have associated temperature signals (Marmorino et al. 2013; Hally-Rosendahl et al. 2014), suggesting their role in inner-shelf temperature evolution. However, for a stratified inner shelf, the effects of Stokes drift and TRCs on mean Lagrangian circulation, inner-shelf eddies, temperature evolution, mixing, momentum dynamics, and cross-shelf exchange is poorly understood.

Three-dimensional (3D) transient rip currents and Stokes drift–driven flow on a stratified inner shelf have never been modeled before. In Part I, the wave-resolving Boussinesq model funwaveC is coupled to a wave-averaged, depth- and stratification-resolving model, Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST), to allow both Stokes drift effects and 3D TRCs on an unstratified inner shelf. Relative to simulations without TRCs, TRCs induced significant changes in the mean overturning Lagrangian circulation, velocity variability, mean eddy viscosity, momentum balances, and exchange velocity out to 5LSZ offshore.

Here in Part II, the work in Part I is extended to a stratified inner shelf with a single-case example having typical bathymetry, stratification, and waves but no wind. Two simulations are analyzed and contrasted, one without (R3) and one with (R4) TRC effects, analogous to R1 and R2 in the unstratified Part I. Here, the focus is on the effect of Stokes drift and TRCs on the mean overturning Lagrangian circulation, velocity variability, temperature evolution, mixing, momentum balances, and a cross-shelf exchange velocity on the stratified inner shelf.

A detailed model description and validation is provided in Part I and briefly described for a stratified inner shelf (section 2). The surfzone and inner-shelf temperature evolution over 48 h both without (R3) and with (R4) TRCs is described in section 3. The effect of TRCs on the inner-shelf mean Lagrangian overturning streamfunction, eddy variability, mean vertical eddy viscosity, and mean temperature are examined in section 4. The discussion (section 5) examines mean cross-shore momentum balances, irreversible mixing, and cross-shore exchange velocity both with and without TRCs. Last, the development of an inner-shelf, alongshore, geostrophic jet is explained. The results are summarized in section 6.

2. Methods

a. funwaveC model description and configuration

The open-source, wave-resolving Boussinesq model funwaveC (e.g., Feddersen 2007; Spydell and Feddersen 2009; Feddersen et al. 2011; Clark et al. 2012; Guza and Feddersen 2012; Feddersen 2014; Suanda and Feddersen 2015; Hally-Rosendahl and Feddersen 2016) is described in detail elsewhere (Feddersen et al. 2011). Model configuration is discussed in Part I, with a quick summary provided here. The funwaveC simulation generates a transient rip current due to normally incident random directional waves on an alongshore, uniform planar beach with slope 0.025 to depth h = 7 m and constant h farther offshore. The total cross-shore x and alongshore y domain lengths are 500 and 1000 m, respectively, with cross- and alongshore grid sizes of Δx = 1.25 m and Δy = 1 m. The alongshore boundary conditions are periodic. Random directionally spread waves with significant wave height Hs = 1 m, peak period Tp = 10 s, bulk (mean) wave angle = 0°, and directional spread σθ = 10° allow vorticity generation due to finite-crested wave breaking (Peregrine 1998). Model variables are output at 1 Hz. The funwaveC-simulated curl of the breaking wave that generates surfzone vertical vorticity (eddies) on a variety of length scales (Feddersen 2014) is expressed using a scalar forcing streamfunction ψF:
e2
The streamfunction ψF is solved for at 1 Hz (e.g., Spydell and Feddersen 2009) and stored for input to COAWST (section 2b).

b. COAWST model description and configuration

The COAWST model (Warner et al. 2010) couples the circulation model ROMS and the wave model Simulating Waves Nearshore (SWAN) and has been validated in a range of surfzone, estuary, and inner-shelf scenarios in the subtidal and tidal band (e.g., Kumar et al. 2012; Olabarrieta et al. 2011; Kumar et al. 2015b,a). However, wave-averaged COAWST cannot simulate transient rip currents through surfzone eddies generated by finite-crested wave breaking, motivating the coupling with funwaveC.

Detailed model setup is provided in Part I. The bathymetry h(x) is alongshore uniform. The cross-shore profile (thick solid black line, Fig. 1) is planar with slope 0.025 to h = 7-m depth, matching the funwaveC bathymetry. Farther offshore the bathymetry is concave and the slope reduces to the typical, Southern California, inner-shelf bathymetry profiles (Kumar et al. 2015b). The COAWST model domain is 1000 m in the alongshore and 800 m in the cross shore with grid resolution of Δx = 1.25 m and Δy = 2 m. The alongshore boundary conditions are periodic. Here, ROMS has 10 bathymetry-following vertical levels and the Coriolis parameter f = 8.09 × 10−5 s−1 is typical for Southern California. In contrast to Part I, the simulations here are run for 48 h (2 days) with a ROMS baroclinic time step of 0.25 s and barotropic time step of 0.0125 s.

Fig. 1.
Fig. 1.

(a) COAWST cross-shore bathymetry (thick solid black line) and temperature initial condition (colors and contours). The stratification of ∂T/∂z = 0.25°C m−1 corresponds to N2 = 6 × 10−4 s−2, typical for the Southern California inner shelf. (b) Significant wave height Hs for SWAN (red) and funwaveC (black dashed). In (a) and (b), the vertical dashed–dotted line delimits the surfzone x = −LSZ (where LSZ = 100 m).

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

At the SWAN offshore boundary, the wave field (Hs = 0.95 m, peak period Tp = 10 s, mean wave direction = 0°, and a directional spread σθ = 10°) is prescribed to match funwaveC at x > −280 m. SWAN cross shore evolves the wave field with standard parameters. SWAN- and funwaveC-modeled Hs compare well (Part I), indicating that the two models evolve waves consistently. SWAN-derived, vertically varying Stokes drift drives circulation in ROMS with one-way coupling (Part I). COAWST surfzone eddy generation is given by the funwaveC rotational wave forcing [i.e., ∇×ψF(x, y, t)k, where k is the upward unit vector] as a depth-uniform body force at 1 Hz. As the funwaveC simulation is for 12 h, this body force is symmetric from 1 to 12 h and from 12 to 24 h, and similarly from 24 to 36 h and from 36 to 48 h.

The ROMS temperature initial condition is T = 20°C at z = 0 m, with constant stratification of ∂T/∂z = 0.25°C m−1 everywhere (surfzone and inner shelf) such that at z = −12 m, T = 17°C (Fig. 1). This stratification corresponds to N2 = 6 × 10−4 s−2, which is typical for the Southern California Bight (Omand et al. 2012; Hally-Rosendahl et al. 2014; Kumar et al. 2015b). At the offshore boundary (x = −800 m), temperature is kept fixed to the initial condition. Solar heating and air–sea fluxes, which can also modify water column temperature, are not considered in this study.

c. COAWST stratified simulations R3 and R4: Without and with transient rip currents

In Part I, two unstratified COAWST simulations R1 and R2 were conducted without (R1) and with (R2) transient rip currents generated by the coupling to funwaveC. Similarly, here, two stratified COAWST simulations are performed, one without (R3) and one with (R4) transient rip currents generated by coupling to funwaveC. Both simulations have identical initial stratifications (Fig. 1).

3. Results: Effects of Stokes drift and transient rip currents on a stratified inner shelf—Temperature evolution

Here, the stratified inner-shelf temperature and vorticity evolution over 48 h is examined for the case without (R3) and with (R4) transient rip currents.

a. Inner-shelf temperature evolution without transient rip currents: R3

Here, R3 instantaneous temperature evolution is examined after 6 (0 h is model start time) and 48 h have elapsed (Fig. 2). After 6 h, the surfzone (x > −LSZ) temperature is vertically well mixed (unstratified; Fig. 2a) from its stratified initial condition. Offshore of the surfzone at x = −2LSZ and x = −3LSZ, the stratification is reduced 25% and 15%, respectively, of the initial ∂T/∂z = 0.25°C m−1 (Fig. 2a). Farther offshore at x < −4LSZ, stratification is close to the initial stratification. This slow temperature evolution is driven by the mean Lagrangian circulation (onshore Stokes drift and offshore undertow) coupled with strong surfzone vertical mixing. Thus, relatively warmer near-surface waters enter the surfzone and relatively colder well-mixed water leaves the surfzone. At 48 h stratification is reduced (Fig. 2b). At x = −2LSZ and x = −3LSZ, ∂T/∂z is reduced 60% and 36%, respectively, relative to the original stratification. However, farther offshore at x ≤ −4LSZ, the stratification remains close to the initial stratification (Fig. 2b2). The processes driving this R3 (no TRCs) temperature evolution pattern are discussed later.

Fig. 2.
Fig. 2.

