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Effects of Wave Streaming and Wave Variations on Nearshore Wave-Driven Circulation
PENG WANG,a JAMES C. MCWILLIAMS,a YUSUKE UCHIYAMA,b MICKAËL D. CHEKROUN,a AND DALING LI YIcaDepartment of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California; b Department ofCivil Engineering, Kobe University, Kobe, Japan; c International Pacific Research Center, School of Ocean and Earth Science andTechnology, University of Hawaiʻi at Manoa, Honolulu, Hawaii(Manuscript received 3 December 2019, in final form 30 June 2020)

KEYWORDS: Bottom currents; Coastal flows; Waves, oceanic; Lagrangian circulation/transport

1. Introduction
The nearshore region that includes a surf zone and an innershelf 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 toshallow depths (Battjes 1988; MacMahan et al. 2006). Next tothe surf zone seaward, the inner shelf is where the surface andbottom 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 thesurf 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 thepersistent horizontal recirculations (MacMahan et al. 2010;
Reniers et al. 2010; Brown et al. 2015). Besides, internalbores 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 materialtransport is the wave streaming. In shallow water, surface
waves are subject to bottom drag, which induces a nearbottom current along the wave propagation. This current isreferred 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 dragin deeper water, and subsequently the wave bottom streaming

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

commences there. Nonetheless, the wave bottom streaming
may be reduced under nonlinear waves (Trowbridge andMadsen 1984; Davies and Villaret 1999; van der A et al. 2011;
Kranenburg et al. 2012; Henriquez et al. 2014). In addition, theviscous 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 isnegligible in this study, in agreement with Uchiyama et al.
(2010). Thus, we focus on the wave bottom streaming only andfor simplicity call it ‘‘wave streaming.’’
Wave streaming is important to the sediment transport andcan 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 tothe 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 awater depth of 12 m, Lentz et al. (2008) measured an onshore
current of 0.5 cm s21 at 2 m above the bottom (cf. theirFig. 10a), and they claimed that this current may be related totides; in addition to tides, however, other processes like theonshore wave streaming may also contribute to this current.
Further, wave streaming decreases rapidly away from thebottom, 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 theimportance and poor understanding of wave streaming, more
field observations are needed, especially in the inner shelf.
In addition, realistic surface waves contain variations inwave 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 beas important as that due to shear instabilities of surf-zone
alongshore currents (Long and Özkan-Haller 2009; Feddersen2014). Further, wave variations inherently reside in wave

DOI: 10.1175/JPO-D-19-0304.1
Ó 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).

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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 theeffects 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 crossshelf material transport.

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groups, and surf eddies due to wave-group variations candevelop 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 thewave streaming and wave variations to the nearshore wavedriven circulation? How do they together impact the crossshelf material transport? Here we aim to answer thesequestions using numerical simulations. The structure of thisstudy is as follows. The numerical model and experimental
setup are described in section 2. The major results arepresented 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 thestudy 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 oncurrents (WEC) and current effects on waves (CEW) aretaken into account. The WEC contain two parts, i.e., conservative and nonconservative. The conservative WEC arebased on the wave-averaged theory by McWilliams et al.
(2004), including the Stokes vortex force (Craik and Leibovich1976), Stokes–Coriolis force (Hasselmann 1970), and pressureadjustments. The nonconservative WEC are parameterized interms 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 parametersof 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 slopeof 0.005 (Fig. 1a), which is a typically gentle slope for the innershelf. 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

TABLE 1. Types of incident waves.
Incident waves

Uniform waves

Stochastic waves

Normal incidence (08)Oblique incidence (308)

UN
UO

SN
SO

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FIG. 1. (a) A schematic illustration of the numerical experiments. (b) A time series of stochastic wave heights atthe offshore boundary, with a mean height of H0 5 0.8 m. (c),(d) The wave heights averaged in each period bandfrom 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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vertical layers shows that 12 vertical layers are good enough toresolve the boundary layers and velocity profiles.
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 andeast boundaries have zero net fluxes. The Coriolis parameter is
8.13 3 1025 s21 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 uniformor 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 thewaves heights and periods have stochastic variations of 10% of
the mean. Stochastic waves (Fig. 1b) are constructed with thesolutions 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 bandaround 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 towardrealistic 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 308 (northwestward)
relative to the shore normal. The stochastic waves also have a
directional spread of 108 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.

