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A Nearshore Oceanic Front Induced by Wave Streaming
PENG WANG,a JAMES C. MCWILLIAMS,a AND YUSUKE UCHIYAMAba

Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, CaliforniabDepartment of Civil Engineering, Kobe University, Kobe, Japan(Manuscript received 6 January 2021, in final form 23 March 2021)

ABSTRACT: Coastal fronts impact cross-shelf exchange of materials, such as plankton and nutrients, that are important to
the ecosystems in continental shelves. Here, using numerical simulation we demonstrate a nearshore front induced by wave
streaming. Wave streaming is a bottom Eulerian current along the surface wave direction, and it is caused by the wave
bottom dissipation. Wave streaming drives a Lagrangian overturning circulation in the inner shelf and pumps up deep andcold water into the overturning circulation. The water inside the overturning circulation is quickly mixed and cooled becauseof the wave-streaming-enhanced viscosity. However, the offshore water outside the overturning circulation remains
stratified and warmer. Hence, a front develops between the water inside and outside the overturning circulation. The front isunstable and generates submesoscale shelf eddies, which cause the offshore transport across the front. This study presents a
new mechanism for coastal frontogenesis.
SIGNIFICANCE STATEMENT: Cross-shelf exchange of materials, such as plankton and nutrients, is important to thehealth of ecosystems in the continental shelves. Such material exchange is affected by various coastal fronts that arecharacterized by a sharp change of water properties (e.g., density, temperature, or salinity) in a narrow distance. Here wefind a novel nearshore front caused by the bottom drag on surface waves. The front is located in water that is ;10 m deep,roughly parallel to the shore, and it extends from the surface to the bottom. The front acts as a barrier that limits thematerial transport across the front. However, the front is unstable and generates eddies spreading offshore. These eddies
break the frontal barrier, causing the offshore transport across the front. This study presents a new mechanism forcoastal frontogenesis.
KEYWORDS: Coastal flows; Frontogenesis/frontolysis; Fronts; Waves, oceanic

1. Introduction
The health of marine ecosystems in continental shelves relies on the cross-shelf exchange of materials, such as nutrients,
pollutants, and plankton (Nittrouer and Wright 1994; Shanks
et al. 2015; Fujimura et al. 2018). The cross-shelf exchange iscontrolled by numerous processes (Lentz and Fewings 2012;
Brink 2016), and one of them is coastal fronts that are characterized by a sharp change of water properties (e.g., density,
temperature, or salinity) in a narrow distance (McWilliams
2021). Fronts can act as a barrier that limits the cross-shelfexchange (Brink 2016). For example, Fig. 1 shows a nearshore
front that is located at about 2 km from the shore, roughly
parallel to the shoreline. The front gathers floating yellow algae into an alongshore band, limiting the cross-shelf dispersionof the algae.
There are several types of coastal fronts (McWilliams 2021).
Along-shelf winds drive a cross-shelf Ekman transport, causing
an upwelling or downwelling front (Allen et al. 1995; Austin
and Lentz 2002). Tidal currents enhance vertical mixing inshallow water, resulting in a tidal mixing front that separates
mixed shallow water from its surroundings (Simpson andHunter 1974; Chen et al. 2003). Around a shelf break, cold
fresh shelf water is separated from warm salty slope water by ashelf-break front (Houghton et al. 1988; Chapman and Lentz

Corresponding author: Peng Wang, wangpeng@ucla.edu

1994; Gawarkiewicz et al. 2004). Internal waves can create
fronts parallel to wave crests, and such fronts usually movewith the waves (Shanks 1983; Pineda 1994).
Surface gravity waves affect oceanic frontogenesis (Suzuki
et al. 2016; McWilliams 2018). Surface waves may sharpen or
widen a front depending on the alignment between the front
and Stokes drift (Suzuki et al. 2016; Sullivan and McWilliams
2019). Moreover, depth-induced wave breaking enhances
vertical mixing in a surf zone (MacMahan et al. 2006; Reniers
et al. 2009; Brown et al. 2015; Uchiyama et al. 2017; Akan et al.
2020; Wang et al. 2020); in contrast to the offshore weakermixing, a front may occur around the surf-zone edge (Kumar
and Feddersen 2017).
Surface waves also induce wave streaming that is a waveaveraged Eulerian current along the wave direction (LonguetHiggins 1953). Wave streaming includes surface streaming andbottom streaming. The surface streaming is driven by a viscousstress with wave velocity shear in a wave surface boundary
layer (Xu and Bowen 1994). Surface streaming may affectocean surface transport and Langmuir circulation (Weber et al.
2006; Wang and Özgökmen 2018). The bottom streaming isdriven by a stress due to wave velocity fluctuations in a wave
bottom boundary layer (Longuet-Higgins 1953). The wave
velocity fluctuations are generated when surface waves experience bottom drag in shallow water. Bottom streaming may beaffected by asymmetric wave shapes (Trowbridge and Madsen
1984; Kranenburg et al. 2012). In this study we neglect the surface streaming because of its smallness (Uchiyama et al. 2010)

DOI: 10.1175/JPO-D-21-0004.1
Ó 2021 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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FIG. 1. A satellite image of algae (Trichodesmium) blooms off Nine Mile Beach, Queensland, Australia, on 15
Sep 2018. The image was collected by the USGS–NASA Landsat-8 Operational Land Imager and distributed by theNASA Ocean Biology Processing Group. In the far offshore, yellow algae slicks are gathered into shore-normal
windrows by Langmuir circulation. In the nearshore, yellow algae are accumulated into a shore-parallel band
located at about 2 km from the shore. This algae band is more than 10 km alongshore in about 10 m-deep water(inner shelf), and it implies that there exists an alongshore front. The murky water between the shore and the algae
band is likely due to strong mixing by surf eddies and inner-shelf currents.

and focus on the bottom streaming; hereinafter, ‘‘wave streaming’’ denotes the bottom streaming only.
Wave streaming affects inner-shelf circulation (Lentz et al.
2008; Wang et al. 2020), and it is crucial to the transport of
sediments and near-bed fish larvae from the inner shelf to thesurf zone (Reniers et al. 2004; Shanks et al. 2015; Fujimura
et al. 2018). But roles of wave streaming in coastal frontogenesis remain unknown. Wang et al. (2020) report that wave
streaming drives an inner-shelf Lagrangian overturning circulation in unstratified water. By extending that study to stratified
water, we find that wave streaming induces a shore-parallelnearshore front, associated with the inner-shelf Lagrangian
overturning circulation.
This study is organized as follows. The numerical model andexperiment configurations are described in section 2. The experiment results are analyzed in section 3, such as the wavestreaming-induced circulation and front. Further, in section 4
the circulation and front are examined in various environments, including tides and winds. The study is summarized in
section 5.

