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    (a) A schematic illustration of the numerical experiments. (b) A time series of stochastic wave heights at the offshore boundary, with a mean height of H0 = 0.8 m. (c),(d) The wave heights averaged in each period band from the (c) stochastic waves and (d) realistic waves measured by a CDIP wave buoy (station 217 from 9 October 2019 to 9 November 2019).

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    (a),(b) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5); solid lines] and by the surface breaking εb/ρ0¯ [Eq. (A3); dashed lines]. (c),(d) Cross-shelf profiles the mean wave heights (left y axis) and wavenumbers (right y axis). Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. The four colors represent four types of incident waves. (e),(f) Wave rays for (e) SN waves and (f) SO waves both with streaming. The colors indicate wave heights.

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    (a),(b) Mean cross-shelf Eulerian current. (c),(d) Mean vertical eddy viscosity. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

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    Depth-integrated mean momentum balance per unit mass (a),(b) in the cross-shelf direction with UN waves and (c),(d) in the along-shelf direction with UO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important forces are displayed.

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    (a),(b) Mean cross-shelf, depth-averaged Eulerian current and Stokes drift. Asterisks mark the surf-zone edge. Vertical profiles of the mean cross-shelf, depth-dependent Eulerian current and Stokes drift in the (c),(d) surf zone x = 0.2 km, (e),(f) inner shelf x = 2.4 km, and (g),(h) outer shelf x = 20 km. Left and right columns are with and without wave streaming. Solid and dashed lines denote the Eulerian current and Stokes drift.

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    Mean Lagrangian streamfunction ψL¯ defined in Eq. (2), (a) with and (b) without wave streaming. The surface and bottom arrows indicate the near-surface and near-bottom Lagrangian flow directions. Dashed lines delineate the surf-zone edge. Incident waves are UN, and other incident waves (UO, SN, SO) yield the ψL¯ similar to with and without wave streaming in (a) and (b), respectively.

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    Mean along-shelf Eulerian current with (a),(b) UN waves and (c),(d) UO waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

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    Cross-shelf profiles of the depth-integrated (a),(b) MKE and (c)–(f) MKE budgets per unit mass with (c),(d) SN waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed.

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    (a),(b) Instantaneous surface vertical vorticity of surf eddies, normalized by the Coriolis parameter f, with (a) SN waves and (b) SO waves both with streaming. Dashed lines delineate the surf-zone edge. Cross-shelf profiles of the depth-integrated EKE budgets per unit mass with (c),(d) SN waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed.

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    (a),(b) Instantaneous surface vertical vorticity of surf eddies, normalized by the Coriolis parameter f, for UO waves (a) with and (b) without streaming. Dashed lines delineate the surf-zone edge. Cross-shelf profiles of the depth-integrated EKE budgets per unit mass with (c),(d) UO waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed. All results are with reduced current bottom drag, in comparison to Fig. 9 with the default drag defined by Eq. (A7).

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    Mean concentration (no units) of the shore-released passive tracers with (a),(b) UN waves and (c),(d) SN waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

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    Cross-shelf tracer fluxes by the (a)–(d) mean Lagrangian flow with UN waves in (a) and (b), and SN waves in (c) and (d); (e),(f) fluxes by the surf eddies with SN waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

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    Sensitivity to bottom roughness height zo. Cross-shelf profiles of the mean wave dissipation (a) by the bottom drag εf/ρ0¯ [Eq. (A5)] and (b) by the surface breaking εb/ρ0¯ [Eq. (A3)]. (c) Mean cross-shelf bottom Eulerian current. (d) Mean Lagrangian streamfunction ψL¯ defined in Eq. (2) with zo = 10−4 m, in comparison to Fig. 6a with zo = 10−3 m. The asterisks in (a)–(c) and the dashed line in (d) mark the surf-zone edge. The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

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    Sensitivity to bottom slopes. (a) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5), solid lines and left y axis] and by the surface breaking εb/ρ0¯ [Eq. (A3), dashed lines and right y axis]. (b) Mean cross-shelf bottom Eulerian current. (c),(d) Mean Lagrangian streamfunction ψL¯ defined in Eq. (2) with a constant slope of 0.01 in (c) and a nonconstant slope in (d), in comparison to Fig. 6a with a constant slope of 0.005. The slope in (d) decreases from 0.014 in the surf zone to 0.004 in the inner shelf. The asterisks in (a) and (b) and the dashed lines in (c) and (d) mark the surf-zone edge. The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

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    Sensitivity to wave variations, including the wave height variation QHgt, period variation QPrd, and direction variation QDir. The QHgt and QPrd are expressed in percent (%) of their mean, and the QDir is expressed in angle degrees (°). (a) The depth-integrated EKE in the surf zone x = 0.2 km with different wave variations. (b) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5); solid lines] and by the surface breaking εb/ρ0¯ [Eq. (A3); dashed lines]. Asterisks mark the surf-zone edge. (c),(d) Vertical profiles of the cross-shelf mean Eulerian current in the surf zone x = 0.2 km in (c) and the inner shelf x =.2.4 km in (d). The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

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Effects of Wave Streaming and Wave Variations on Nearshore Wave-Driven Circulation

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  • 1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
  • | 2 Department of Civil Engineering, Kobe University, Kobe, Japan
  • | 3 International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaiʻi at Mānoa, Honolulu, Hawaii
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Abstract

Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause alongshore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross-shelf material transport.