R3 (no TRCs) simulated temperature T at (a) 6 and (b) 48 h. Dashed black line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

b. Inner-shelf temperature evolution with transient rip currents: R4

Here, R4 (with TRCs) modeled vorticity and temperature evolution is examined over 48 h and contrasted with R3 (Figs. 35). In Part I, TRC ejection onto the unstratified inner shelf (R2) strongly influenced the mean overturning Lagrangian circulation, vertical eddy viscosity, and cross-shelf exchange flow out to x = −3LSZ. With stratification, the R4, modeled, near-surface (z = −1 m) vorticity is qualitatively similar to R2 (see Part I) both for the first 24 h of simulation (Figs. 3a1–a4) and 24–48 h of simulation (Figs. 4a1–a4). The R4 surfzone vorticity field is highly variable with a range of length scales similar to R2 and funwaveC (Part I; Feddersen 2014). R4 inner-shelf eddy variability extends offshore to x = −3LSZ with vorticity monopoles, dipoles, filaments, and streaks at O(10−2) s−1, much larger than the Coriolis parameter. At 6 h and onward the eddy field has equilibrated at x > −4LSZ (left columns, Figs. 3 and 4).

Fig. 3.
Fig. 3.

R4 (with TRCs) simulated (left) vertical vorticity and (center) temperature at z = 1 m; and (right) temperature at the dashed–dotted line at times (a1),(b1),(c1) 1, (a2),(b2),(c2) 6, (a3),(b3),(c3) 12, and (a4),(b4),(c4) 18 h. Dashed black line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

Fig. 4.
Fig. 4.

As in Fig 3, but at times (a1),(b1),(c1) 24, (a2),(b2),(c2) 30, (a3),(b3),(c3) 42, and (a4),(b4),(c4) 48 h.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

Fig. 5.
Fig. 5.

R4 (with TRCs) simulated temperature at x = −1.6LSZ (−160 m) at (a) 0, (b) 1, (c) 6, (d) 24, and (e) 48 h.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

TRCs and the resulting inner-shelf eddy field strongly affect the R4 temperature evolution over 48 h (center and right columns, Figs. 3 and 4) through stirring and mixing. At 1 h, the R4, near-surface (z = −1 m), inner-shelf temperature T(x, y) is mostly ≥19.6°C with a few cold patches (≤19.4°C) extending out to x = −3LSZ with scales of 100 m (Fig. 3b1), corresponding to TRC ejection locations (Fig. 3a1). At y = 600 m (dashed–dotted line in Fig. 3b1), the surfzone is vertically well mixed with T(x, z) = 19.6°C (Fig. 3c1), similar to R3. Farther offshore at x = −2.5LSZ, the isotherms are elevated up to 3 m at the cold patch location, and just offshore at x = −3.5LSZ the isotherms are depressed up to 2 m (Fig. 3c1). These R4 temperature features starkly contrast to those of R3 (Fig. 2).

At 6 h, R4 near-surface temperature is cooler (down to T = 19.2°C) out to x = −4LSZ with larger 200-m length scales (Fig. 3a2) compared to at 1 h. At 6 h, the R4 (with TRCs) near-surface T is much cooler and more variable. At y = 600 m, the well-mixed surfzone has cooled further with T(x, z) = 19.3°C (Fig. 3c2), much cooler than R3. Farther offshore at x = −3LSZ, the T = 19°C isotherm forms an eddy temperature front (Fig. 3c2), while near x = −6LSZ, R4 temperature is similar to the initial condition and R3. Transient rip currents result in strong vertical mixing, reducing the stratification at x ≥ −3LSZ significantly relative to R3. For example, at x = −2LSZ and x = −3LSZ, the alongshore-averaged ∂T/∂z is reduced 54% and 24%, respectively, relative to the initial stratification, much larger than the 25% and 15% stratification reductions of R3.

The R4 temperature evolution with patchy, near-surface cooling (Figs. 3b24b4) and weakening of the stratification (Figs. 3c24c4) continues throughout the 48-h simulation as transient rip currents deliver eddies to the inner shelf. By 24 h, significant, near-surface (z = −1 m), R4 cooling has reached x = −5LSZ (Fig. 4a2). From 30 to 48 h, a near-surface, cross-shore temperature front has formed near x = −3LSZ, separating the onshore largely homogenized waters in the upper 4 m and the still stratified waters farther offshore. At 48 h, the R4 near-surface T(x, y) ≈ 19.2°C for x > −3LSZ (Fig. 4b4), significantly cooled (ΔT = 0.55°C) relative to R3 (ΔT = 0.30°C; Fig. 2b2). Offshore of x = −3LSZ, the T = 19°C isotherm, originally at z = −4 m, is depressed 1–2 m and slopes upward farther offshore (Fig. 4b4). At x = −2LSZ, R4 stratification is essentially destroyed (16% of initial stratification), and at x = −3LSZ stratification is similarly reduced (26% of initial stratification). Even farther offshore, the R4 upper 4-m stratification is substantially reduced relative to R3, which is explored further in section 4d.

After examining R4 temperature evolution and variability in the (x, y) and (x, z) planes, the vertical and alongshore structure of the TRC-induced R4 temperature variability is examined in the (y, z) plane at x = −1.6LSZ (h = 4 m; Fig. 5). Recall that at 0 h (initial condition) T is alongshore uniform with constant stratification of ∂T/∂z = 0.25°C m−1 (Fig. 5a). At 1 h, T(y, z) is patchy with the 19.5°C isotherm raised and lowered ±2 m at the 50–200-m alongshore length scales (Fig. 5a), consistent with patchy T(x, y) (Fig. 3b1). The alongshore temperature variability is largest at depth with alongshore standard deviation of 0.15°C. At y ≈ 700 m and y ≈ 0 m, the instantaneous water column is essentially unstratified. Net cooling is not yet obvious.

At 6 h, R4 T(y, z) has cooled significantly to an (vertical and alongshore) average of 19.35°C, is alongshore patchy with 50–100-m length scales, and is largely unstratified (isotherms are vertical; Fig. 5c). This temperature structure is consistent with observations over −3 < z < −1 m 80 m offshore of the surfzone with similar Hs (Hally-Rosendahl et al. 2014). This process of net cooling continues but slows down at 24 h with a mean 19.23°C and 48 h with a mean 19.2°C (Figs. 5d,e). Similarly the temperature alongshore standard deviation decreases from 0.1°C at 1 h to 0.05°C at 48 h. The alongshore T standard deviation is 2 times larger near bed (z < −3 m) than in the upper (z > −2 m) water column, also consistent with field observations (Hally-Rosendahl et al. 2014).

4. Results: Effects of transient rip currents on a stratified inner shelf—Circulation and temperature statistics

In Part I, TRCs were shown to have a strong effect on inner-shelf velocity variability, vertical viscosity, and exchange flow. The R3 and R4 qualitative temperature evolution comparison (section 3) demonstrates that TRCs have a strong effect on the inner-shelf temperature and stratification to x = −3LSZ. Here, the effects of TRCs on the inner-shelf circulation and temperature statistics are quantified with statistics that are alongshore averaged and time averaged from 12 to 48 h (or 12 to 24 h, section 4b). Unlike the unstratified cases in Part I, where equilibrium had been reached and statistics were stationary, here the stratification evolves over the averaging period and thus the statistics are not strictly stationary.

a. Lagrangian mean circulation

As in Part I, a Lagrangian streamfunction ψL, defined so that and , where is averaged over 12–48 h and the alongshore, is used to characterize the effects of Stokes drift and TRCs on the stratified inner-shelf mean Lagrangian circulation (Fig. 6). The inner-shelf stratification has a strong effect on the ψL Lagrangian circulation both without (R3) and with (R4) TRCs, relative to an unstratified inner shelf (Part I). Within and just seaward of the surfzone (>−1.8LSZ), the R3 ψL(x, z) streamlines are closed (Fig. 6a), indicating an overturning circulation pattern with near-surface onshore flow and near-bed offshore flow, consistent with the unstratified case R1 (Part I). Only a few streamlines cross x = −2LSZ, and the lower water column ψL streamlines are directed upward into the midwater column (Fig. 6a) in contrast to the along-bed offshore flow of the R1 (see Part I). This ψL pattern indicates that, without TRCs, stratification acts to an exchange barrier between the surfzone and the inner shelf. Farther offshore x < −2.5LSZ, a clockwise ψL circulation cell is onshore-directed near the surface and offshore-directed in the midwater column that is largely disconnected from the surfzone (Fig. 6a). Below this upper ψL cell, a second weaker counterclockwise ψL cell is present with onshore flow near bed. This contrasts with the unstratified no TRC (R1) single inner-shelf ψL circulation cell (Part I).

Fig. 6.
Fig. 6.