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FIG. 2. (a),(b) Cross-shelf profiles of the mean wave dissipation by the bottom drag h«f /r0 i [Eq. (A5); solid lines]
and by the surface breaking h«b /r0 i [Eq. (A3); dashed lines]. (c),(d) Cross-shelf profiles the mean wave heights (lefty axis) and wavenumbers (right y axis). Left and right columns are with and without wave streaming. Asterisks markthe 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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3. Results

denoted by hi; the angle brackets hi denote the alongshore
average, and the overline  denotes the time average.

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 is1) CROSS-SHELF CURRENTS
After the uniform or stochastic waves prescribed at theoffshore boundary propagate into the model domain, they are

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.

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FIG. 3. (a),(b) Mean cross-shelf Eulerian current. (c),(d) Mean vertical eddy viscosity. Left and right columns arewith and without wave streaming. Dashed lines delineate the surf-zone edge.

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FIG. 5. (a),(b) Mean cross-shelf, depth-averaged Eulerian current and Stokes drift. Asterisks mark the surf-zoneedge. Vertical profiles of the mean cross-shelf, depth-dependent Eulerian current and Stokes drift in the (c),(d) surfzone x 5 0.2 km, (e),(f) inner shelf x 5 2.4 km, and (g),(h) outer shelf x 5 20 km. Left and right columns are with andwithout wave streaming. Solid and dashed lines denote the Eulerian current and Stokes drift.

governed by the wave model and obey the fundamental wave
dynamics, including refraction, shoaling, bottom drag, andsurface 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).
The stochastic waves spread directionally and therefore
experience wave ray convergences during the propagation

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VOLUME 50(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 thanSN 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 theonshore mass transport (Fig. 3b). Without wave streaming, thewave 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).

The wave streaming in the inner shelf manifests itself as anear-bottom onshore current (Fig. 3a), which meets theoffshore undertow around the surf-zone edge, creating anear-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 nearbottom 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

FIG. 7. Mean along-shelf Eulerian current with (a),(b) UN waves and (c),(d) UO waves. Left and right columns arewith and without wave streaming. Dashed lines delineate the surf-zone edge.

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FIG. 6. Mean Lagrangian streamfunction hcL i 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 hcL i similar
to with and without wave streaming in (a) and (b), respectively.

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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., windstress and heat flux), the near-surface vertical mixing is weakin 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 CIRCULATIONThe Lagrangian velocity uL is the sum of Eulerian velocity
and Stokes drift (i.e., uL 5 u 1 us). For a steady flow, thecontinuity equation of the depth-averaged Lagrangian velocity
UL (Lentz and Fewings 2012) is›(hU L ) ›(hV L ) ›[h(U 1 U s )] ›[h(V 1 V s )]
1
[
1
5 0.
›x
›y
›x
›y

(1)

Here h is the water depth, (U, V) is the depth-averaged
Eulerian velocity, and (U s , V s ) is the depth-averaged
Stokes drift. When the flow is alongshore uniform (i.e.,›/›y 5 0) and has zero net flux across the shore boundary(i.e., U L jx50 5 0), Eq. (1) implies the depth-averaged
U L (x) [ U 1 U s 5 0 everywhere (Figs. 5a,b). However,the depth-dependent Lagrangian velocity u L (x, z) [ u 1 u s
is nonzero in the surf zone and inner shelf, viz., u(x, z) 6¼
2u s (x, z).
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 thenear-bottom current and decreases rapidly away from thebottom (Fig. 5e); without wave streaming, the inner-shelfcurrent is nearly antiStokes (Fig. 5f). In the outer shelf, thecurrent cancels the Stokes drift at each depth (Figs. 5g,h),leading to the Stokes–Coriolis balance. Due to the deep

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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 markthe surf-zone edge. Only important work is displayed.