2. Methods
a. Numerical model
Numerical experiments are conducted with a coupled wave–
circulation model (Uchiyama et al. 2010), which is a spectrumpeak surface wave model coupled with Regional Oceanic
Modeling System (ROMS). This coupled model has beenvalidated against field measurements of waves and currents(Uchiyama et al. 2010; Marchesiello et al. 2015). The wave
model gives wave effects on currents (WEC) to the circulation
model and, in turn, takes current effects on waves (CEW) fromthe circulation model. Taking CEW into account is crucial for awave model to give accurate WEC, because CEW significantly
modify waves and thereby WEC (Wang and Özgökmen 2018;
Romero et al. 2020). The WEC are separated into conservativeand nonconservative parts. The conservative part contains
Stokes vortex force, Stokes–Coriolis force, and wave-inducedpressure (McWilliams et al. 2004). The nonconservative part isparameterized, including wave breaking, wave streaming, andwave-enhanced bottom drag (see the appendix).

b. Experiment configuration
The numerical topography (Fig. 2a) is an idealized continental shelf, a sloping plane with a slope of 0.005 (’0.38). It is
25.6 km in the cross-shelf direction X and 20.48 km in thealong-shelf direction Y, with a horizontal resolution of 40 m
and 18 terrain-following layers. The water depth is 0.5 m at theshoreline and 128.5 m at the offshore boundary.
The initial water temperature is horizontally uniform
and decreases with depth (Fig. 2b). The salinity is constant.
The buoyancy frequency varies with depth (Fig. 2b) andrepresents a summertime stratification. Tides and winds
are absent unless explicitly stated. Incident surface waves
are prescribed at the offshore (eastern) open boundary.
The western shoreline boundary is enforced with zero massflux. The southern and northern boundaries are periodic.
The K-profile parameterization (KPP; Large et al. 1994) isused for vertical mixing. The Coriolis parameter is 8.13 3
10 25 s 21 for a midlatitude. The experiment is simulated for
41 numerical days.
As explained by Wang et al. (2020), the incident waves arestochastic and mimic a spectrum-peak band of measured surface waves. The waves contain stochastic variations in the wave

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where uL and wL denote the cross-shelf and vertical Lagrangian
velocity, respectively (Wang et al. 2020). The Lagrangian velocity is the sum of an Eulerian velocity and Stokes drift.
The Lagrangian streamfunction reveals two overturning
circulations located in the surf zone and inner shelf (Fig. 3b).
In particular, the inner-shelf overturning circulation isdriven by wave streaming (Wang et al. 2020), and it is narrower than its counterpart in unstratified water (Wang et al.
2020, their Fig. 6a).
Wave breaking and wave streaming enhance vertical eddy
viscosity in the surf zone and inner shelf, respectively (Fig. 3c).
The enhancement of eddy viscosity is incorporated into KPP(Uchiyama et al. 2010), including a bottom KPP. Interestingly,
the viscosity enhanced by wave streaming appears largely inside the inner-shelf overturning circulation, significantly increasing the vertical mixing inside this circulation.
Wave streaming pumps up deep and cold water into theinner-shelf overturning circulation. Because of the enhanced
viscosity, the water inside the overturning circulation isquickly mixed and cooled (Fig. 3d; 0.5 , x , 2 km). By
contrast, the offshore water outside the overturning circulation remains stratified and warmer (Fig. 3d; x . 2 km).
Hence, a front develops between the water inside and outside
the overturning circulation (Fig. 3d; x ’ 2 km). We call it the‘‘wave-streaming front.’’

b. Frontogenetic tendency
FIG. 2. (a) A schematic illustration of experiment configurations.
(b) Initial profiles of temperature and stratification N2.

height, period, and direction. The mean wave height is 0.8 m,
the mean period is 10 s, and the mean direction is normal to theshore yet with a directional spread of 108.
We focus on a nearshore region that consists of a surf zone
and an inner shelf. The surf zone is where surface waves break
due to shallow depth. Offshore next to the surf zone, the innershelf is where surface and bottom boundary layers overlap(Nittrouer and Wright 1994; Lentz and Fewings 2012). In our
experiment, however, the surface boundary layer does not exist
owing to the absence of surface forcing. Instead, we refer to theinner shelf as where the bottom boundary layer roughly occupies the full depth next to the surf zone.

3. Resultsa. Wave-streaming front
In the surf zone, depth-induced wave breaking is parameterized as a near-surface breaker force [Eq. (A2) in the
appendix]. A pressure gradient force that balances the breaker
force drives an offshore undertow. This undertow extends fromthe surf zone to the inner shelf (Fig. 3a; x , 2 km). There isalso a bottom onshore current in the inner shelf (Fig. 3a; 0.5 ,x , 2 km). This current is wave streaming and is generated by aparameterized, near-bottom streaming force [Eq. (A4)].
We define a Lagrangian streamfunction cL as
uL 5 2

›cL
›z

and

wL 5

›cL
,
›x

(1)

The wave-streaming front is featured with large cross-shelftemperature gradients (Fig. 4a), which extend to the bottom
along the downwelling (Fig. 3b, green arrow) beneath thesurface front. The frontogenetic tendency expressed with thecross-shelf temperature gradient is diagnosed with the following equation:  2  
DL 1 ›T›u ›T ›y ›T ›T›w ›T ›T2
1 2
5 2
›x ›x ›x ›y ›x›x ›z ›x
Dt 2 ›x|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
FvelH
FvelV
"
!
!#
›usj ›T ›T
› ›
›T ›T
1 2
1
. (2)
k
›xj
›x
›x ›xj ›x ›x ›xj|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
FStk