(a) R3 (no TRCs) and (b) R4 (with TRCs) Lagrangian overturning streamfunction ψL (colors and contours at 10−3 m2 s−1 intervals). Arrows on the contours indicate the direction of the mean Lagrangian velocity, that is, and . The averaging is over 12–48 h and the alongshore direction. Dashed vertical yellow line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

Near the surfzone x > −1.5LSZ, R4 (with TRCs) ψL is similar to R3 (Fig. 6b). As with R3, few streamlines cross x = −2LSZ, also indicating an exchange barrier to the mean . However, the offshore (x < −2.5LSZ) upper clockwise circulation cell is almost twice as strong for R4 as for R3 and extends deeper into the water column (Fig. 6b). The lower counterclockwise circulation is qualitatively similar to R3. This two circulation cell system also contrasts with the unstratified with TRC (R2) ψL circulation cell (Part I). The implications of R3 and R4 ψL on momentum balances and their relation to observations of Lentz et al. (2008) are discussed in section 5a.

b. Velocity variability

In Part I, the effect of surfzone-generated TRCs on the unstratified inner-shelf velocity variability is quantified with Eulerian cross-shore and vertical velocity standard deviation [σu(x, z) and σw(x, z)] in simulation R2. Similar to simulation R1 (Part I), R3 is essentially steady. Here, unstratified R2 and stratified R4 σu(x, z) and σw(x, z) (averaged over the alongshore and 12–24 h, the duration of R2) are compared to determine the effect of stratification on inner-shelf eddy velocity variability. The R2 σu(x, z) is described in Part I, and the R4 σu(x, z) is similar to R2 but slightly reduced (not shown) and is not described further.

In the surfzone (x = −LSZ), R2 and R4 σw(x, z) are similar. In contrast, the unstratified R2 and stratified R4 σw(x, z) are quite different on the inner shelf at −4LSZ < x < −LSZ (Fig. 7). In this region, R2 σw varies from (0.5–2) × 10−3 m s−1 (Fig. 7a), with vertical velocities associated with cyclostrophically balanced horizontal eddies (e.g., Burgers 1948; Rott 1958; Sullivan 1959). However, in this region the R4 σw is 2 to 3 times stronger than R2, also at maximum in the midwater column (Fig. 7b). Relative to R2, the R4 elevated σw is in part due to hydrostatic (i.e., shallow water) internal gravity wave motions that are generated through adjustment to the cyclostrophic eddy-induced raising and lowering of isotherms (e.g., Figs. 3c1, 5b).

Fig. 7.
Fig. 7.

(a) R2 (no stratification) and (b) R4 (with stratification) simulated vertical Eulerian velocity standard deviation σw. The averaging is over 12–24 h and the alongshore direction. Dashed black line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

c. Vertical eddy viscosity

Stratification inhibits vertical mixing (e.g., Lentz 2001). The ROMS vertical eddy diffusivity KT and eddy viscosity Kυ represent turbulence effects on vertical temperature and momentum mixing. In R3 and R4, the turbulent Prandtl number Kυ/KT is near one and approximately constant. Thus, here the mean eddy viscosity (12–48-h time average and alongshore average) is used to examine the role of TRCs in vertical mixing on the stratified inner shelf (Fig. 8).

Fig. 8.
Fig. 8.

(a) R3 (no TRCs) and (b) R4 (with TRCs) simulated mean vertical eddy viscosity (colored). The averaging is over 12–48 h and the alongshore direction. Dashed black line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

As in R1 and R2 (Part I), surfzone turbulence in R3 and R4 is principally generated by depth-limited wave breaking. In both R3 and R4, the surfzone (x > −LSZ) is maximum at ≈2 × 10−2 m2 s−1 (Fig. 8). Near x ≈ −2LSZ, the R3 is near-surface (z < −1 m) intensified at 10−3 m2 s−1 and much weaker O(10−5) m2 s−1 in the lower water column (Fig. 8). This region separates the two R3 Lagrangian overturning circulation cells (Fig. 6a). Farther offshore near x ≈ −3LSZ, R3 is elevated near 10−3 m2 s−1, extending from the surface to the midwater column (z = −3 m), and is much weaker O(10−5) m2 s−1 below (Fig. 6a). This enhanced corresponds to the downward-directed component of the offshore circulation cell (Fig. 6a).

In contrast, the R4 is quasi depth uniform and decays slowly out to x ≈ −3LSZ (Fig. 8b). At x = −2LSZ, below the surface layer, z < −1 m; R4 is 100 times stronger than R3 (Fig. 8b). Farther offshore at x < −4LSZ, R4 and R3 are similar over most of the water column (Fig. 8). However at x = −4LSZ at z ≈ −2.5 m, R4 is weakly elevated due to the Eulerian velocity shear associated with the offshore overturning circulation cell (Fig. 6b). Relative to the unstratified R2 (Part I), the R4 is weaker and decays offshore more rapidly, particularly at x < −3LSZ.

The R4 elevated , relative to R3, is in part due to density overturns induced by the inner-shelf eddies ejected by TRCs. R4 eddy vertical velocities raise cold water and spread it laterally (Figs. 3, 4) causing density (temperature) overturns and strongly elevating (e.g., Burchard and Bolding 2001). For example, at x = −2LSZ, temperature overturns of ∂T/∂z < 10−3 °C m−1 occurred over the water column 5% of the time for R4 and 0% of the time for R3. This leads to a R4 instantaneous Kυ upper range of 10−1 m2 s−2 that rapidly mixes the overturn analogous to mixed layer deepening by convective mixing (Burchard and Bolding 2001). At x = −3LSZ, R3 has 4% overturning only at z = −2 m (and essentially 0% elsewhere in z), which is associated with the downward component of the Lagrangian circulation (Fig. 6a) cell, explaining the elevated branch (Fig. 8a). At x = −3LSZ, R4 has 2%–4% overturning throughout the water column, also explaining the elevated over most of depth (Fig. 8). Overall, TRCs can significantly impact stratified inner-shelf vertical mixing out to x = −3LSZ, which likely has implications for tracer exchange across the inner shelf.

d. Mean temperature evolution

Temperature T(x, z) snapshots of R3 (without TRCs, Fig. 2) and R4 (with TRCs, Figs. 35) highlight the influence of Lagrangian overturning circulation and TRCs on the evolving temperature field. Here, the (alongshore and 0.5-h time averaged) mean temperature for R3 and R4 is examined at 24 and 48 h (Fig. 9) to quantify changes induced by the TRCs. At 24 h, the R3 and isotherms at deeper depths (z < −6 m, Fig. 9a) are largely unchanged from their initial location (Fig. 1a), except near where they contact the seabed. The R3 isotherm is largely horizontal and is quasi-uniformly depressed downward 0.4 m (Fig. 9a). However, between −6LSZ < x < −3LSZ, the isotherm is tilted upward, strongly raising 1.5 m over 300 m cross shore, eventually outcropping between −3LSZ < x < −2LSZ (Fig. 9a). Farther onshore (x > −2LSZ), the isotherm falls 3 m over 80 m.

Fig. 9.
Fig. 9.

R3 (no TRCs) and R4 (with TRCs) alongshore-averaged temperature at time t = (a),(b) 24 and (c),(d) 48 h. Solid black and white lines (labeled) are temperature contours.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

At 24 h, the R4 lower water column and isotherms are largely similar to R3 (Fig. 9b). In contrast to R3, the R4 isotherm is depressed between 0.4 and 1.2 m, sloping downward (from offshore to onshore) with a Δz = 0.8 m over Δx = 250 m. The R4 isotherm is raised significantly from its original location (at z = −2 m) and slopes upward onshore from z = −1.2 m at x = −6LSZ to z = 0 m at x = −3.2LSZ. At x > −3LSZ, the R4 water column is significantly more well mixed than for R3 (Figs. 9a,b).

At 48 h, the R3 and R4 temperature evolution continues although less rapidly than in the first 24 h (Figs. 9c,d). The R3 , , and isotherms at 48 h is similar at 24 h (Figs. 9a,c). Farther up in the water column, the R3 isotherm has continued to raise and outcrop 50 m farther offshore than at 24 h due to the stirring of the Lagrangian circulation cell’s upper branch (Fig. 6a). In contrast, the R4 isotherms have evolved significantly at 48 h (cf. Figs. 9b,d). At 48 h, the R4 isotherm slopes downward from z = −6.4 m at x = −6LSZ to z ≈ −6.8 m at x = −3.5LSZ (Fig. 9d) where no isotherm slope was evident at 24 h (Fig. 9b). At 48 h, the R4 isotherm is on average deeper and is slightly more sloped than at 24 h (Figs. 9b,d). The 48-h R4 isotherm was raised higher and outcrops farther offshore than at 24 h. At x > −6LSZ, almost all of the water initially at >19.5°C, with large volume contribution, has been transformed via TRC-driven mixing to lower temperature. Above the isotherm at 48 h, the R4 stratification is strongly reduced relative to R3 at x > −6LSZ.