Corresponding author: Peng Wang, pwang@atmos.ucla.edu

Abstract

Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause alongshore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross-shelf material transport.

Corresponding author: Peng Wang, pwang@atmos.ucla.edu

1. Introduction

The nearshore region that includes a surf zone and an inner shelf is important to the marine ecosystem and human health (Ryan et al. 2005; Pineda et al. 2007; Kumar and Feddersen 2017a). The surf zone is where surface waves break due to shallow depths (Battjes 1988; MacMahan et al. 2006). Next to the surf zone seaward, the inner shelf is where the surface and bottom boundary layers overlap (Nittrouer and Wright 1994; Lentz 1995).

One of the critical processes to sustain the nearshore ecosystem is the cross-shelf material transport, such as the transport of nutrients, pollutants, and plankton (Shanks et al. 2015; Fujimura et al. 2018; Morgan et al. 2018). The cross-shelf material transport is controlled by many processes (Dalrymple et al. 2011; Lentz and Fewings 2012; Brink 2016; Castelle et al. 2016; Trowbridge and Lentz 2018). For example, rip currents and surf eddies dominate the material transport between the surf zone and inner shelf (Hally-Rosendahl et al. 2014; Suanda and Feddersen 2015; Spydell et al. 2019); particularly, bathymetric rip currents due to alongshore-variable bathymetry (e.g., rip channels) can trap materials within the persistent horizontal recirculations (MacMahan et al. 2010; Reniers et al. 2010; Brown et al. 2015). Besides, internal bores can carry materials within a water mass (e.g., heat, salt, and larvae) from the inner shelf to the surf zone (Pineda 1994; Sinnett et al. 2018).

Another process important to the cross-shelf material transport is the wave streaming. In shallow water, surface waves are subject to bottom drag, which induces a near-bottom current along the wave propagation. This current is referred to as wave bottom streaming (Longuet-Higgins 1953). The onset of wave bottom streaming depends on the wavelength; long waves like swells are able to feel bottom drag in deeper water, and subsequently the wave bottom streaming commences there. Nonetheless, the wave bottom streaming may be reduced under nonlinear waves (Trowbridge and Madsen 1984; Davies and Villaret 1999; van der A et al. 2011; Kranenburg et al. 2012; Henriquez et al. 2014). In addition, the viscous stress within the wave surface boundary layer can cause wave surface streaming (Longuet-Higgins 1953), which might be important to the upper ocean transport (Madsen 1978; Weber et al. 2006; Wang and Özgökmen 2018); however, it is negligible in this study, in agreement with Uchiyama et al. (2010). Thus, we focus on the wave bottom streaming only and for simplicity call it “wave streaming.”

Wave streaming is important to the sediment transport and can shape the bed forms (Reniers et al. 2004a; Nielsen 2006; Ruessink et al. 2009; Reniers et al. 2013). Besides, wave streaming delivers plankton and larvae from the inner shelf to the surf zone, which is a crucial migration for some marine species (Pineda et al. 2007; Shanks et al. 2015; Fujimura et al. 2018; Morgan et al. 2018). Moreover, using observations in a water depth of 12 m, Lentz et al. (2008) measured an onshore current of 0.5 cm s−1 at 2 m above the bottom (cf. their Fig. 10a), and they claimed that this current may be related to tides; in addition to tides, however, other processes like the onshore wave streaming may also contribute to this current. Further, wave streaming decreases rapidly away from the bottom, and the deepest measurement in Lentz et al. (2008) is still far away from the bottom; thus, their measurements cannot adequately resolve the wave streaming profiles. Due to the importance and poor understanding of wave streaming, more field observations are needed, especially in the inner shelf.

In addition, realistic surface waves contain variations in wave heights, periods, and directions. These wave variations cause alongshore-variable wave breaking that generates surf eddies (Bowen 1969; Peregrine 1998; Clark et al. 2012). The surf-eddy generation mechanism due to wave variations can be as important as that due to shear instabilities of surf-zone alongshore currents (Long and Özkan-Haller 2009; Feddersen 2014). Further, wave variations inherently reside in wave groups, and surf eddies due to wave-group variations can develop spatial and temporal scales related to the wave groups (Reniers et al. 2004b; Long and Özkan-Haller 2009; MacMahan et al. 2010).

In the nearshore region, wave streaming and wave variations exist simultaneously. Then, how important are the wave streaming and wave variations to the nearshore wave-driven circulation? How do they together impact the cross-shelf material transport? Here we aim to answer these questions using numerical simulations. The structure of this study is as follows. The numerical model and experimental setup are described in section 2. The major results are presented in section 3, which are the mean flow, surf eddies, and the associated material transport. Moreover, sensitivity tests of the circulation are reported in section 4, and the study is concluded in section 5.

2. Methods

a. Numerical model

The experiments are simulated with a coupled wave–circulation model, which is the Regional Oceanic Modeling System (ROMS) fully coupled with a phase-averaged wave model (Uchiyama et al. 2010); thus, both wave effects on currents (WEC) and current effects on waves (CEW) are taken into account. The WEC contain two parts, i.e., conservative and nonconservative. The conservative WEC are based on the wave-averaged theory by McWilliams et al. (2004), including the Stokes vortex force (Craik and Leibovich 1976), Stokes–Coriolis force (Hasselmann 1970), and pressure adjustments. The nonconservative WEC are parameterized in terms of the wave dissipation, such as the wave breaking, wave streaming, and wave-enhanced current bottom drag. The model is briefly described in appendix A.