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water there, the bottom drag dissipation and wave streaming
are negligible.
To describe the Lagrangian overturning circulation in the depthand cross-shelf section, we define a Lagrangian streamfunction
cL(x, z) so thatuL 5 2›cL›cL, wL 5
.
›z
›x

(2)

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 surfzone edge.

3) ALONG-SHELF CURRENTS
With a normal wave incidence, there are no along-shelfcomponents 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.
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).

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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 profilesof the depth-integrated EKE budgets per unit mass with (c),(d) SN waves and (e),(f) SO waves. Left and rightcolumns are with and without wave streaming. Asterisks mark the surf-zone edge. Only important work is
displayed.

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4) MEAN KINETIC ENERGY
The mean kinetic energy (MKE) is defined as (1/2)(hui 1
hyi2 1 hwi2 ), where hui, hyi, hwi 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 analongshore current.
The budgets of depth-integrated MKE is summarizedin Eq. (3):ðdzd(MKE) 5 Wbrk 1 Wstrm 1 Wdrag 1 WStkCor
dt
1 WStk 1 Wcurr 1 Wres .

The r.h.s. work is denoted by the symbols:Ð
Wbrk 5 dz(Fib ui ) with the mean wave breaker force Fib (A4);Ðfd W
strm 5 dz(Fi ui ) with the mean wave streaming forcef
Fi (A6);ddd Wdrag 5 F ui with the mean current bottom drag F (A7);
Ð
s
^
d W
StkCor 5 dz(2f k 3 uj )ui with the mean Stokes–Corioliss^force (2fÐ k 3 uj ); sd W
Stk 5 dz[2ui uj (›uj /›xi )] with the mean Stokes shear
force [2uj (›usj /›xi )];
Ð
0 L0
d W
curr 5 dz[ui uj (›ui /›xj )] with the mean-flow shear ›ui /›xj ;d W
res the residual including the transport divergence and thedissipation by viscosity.
d

2

(3)

The major work of interest is Wbrk by the wave breaker force,
Wstrm by the wave streaming force, and Wdrag by the waveenhanced bottom drag. Without wave streaming (Figs. 8d,f), most
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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 thedepth-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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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 adjacenteddy of opposite vorticity to form a transient rip current(Peregrine 1998). Here surf eddies are revealed by the turbulent vertical vorticity calculated as z0 5 (›y 0 /›x) 2 (›u0 /›y)(Figs. 9a,b), where (u 0 , y0 ) are the turbulent horizontal
velocities.
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 andare 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):
ð
dz

d
0
0
0
0
0
(EKE) 5 Wbrk
1 Wstrm
1 Wdrag
1 WStkCor
1 WStkdt
0
0
1 Wcurr
1 Wres.

(4)

The r.h.s. work is denoted by the symbols:
Ð
0
b0 0
d W
brk 5 dz(Fi ui ) with the turbulent wave breaker forceb0
Fi (A4);

Ð
0
0
Wstrm
5 dz(Fif u0i ) with the turbulent wave streaming force0
Fif (A6);
0
d0 0
d W
drag 5 F ui with the turbulent current bottom dragd0
F (A7); Ð
0
s0 0
^
d W
StkCor 5 dz(2f k 3 uj )ui with the turbulent Stokes–Coriolis
s0^
force (2fÐ k 3 uj 0 );0f2ui [uj (›usj /›xi )]0 g with the turbulent Stokes sheard W
Stk 5 dz
0(›us /›xi )]0 g;
force f2u
Ð i [uj 0 j L00
d W
curr 5 dz[2ui uj (›ui /›xj )] with the mean-flow shear ›ui /›xj ;0d W
res the residual including the transport divergence and thedissipation by viscosity.
d

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 INSTABILITIESSurf eddies can also be generated by the shear instabilityof 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 andevolve to surf eddies (Figs. 10a,b).
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

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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.

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waves are uniform, alongshore-variable wave breaking may
occur and produce EKE (Wbrk, Fig. 10d).
For SO waves with streaming, the shear production andwave-breaking production are equally important (Fig. 10e),
while the shear production prevails without wave streaming(Fig. 10f). Moreover, surf eddies are suppressed by thewave streaming (W strm ) and Stokes shear force (W stk )(Figs. 10c–f).