Fmix

Equation (2) is derived by taking the derivative ›/›x of theadvection–diffusion equation for temperature with Stokes drift
included, and then multiplying it by ›T/›x (Hoskins 1982;
McWilliams 2021). The operator DL/Dt 5 ›/›t 1 (uL  =) is a
total derivative with Lagrangian velocity uL, k is the temperature diffusivity, and j 5 1, 2, 3. The l.h.s. of Eq. (2) is thefrontogenetic tendency. A positive or negative frontogenetic
tendency respectively means that temperature gradients acrossthe front will be intensified or reduced with time. The r.h.s. of
Eq. (2) is grouped as follows: FvelH is strain by horizontal
Eulerian velocity (u, y), FvelV is strain by vertical Eulerianvelocity w, FStk is strain by Stokes drift (us, y s, ws), and Fmix isdue to differential mixing.
The strains of Eulerian current FvelH (Fig. 4b) and FvelV(Fig. 4c) are the largest terms, yet the two compete with eachother. Differential mixing Fmix is also significant, especially for

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FIG. 3. (a) Cross-shelf Eulerian velocity. (b) Lagrangian streamfunction defined in Eq. (1). The arrows indicate
directions of Lagrangian currents. (c) Vertical eddy viscosity. (d) Temperature. The angle brackets with overbar
indicate averaging over both time and alongshore. A temporal average is made over the second day. The dashedlines mark the surf-zone edge.

the near-bottom frontogenetic tendency (Fig. 4d). Nevertheless,the Stokes drift effect FStk is weaker by a factor of O(10) thanothers and therefore is neglected. Hence, the frontogenetic
tendency here is determined by net effects of Eulerian current
and mixing (Fig. 4e), and it indicates that the surface front isintensified with time while the bottom front is reduced.
Intensification of the surface front may destabilize the front,
leading to a frontal instability.

c. Frontal instability
In the surf zone, alongshore-variable wave breaking generates surf eddies (Fig. 5a, inset); they can burst out of the surfzone and reach and perturb the wave-streaming front at x ’
2 km. About 5 days later, the front destabilizes at the surface
and generates small eddies of O(100) m with Rossby number

Ro [ jzj/f ; 5 (Fig. 5a; x ’ 3 km). While spreading offshore(Figs. 5b,c), these eddies grow bigger to O(1) km with Ro of
approximately 2. We call the offshore eddies generated by thefront ‘‘shelf eddies.’’
The offshore spreading of shelf eddies is further demonstrated by eddy kinetic energy (EKE) in a Hovmöller diagram(Fig. 5d). Offshore from the front (x . 2 km), EKE grows withtime and spreads over the shelf. The offshore EKE is largely
confined within the top 10 m (Fig. 5e; x . 3 km). Especially at
the front (x ’ 2 km), EKE concentrates at the surface rather
than through full depth; this is consistent with the result thatthe frontal instability occurs at the surface.
From wave-averaged momentum equations that include
wave forces (Uchiyama et al. 2010; Suzuki and Fox-Kemper2016), we derive the following EKE equation:

" 
!
!


 #
0
0
›us 0
DL 1 020 ›ui^ 3 us0 ) u0 1 2 u j u0 1 F b0 u0 1 F f 0 u0 1 2n ›ui ›ui
ui 5 2u0i uLj
1 w0 b0 1 (2f kii ij|ﬄ{zﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ|ﬄﬄi{zﬄﬄ}i|ﬄﬄi{zﬄﬄ}i
Dt 2
›xj
›xi
›xj ›xj
ﬄ}
Wbuoy
Wbrk
Wstrm
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
WStkCor
Wflow



1

0



"



#

WStk

"



#

›
p
›
1 02
›
›
› 1 020u
2 u0i 1
1 [(usj uj )0 u0i ] 1
u
2uLj
n
›xi
›xj
2 i
›xi
›xj ›xj 2 i
r0|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

Wdiss

(3)

WtransAgain, the operator DL/Dt 5 ›/›t 1 (uL  =) is a total derivativewith Lagrangian velocity uL. The eddy component is denotedby the prime symbol and is defined as deviation from thealongshore average; for example, u0i 5 ui 2 hui i, where angle

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FIG. 4. (a) Cross-shelf temperature gradient. Also shown are the r.h.s. terms of Eq. (2) for frontogenetic tendency:(b) strain by horizontal Eulerian velocity, (c) strain by vertical Eulerian velocity, and (d) differential mixing. (e) The
sum of the terms in (b)–(d). A temporal average is made over the second day. The dashed lines mark the surfzone edge.

brackets denote an alongshore average. The overline above
each term denotes a temporal average. The r.h.s. of Eq. (3) aregrouped as follows:
Wflow is mean-flow
shear production,
horizontal including›ui ›ui›ui,
and vertical shear
;
shear
›x ›y
›z
0
d W
is
buoyancybproduction;
buoy
0
d W
z 3us ;StkCor is work by Stokes–Coriolis force 2f ^›usj 0
1
d W
2u
shear
force
5
j
 S t k iss work by Stokes

›xi
›uj
›usj
2uj
2 2uj
;
›xi
›xi
b0
b
b
d W
brk is work by wave-breaker force Fi 5 Fi 2 hFi i[appendix Eq. (A2)];d

1
Here the Stokes shear force is defined as [2uj (›usj /›xi )], slightly
different than the Suzuki and Fox-Kemper (2016) definition of[2uLj (›usj /›xi )] because of different algebraic manipulations.

0

WStrm is work by wave-streaming force Fif 5 Fif 2 hFif i[Eq. (A4)];d W
diss is turbulence dissipation;d W
trans is divergence of EKE transport by turbulence.
dEquation (3)’s r.h.s. contains four terms related to wave
forces, namely, WStkCor, WStk, Wbrk, and WStrm. The first twoare linked to Stokes drift, and the last two arise from wave
dissipation. These four terms represent wave effects on EKE
and are primary energy passages from waves to currents.
In addition to the above four wave-related terms, there
is another one in the mean-flow shear production W flow ,0that is, the Stokes-drift-modified Reynolds stress u0i uLj 5
0 0
0 s0
0 s0
ui uj 1 ui uj . The term ui uj is generally nonzero because u 00
and us are correlated by wave–current interaction. This
Reynolds stress around the front is enhanced by surf eddies
that reach the front. The enhanced Reynolds stress thenamplifies the shear production at the front, accelerating
frontal instability.

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FIG. 5. (a)–(c) Evolution of the surface vertical vorticity normalized by the Coriolis parameter. (d) Hovmöller
diagram of EKE (per unit mass) integrated over the depth and averaged alongshore. (e) EKE averaged over the last
10 days and alongshore. The dashed lines mark the surf-zone edge.