5. Discussion

Both Stokes drift only (R3) and Stokes drift and TRCs (R4) modify the inner-shelf temperature. However, TRCs (R4) induce substantially larger inner-shelf mean temperature changes out to x = −6LSZ because of differences in the mean Lagrangian circulation (Fig. 6), the presence of eddies (Fig. 7), and enhanced vertical mixing at x > −3LSZ (Fig. 8). Even though the R4 eddies and elevated mixing is mostly confined to x > −3LSZ, the strong Lagrangian circulation cell continually brings in offshore water to be transformed and exported, explaining the temperature evolution farther offshore. The implications of Stokes drift and TRCs on stratified inner-shelf momentum balances, mixing, exchange velocity, and along-shelf flow are explored next.

a. Mean momentum balances

As in the unstratified cases in Part I, the effect of Stokes drift and TRCs on the stratified R3 and R4 mean cross-shore and alongshore momentum dynamics is examined at two locations x = −6LSZ and x = −3LSZ. At x = −6LSZ (h = 12 m), TRCs effects (R4) on mean Lagrangian circulation ψL, eddy velocity σw, and mean vertical eddy viscosity are weak (Figs. 68), whereas at x = −3LSZ (h = 7.4 m), TRCs strongly influence σu, σw, , and ψL. Mean (represented by 〈⋅〉) momentum dynamics terms are estimated by an alongshore average and 42–48-h simulation time average. Mean, cross-shore, momentum balance terms examined include mean pressure gradient (PG), Coriolis term, combined advective terms, and vertical mixing (VM). The combined horizontal and vertical advective (ADV) terms are
e3
where ue and we are Eulerian velocities consisting of mean and eddy contributions and . The mean VM is defined as
e4
The Coriolis term is , where is the mean, Eulerian, alongshore current.

At x = −6LSZ (h = 12 m), both R3 and R4 are in a geostrophic balance over depth, that is, the cross-shore PG and the Coriolis term largely balance through the water column (not shown) with magnitude ≈10−6 m s−2, 4 times stronger than the next largest term. The PG is dominated by a baroclinic component due to the tilting isotherms (Fig. 9), and the associated geostrophic balance is discussed further in section 5d. In contrast to the unstratified R1 and R2 (see Part I), at x = −6LSZ R3 and R4 are not in a Stokes–Coriolis balance [(1)], due to the enhanced Lagrangian circulation (Fig. 6) forced by the nongeostrophic residual PG.

For the stratified simulations R3 and R4, the cross-shore momentum balance at x = −3LSZ is different than at x = −6LSZ. It is also different from the unstratified R1 and R2 at x = −3LSZ (Part I). For R3 (no TRCs), the cross-shore PG approximately balances ADV throughout the water column (solid black and red, Fig. 10) with magnitude ≈2 × 10−6 m s−2. The PG has a significant baroclinic component and changes sign at z = −2.2 m (Fig. 10). At z > −2 m, the negative PG decelerates the onshore current leading to the closed ψL circulation cell (Fig. 6). The vertical mixing term VM is weak (not shown) as stratification reduces the vertical eddy viscosity (Fig. 8a), and the Coriolis term is weak.

Fig. 10.
Fig. 10.

R3 (solid, no TRCs) and R4 (dashed, with TRCs) mean cross-shore momentum balance terms (PG and ADV) at x = −3LSZ. PG (black) represents the cross-shore pressure gradient. ADV (red) is the sum of horizontal and vertical momentum advection that includes both Eulerian mean and eddy contributions ( and ). The averaging is over 42–48 h and the alongshore direction.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

At x = −3LSZ, the R4 (with TRCs) momentum balance has cross-shore PG, which balances ADV throughout the water column (dashed red and black, Fig. 10) but with a relatively more barotropic vertical structure that does not change sign. This is because TRCs, as in R2, induce a barotropic PG. The R4 vertical advective term ∂〈uewe〉/∂z is about half that of . In contrast to the unstratified R2 where VM [(4)] was as large as ADV, the stratified R4 VM term is one-quarter of the ADV term, due to the reduced (Part I). Between −6LSZ < x < −3LSZ, the R3 and R4 mean cross-shore momentum dynamics changes from advection dominated to geostrophy dominated regime. At x = −3LSZ, a Stokes–Coriolis balance does not occur in either R3 or R4.

For the stratified R3 and R4, the lack of a Stokes–Coriolis balance [(1)] either at x = −3LSZ or x = −6LSZ contrasts with the Stokes–Coriolis balance at both locations for the unstratified R1 (without TRCs) and at x = −6LSZ for the unstratified R2 (with TRCs). This result is consistent with h = 12-m depth observations for weak winds where Stokes–Coriolis balance [(1)] with held for unstratified (winter) but did not hold for stratified (summer) conditions (Lentz et al. 2008). The lack of Stokes–Coriolis balance in stratified conditions was unexpected as stratification reduces vertical mixing and should allow for a Stokes–Coriolis balance to dominate. For stratified conditions, observed onshore was enhanced (onshore ) at z < −5 m and offshore was enhanced (offshore ) at −4 < z < −2 m, relative to a Stokes–Coriolis balance (Lentz et al. 2008). The observed pattern is consistent with the R3 and R4 mean Lagrangian circulation pattern around x = −6LSZ (Fig. 6). This suggests that the observed summertime inner-shelf deviations from a Stokes–Coriolis balance is due to cross-shore baroclinic pressure gradients induced by surfzone processes.

b. Potential energy and irreversible mixing

In both R3 and R4, the inner-shelf stratification weakens and for x > −3LSZ temperature cools (Fig. 9), indicating potentially both irreversible mixing and offshore heat (or buoyancy) export. Water column buoyancy can be quantified with a depth-integrated, time- and cross shore–dependent potential energy ep(x, t):
e5
where is the alongshore-averaged density derived from temperature, ρ0 = 1024.5 kg m−3, and is the mean sea surface elevation. The change in water column potential energy Δep(x, t) = ep(x, t) − ep(x, t = 0) occurs due to both diabatic (irreversible) mixing and cross-shelf buoyancy fluxes induced by Stokes drift and TRCs. Here, R3 and R4 Δep(x) at times 24 and 48 h are considered (Fig. 11) from the edge of the surfzone (x = −LSZ) to x = −6LSZ.
Fig. 11.
Fig. 11.

R3 (black, no TRCs) and R4 (red, with TRCs) alongshore-averaged potential energy change Δep after 24 (solid) and 48 h (dashed) of simulation. Gray curve (mixed IC) represents the potential energy increase if the initial temperature profile was fully vertically mixed.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

After 24 h, the R3 Δep is ≈3.3 J m−2 at x = −LSZ with maximum ≈6 J m−2 at −3LSZ < x < −2.2LSZ (solid black, Fig. 11). The Δep increases partly because of the increased water depth. Farther offshore the 24 h R3 Δep decreases to ≈3 J m−2 at x = −6LSZ, even though water depth continues to increase. The R3 positive Δep is consistent with the R3 stratification reduction that is strongest for x > −3LSZ (Fig. 9a). After 48 h, R3 Δep (black dashed in Fig. 11) is about 1.5 times the 24-h R3 Δep, indicating that the rate of potential energy increase has slowed relative to the first 24 h. The 48-h R3 maximum Δep ≈ 9 J m−2 is near x = −3LSZ (Fig. 11), where the Lagrangian circulation is downward (Fig. 6a) and mean eddy diffusivity is enhanced (Fig. 8a), reducing the stratification.

In contrast, the R4 Δep is substantially enhanced at all cross-shore locations relative to R3 at both 24 and 48 h. After 24 h, the R4 Δep is 1.5 to 2 times larger than R3 between −3LSZ < x < −LSZ (solid red line, Fig. 11), where TRCs have the most impact on vertical velocity variability (Fig. 7b) and (Fig. 8b). This region (x > −3LSZ) corresponds to a direct TRC influence region. Farther offshore, the R4 Δep also decays and at x = −6LSZ is similar to the R3 Δep, indicating that TRC indirect influence is not yet strong this far offshore (i.e., x < −3LSZ).