Our model has been validated against the nearshore measurements; it is able to reproduce the measured current profiles and wave characteristics (Uchiyama et al. 2010; Marchesiello et al. 2015). In our experiments, the parameters of wave forces are chosen based on those model validations, so that the simulated wave-driven circulation is reasonable compared to reality.

b. Experimental setup

The bathymetry is an alongshore-uniform shelf with a slope of 0.005 (Fig. 1a), which is a typically gentle slope for the inner shelf. The depth increases linearly from 0.3 to 205.1 m offshore. The horizontal lengths in the cross-shelf (X) and along-shelf (Y) directions are both equal to 40.96 km, with a horizontal resolution of 40 m. There are 12 vertical terrain-following layers that are denser near the surface and bottom to better resolve the boundary layers. Also, a comparison with 24 vertical layers shows that 12 vertical layers are good enough to resolve the boundary layers and velocity profiles.

Fig. 1.
Fig. 1.

(a) A schematic illustration of the numerical experiments. (b) A time series of stochastic wave heights at the offshore boundary, with a mean height of H0 = 0.8 m. (c),(d) The wave heights averaged in each period band from the (c) stochastic waves and (d) realistic waves measured by a CDIP wave buoy (station 217 from 9 October 2019 to 9 November 2019).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

Initially the water is at rest with uniform density. Incident surface waves are prescribed the offshore (east) boundary, where the current velocity is nudged to the anti-Stokes drift. The north and south boundaries are periodic, and the west and east boundaries have zero net fluxes. The Coriolis parameter is 8.13 × 10−5 s−1 for a midlatitude.

The wave model is forced by the offshore boundary condition, which is the prescribed wave heights, periods, and directions of incident waves. The prescribed waves can be uniform or stochastic. Uniform waves have a constant height of 0.8 m and a constant period of 10 s; stochastic waves have the mean wave height and period same as the uniform waves, but the waves heights and periods have stochastic variations of 10% of the mean. Stochastic waves (Fig. 1b) are constructed with the solutions of a stochastic partial differential equation (appendix B). The total wave energy input at the offshore boundary is approximately equal between the stochastic and uniform waves. Here the stochastic waves represent a narrow period band around the wave-spectrum peak of the realistic waves recorded by a wave buoy (Figs. 1c,d). Although the stochastic waves are not full spectrum, they advance beyond uniform waves toward realistic ones. In addition, the wave incidence can be normal or oblique. The normal incidence is perpendicular to the shore, and the oblique incidence is at an angle of 30° (northwestward) relative to the shore normal. The stochastic waves also have a directional spread of 10° about their mean incidence. By combining uniform or stochastic waves with the normal or oblique incidence, we obtain four types of incident waves (UN, UO, SN, and SO; see Table 1). Then we conduct experiments for all wave types with and without wave streaming.

Table 1.

Types of incident waves.

Table 1.

3. Results

a. Mean circulation

We define the “mean” as the average both over the alongshore direction and over the time period from day 5 to 10, when the mean circulation reaches a quasi-steady state. The mean quantity is denoted by ¯; the angle brackets ⟨⋅⟩ denote the alongshore average, and the overline ¯ denotes the time average.

1) Cross-shelf currents

After the uniform or stochastic waves prescribed at the offshore boundary propagate into the model domain, they are governed by the wave model and obey the fundamental wave dynamics, including refraction, shoaling, bottom drag, and surface breaking (Fig. 2a). The shoaling increases the wavenumber and wave heights (Fig. 2c). Without bottom drag dissipation, the wave heights increase more via shoaling (Fig. 2d); besides, wave energy is dissipated by surface breaking only because of no bottom drag, resulting in larger breaking dissipation and a wider surf zone (Fig. 2b).

Fig. 2.
Fig. 2.

(a),(b) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5); solid lines] and by the surface breaking εb/ρ0¯ [Eq. (A3); dashed lines]. (c),(d) Cross-shelf profiles the mean wave heights (left y axis) and wavenumbers (right y axis). Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. The four colors represent four types of incident waves. (e),(f) Wave rays for (e) SN waves and (f) SO waves both with streaming. The colors indicate wave heights.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

The stochastic waves spread directionally and therefore experience wave ray convergences during the propagation (Figs. 2e,f). In the experiments with SN waves, the wave ray convergence is stronger, creating bigger waves (Fig. 2e). By contrast, SO wave rays take a longer path to the shore than SN wave rays due to the oblique incidence (Fig. 2f); thus, SO waves are dissipated more by bottom drag, lacking big waves as exist in SN waves. Because big waves break farther offshore, the surf zone with SN waves is wider (Figs. 2a,b).

Wave surface breaking induces a near-surface wave breaker force [Eq. (A4)] that can drive a near-surface onshore current in the surf zone. Meantime, the pressure gradient force generates a near-bottom offshore undertow to balance the onshore mass transport (Fig. 3b). Without wave streaming, the wave breaker force is balanced by the pressure gradient force in the cross-shelf direction (Fig. 4b) and by the current bottom drag in the along-shelf direction (Fig. 4d).

Fig. 3.
Fig. 3.

(a),(b) Mean cross-shelf Eulerian current. (c),(d) Mean vertical eddy viscosity. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

Fig. 4.
Fig. 4.