Most positive mean-flow shear work Wflow appears near thesurface front (Fig. 6b; x , 2.5 km), while negative work appears
near the inner-shelf bottom (Fig. 6b; 1 , x , 3 km). A close
examination decomposes the shear work into work by thehorizontal shear and vertical shear. Specifically, it is the horizontal shear, instead of vertical shear, that makes the total
shear work negative near the bottom. On the other hand, thehorizontal shear is a leading contributor to the positive total
shear work near the surface.
Large buoyancy production Wbuoy around the front (Fig. 6a;2 , x , 2.5 km) supplies energy to the growth of frontal instability that evolves to shelf eddies. Besides, within the top
10 m offshore from the front (Fig. 6a; 3 , x , 5 km), there
exists another important buoyancy production that sustains thegrowth and offshore spreading of shelf eddies.
Stokes shear work WStk is significant near the surface front
where the Stokes shear is strong (Fig. 6c). However, there thesurface current is directed against the Stokes drift, making
Stokes shear work compete with mean-flow shear work
(Suzuki and Fox-Kemper 2016). Here the negative Stokes
shear work removes EKE and weakens frontal instability,
opposing the positive mean-flow shear work near the surface.
Wave-streaming work WStrm is confined to the inner-shelfbottom because of the rapid decay of streaming force awayfrom the bottom (Fig. 6d). Nonetheless, the frontal instability
and shelf eddies largely appear near the surface away from thebottom (Fig. 5e; x . 2 km). Hence, the wave-streaming work is
not considered to be a major energy source for the frontal instability and shelf eddies.
In brief, the shear production (Wflow) and buoyancy production (Wbuoy) are major energy sources for the frontal instability and shelf eddies (Fig. 6e). This indicates that thefrontal instability is a mixed barotropic–baroclinic instability(Cushman-Roisin and Beckers 2011).
Further, rotation is essential to the development ofbaroclinic instability (Cushman-Roisin and Beckers2011), which makes the deformation radius relevant to thesize of shelf eddies. Here the deformation radius 2 is about
1.23 km, close to the shelf-eddy size. In a comparison experiment without rotation, wave streaming still induces aninner-shelf overturning circulation and an associatedfront, similar to the case with rotation. But without rotation the front is stable and does not generate shelf eddies.
Moreover, with rotation there is an alongfront current
roughly in geostrophy, but without rotation this current
disappears.

2
The deformation radius is estimated as Nh/f with the buoyancy
frequency N 5 0.01 s21, frontal height h 5 10 m, and Coriolis parameter f 5 8.13 3 1025 s21.

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FIG. 6. The r.h.s. terms of Eq. (3) for eddy kinetic energy (per unit mass): work by (a) mean-flow shear,(b) buoyancy, (c) Stokes shear force, and (d) wave-streaming force. (e) Depth integrals of Eq. (3)’s r.h.s. terms.
Note that the terms WStkCor and Wbrk are not shown in this plot because WStkCor is too small and Wbrk diminishes
outside the surf zone. A temporal average is made over the last 10 days.

d. Cross-shelf transport
We use shore-released passive tracers to examine how the
wave-streaming front (Fig. 7a) affects the cross-shelf transport.
The tracers are governed by an advection–diffusion equation,in which the Stokes-drift advection is added (McWilliams et al.
2004; Uchiyama et al. 2010). Passive tracers are released in the
shore-side water at the beginning of simulation. The initialtracer concentration in water shallower than 0.6 m is uniform
and equals to 1 (dimensionless), and no tracer is released in therest of water. Such tracers may represent plankton, nutrients,
and pollutants in the surf zone.
The shore-released tracers are rapidly mixed within the surfzone (Fig. 7b; x , 0.5 km) owing to the wave-breakingenhanced viscosity. Meantime, tracers are transported out of
the surf zone into the inner shelf (0.5 , x , 2 km) by the mean
Lagrangian current (Fig. 7c) and surf eddies (Fig. 7d).
Surface tracers are subducted at the front by the downwelling. One part (;65%) of the subducted tracers is then sent

back to the surf zone by the bottom onshore mean Lagrangian
current (Fig. 7c; 0.5 , x , 3 km). The other part (;35%) isfarther dispersed offshore mainly by shelf eddies (Fig. 7d; x .
3 km). In other words, shelf eddies cause the offshore transport
across the front.
Because of the mutual advection by frontal downwelling andshelf eddies, tracers that cross the front into the offshore areconcentrated below the surface about 10 m deep (Fig. 7b; x .
3 km). In unstratified water (no front), Wang et al. (2020) do
not observe such offshore transport centered below thesurface.

4. Discussion
The nearshore region is a complex environment that includes various dynamical processes and topography. Hence, itis necessary to explore how the wave-streaming-induced
overturning circulation and front may be modified by various environments, such as topography, tides, and winds.

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FIG. 7. (a) Temperature. Here the late-stage surface front is a few hundreds of meters wider than its early stage(Fig. 3d) but remains at similar strength. The curved dashed line is a contour of tracer concentration hci 5 0:01 (nounits), and it roughly follows an isotherm (isopycnal). (b) Concentrations of shore-released passive tracers. The
curved dashed line is the same as in (a). Also shown are cross-shelf tracer flux by the (c) mean Lagrangian currentsand (d) eddies. A temporal average is made over the last 10 days. The vertical dashed lines mark the surf-zone edge.

a. Sensitivity of wave-streaming front
Sensitivities of the wave-streaming front are examinedagainst wave energy, bottom roughness, shelf slope, andstratification, as they can affect the front via the parameterized
streaming force [Eq. (A4) in the appendix].

1) WAVE ENERGY
We use wave height as a proxy of wave energy. In this study
the offshore incident waves have a default mean height of
0.8 m. Now we use two additional mean wave heights of 0.4 and1.2 m that represent smaller and bigger wave energy, respectively. The wave-streaming-induced overturning circulation(Figs. 8a,b) and front (Figs. 8c,d) can occur with either smaller
or bigger waves but are different in size, strength, and location.
The surf zone is wider with bigger waves (Fig. 8b; x , 1 km)
than with smaller waves (Fig. 8a; x , 0.3 km), because depthinduced breaking of bigger waves begins at a farther offshore
location. Moreover, bigger waves induce a stronger breaker
force [Eq. (A2)] and thereby drive a stronger surf-zone circulation (Fig. 8b) than smaller waves do (Fig. 8a).
Bigger waves also induce a stronger streaming force[appendix Eq. (A4)] and thus generate stronger wave streaming than smaller waves do (Fig. 8e). The wave streaming withbigger waves starts at a farther offshore location; thus, it
drives a wider and stronger inner-shelf overturning circulation(Fig. 8b) than the streaming with smaller waves does. Namely,
as wave energy increases, the inner-shelf overturning circulation expands offshore and pushes the front offshore (Fig. 8f).
Furthermore, the wave streaming with bigger waves pumps

up deeper and colder water, making the nearshore waterwith bigger waves cooler (Fig. 8d) than that with smaller
waves (Fig. 8c).