After 48 h, the R4 Δep is only moderately larger than at 24 h for x > −1.5LSZ, indicating that the processes responsible for stirring in this region are on shorter time scales and thus mostly saturate within 24 h. However, the 48-h R4 Δep is substantially larger (1.3 to 1.5 times) at x < −3LSZ (cf. red solid and dashed, Fig. 11), indicating a stratification weakening process occurring on longer time scales. The 48-h R4 Δep is also substantially larger than the 48-h R3 Δep (cf. red and black dashed, Fig. 11), particularly from offshore of direct TRC influence x < −3LSZ. Thus, TRCs have strong yet indirect impacts on the inner-shelf temperature (density) field even out to x = −6LSZ. This occurs because of the strong mixing at x > −3LSZ (Fig. 8b, particularly in the surfzone on shorter time scales), creating cross-shore temperature gradients (Fig. 9b), which induce enhanced ψL (Fig. 6b) and cross-shelf buoyancy fluxes at longer non-TRC time scales.

For reference, the R3 and R4 increases in potential energy Δep are compared to the potential energy increase that would occur if the initial (constant ) temperature profile (Fig. 1a) were completely vertically mixed at that cross-shore location (denoted as mixed IC, solid gray in Fig. 11). The intersection of this curve with Δep(x) indicates the cross-shore location where the Δep is equivalent to fully mixed initial condition. For R3, this intersection occurs at x = −1.7LSZ at 24 h and x = −2.1LSZ at 48 h (Fig. 11). Reflecting the enhanced stirring and mixing effects of TRCs, the R4 intersection occurs farther offshore at x = −2.3LSZ (h = 5.75 m) at 24 h and x = −2.6LSZ (h = 6.5 m) at 48 h (Fig. 11). For both R3 and R4, intersection locations are not unstratified (Fig. 9), demonstrating that heat (buoyancy) is being exported offshore by TRCs and the mean Lagrangian circulation, consistent with field observations (Hally-Rosendahl et al. 2014; Sinnett and Feddersen 2014).

Although R4 Δep is larger than R3 at all cross-shore locations, the sloping isotherms (Fig. 9) imply that a part of the potential energy increase is due to adiabatic processes and not irreversible mixing. Here, the role of TRCs and Stokes drift in inducing irreversible mixing is examined via changes in the background potential energy Eb, the minimum potential energy attainable by an adiabatic redistribution of density (Winters et al. 1995). In a closed system, only an irreversible mixing changes the background potential energy. Here, the alongshore normalized Eb (joules per meter) is estimated by sorting the model density and the associated grid volume and calculating (Winters et al. 1995)
e6
where , , and are the sorted density, grid cell volume, and vertical coordinate, respectively; ρ0 = 1024.5 kg m−3; and the integral is over the entire sorted model volume. For R3 and R4, Eb(t) is estimated every five minutes and the initial Eb(t = 0) is subtracted to give ΔEb(t) (Fig. 12). Similarly, the total potential energy ΔEp is the cross shore–integrated ep,
e7
over the model domain. The available potential energy ΔEa reflects adiabatic potential energy increases and is the difference between total and background potential energy (i.e., ΔEa = ΔEp − ΔEb).
Fig. 12.
Fig. 12.

R3 (black, no TRCs) and R4 (red, with TRCs) alongshore normalized change in background potential energy ΔEb.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

The R3 ΔEb increases monotonically to 8125 J m−1 at 48 h (solid black in Fig. 12). Prior to 1 h, R3 ΔEb is near zero as the model spins up and enhanced total Δep is mostly all available potential energy. Over the next 19 h, ΔEb(t) increases quasi linearly at a rate of 210 J m−1 h−1 as the surfzone (x ≥ −LSZ) strongly mixes (Fig. 8b) and the Stokes drift–driven Lagrangian circulation cell develops exchanging water with the surfzone. At 24 h, the available potential energy is 4% of the total, indicating that most of the Δep increase (Fig. 11) is due to irreversible mixing. Thereafter, from 30 to 48 h, ΔEb(t) increases relatively slowly at a rate of 120 J m−1 h−1. This ΔEb(t) deceleration occurs because the near surfzone (x > −1.5LSZ) with enhanced (Fig. 8) is already overmixed (Fig. 11). In addition, cross-shore exchange with offshore is limited as very few Lagrangian streamlines cross x = −2.3LSZ (Fig. 6a). Thus, most of the later R3 irreversible mixing (requiring elevated Kυ) occurs near x = −3LSZ, where R3 is enhanced (Fig. 8). At 48 h, the available potential energy is 1% of the total, indicating nearly all of the Δep increase is due to mixing.

The R4 ΔEb increases to 11 420 J m−1 at 48 h (solid red in Fig. 12), overall much more rapidly than for R3, indicating stronger irreversible mixing. Initially R4 ΔEb is near zero for 1 h as the model spins up, and almost all Δep is adiabatic (reversible mixing). However, from 2 to 8 h, ΔEb increases rapidly (Fig. 12) at 438 J m−1 h−1, twice that of R3 as TRCs exchange surfzone and inner-shelf water (Hally-Rosendahl et al. 2015; Hally-Rosendahl and Feddersen 2016) and export elevated surfzone turbulence. Thereafter, between 10 and 24 h, a transition occurs and ΔEb increases more slowly at 290 J m−1 h−1, still 1.5 times more rapidly than R3. This occurs as TRCs overmix out to x = −2.4LSZ (red solid, Fig. 11). At 24 h, the available potential energy is 7% of the total, again indicating most ΔEp increase is due to mixing.

Later, from 30 to 48 h, ΔEb continues to decelerate at 145 J m−1 h−1 (Fig. 12), only 1.2 times that of R3. The 30–48-h similar R3 and R4 ΔEb increase masks the spatial differences in increased Δep (Fig. 11). For example, from 24 to 28 h, the region x > −2LSZ is still increasing in potential energy in R3 but not in R4 as TRCs have shortened the mixing time scale (Fig. 11). In the region −6LSZ < x< −3LSZ, the 24–48-h R4 Δep increase is much larger than R3, reflecting the enhanced Lagrangian circulation and mixing both directly and indirectly induced by TRCs. At 48 h, nearly all of the Δep increase is due to irreversible mixing, as the available potential energy is 2% of the total.

c. Cross-shelf exchange velocity

On alongshore, uniform bathymetry, both transient rip currents (Hally-Rosendahl et al. 2015; Suanda and Feddersen 2015) and the mean Lagrangian overturning circulation are potential cross-shore exchange mechanisms across the unstratified (Part I) and stratified inner shelves. In Part I, cross-shelf exchange across an unstratified inner shelf was quantified with a cross-shelf exchange velocity Uex (e.g., MacCready 2011; Hally-Rosendahl et al. 2014; Suanda and Feddersen 2015), defined as
e8
where is the instantaneous offshore directed uL (and is zero for onshore uL), η is the sea surface elevation, and 〈⋅〉 represents a time and alongshore average. On the unstratified inner shelf, TRC contributions (simulation R2) to cross-shelf exchange dominated over the mean Lagrangian circulation (simulation R1) out to 6LSZ. Here, the relative importance of these mechanisms is quantified on the stratified inner shelf, where Uex(x) is calculated over 12–48 h for R3 and R4.

As in the unstratified inner shelf without TRCs (simulation R1; see Part I), the stratified R3 mean Lagrangian circulation (Fig. 6a) results in a nonzero cross-shore exchange velocity (black curves, Fig. 13). For x > −2LSZ, the stratified R3 and unstratified R1 Uex(x) are similar with surfzone maximum and offshore decay. Near x = −2LSZ, is slightly weaker than , corresponding to the few ψL streamlines crossing this region (Fig. 6a). However, farther offshore (−6LSZ < x < −2.5LSZ), the stratified is enhanced 2–3 times over as both decay offshore (black solid and dashed curves, Fig. 13), indicating that without TRCs, stratification enhances the Stokes drift–induced cross- and inner-shelf exchange. The mean momentum balance analysis (section 5a) indicates that is enhanced because the cross-shore PG prevents a Stokes–Coriolis balance [(1)] from developing.

Fig. 13.
Fig. 13.