Depth-integrated mean momentum balance per unit mass (a),(b) in the cross-shelf direction with UN waves and (c),(d) in the along-shelf direction with UO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important forces are displayed.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

The wave streaming in the inner shelf manifests itself as a near-bottom onshore current (Fig. 3a), which meets the offshore undertow around the surf-zone edge, creating a near-bottom convergence zone. Also, the wave streaming force [Eq. (A6)] joins the momentum balance with breaker force (Figs. 4a,c).

The Stokes vortex force and Stokes–Coriolis force are insignificant in the surf zone with a normal wave incidence. Nonetheless, with an oblique incidence (Figs. 4c,d), the Stokes vortex force becomes significant and is balanced by the nonlinear advection (Newberger and Allen 2007; Uchiyama et al. 2009; Kumar et al. 2012).

Wave streaming enhances the vertical shear and near-bottom turbulence in the inner shelf; hence, the vertical eddy viscosity and bottom boundary layer thickness are increased (Fig. 3c). Without wave streaming, the vertical eddy viscosity is weak in the inner shelf (Fig. 3d), and the inner-shelf bottom boundary layer is thin with a short offshore span. In addition, owing to the lack of surface forcing (e.g., wind stress and heat flux), the near-surface vertical mixing is weak in the inner shelf. In the surf zone, the vertical mixing is dominated by breaking waves (Battjes 1975), and the eddy viscosity without wave streaming is larger due to stronger wave breaking (Fig. 2b).

2) Lagrangian overturning circulation

The Lagrangian velocity uL is the sum of Eulerian velocity and Stokes drift (i.e., uL = u + us). For a steady flow, the continuity equation of the depth-averaged Lagrangian velocity UL (Lentz and Fewings 2012) is

(hUL)x+(hVL)y[h(U+Us)]x+[h(V+Vs)]y=0.

Here h is the water depth, (U, V) is the depth-averaged Eulerian velocity, and (Us, Vs) is the depth-averaged Stokes drift. When the flow is alongshore uniform (i.e., ∂/∂y = 0) and has zero net flux across the shore boundary (i.e., UL|x=0 = 0), Eq. (1) implies the depth-averaged UL(x) ≡ U + Us = 0 everywhere (Figs. 5a,b). However, the depth-dependent Lagrangian velocity uL(x, z) ≡ u + us is nonzero in the surf zone and inner shelf, viz., u(x, z) ≠ −us(x, z).

Fig. 5.
Fig. 5.

(a),(b) Mean cross-shelf, depth-averaged Eulerian current and Stokes drift. Asterisks mark the surf-zone edge. Vertical profiles of the mean cross-shelf, depth-dependent Eulerian current and Stokes drift in the (c),(d) surf zone x = 0.2 km, (e),(f) inner shelf x = 2.4 km, and (g),(h) outer shelf x = 20 km. Left and right columns are with and without wave streaming. Solid and dashed lines denote the Eulerian current and Stokes drift.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

In the surf zone, the undertow u(x, z) with and without wave streaming exhibits different profiles (Figs. 5c,d). In the inner shelf, the onshore wave streaming dominates the near-bottom current and decreases rapidly away from the bottom (Fig. 5e); without wave streaming, the inner-shelf current is nearly antiStokes (Fig. 5f). In the outer shelf, the current cancels the Stokes drift at each depth (Figs. 5g,h), leading to the Stokes–Coriolis balance. Due to the deep water there, the bottom drag dissipation and wave streaming are negligible.

To describe the Lagrangian overturning circulation in the depth- and cross-shelf section, we define a Lagrangian streamfunction ψL(x, z) so that

uL=ψLz,wL=ψLx.

Without wave streaming (Fig. 6b), only one Lagrangian overturning circulation exists and occupies the entire surf zone. With wave streaming (Fig. 6a), the surf-zone Lagrangian overturning circulation gets smaller and weaker; meantime, an additional Lagrangian overturning circulation arises in the inner shelf and crosses the surf-zone edge.

Fig. 6.
Fig. 6.

Mean Lagrangian streamfunction ψL¯ defined in Eq. (2), (a) with and (b) without wave streaming. The surface and bottom arrows indicate the near-surface and near-bottom Lagrangian flow directions. Dashed lines delineate the surf-zone edge. Incident waves are UN, and other incident waves (UO, SN, SO) yield the ψL¯ similar to with and without wave streaming in (a) and (b), respectively.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

3) Along-shelf currents

With a normal wave incidence, there are no along-shelf components of the wave breaker force and streaming force. Nonetheless, the along-shelf Coriolis force fuL associated with the Lagrangian velocity can drive an along-shelf current in the inner shelf (Figs. 7a,b), and the current is stronger with wave streaming due to stronger uL.

Fig. 7.
Fig. 7.

Mean along-shelf Eulerian current with (a),(b) UN waves and (c),(d) UO waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

With an oblique wave incidence, the along-shelf wave breaker force and streaming force arise. In the surf zone, the along-shelf breaker force drives an alongshore current (Figs. 7c,d), which is stronger without wave streaming. In the inner shelf, the along-shelf streaming force dominates over the Coriolis force and drives an along-shelf current (Fig. 7c), which is opposite to and stronger than that without wave streaming (Fig. 7d).