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. Discussionc. Cross-shelf transport
Passive tracers are released at the shore continuously forthe 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 surfzone 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 thesurf-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 backto the surf zone (Fig. 12a).

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 z o in our experiments is 10 23 m, and here we
explore how the different z o 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 1022 m, the maximum bottom dragdissipation is less than that with zo 5 5 3 1023 m (Fig. 13a).
Because the bottom drag dissipation is also proportional to the

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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.

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cubic wave height [Eq. (A5)], which is dampened greatly by avery large zo.
The surface breaking dissipation decreases with increasingzo (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, theonshore 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 5 1024 m, the wave streaming canstill induce an inner-shelf Lagrangian overturning circulation(Fig. 13d), though weaker than that with zo 5 1023 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 thedifferent slopes change the wave-driven circulation. When thebottom 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 thelocation 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 surfzone narrows (Fig. 14a).
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 instronger 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 innershelf 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 insuch a wide range of bottom slopes, the wave streaming and its
induced inner-shelf Lagrangian overturning circulation cansurvive.

c. Sensitivity to wave variations
The default wave variations in our experiments include aheight variation QHgt 5 10% of its mean, a period variation
QPrd 5 10% of its mean, and a direction variation QDir 5 108
about its mean incident angle. Now we examine how thedifferent QHgt, QPrd, and QDir change the wave-driven circulation, respectively; for example, when QDir varies, both
QHgt and QPrd stay at zero.

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FIG. 13. Sensitivity to bottom roughness height zo. Cross-shelf profiles of the mean wave dissipation (a) by thebottom drag h«f /r0 i [Eq. (A5)] and (b) by the surface breaking h«b /r0 i [Eq. (A3)]. (c) Mean cross-shelf bottom
Eulerian current. (d) Mean Lagrangian streamfunction hcL i defined in Eq. (2) with zo 5 1024 m, in comparison to
Fig. 6a with zo 5 1023 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.

OCTOBER 2020

WANG ET AL.

3037

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 toinduce strong surf eddies. Then we use the QDir as a representative for wave variations.
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 theeddy Reynolds stress, which enhances the near-surface mean
flow (Fig. 15d).

5. Conclusions
Using numerical simulations in a nearshore region (surfzone and inner shelf), we examine how the wave streaming andwave variations affect the wave-driven circulation and thecross-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 tworegions. 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 overthe shelf.
Increasing the bottom roughness or decreasing the bottom
slope can enhance wave streaming, and increasing wave variations can intensify surf eddies. Particularly, the directionallyspread waves (direction variations) are effective to induce
strong surf eddies.
Alongshore-variable bathymetry, such as rip channels, embayments, headlands, and submarine canyons, can impact thewave-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 alongshorevariable 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

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FIG. 14. Sensitivity to bottom slopes. (a) Cross-shelf profiles of the mean wave dissipation by the bottom dragh«f /r0 i [Eq. (A5), solid lines and left y axis] and by the surface breaking h«b /r0 i [Eq. (A3), dashed lines and right yaxis]. (b) Mean cross-shelf bottom Eulerian current. (c),(d) Mean Lagrangian streamfunction hcL i 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 constantslope 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 shadedwith gray. The incident waves are SN with streaming.

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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) andby 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)



›u
›u
›
›u
1 (u  =h )u 1 w 5 2f ^z 3 u 2 =h f 1
Ky
2 =h K 1 J 1 F w ,
›t
›z
›z
›z
›f gr
›K
1 52
1 K,
›z r0
›z
›w
5 0,
=h  u 1
›z