2) BOTTOM ROUGHNESS
Sea bed roughness is measured by a bottom roughness
height. In this study the default bottom roughness height is
zo 5 1023 m, commonly used in nearshore ocean simulations.
Now we use two additional bottom roughness heights of zo 51024 m and zo 5 1022 m that represent smooth and rough beds,
respectively. The wave-streaming-induced overturning circulation (Figs. 9a,b) and front (Figs. 9c,d) can form above either asmooth or a rough bed but differs in size, strength, andlocation.
Before depth-induced breaking, wave energy is dissipatedby bottom drag first. Less wave energy is removed by bottom
dissipation above a smooth bed than above a rough bed; thus,
more wave energy is left for breaking above a smooth bed,leading to a stronger surf-zone circulation (Fig. 9a). In fact, in
this rough-bed case nearly all the wave energy is dissipated bybottom drag, making wave breaking trivial; thus, there is nosurf zone or related circulation (Fig. 9b).
Wave streaming is stronger and starts at a farther offshore
location above a rough bed than above a smooth bed (Fig. 9e).
Hence, the wave streaming above a rough bed drives a stronger
and wider inner-shelf overturning circulation (Fig. 9b) than it
does above a smooth bed (Fig. 9a). Namely, as bed roughness
increases, the inner-shelf overturning circulation expands offshore, pushing the front offshore (Fig. 9f). Furthermore, theBrought to you by KOBE UNIVERSITY | Unauthenticated | Downloaded 08/02/22 01:37 AM UTC

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FIG. 8. Sensitivity to wave energy: Lagrangian streamfunction with wave height H 5 (a) 0.4 and (b) 1.2 m. The
arrows indicate directions of Lagrangian currents. Temperature with H 5 (c) 0.4 and (d) 1.2 m. Alongshoreaveraged (e) bottom cross-shelf Eulerian velocity and (f) surface cross-shelf temperature gradient. A temporal
average is made over the second day. The dashed lines in (a)–(d) and the asterisks in (e) and (f) mark the surfzone edge.

wave streaming above a rough bed brings up deeper and colder
water, making the nearshore water above a rough bed cooler(Fig. 9d) than that above a smooth bed (Fig. 9c).

3) SHELF SLOPE
Shelf slopes affect the locations where wave breaking andstreaming start. In this study the default shelf slope is 0.005.
Now we use two additional shelf slopes of 0.0025 and 0.01 that
represent gentle and steep shelves, respectively. The wavestreaming-induced overturning circulation (Figs. 10a,b) andfront (Figs. 10c,d) can form in either a gentle or a steep shelf
but vary in size, strength, and location.
The location where surface waves begin to experience bottom drag is farther offshore in a gentle shelf than in a steep
shelf. Thus, wave streaming starts farther offshore in a gentle
shelf, leading to a wider inner-shelf overturning circulation(Fig. 10a) than that in a steep shelf (Fig. 10b). Namely, as shelfslopes increase, the starting location of wave streaming shifts
onshore (Fig. 10e). This narrows the inner-shelf overturning
circulation, pulling the front onshore (Fig. 10f).
The distance over which surface waves experience bottom
drag is shorter in a steep shelf than in a gentle shelf. Hence, lesswave energy is removed by bottom dissipation in a steep shelf,
and more energy is left for depth-induced breaking. This
yields a stronger surf-zone circulation in a steep shelf (Fig. 10b)than that in a gentle shelf (Fig. 10a). Moreover, the surf zone isnarrower in a steep shelf (Fig. 10b; x , 0.3 km) than in a gentle
shelf (Fig. 10a; x , 0.7 km), because waves in a steep shelf haveto be closer to the shore in order to trigger breaking caused byshallow depth.
A realistic shelf slope usually varies with locations. By
mimicking the shelf shown in Fig. 1, we adopt a variable shelfslope that decreases from 0.02 in the surf zone to 0.002 in thefar offshore. In this more realistic shelf, wave streaming is still

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FIG. 9. Sensitivity to bottom roughness: Lagrangian streamfunction with bottom roughness height zo 5 (a) 1024
and (b) 1022 m. The arrows indicate directions of Lagrangian currents. Temperature with bottom roughness heightzo 5 (c) 1024 and (d) 1022 m. Alongshore-averaged (e) bottom cross-shelf Eulerian velocity and (f) surface crossshelf temperature gradient. A temporal average is made over the second day. The dashed lines in (a) and (c) and theasterisks in (e) and (f) mark the surf-zone edge. In this rough-bed case (zo 5 1022 m), wave breaking is trivial andthus no surf zone is identified.

able to drive an inner-shelf overturning circulation (Fig. 10g)
and an associated front (Fig. 10h), showing the robustness ofthese wave-streaming-induced phenomena.

4) STRATIFICATION
For simplicity, we now use a linear and depth-uniform
stratification, rather than the depth-dependent one shown in
Fig. 2b. Three kinds of initial stratification are used, namely,
N2 5 0.25 3 1024 s22, 1 3 1024 s22, and 4 3 1024 s22. The
wave-streaming-induced overturning circulation (Figs. 11a,b)
and front (Figs. 11c,d) take place in all cases but are different insize, strength, and location.
As stratification increases, the inner-shelf overturning circulation is narrowed, pulling the front onshore (Fig. 11f). Thisnarrowing of overturning circulation is not due to wave streaming, which stays nearly the same with different stratification(Fig. 11e). Instead, the narrowing is linked to the downwelling(Fig. 11b, green arrow) beneath the front. The front is intensified
with increasing stratification (Fig. 11f), and so is the downwelling.
The intensified downwelling makes the inner-shelf surface watersink fast and thus shortens the offshore displacement completed
by the surface water at the time of sinking to the bottom. Thisnarrows the inner-shelf overturning circulation.