Exchange velocity Uex [(8) in Part I] for unstratified simulations R1 (black, no TRCs) and R2 (red, with TRCs) and the stratified simulations R3 (dashed black, no TRCs) and R4 (dashed red, with TRCs). Dashed black line delimits the surfzone x = −LSZ.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

As with the unstratified inner shelf with TRCs (R2; see Part I), the stratified R4 with TRCs results in an elevated Uex(x) (Fig. 13) that is at maximum within the surfzone and decays quasi-exponentially offshore. For x > −4LSZ, the stratified and unstratified are similar (red curves, Fig. 13). However, farther offshore x < −5LSZ; is elevated over because of the stratification-enhanced mean Lagrangian circulation (Fig. 6b). Analogous to the unstratified R1 and R2, the R4 with TRCs results in that is an order of magnitude stronger than between −2.5LSZ < x < −1.5LSZ (cf. red and black dashed curves, Fig. 13). This demonstrates that, as with unstratified inner shelves (Part I), neglecting TRCs on the stratified inner shelf will result in substantially weaker simulated surfzone to inner-shelf exchange of larvae, sediment, or pollutants. Farther offshore x < −4LSZ, is about 2 times larger than as the exchange is dominated by the mean Lagrangian circulation set up by the baroclinic cross-shelf pressure gradients (Fig. 10), which are larger with TRCs. Overall, this shows the importance of TRCs of cross-shelf exchange across the inner shelf.

d. Thermal wind balance

The R4 mean (alongshore and 42–48-h averaged) alongshore Eulerian velocity without rotation (i.e., f = 0) is expected to be zero as there are no net alongshore forcing mechanisms. However, with rotation, a R4 alongshore jet forms (Fig. 14a) between −7LSZ < x < −4.5LSZ that is 4 m thick centered at z = −2 m with magnitude of 0.03 m s−1, stronger than . Farther onshore (x > −4.5LSZ), the < 0.01 m s−1 is weak and of opposite sign (Fig. 14a). A weaker R3 jet also forms (not shown). The alongshore jet results from the largely geostrophic balance, as discussed in section 5a at x = −6LSZ, as R3 and R4 mean cross-shore temperature gradients (Fig. 9) lead to baroclinic pressure gradients that are stronger in R4. If the mean alongshore current is in geostrophic balance, then the thermal wind relationship is expected to hold
e9
where αT is the thermal expansion coefficient, and is the thermal wind–based geostrophic alongshore current. The thermal wind relationship [(9)] has been observed on the northern California outer shelf (e.g., Winant et al. 1987). On the North Carolina mid- to inner shelf in summer and early fall, the thermal wind balance was found to approximately hold (r2 = 0.43) in the midwater column across 26–8-m water depth (Lentz et al. 1999).
Fig. 14.
Fig. 14.

R4 (a) modeled mean alongshore velocity and (b) mean temperature and thermal wind–derived [(9)] geostrophic alongshore velocity . The mean is defined as an alongshore and 42–48-h average.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-16-0077.1

The ability of the thermal wind relationship to predict the jet is examined by estimating the R4 thermal wind–based geostrophic alongshore velocity by vertically integrating (9), assuming near-bed (Fig. 14b) and using the R4 mean (alongshore and 42–48 h averaged) temperature . In the region of direct TRC influence (x > −3LSZ), the cross-shore momentum dynamics are complex (Fig. 10b), with time scales much shorter than inertial. Thus, in this region a thermal wind balance (9) is not appropriate, and the thermal wind–derived does not resemble (Fig. 14). However, farther offshore (x < −4.5LSZ), the thermal wind–derived has an alongshore jet with width, vertical location, and magnitude similar although somewhat stronger than the modeled (Fig. 14). The modeled may be weaker than because insufficient time relative to rotational time scales has elapsed.

This agreement between and indicates that the inner shelf at x < −4.5LSZ is in a thermal wind balance whose cross-shore pressure gradients are induced by surfzone processes. Surface wave breaking induces vertical mixing and generates TRCs ejected onto the inner shelf. The presence of a shoreline induces a return Eulerian flow that is not in Stokes–Coriolis balance. This work suggests that surfzone processes must be considered in analysis of stratified inner shelves even in 12-m water depth.

e. Closing thoughts

The inner shelf is a complex region with many forcing mechanisms. These results show the importance of Stokes drift and TRCs on the unstratified and stratified inner shelf. Models for larval, nutrient, or pollution transport that do not include TRC effects will not accurately simulate surfzone to inner-shelf tracer exchange. Furthermore, models that do not include the surfzone and at a minimum Stokes drift, will not accurately simulate even the stratified inner-shelf region at x = −6LSZ. These simulations are simplified by neglecting the effects of wind (e.g., Lentz and Fewings 2012), diurnal solar heating and cooling, Langmuir cell circulations (e.g., Gargett and Wells 2007; Tejada-Martínez and Grosch 2007), internal tides and nonlinear internal waves (e.g., Lucas et al. 2011; Wong et al. 2012; Walter et al. 2012; Sinnett and Feddersen 2014; Kumar et al. 2015a; Arthur and Fringer 2016), and rip channel bathymetry (Brown et al. 2015), which can contribute to inner-shelf velocity and temperature variability. Only a single, steady, incident wave field (Hs, , and σθ) is considered. For normally incident waves, varying Hs and σθ have a strong impact on TRC-induced exchange on the inner shelf (Suanda and Feddersen 2015). Last, here in Part II, the initial stratification is also not varied. Varying these parameters will strongly affect the circulation, eddies, mixing, and stratification evolution on the inner shelf.

6. Summary

In this two-part study, a depth-integrated, wave-resolving, Boussinesq model funwaveC is coupled to a 3D, wave-averaged ocean circulation and wave propagation model COAWST to diagnose Stokes drift and transient rip current (TRC) effects on an unstratified (Part I) and a stratified (this work, Part II) inner shelf. In Part I, two unstratified simulations were performed without (R1) and with (R2) TRC effects. Here, analogous stratified simulations without (R3) and with (R4) TRC effects are performed; R3 cross-shore temperature evolves slowly due to mean Lagrangian circulation (Stokes drift and offshore-directed undertow). R4 has TRCs that eject onto the inner shelf with eddies out to 4 times the surfzone width LSZ, inducing patchy, near-surface cooling and vertical isotherm displacement. For both R3 and R4, the mean Lagrangian circulation has two clockwise circulation cells: one surfzone centered and one centered farther offshore of the inner shelf that is stronger in R4. Very few R3 and R4 streamlines cross near x = −2LSZ, connecting these two cells, indicating a stratified mean circulation barrier to surfzone to inner-shelf exchange, in contrast to the unstratified R1 and R2 simulations. Inner-shelf vertical velocity variability for stratified R4 is 2 to 3 times stronger than unstratified R2 (with TRC). The R4 mean vertical eddy diffusivity is much larger than for R3 largely due to TRC eddy-induced density overturns. The R4 TRC-enhanced stirring and mixing leads to more rapid mean temperature evolution and more strongly sloped isotherms than with the non-TRC R3.

At x = −6LSZ, R3 and R4 mean cross-shore momentum balances are geostrophic, driven by cross-shore baroclinic pressure gradient–induced sloping isotherms. This contrasts with the unstratified (R1 and R2) x = −6LSZ Stokes–Coriolis balance, explaining the summertime (stratified) inner-shelf observed deviation from Stokes–Coriolis balance. An alongshore geostrophic jet develops out to x = −7LSZ that is strongest in R4. Farther onshore, the geostrophic balance transitions to (at x = −3LSZ) an R3 and R4 balance between the cross-shore pressure gradient and advective terms, with weaker vertical momentum mixing than the unstratified R1 and R2. The R4 increase in potential energy due to both irreversible mixing and cross-shelf buoyancy fluxes is 1.3 to 2 times stronger than R3 across the inner shelf over 48 h. This indicates that TRCs strongly influence inner-shelf cross-shelf heat flux and mixing. TRCs induce an enhanced R4 cross-shore exchange velocity across the entire inner shelf relative to R3 due to both eddy stirring and the enhanced mean Lagrangian circulation. These results demonstrate the direct and indirect effect that TRCs have on the stratified inner shelf and show that accurate inner-shelf simulations should incorporate these wave-driven surfzone processes.

Acknowledgments

Support for N. Kumar and F. Feddersen was provided by Office of Naval Research (ONR) Grant N00014-14-1-0553. Computational support was provided by the COMPAS/ATLAS cluster maintained by Caroline Papadopoulos and Bruce Cornuelle. K. Winters, S. H. Suanda, P. MacCready, and M. S. Spydell provided useful feedback.

REFERENCES

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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S. J., and M. R. Fewings, 2012: The wind- and wave-driven inner-shelf circulation. Annu. Rev. Mar. Sci., 4, 317343, doi:10.1146/annurev-marine-120709-142745.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S., R. T. Guza, S. Elgar, F. Feddersen, and T. H. C. Herbers, 1999: Momentum balances on the North Carolina inner shelf. J. Geophys. Res., 104, 18 20518 240, doi:10.1029/1999JC900101.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S., M. Fewings, P. Howd, J. Fredericks, and K. Hathaway, 2008: Observations and a model of undertow over the inner continental shelf. J. Phys. Oceanogr., 38, 23412357, doi:10.1175/2008JPO3986.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lucas, A., P. Franks, and C. Dupont, 2011: Horizontal internal-tide fluxes support elevated phytoplankton productivity over the inner continental shelf. Limnol. Oceanogr. Fluids Environ., 1, 5674, doi:10.1215/21573698-1258185.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • MacCready, P., 2011: Calculating estuarine exchange flow using isohaline coordinates. J. Phys. Oceanogr., 41, 11161124, doi:10.1175/2011JPO4517.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marmorino, G. O., G. B. Smith, and W. D. Miller, 2013: Infrared remote sensing of surf-zone eddies. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 6, 17101718, doi:10.1109/JSTARS.2013.2257695.