›c
›c
›c ›
›c
1 (u  =h )c 1 w 1 (uS  =h )c 1 wS 5
Kt
1S .
›t
›z
›z ›z
›z

Here (u, w) is the 3D Eulerian velocity with u 5 (u, y), and (us, ws)
is the 3D Stokes drift, =h 5 (›/›x, ›/›y), f is the Coriolis parameter,and ^z is a unit vertical vector. Also, f is the dynamic pressure, andK is the Bernoulli head. The Stokes vortex force is (J, K), with the(A1)

horizontal vector J incorporating the Stokes–Coriolis force. The
variable F w represents the nonconservative wave forces, r0 is
the mean density, and r is the density anomaly. The term c is thetracer concentration, and S is the tracer source; Ky (Kt) is the

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FIG. 15. Sensitivity to wave variations, including the wave height variation QHgt, period variation QPrd, anddirection variation QDir. The QHgt and QPrd are expressed in percent (%) of their mean, and the QDir is expressed inangle degrees (8). (a) The depth-integrated EKE in the surf zone x 5 0.2 km with different wave variations.
(b) Cross-shelf profiles of the mean wave dissipation by the bottom drag h«f /r0 i [Eq. (A5); solid lines] and by thesurface breaking h«b /r0 i [Eq. (A3); dashed lines]. Asterisks mark the surf-zone edge. (c),(d) Vertical profiles of thecross-shelf mean Eulerian current in the surf zone x 5 0.2 km in (c) and the inner shelf x 5.2.4 km in (d). The 95%
confidence intervals are shaded with gray. The incident waves are SN with streaming.

OCTOBER 2020

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WANG ET AL.

vertical eddy viscosity (diffusivity), which includes the viscosity(diffusivity) induced by breaking waves (Battjes 1975).
The WKB wave model is (Uchiyama et al. 2010)
›k
ks
~= e
1 [(u 1 cg )  =h ]k 5 2k
= D,
u2
h
›t
sinh2kD h›A«w
1 =h  [(u 1 cg )A] 5 2 .
›t
s

(A2)



 
pﬃﬃﬃﬃ
Hrms
3 p
B3 s 321
1 1 tanh 8
r0 g b Hrms
«b 5
2pD
gb D
16
(
"
2 #25/2 )Hrms
3 12 11
,
gb D

(A3)

(A4)

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 anempirical parameterization (Reniers et al. 2004a),1« f 5 pﬃﬃﬃﬃ r0 fw juworb j3 ,
2 p

(A5)

where juworb j 5 sHrms /2 sinh(kD) is the near-bottom wave orbital velocity, fw 5 1:39(szo /juworb j)0:52 is the wave friction factor(Soulsby 1997), and zo is the bottom roughness height. The
wave streaming force is given by
Ff 5

«f
k.
s

2 #1/2
 1/3 "kajuj
juwrms ju,(1:16)2 1
juwrms j
D

(A7)

pﬃﬃﬃ
where uwrms 5 uworb / 2 is the near-bottom RMS wave orbital velocity,
ka 5 0.0125 m is the apparent roughness (Ruessink et al. 2001).

APPENDIX B
Stochastic Waves

where Hrms is the RMS wave height and Bb and g b are empirical constants depending on wave breaking type and bathymetry. The wave breaker force is given by«b
Fb 5 k.
s

Fd 5 0:015r0(A6)

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 andBowen (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 marginallyresolve the wave bottom boundary layer, making the Xu–Bowen

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),
›2 F
dF(y, t) 5 2lF(y, t) 1 g 2 dt 1 dW(y, t).
(B1)›yHere l and g can be tuned to obtain a proper spatiotemporalcorrelation, 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 Gaussianstochastic process with spatiotemporal correlations for ta, tb $ 0,
given by
E[dW(ya , ta )dW(yb , tb )] 5 d(ta 2 tb )d(ya 2 yb ).

(B2)Equation (B1) is solved as follows: Over a grid of mesh size dy 5
Ly/Ny, the discrete approximation Fjn of F(jdy, ndt) is obtained via
an explicit Euler–Maruyama scheme with a periodic boundary
pﬃﬃﬃﬃﬃ
condition in y. The noise term dW(y, t) is estimated by jnj dt ateach time step ndt, and jnj is a Ny-dimensional vector of randomvariables each drawn independently from the standard normal distribution N (0, 1) with respect to 1 # j # Ny at each time step ndt.
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