b. No wave streaming
To highlight wave-streaming effects, we design a comparison experiment in which wave streaming is deactivatedby turning off wave bottom dissipation. Without wave
streaming, the inner shelf no longer has the bottom onshore
current (Fig. 12a) and overturning circulation (Fig. 12b). In
addition, because wave bottom dissipation is turned off,wave energy is dissipated only by depth-induced breaking,

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FIG. 10. Sensitivity to shelf slopes: Lagrangian streamfunction with shelf slope s 5 (a) 0.0025 and (b) 0.01.
The arrows indicate directions of Lagrangian currents. Temperature with shelf slope s 5 (c) 0.0025 and(d) 0.01. Note that the x-axis range differs between (a) and (b) and between (c) and (d). Alongshore-averaged(e) bottom cross-shelf Eulerian velocity and (f) surface cross-shelf temperature gradient. (g) Lagrangianstreamfunction and (h) temperature, with a more realistic shelf slope that decreases from 0.02 in the surf zone
to 0.002 in the far offshore. Streamline jitters in (b) and (g) are due to computational errors. A temporal
average is made over the second day. The dashed lines in (a)–(d), (g), and (h) and the asterisks in (e) and(f) mark the surf-zone edge.

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FIG. 11. Sensitivity to stratification: Lagrangian streamfunction with initial stratification N2 5 (a) 0.25 3 1024 and(b) 4 3 1024 s22. The arrows indicate directions of Lagrangian currents. Temperature with initial stratification N2 5(c) 0.25 3 1024 and (d) 4 3 1024 s22. Note that the scale of the temperature color bar differs between (c) and (d).
Alongshore-averaged (e) bottom cross-shelf Eulerian velocity and (f) surface cross-shelf temperature gradient. A
temporal average is made over the second day. The dashed lines in (a)–(d) and the asterisks in (e) and (f) mark thesurf-zone edge.

resulting in a stronger surf-zone circulation than that withwave streaming (Fig. 3b).
Without wave streaming, the inner-shelf vertical eddy viscosity is greatly reduced (Fig. 12c), smaller by a factor of approximately O(10) than that with wave streaming (Fig. 3c).
Nonetheless, wave breaking still increases the surf-zone eddy
viscosity, causing differential mixing between the inside andoutside of the surf zone. This gives rise to a front around thesurf-zone edge (Fig. 12d; x ’ 1 km), called ‘‘wave-breaking
front.’’ Its strength is only about 1/7 of that of the farther offshore, wave-streaming front (Fig. 3d; x ’ 2 km).
Kumar and Feddersen (2017) also report a wave-breaking
front caused by alongshore-variable wave breaking, and thefront lies between two Lagrangian overturning circulations
that are located in the surf zone and inner shelf. Nevertheless,although they include wave bottom dissipation, they ignore thewave streaming, missing its induced overturning circulationand front. In the nearshore, however, surfaces waves are usually subject to bottom drag, inducing wave streaming. Hence,
the wave-streaming front is expected, and because of its larger
strength, it is likely to dominate over the wave-breaking front.

c. Tide included
Tides are common in the nearshore and they can shift positions of the surf zone and inner shelf onshore/offshore bychanging the sea level height (Masselink and Short 1993).
Also, tidal currents enhance vertical mixing, leading to a tidal
mixing front (Simpson and Hunter 1974). Then, how do tides
modify the wave-streaming-induced overturning circulationand front? Here we provide a first glimpse of this question.
We adopt analytic, semidiurnal barotropic tides at the offshore boundary. The tidal range is 4 m; to avoid dry cells at

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FIG. 12. Without wave streaming (cf. Fig. 3 with wave streaming): (a) cross-shelf Eulerian velocity,(b) Lagrangian streamfunction (the arrows indicate directions of Lagrangian currents), (c) vertical eddy viscosity,and (d) temperature. Note that the scale of the temperature color bar differs from that in Fig. 3. A temporal average
is made over the second day. The dashed lines mark the surf-zone edge.

low tide, the shoreline is shifted offshore by 0.4 km. The tidal
current is nearly equal to the wave streaming in magnitude,
making the case most interesting. If the tidal current is too
weak or too strong relative to the wave streaming, the strong
one dominates and rules the inner-shelf circulation.
With tides the wave streaming drives a tidally averaged,
inner-shelf Lagrangian overturning circulation (Fig. 13a). This
overturning circulation is weakened by tides in comparison
with that without tides (Fig. 3b), however, because the wave
streaming is reduced by tides (Fig. 13d).
Tidal current and wave streaming together enhance thevertical eddy viscosity (Fig. 13b), making the inner-shelf vertical viscosity larger than that without tides (Fig. 3c). The larger
viscosity causes more mixing, making the inner-shelf stratification weaker than that without tides. With weaker stratification, as explained in section 4, the inner-shelf overturning
circulation becomes wider.
The enhanced mixing in the inner shelf also results in a front(Fig. 13c; x ’ 2.5 km) in contrast to the offshore weak mixing.
The front arises from the mutual action of tidal current andwave streaming. Relative to the front without tides (Fig. 3d),
this front is weaker and wider (Fig. 13e) and is pushed farther
offshore by the wider inner-shelf overturning circulation. Still,
it is unstable and generates shelf eddies.
In addition to the above tidally averaged results, there arealso interesting results at instant moments due to the changeof tidal current and sea level in each tidal cycle. For example,
when the tidal current reaches its maximum, the wavestreaming overturning circulation disappears at that moment. Nevertheless, the front always exists because it can be

maintained by either tidal current alone or wave streaming
alone. Furthermore, the rising sea level during flood tides
moves the overturning circulation and front onshore by several hundreds of meters, and, conversely, the falling sea level
during ebb tides moves the overturning circulation and front
offshore by several hundreds of meters.

d. Wind included
Winds are a common driving force to ocean circulation.
Then, how do winds modify the wave-streaming-induced
overturning circulation and front? Here we give a first glanceat this question. We add a constant wind stress of 0.03 N m22 in
the cross-shelf direction, which drives a current comparable to
the wave streaming in magnitude.
Because of the wind, a surface boundary layer forms, about
10 m deep, and extends from the inner shelf till the offshore
boundary. In the inner shelf, it overlaps with the bottom
boundary layer created by wave streaming. In both boundary
layers, the vertical eddy viscosity is enhanced and has
similar values.
An offshore-directed wind drives an overturning circulation
across the surface boundary layer (Fig. 14a), next to the surfzone overturning circulation generated by wave breaking. The
water inside the wind-driven overturning circulation is well
mixed due to enhanced vertical viscosity. This makes thetemperature nearly uniform within the surface boundary layer(Fig. 14b).
Wave streaming again induces an overturning circulationin the inner shelf, pushing the wind-driven overturning circulation offshore (Fig. 14c). Also, wave streaming pumps up