    • Crossref
    • Search Google Scholar
    • Export Citation
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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Omand, M. M., F. Feddersen, P. J. S. Franks, and R. T. Guza, 2012: Episodic vertical nutrient fluxes and nearshore phytoplankton blooms in Southern California. Limnol. Oceanogr., 57, 16731688, doi:10.4319/lo.2012.57.6.1673.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Peregrine, D. H., 1998: Surf zone currents. Theor. Comput. Fluid Dyn., 10, 295309, doi:10.1007/s001620050065.

  • Rott, N., 1958: On the viscous core of a line vortex. Z. Angew. Math. Phys. ZAMP, 9, 543553, doi:10.1007/BF02424773.

  • Sinnett, G., and F. Feddersen, 2014: The surf zone heat budget: The effect of wave heating. Geophys. Res. Lett., 41, 72177226, doi:10.1002/2014GL061398.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Spydell, M. S., and F. Feddersen, 2009: Lagrangian drifter dispersion in the surf zone: Directionally spread, normally incident waves. J. Phys. Oceanogr., 39, 809830, doi:10.1175/2008JPO3892.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Suanda, S. H., and F. Feddersen, 2015: A self-similar scaling for cross-shelf exchange driven by transient rip currents. Geophys. Res. Lett., 42, 54275434, doi:10.1002/2015GL063944.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sullivan, R. D., 1959: A two-cell vortex solution of the Navier–Stokes equations. J. Aerosp. Sci., 26, 767768, doi:10.2514/8.8303.

  • Tejada-Martínez, A. E., and C. E. Grosch, 2007: Langmuir turbulence in shallow water. Part 2. Large-eddy simulation. J. Fluid Mech., 576, 63108, doi:10.1017/S0022112006004587.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tilburg, C., 2003: Across-shelf transport on a continental shelf: Do across-shelf winds matter? J. Phys. Oceanogr., 33, 26752688, doi:10.1175/1520-0485(2003)033<2675:ATOACS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Walter, R. K., C. B. Woodson, R. S. Arthur, O. B. Fringer, and S. G. Monismith, 2012: Nearshore internal bores and turbulent mixing in southern Monterey Bay. J. Geophys. Res., 117, C07017, doi:10.1029/2012JC008115.

    • Search Google Scholar
    • Export Citation
  • Warner, J. C., B. Armstrong, R. He, and J. B. Zambon, 2010: Development of a Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. Ocean Modell., 35, 230244, doi:10.1016/j.ocemod.2010.07.010.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Winant, C., R. Beardsley, and R. Davis, 1987: Moored wind, temperature, and current observations made during coastal ocean dynamics experiments 1 and 2 over the northern California continental shelf and upper slope. J. Geophys. Res., 92, 15691604, doi:10.1029/JC092iC02p01569.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. D’Asaro, 1995: Available potential energy and mixing in density-stratified fluids. J. Fluid Mech., 289, 115128, doi:10.1017/S002211209500125X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wong, S. H. C., A. E. Santoro, N. J. Nidzieko, J. L. Hench, and A. B. Boehm, 2012: Coupled physical, chemical, and microbiological measurements suggest a connection between internal waves and surf zone water quality in the Southern California Bight. Cont. Shelf Res., 34, 6478, doi:10.1016/j.csr.2011.12.005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Xu, Z., and A. J. Bowen, 1994: Wave- and wind-driven flow in water of finite depth. J. Phys. Oceanogr., 24, 18501866, doi:10.1175/1520-0485(1994)024<1850:WAWDFI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Arthur, R. S., and O. B. Fringer, 2016: Transport by breaking internal gravity waves on slopes. J. Fluid Mech., 789, 93126, doi:10.1017/jfm.2015.723.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brown, J. A., J. H. MacMahan, A. J. H. M. Reniers, and E. B. Thornton, 2015: Field observations of surf zone–inner shelf exchange on a rip-channeled beach. J. Phys. Oceanogr., 45, 23392355, doi:10.1175/JPO-D-14-0118.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Burchard, H., and K. Bolding, 2001: Comparative analysis of four second-moment turbulence closure models for the oceanic mixed layer. J. Phys. Oceanogr., 31, 19431968, doi:10.1175/1520-0485(2001)031<1943:CAOFSM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Burgers, J. M., 1948: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech., 1, 171199, doi:10.1016/S0065-2156(08)70100-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clark, D. B., S. Elgar, and B. Raubenheimer, 2012: Vorticity generation by short-crested wave breaking. Geophys. Res. Lett., 39, L24604, doi:10.1029/2012GL054034.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feddersen, F., 2007: Breaking wave induced cross-shore tracer dispersion in the surfzone: Model results and scalings. J. Geophys. Res., 112, C09012, doi:10.1029/2006JC004006.

    • Search Google Scholar
    • Export Citation
  • Feddersen, F., 2014: The generation of surfzone eddies in a strong alongshore current. J. Phys. Oceanogr., 44, 600617, doi:10.1175/JPO-D-13-051.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feddersen, F., D. B. Clark, and R. T. Guza, 2011: Modeling of surfzone tracer plumes: 1. Waves, mean currents, and low-frequency eddies. J. Geophys. Res., 116, C11027, doi:10.1029/2011JC007210.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fewings, M., S. J. Lentz, and J. Fredericks, 2008: Observations of cross-shelf flow driven by cross-shelf winds on the inner continental shelf. J. Phys. Oceanogr., 38, 23582378, doi:10.1175/2008JPO3990.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gargett, A. E., and J. R. Wells, 2007: Langmuir turbulence in shallow water. Part 1. Observations. J. Fluid Mech., 576, 2761, doi:10.1017/S0022112006004575.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guza, R. T., and F. Feddersen, 2012: Effect of wave frequency and directional spread on shoreline runup. Geophys. Res. Lett., 39, L11607, doi:10.1029/2012GL051959.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hally-Rosendahl, K., and F. Feddersen, 2016: Modeling surfzone to inner-shelf tracer exchange. J. Geophys. Res. Oceans, 121, 40074025, doi:10.1002/2015JC011530.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hally-Rosendahl, K., F. Feddersen, and R. T. Guza, 2014: Cross-shore tracer exchange between the surfzone and inner-shelf. J. Geophys. Res. Oceans, 119, 43674388, doi:10.1002/2013JC009722.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hally-Rosendahl, K., F. Feddersen, D. B. Clark, and R. Guza, 2015: Surfzone to inner-shelf exchange estimated from dye tracer balances. J. Geophys. Res. Oceans, 120, 62896308, doi:10.1002/2015JC010844.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Horwitz, R., and S. J. Lentz, 2014: Inner-shelf response to cross-shelf wind stress: The importance of the cross-shelf density gradient in an idealized numerical model and field observations. J. Phys. Oceanogr., 44, 86103, doi:10.1175/JPO-D-13-075.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Johnson, D., and C. Pattiaratchi, 2006: Boussinesq modelling of transient rip currents. Coastal Eng., 53, 419439, doi:10.1016/j.coastaleng.2005.11.005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, N., and F. Feddersen, 2016: The effect of Stokes drift and transient rip currents on the inner shelf. Part I: No stratification. J. Phys. Oceanogr., 47, 227241, doi:10.1175/JPO-D-16-0076.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, N., G. Voulgaris, J. C. Warner, and M. Olabarrieta, 2012: Implementation of the vortex force formalism in the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system for inner shelf and surf zone applications. Ocean Modell., 47, 6595, doi:10.1016/j.ocemod.2012.01.003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, N., F. Feddersen, S. Suanda, Y. Uchiyama, and J. McWilliams, 2015a: Mid- to inner-shelf coupled ROMS–SWAN model–data comparison of currents and temperature: Diurnal and semi-diurnal variability. J. Phys. Oceanogr., 46, 841862, doi:10.1175/JPO-D-15-0103.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, N., F. Feddersen, Y. Uchiyama, J. McWilliams, and W. O’Reilly, 2015b: Midshelf to surfzone coupled ROMS–SWAN model data comparison of waves, currents, and temperature: Diagnosis of subtidal forcings and response. J. Phys. Oceanogr., 45, 14641490, doi:10.1175/JPO-D-14-0151.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S. J., 2001: The influence of stratification on the wind-driven cross-shelf circulation over the North Carolina shelf. J. Phys. Oceanogr., 31, 27492760, doi:10.1175/1520-0485(2001)031<2749:TIOSOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S. J., and M. R. Fewings, 2012: The wind- and wave-driven inner-shelf circulation. Annu. Rev. Mar. Sci., 4, 317343, doi:10.1146/annurev-marine-120709-142745.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S., R. T. Guza, S. Elgar, F. Feddersen, and T. H. C. Herbers, 1999: Momentum balances on the North Carolina inner shelf. J. Geophys. Res., 104, 18 20518 240, doi:10.1029/1999JC900101.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lentz, S., M. Fewings, P. Howd, J. Fredericks, and K. Hathaway, 2008: Observations and a model of undertow over the inner continental shelf. J. Phys. Oceanogr., 38, 23412357, doi:10.1175/2008JPO3986.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lucas, A., P. Franks, and C. Dupont, 2011: Horizontal internal-tide fluxes support elevated phytoplankton productivity over the inner continental shelf. Limnol. Oceanogr. Fluids Environ., 1, 5674, doi:10.1215/21573698-1258185.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • MacCready, P., 2011: Calculating estuarine exchange flow using isohaline coordinates. J. Phys. Oceanogr., 41, 11161124, doi:10.1175/2011JPO4517.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marmorino, G. O., G. B. Smith, and W. D. Miller, 2013: Infrared remote sensing of surf-zone eddies. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 6, 17101718, doi:10.1109/JSTARS.2013.2257695.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Olabarrieta, M., J. C. Warner, and N. Kumar, 2011: Wave-current interaction in Willapa Bay. J. Geophys. Res., 116, C12014, doi:10.1029/2011JC007387.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Omand, M. M., F. Feddersen, P. J. S. Franks, and R. T. Guza, 2012: Episodic vertical nutrient fluxes and nearshore phytoplankton blooms in Southern California. Limnol. Oceanogr., 57, 16731688, doi:10.4319/lo.2012.57.6.1673.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Peregrine, D. H., 1998: Surf zone currents. Theor. Comput. Fluid Dyn., 10, 295309, doi:10.1007/s001620050065.