4) Mean kinetic energy

The mean kinetic energy (MKE) is defined as (1/2)(u¯2+υ¯2+w¯2), where u¯, υ¯, w¯ denote the mean-flow components in the x, y, z directions. In terms of the MKE (Figs. 8a,b), the inner-shelf mean current is enhanced greatly by wave streaming, and the surf-zone mean current is increased significantly owing to the oblique wave incidence, which causes an alongshore current.

Fig. 8.
Fig. 8.

Cross-shelf profiles of the depth-integrated (a),(b) MKE and (c)–(f) MKE budgets per unit mass with (c),(d) SN waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

The budgets of depth-integrated MKE is summarized in Eq. (3):

dzddt(MKE)=Wbrk+Wstrm+Wdrag+WStkCor+WStk+Wcurr+Wres.

The r.h.s. work is denoted by the symbols:

  • Wbrk=dz(Fib¯u¯i) with the mean wave breaker force Fib¯ (A4);
  • Wstrm=dz(Fif¯u¯i) with the mean wave streaming force Fif¯ (A6);
  • Wdrag=Fd¯u¯i with the mean current bottom drag Fd¯ (A7);
  • WStkCor=dz(fk^×us¯j)u¯i with the mean Stokes–Coriolis force (fk^×us¯j);
  • WStk=dz[u¯iuj(ujs/xi)¯] with the mean Stokes shear force [uj(ujs/xi)¯];
  • Wcurr=dz[uiujL¯(u¯i/xj)] with the mean-flow shear u¯i/xj;
  • Wres the residual including the transport divergence and the dissipation by viscosity.

The major work of interest is Wbrk by the wave breaker force, Wstrm by the wave streaming force, and Wdrag by the wave-enhanced bottom drag. Without wave streaming (Figs. 8d,f), most MKE is produced by the wave breaker force and is dissipated by the current bottom drag. With wave streaming (Figs. 8c,e), MKE is primarily produced by the wave streaming force, whereas the wave breaker force may even decrease the MKE, especially with a normal wave incidence (Fig. 8c).

b. Surf eddies

1) Eddies by breaking waves

Surf eddies originate in the surf zone but can reach beyond the surf-zone edge. A surf eddy may combine with its adjacent eddy of opposite vorticity to form a transient rip current (Peregrine 1998). Here surf eddies are revealed by the turbulent vertical vorticity calculated as ζ′ = (∂υ′/∂x) − (∂u′/∂y) (Figs. 9a,b), where (u′, υ′) are the turbulent horizontal velocities.

Fig. 9.
Fig. 9.

(a),(b) Instantaneous surface vertical vorticity of surf eddies, normalized by the Coriolis parameter f, with (a) SN waves and (b) SO waves both with streaming. Dashed lines delineate the surf-zone edge. Cross-shelf profiles of the depth-integrated EKE budgets per unit mass with (c),(d) SN waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

SN waves induce alongshore-variable wave breaking that produces surf eddies and transient rip currents (Fig. 9a). By contrast, surf eddies with SO waves are much weaker and are mostly restricted within the surf zone without forming transient rip currents (Fig. 9b). Namely, an oblique wave incidence suppresses the surf eddies that are generated by the wave variation–induced, alongshore-variable wave breaking.

The budgets of depth-integrated eddy kinetic energy (EKE) is summarized in the Eq. (4):

dzddt(EKE)=Wbrk+Wstrm+Wdrag+WStkCor+WStk+Wcurr+Wres.

The r.h.s. work is denoted by the symbols:

  • Wbrk=dz(Fibui) with the turbulent wave breaker force Fib (A4);
  • Wstrm=dz(Fifui) with the turbulent wave streaming force Fif (A6);
  • Wdrag=Fdui with the turbulent current bottom drag Fd (A7);
  • WStkCor=dz(fk^×ujs)ui with the turbulent Stokes–Coriolis force (fk^×ujs);
  • WStk=dz{ui[uj(ujs/xi)]} with the turbulent Stokes shear force {ui[uj(ujs/xi)]};
  • Wcurr=dz[uiujL¯(u¯i/xj)] with the mean-flow shear u¯i/xj;
  • Wres the residual including the transport divergence and the dissipation by viscosity.

In all cases, wave breaking produces nearly all the surf-zone EKE (Wbrk, Figs. 9c–f). The EKE production with SO waves (Figs. 9e,f) is merely 20%–30% of that with SN waves (Figs. 9c,d), which explains the weak surf eddies with SO waves (Fig. 9b). Besides, wave streaming generally suppresses surf eddies (Wstrm, Fig. 9e), and the EKE production with wave streaming (Figs. 9c,e) is only about half of that without wave streaming (Figs. 9d,f).

2) Eddies by shear instabilities

Surf eddies can also be generated by the shear instability of surf-zone alongshore currents (Oltman-Shay et al. 1989; Özkan-Haller and Kirby 1999). When the current bottom drag is reduced [e.g., by decreasing the drag coefficient in Eq. (A7) to 0.0015], the shear instability may develop and evolve to surf eddies (Figs. 10a,b).

Fig. 10.
Fig. 10.