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FIG. 13. With tides: (a) Lagrangian streamfunction (the arrows indicate directions of Lagrangian currents),(b) vertical eddy viscosity, (c) temperature, and alongshore-averaged (d) bottom cross-shelf Eulerian velocity and(e) surface cross-shelf temperature gradient. A temporal (tidal) average is made over the second day. The dashedlines in (a)–(c) mark the surf-zone edge.

deep and cold water into its induced overturning circulation,
creating a front between the water inside and outside theoverturning circulation (Fig. 14d; x ’ 2 km). The frontal
strength is similar to that without winds, and the front is also
unstable and generates shelf eddies. Nonetheless, the shelfeddies are weakened by Langmuir circulation3 that appears
because of the wind.
By contrast, an onshore-directed wind drives an overturning
circulation in opposite direction to the wave-streaming overturning circulation (Fig. 14e). This considerably suppresses thewave-streaming overturning circulation and its capability ofbringing up deep and cold water. Thus, the wave-streaming

3
Our hydrostatic, coupled wave–circulation model can partly
resolve large-scale Langmuir circulation that is O(1) km long,O(100) m wide, and O(10) m deep. The small aspect ratio (;0.1)
justifies the hydrostatic approximation.

front is reduced dramatically (Fig. 14f; x ’ 2 km), and as aresult, the frontal instability cannot develop.

5. Conclusions
We demonstrate a nearshore front induced by wave
streaming. Horizontally the wave-streaming front is parallelto the shore, and vertically it extends from the surface to the bottom. Wave streaming plays two roles in the frontogenesis. First,
wave streaming drives an inner-shelf Lagrangian overturning circulation and pumps up deep and cold water into the overturning
circulation. The water inside the overturning circulation is rapidly
mixed and cooled because of the wave-streaming-enhanced viscosity. However, the offshore water outside the overturning circulation remains stratified and warmer. Hence, a front develops
between the water inside and outside the overturning circulation.
Second, to balance the onshore transport by bottom wave streaming, the inner-shelf offshore current is intensified near the surface,

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FIG. 14. With winds: (left) Lagrangian streaming function and (right) temperature for (a),(b) offshore windwithout wave streaming; (c),(d) offshore wind with wave streaming; and (e),(f) onshore wind with wave streaming.
The gray arrows denote wind directions, and other arrows in (a), (c), and (e) indicate directions of Lagrangian
currents. A temporal average is made over the second day. The dashed lines mark the surf-zone edge.

sharpening the surface front. The schematic Fig. 15 outlines wavestreaming effects on the nearshore ocean circulation.
The wave-streaming front destabilizes at the surface andgenerates submesoscale shelf eddies spreading offshore. The
frontal instability is accelerated by surf eddies that are able toreach and perturb the front. Using shore-released passive
tracers, we show that shelf eddies cause the offshore transport
across the front. Particularly, after crossing the front into theoffshore, tracers are concentrated below the surface.
Sensitivities of the wave-streaming front are examined withrespect to the wave energy, bottom roughness, shelf slope, andstratification. As the wave energy or bottom roughness increases, the wave-streaming overturning circulation expands
offshore, pushing the front offshore. As the shelf slope or
stratification increases, the overturning circulation narrows
onshore, pulling the front onshore.
In the presence of tides, wave streaming drives a tidally
averaged, inner-shelf Lagrangian overturning circulation andan associated front. In fact, the front is generated by the mutual
action of wave streaming and tides. In comparison with thecase without tides, both the overturning circulation and thefront are wider and weaker, and they are shifted back and forthin the cross-shelf direction due to the sea level change in eachtidal cycle.
With offshore-directed winds, wave streaming can still induce an inner-shelf overturning circulation and an associatedfront. With onshore-directed winds, however, the overturning
circulation and front are suppressed remarkably by an onshorewind-driven overturning circulation that is opposite to thewave-streaming overturning circulation.
The wave-streaming-induced overturning circulation andfront play an important role in regulating the cross-shelftransport among the surf zone, inner shelf and the farther
offshore, especially when tides and winds are negligible or
weak relative to the wave streaming. The wave-streaming
overturning circulation connects the inner shelf with the surf

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APPENDIX
Parameterizations of Nonconservative Wave Forces
The wave breaking dissipation «b is determined by the parameterization (Church and Thornton 1993),«b 5

FIG. 15. A schematic illustration of the wave-streaming-induced
Lagrangian overturning circulation and front in the inner shelf, as
well as offshore-spreading shelf eddies generated by the frontal
instability.

zone and promotes material exchange between the two regions(Wang et al. 2020). For example, wave streaming can transport
near-bottom fish larvae from the inner shelf to the surf zone
(Shanks et al. 2015; Fujimura et al. 2018). On the other hand,the overturning circulation can bring surf-zone fish larvae tothe inner shelf. Therefore, some marine species may utilize theoverturning circulation to help them migrate between the surfzone and inner shelf.
The wave-streaming front acts as a barrier that limits the
cross-shelf transport between the inner shelf and the farther
offshore, though frontal instabilities and shelf eddies can breakthe frontal barrier. Once materials such as plankton and nutrients cross the front into the offshore, they are likely toconcentrate below the surface. That is, in the offshore next to
the front, the subsurface layer may contain a higher concentration of some plankton and may show a higher biologicalproduction than the surface.
In brief, wave streaming and its induced front and overturning circulation modify the nearshore ocean circulation,
regulate the cross-shelf transport, and further affect theecosystem’s health. Hence, we call for more attention to thewave streaming, in both numerical simulations and fieldobservations.
Acknowledgments. Authors Wang and McWilliams are
grateful for the support of the Office of Naval Research(N00014-15-1-2645) and the National Science Foundation(OCE-1355970). Author Uchiyama appreciates the support ofJapan Society for the Promotion of Science (JSPS) Grants-inAid for Scientific Research (18H03798) at Kobe University.
The numerical experiments are done at the Extreme Scienceand Engineering Discovery Environment (XSEDE). The authors also appreciate helpful suggestions from two anonymous
reviewers.
Data availability statement. The dataset on which this paper
is based is too large to be retained or publicly archived withavailable resources. Documentation and methods used tosupport this study are available from the corresponding
author Peng Wang at the University of California, Los
Angeles.