  • Rott, N., 1958: On the viscous core of a line vortex. Z. Angew. Math. Phys. ZAMP, 9, 543553, doi:10.1007/BF02424773.

  • Sinnett, G., and F. Feddersen, 2014: The surf zone heat budget: The effect of wave heating. Geophys. Res. Lett., 41, 72177226, doi:10.1002/2014GL061398.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Spydell, M. S., and F. Feddersen, 2009: Lagrangian drifter dispersion in the surf zone: Directionally spread, normally incident waves. J. Phys. Oceanogr., 39, 809830, doi:10.1175/2008JPO3892.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Suanda, S. H., and F. Feddersen, 2015: A self-similar scaling for cross-shelf exchange driven by transient rip currents. Geophys. Res. Lett., 42, 54275434, doi:10.1002/2015GL063944.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sullivan, R. D., 1959: A two-cell vortex solution of the Navier–Stokes equations. J. Aerosp. Sci., 26, 767768, doi:10.2514/8.8303.

  • Tejada-Martínez, A. E., and C. E. Grosch, 2007: Langmuir turbulence in shallow water. Part 2. Large-eddy simulation. J. Fluid Mech., 576, 63108, doi:10.1017/S0022112006004587.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tilburg, C., 2003: Across-shelf transport on a continental shelf: Do across-shelf winds matter? J. Phys. Oceanogr., 33, 26752688, doi:10.1175/1520-0485(2003)033<2675:ATOACS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Walter, R. K., C. B. Woodson, R. S. Arthur, O. B. Fringer, and S. G. Monismith, 2012: Nearshore internal bores and turbulent mixing in southern Monterey Bay. J. Geophys. Res., 117, C07017, doi:10.1029/2012JC008115.

    • Search Google Scholar
    • Export Citation
  • Warner, J. C., B. Armstrong, R. He, and J. B. Zambon, 2010: Development of a Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. Ocean Modell., 35, 230244, doi:10.1016/j.ocemod.2010.07.010.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Winant, C., R. Beardsley, and R. Davis, 1987: Moored wind, temperature, and current observations made during coastal ocean dynamics experiments 1 and 2 over the northern California continental shelf and upper slope. J. Geophys. Res., 92, 15691604, doi:10.1029/JC092iC02p01569.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. D’Asaro, 1995: Available potential energy and mixing in density-stratified fluids. J. Fluid Mech., 289, 115128, doi:10.1017/S002211209500125X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wong, S. H. C., A. E. Santoro, N. J. Nidzieko, J. L. Hench, and A. B. Boehm, 2012: Coupled physical, chemical, and microbiological measurements suggest a connection between internal waves and surf zone water quality in the Southern California Bight. Cont. Shelf Res., 34, 6478, doi:10.1016/j.csr.2011.12.005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Xu, Z., and A. J. Bowen, 1994: Wave- and wind-driven flow in water of finite depth. J. Phys. Oceanogr., 24, 18501866, doi:10.1175/1520-0485(1994)024<1850:WAWDFI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
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  • Fig. 1.

    (a) COAWST cross-shore bathymetry (thick solid black line) and temperature initial condition (colors and contours). The stratification of ∂T/∂z = 0.25°C m−1 corresponds to N2 = 6 × 10−4 s−2, typical for the Southern California inner shelf. (b) Significant wave height Hs for SWAN (red) and funwaveC (black dashed). In (a) and (b), the vertical dashed–dotted line delimits the surfzone x = −LSZ (where LSZ = 100 m).

  • Fig. 2.

    R3 (no TRCs) simulated temperature T at (a) 6 and (b) 48 h. Dashed black line delimits the surfzone x = −LSZ.

  • Fig. 3.

    R4 (with TRCs) simulated (left) vertical vorticity and (center) temperature at z = 1 m; and (right) temperature at the dashed–dotted line at times (a1),(b1),(c1) 1, (a2),(b2),(c2) 6, (a3),(b3),(c3) 12, and (a4),(b4),(c4) 18 h. Dashed black line delimits the surfzone x = −LSZ.

  • Fig. 4.

    As in Fig 3, but at times (a1),(b1),(c1) 24, (a2),(b2),(c2) 30, (a3),(b3),(c3) 42, and (a4),(b4),(c4) 48 h.

  • Fig. 5.

    R4 (with TRCs) simulated temperature at x = −1.6LSZ (−160 m) at (a) 0, (b) 1, (c) 6, (d) 24, and (e) 48 h.

  • Fig. 6.

    (a) R3 (no TRCs) and (b) R4 (with TRCs) Lagrangian overturning streamfunction ψL (colors and contours at 10−3 m2 s−1 intervals). Arrows on the contours indicate the direction of the mean Lagrangian velocity, that is, and . The averaging is over 12–48 h and the alongshore direction. Dashed vertical yellow line delimits the surfzone x = −LSZ.

  • Fig. 7.

    (a) R2 (no stratification) and (b) R4 (with stratification) simulated vertical Eulerian velocity standard deviation σw. The averaging is over 12–24 h and the alongshore direction. Dashed black line delimits the surfzone x = −LSZ.

  • Fig. 8.

    (a) R3 (no TRCs) and (b) R4 (with TRCs) simulated mean vertical eddy viscosity (colored). The averaging is over 12–48 h and the alongshore direction. Dashed black line delimits the surfzone x = −LSZ.

  • Fig. 9.

    R3 (no TRCs) and R4 (with TRCs) alongshore-averaged temperature at time t = (a),(b) 24 and (c),(d) 48 h. Solid black and white lines (labeled) are temperature contours.

  • Fig. 10.

    R3 (solid, no TRCs) and R4 (dashed, with TRCs) mean cross-shore momentum balance terms (PG and ADV) at x = −3LSZ. PG (black) represents the cross-shore pressure gradient. ADV (red) is the sum of horizontal and vertical momentum advection that includes both Eulerian mean and eddy contributions ( and ). The averaging is over 42–48 h and the alongshore direction.

  • Fig. 11.

    R3 (black, no TRCs) and R4 (red, with TRCs) alongshore-averaged potential energy change Δep after 24 (solid) and 48 h (dashed) of simulation. Gray curve (mixed IC) represents the potential energy increase if the initial temperature profile was fully vertically mixed.

  • Fig. 12.

    R3 (black, no TRCs) and R4 (red, with TRCs) alongshore normalized change in background potential energy ΔEb.

  • Fig. 13.

    Exchange velocity Uex [(8) in Part I] for unstratified simulations R1 (black, no TRCs) and R2 (red, with TRCs) and the stratified simulations R3 (dashed black, no TRCs) and R4 (dashed red, with TRCs). Dashed black line delimits the surfzone x = −LSZ.

  • Fig. 14.

    R4 (a) modeled mean alongshore velocity and (b) mean temperature and thermal wind–derived [(9)] geostrophic alongshore velocity . The mean is defined as an alongshore and 42–48-h average.

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