(a),(b) Instantaneous surface vertical vorticity of surf eddies, normalized by the Coriolis parameter f, for UO waves (a) with and (b) without streaming. Dashed lines delineate the surf-zone edge. Cross-shelf profiles of the depth-integrated EKE budgets per unit mass with (c),(d) UO waves and (e),(f) SO waves. Left and right columns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is displayed. All results are with reduced current bottom drag, in comparison to Fig. 9 with the default drag defined by Eq. (A7).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

Surf eddies are stronger without wave streaming (Fig. 10b), because the shear production is stronger (Wcurr, Fig. 10d). Particularly with uniform waves, the shear production contributes to nearly all the EKE (Figs. 10c,d). By wave–current interaction, surf eddies from the shear production introduce variations to uniform waves; thus, even if the offshore incident waves are uniform, alongshore-variable wave breaking may occur and produce EKE (Wbrk, Fig. 10d).

For SO waves with streaming, the shear production and wave-breaking production are equally important (Fig. 10e), while the shear production prevails without wave streaming (Fig. 10f). Moreover, surf eddies are suppressed by the wave streaming (Wstrm) and Stokes shear force (Wstk) (Figs. 10c–f).

c. Cross-shelf transport

Passive tracers are released at the shore continuously for the study of cross-shelf material transport. With uniform waves, surf eddies are absent and the tracers are transported by the mean Lagrangian flow only. Without wave streaming, the tracers are almost trapped within the surf zone and barely enters the inner shelf (Figs. 11b and 12b). With wave streaming, the inner-shelf Lagrangian overturning circulation forms and carries the tracers across the surf-zone edge into the inner shelf by the upper circulation segment (Figs. 11a and 12a); meantime, the lower circulation segment recycles the tracers from the inner shelf back to the surf zone (Fig. 12a).

Fig. 11.
Fig. 11.

Mean concentration (no units) of the shore-released passive tracers with (a),(b) UN waves and (c),(d) SN waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

Fig. 12.
Fig. 12.

Cross-shelf tracer fluxes by the (a)–(d) mean Lagrangian flow with UN waves in (a) and (b), and SN waves in (c) and (d); (e),(f) fluxes by the surf eddies with SN waves. Left and right columns are with and without wave streaming. Dashed lines delineate the surf-zone edge.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

With stochastic waves, surf eddies develop and carry the tracers across the surf-zone edge into the inner shelf (Figs. 12e,f); consequently, the inner-shelf tracer concentration is elevated remarkably (Figs. 11c,d). Particularly with wave streaming, the tracers are transported farther offshore by the inner-shelf Lagrangian overturning circulation (Figs. 11c and 12c).

4. Discussion

a. Sensitivity to bottom roughness

Wave streaming is parameterized with a streaming force given by Eqs. (A5) and (A6), in which the bottom roughness height zo is crucial to determine the force magnitude. The default zo in our experiments is 10−3 m, and here we explore how the different zo changes the wave-driven circulation.

In general, the bottom drag dissipation (Fig. 13a) and the wave streaming (Fig. 13c) are elevated by increasing zo. However, with a very large zo up to 10−2 m, the maximum bottom drag dissipation is less than that with zo = 5 × 10−3 m (Fig. 13a). Because the bottom drag dissipation is also proportional to the cubic wave height [Eq. (A5)], which is dampened greatly by a very large zo.

Fig. 13.
Fig. 13.

Sensitivity to bottom roughness height zo. Cross-shelf profiles of the mean wave dissipation (a) by the bottom drag εf/ρ0¯ [Eq. (A5)] and (b) by the surface breaking εb/ρ0¯ [Eq. (A3)]. (c) Mean cross-shelf bottom Eulerian current. (d) Mean Lagrangian streamfunction ψL¯ defined in Eq. (2) with zo = 10−4 m, in comparison to Fig. 6a with zo = 10−3 m. The asterisks in (a)–(c) and the dashed line in (d) mark the surf-zone edge. The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

The surface breaking dissipation decreases with increasing zo (Fig. 13b), because with a larger zo, more wave energy is dissipated by the bottom drag before breaking. Weak wave breaking leads to a weak breaker force [Eqs. (A3) and (A4)] and a small surf-zone undertow. Thus with increasing zo, the onshore wave streaming is able to overwhelm the offshore undertow, turning the surf-zone, near-bottom current onshore (Fig. 13c).

Even with a very small zo = 10−4 m, the wave streaming can still induce an inner-shelf Lagrangian overturning circulation (Fig. 13d), though weaker than that with zo = 10−3 m (Fig. 6a). Besides, the wave breaking (Fig. 13b) and the surf-zone Lagrangian overturning circulation (Fig. 13d) are enhanced by decreasing zo.

b. Sensitivity to bottom slopes

The default bottom slope in our experiments is 0.005, representing a gently sloping shelf; here we explore how the different slopes change the wave-driven circulation. When the bottom slope increases, the water depth at a fixed offshore location increases; accordingly, the location where surface waves start feeling the bottom moves shoreward, and so do the location where surface waves start breaking due to shallow depths (Fig. 14a). Namely, the offshore spans of the wave streaming (Fig. 14b) and the associated inner-shelf Lagrangian overturning circulation are shortened (Fig. 14c), and the surf zone narrows (Fig. 14a).

Fig. 14.
Fig. 14.

Sensitivity to bottom slopes. (a) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5), solid lines and left y axis] and by the surface breaking εb/ρ0¯ [Eq. (A3), dashed lines and right y axis]. (b) Mean cross-shelf bottom Eulerian current. (c),(d) Mean Lagrangian streamfunction ψL¯ defined in Eq. (2) with a constant slope of 0.01 in (c) and a nonconstant slope in (d), in comparison to Fig. 6a with a constant slope of 0.005. The slope in (d) decreases from 0.014 in the surf zone to 0.004 in the inner shelf. The asterisks in (a) and (b) and the dashed lines in (c) and (d) mark the surf-zone edge. The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

With a larger bottom slope, less wave energy is dissipated by the bottom drag before breaking (Fig. 14a); therefore, the wave heights remain bigger when waves start breaking, resulting in stronger breaking dissipation (Fig. 14a). Strong wave breaking enhances the offshore undertow and suppresses the onshore wave streaming (Fig. 14b).

Further, we design a more realistic bathymetry with a nonconstant slope that decreases from 0.014 in the surf zone to 0.004 in the inner shelf (Fig. 14d). The surf-zone and inner-shelf Lagrangian overturning circulations form again (Fig. 14d) and resemble those in the bathymetry with a constant slope (Figs. 14c and 6a). So far we have explored various bottom slopes in the range of 0.004–0.02 that exist in the nature, and in such a wide range of bottom slopes, the wave streaming and its induced inner-shelf Lagrangian overturning circulation can survive.

c. Sensitivity to wave variations

The default wave variations in our experiments include a height variation QHgt = 10% of its mean, a period variation QPrd = 10% of its mean, and a direction variation QDir = 10° about its mean incident angle. Now we examine how the different QHgt, QPrd, and QDir change the wave-driven circulation, respectively; for example, when QDir varies, both QHgt and QPrd stay at zero.

The surf-zone EKE is elevated by increasing wave variations, and it is the largest with the wave direction variation QDir (Fig. 15a); namely, directional-spread waves are inclined to induce strong surf eddies. Then we use the QDir as a representative for wave variations.

Fig. 15.
Fig. 15.

Sensitivity to wave variations, including the wave height variation QHgt, period variation QPrd, and direction variation QDir. The QHgt and QPrd are expressed in percent (%) of their mean, and the QDir is expressed in angle degrees (°). (a) The depth-integrated EKE in the surf zone x = 0.2 km with different wave variations. (b) Cross-shelf profiles of the mean wave dissipation by the bottom drag εf/ρ0¯ [Eq. (A5); solid lines] and by the surface breaking εb/ρ0¯ [Eq. (A3); dashed lines]. Asterisks mark the surf-zone edge. (c),(d) Vertical profiles of the cross-shelf mean Eulerian current in the surf zone x = 0.2 km in (c) and the inner shelf x =.2.4 km in (d). The 95% confidence intervals are shaded with gray. The incident waves are SN with streaming.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-19-0304.1

As the QDir increases, the mean wave breaking dissipation decreases (Fig. 15b), which then reduces the near-bottom undertow in the surf zone (Fig. 15c). In the inner shelf, however, the stronger surf eddies owing to a larger QDir strengthen the eddy Reynolds stress, which enhances the near-surface mean flow (Fig. 15d).

5. Conclusions

Using numerical simulations in a nearshore region (surf zone and inner shelf), we examine how the wave streaming and wave variations affect the wave-driven circulation and the cross-shelf material transport. Also, sensitivities of the circulation to the bottom roughness, bottom slopes, wave variations are explored.

Wave streaming induces an inner-shelf Lagrangian overturning circulation, which facilitates the material exchange between the inner shelf and the surf zone by linking the two regions. Wave variations cause alongshore-variable wave breaking that generates surf eddies; nonetheless, such eddies can be suppressed by wave streaming or an oblique wave incidence. Moreover, surf eddies actively transport materials across the surf zone, beyond which the inner-shelf Lagrangian overturning circulation further spreads the material over the shelf.

Increasing the bottom roughness or decreasing the bottom slope can enhance wave streaming, and increasing wave variations can intensify surf eddies. Particularly, the directionally spread waves (direction variations) are effective to induce strong surf eddies.

Alongshore-variable bathymetry, such as rip channels, embayments, headlands, and submarine canyons, can impact the wave-driven circulation (Akan et al. 2020; MacMahan et al. 2006; Reniers et al. 2009; Dalrymple et al. 2011; Castelle et al. 2016), and it is worth exploring wave streaming in alongshore-variable bathymetry.

Density stratification can affect the nearshore wave-driven circulation (Kumar and Feddersen 2017b,c); in this study, however, stratification is not considered for simplification. In the companion study that includes stratification, Wang and McWilliams (2020, manuscript submitted to Geophys. Res. Lett.) discover an inner-shelf front induced by wave streaming.

The wave streaming in our experiments is parameterized by the Eqs. (A5) and (A6). The parameterization proposed by Xu and Bowen (1994) is also tried, and it gives similar but stronger wave streaming, possibly because the centimeter-thick wave bottom boundary layer is marginally resolved here. The requirement of vertical resolution for using the Xu–Bowen parameterization is discussed in appendix A [following Eq. (A6)]. Further, we suggest that more field measurements, particularly in the inner shelf, are needed to assess the wave streaming, parameterizations, and the induced Lagrangian overturning circulation.

Acknowledgments

P.W. and J.M. appreciate the support by the Office of Naval Research (Grant N00014-15-1-2645) and by the National Science Foundation (Grant OCE-1355970). Y.U. is grateful for the support by the JSPS Grant-in-Aid for Scientific Research (Grant 15 KK0207, 17 K00653, 18H03798, and 19H00782) at Kobe University. The numerical simulations are done with the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the National Science Foundation. Also, the authors sincerely thank the two reviewers for their constructive suggestions to improve this study.

APPENDIX A

Coupled Wave–Circulation Model

Equations of the momentum and passive tracers are (Uchiyama et al. 2010)

ut+(uh)u+wuz=fz^×uhϕ+z(Kυuz)hK+J+Fw,ϕz+gρρ0=Kz+K,hu+wz=0,ct+(uh)c+wcz+(uSh)c+wScz=z(Ktcz)+S.

Here (u, w) is the 3D Eulerian velocity with u = (u, υ), and (us, ws) is the 3D Stokes drift, ∇h = (∂/∂x, ∂/∂y), f is the Coriolis parameter, and z^ is a unit vertical vector. Also, ϕ is the dynamic pressure, and K is the Bernoulli head. The Stokes vortex force is (J, K), with the horizontal vector J incorporating the Stokes–Coriolis force. The variable Fw represents the nonconservative wave forces, ρ0 is the mean density, and ρ is the density anomaly. The term c is the tracer concentration, and S is the tracer source; Kυ (Kt) is the vertical eddy viscosity (diffusivity), which includes the viscosity (diffusivity) induced by breaking waves (Battjes 1975).

The WKB wave model is (Uchiyama et al. 2010)

kt+[(u¯+cg)h]k=k˜hu¯˜kσsinh2kDhD,At+h[(u¯+cg)A]=εwσ.

Here k is the horizontal wavenumber vector, cg is the group velocity, and u¯ is the depth-averaged horizontal current velocity. The tilde symbol denotes the conjoined vector dot product. The term D is the water depth, σ is the intrinsic wave frequency, and A is the wave action. The term εw represents the wave dissipation, including the surface breaking and bottom drag.

The wave dissipation by surface breaking is determined by a quasi-empirical parameterization (Church and Thornton 1993),

εb=3π16ρ0gBb3σ2πDHrms3{1+tanh[8(HrmsγbD1)]}×{1[1+(HrmsγbD)2]5/2},

where Hrms is the RMS wave height and Bb and γb are empirical constants depending on wave breaking type and bathymetry. The wave breaker force is given by

Fb=εbσk.

Optionally, the wave breaker force Fb can be distributed vertically with a prescribed shape function to represent shallow or deep breaking (Uchiyama et al. 2010).

The wave dissipation by bottom drag is determined by an empirical parameterization (Reniers et al. 2004a),

εf=12πρ0fw|uorbw|3,

where |uorbw|=σHrms/2sinh(kD) is the near-bottom wave orbital velocity, fw=1.39(σzo/|uorbw|)0.52 is the wave friction factor (Soulsby 1997), and zo is the bottom roughness height. The wave streaming force is given by

Ff=εfσk.

The wave streaming parameterization with Eqs. (A5) and (A6) has been widely used in coastal studies (Uchiyama et al. 2010; Kumar et al. 2012; Marchesiello et al. 2015; Shanks et al. 2015; Morgan et al. 2017; Fujimura et al. 2018). Moreover, Xu and Bowen (1994) proposed a different parameterization for wave streaming [their Eq. (63)], which was used by Lentz et al. (2008) and Kumar et al. (2012) for inner-shelf studies. To apply the Xu–Bowen parameterization, a high vertical resolution of O(1) cm that is able to resolve the centimeter-thick wave bottom boundary layer is required (Lentz et al. 2008; Kumar et al. 2012). However, our simulations marginally resolve the wave bottom boundary layer, making the Xu–Bowen parameterization inappropriate here. Also, an unresolved wave bottom boundary layer may amplify the wave streaming and the associated shelf circulation. As noted by Kumar et al. (2012), the parameterization with Eqs. (A5) and (A6) is more suitable if the vertical resolution is not high enough to resolve the wave bottom boundary layer, while the Xu–Bowen parameterization is preferred with a high vertical resolution.

The current bottom drag is enhanced according to an empirical formula that combines the wave orbital velocity (Feddersen et al. 2000),

Fd=0.015ρ0(kaD)1/3[(1.16)2+(|u||urmsw|)2]1/2|urmsw|u,

where urmsw=uorbw/2 is the near-bottom RMS wave orbital velocity, ka = 0.0125 m is the apparent roughness (Ruessink et al. 2001).

APPENDIX B

Stochastic Waves

Stochastic waves are constructed with a space–time correlated stochastic process F(y, t) that is a solution of a stochastic partial differential equation (Gillespie 1996),

dF(y,t)=[λF(y,t)+γ2Fy2]dt+dW(y,t).

Here λ and γ can be tuned to obtain a proper spatiotemporal correlation, which controls the spatial and temporal scales of surf eddies induced by stochastic waves; here the correlation is 10 wavelengths and 10 wave periods. The term dW(y, t) denotes a “space–time” white noise process, roughly speaking, a Gaussian stochastic process with spatiotemporal correlations for ta, tb ≥ 0, given by

E[dW(ya,ta)dW(yb,tb)]=δ(tatb)δ(yayb).

Equation (B1) is solved as follows: Over a grid of mesh size dy = Ly/Ny, the discrete approximation Fjn of F(jdy, ndt) is obtained via an explicit Euler–Maruyama scheme with a periodic boundary condition in y. The noise term dW(y, t) is estimated by ξjndt at each time step ndt, and ξjn is a Ny-dimensional vector of random variables each drawn independently from the standard normal distribution N(0,1) with respect to 1 ≤ jNy at each time step ndt.

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

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