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

(A1)

where r0 is the water density, g is the gravitational acceleration,
D is the water depth, Hrms is the RMS wave height, Bb and g b
are constants, s is the wave intrinsic frequency, and k is thewavenumber. The wave breaker force is then given by
Fb 5 («b /s)k.

(A2)

The wave bottom dissipation « f is determined by the parameterization (Reniers et al. 2004),
1« f 5 pﬃﬃﬃﬃ r0 fw juworb j3 ,
2 p

(A3)

where fw 5 1:39(szo /juworb j)0:52 is the wave friction factor(Soulsby 1997), zo is the bottom roughness height, juworb j 5
sHrms /[2 sinh(kD)] is the near-bottom wave orbital velocity,
Hrms is the RMS wave height, and D is the water depth. The
wave-streaming force is then given by
Ff 5 (« f /s)k .

(A4)

Surface waves enhance bottom drag on currents, and this dragis parameterized as (Feddersen et al. 2000)
2 1/2
 1 / 3 kajuj(1:16)2 1
juwrms ju, (A5)juwrms jD
pﬃﬃﬃ
where uwrms 5 uworb / 2 is the RMS near-bottom wave orbital
velocity and ka 5 0.0125 m is apparent roughness height(Ruessink et al. 2001).
Fd 5 0:015r0

REFERENCES
Akan, Ç., J. C. McWilliams, and Y. Uchiyama, 2020: Topographic
and coastline influences on surf eddies. Ocean Model., 147,
101565, https://doi.org/10.1016/j.ocemod.2019.101565.
Allen, J., P. Newberger, and J. Federiuk, 1995: Upwelling circulationon the Oregon continental shelf. Part I: Response to idealized
forcing. J. Phys. Oceanogr., 25, 1843–1866, https://doi.org/
10.1175/1520-0485(1995)025,1843:UCOTOC.2.0.CO;2.
Austin, J. A., and S. J. Lentz, 2002: The inner shelf response to winddriven upwelling and downwelling. J. Phys. Oceanogr., 32,
2171–2193, https://doi.org/10.1175/1520-0485(2002)032,2171:
TISRTW.2.0.CO;2.
Brink, K. H., 2016: Cross-shelf exchange. Annu. Rev. Mar. Sci., 8, 59–
78, https://doi.org/10.1146/annurev-marine-010814-015717.
Brown, J. A., J. H. MacMahan, A. J. Reniers, and E. B. Thornton,
2015: Field observations of surf zone–inner shelf exchange
on a rip-channeled beach. J. Phys. Oceanogr., 45, 2339–2355,
https://doi.org/10.1175/JPO-D-14-0118.1.

Brought to you by KOBE UNIVERSITY | Unauthenticated | Downloaded 08/02/22 01:37 AM UTC

JUNE 2021

WANG ET AL.

Chapman, D. C., and S. J. Lentz, 1994: Trapping of a coastal density
front by the bottom boundary layer. J. Phys. Oceanogr., 24,
1464–1479, https://doi.org/10.1175/1520-0485(1994)024,1464:
TOACDF.2.0.CO;2.
Chen, C., R. J. Schlitz, R. G. Lough, K. W. Smith, R. Beardsley, andJ. P. Manning, 2003: Wind-induced, cross-frontal exchange on
Georges Bank: A mechanism for early summer on-bank biological particle transport. J. Geophys. Res., 108, 8011, https://doi.org/10.1029/2002JC001358.
Church, J. C., and E. B. Thornton, 1993: Effects of breaking wave
induced turbulence within a longshore current model. Coastal
Eng., 20, 1–28, https://doi.org/10.1016/0378-3839(93)90053-B.
Cushman-Roisin, B., and J.-M. Beckers, 2011: Introduction toGeophysical Fluid Dynamics: Physical and Numerical Aspects.
Academic Press, 875 pp.
Feddersen, F., R. Guza, S. Elgar, and T. Herbers, 2000: Velocity
moments in alongshore bottom stress parameterizations.
J. Geophys. Res., 105, 8673–8686, https://doi.org/10.1029/
2000JC900022.
Fujimura, A. G., A. J. Reniers, C. B. Paris, A. L. Shanks, J. H.
MacMahan, and S. G. Morgan, 2018: Mechanisms of crossshore transport and spatial variability of phytoplankton on arip-channeled beach. Front. Mar. Sci., 5, 183, https://doi.org/
10.3389/fmars.2018.00183.
Gawarkiewicz, G., K. H. Brink, F. Bahr, R. C. Beardsley,
M. Caruso, J. F. Lynch, and C.-S. Chiu, 2004: A largeamplitude meander of the shelfbreak front during summer
south of New England: Observations from the shelfbreakprimer experiment. J. Geophys. Res., 109, C03006, https://doi.org/10.1029/2002JC001468.
Hoskins, B. J., 1982: The mathematical theory of frontogenesis.
Annu. Rev. Fluid Mech., 14, 131–151, https://doi.org/10.1146/
annurev.fl.14.010182.001023.
Houghton, R., F. Aikman III, and H. Ou, 1988: Shelf-slope frontal
structure and cross-shelf exchange at the New England shelfbreak. Cont. Shelf Res., 8, 687–710, https://doi.org/10.1016/
0278-4343(88)90072-6.
Kranenburg, W. M., J. S. Ribberink, R. E. Uittenbogaard, andS. J. Hulscher, 2012: Net currents in the wave bottom
boundary layer: On waveshape streaming and progressive
wave streaming. J. Geophys. Res., 117, F03005, https://doi.org/10.1029/2011JF002070.
Kumar, N., and F. Feddersen, 2017: The effect of Stokes drift andtransient rip currents on the inner shelf. Part II: With stratification. J. Phys. Oceanogr., 47, 243–260, https://doi.org/
10.1175/JPO-D-16-0077.1.
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic
vertical mixing: A review and a model with a nonlocal
boundary layer parameterization. Rev. Geophys., 32, 363–403,
https://doi.org/10.1029/94RG01872.
Lentz, S. J., and M. R. Fewings, 2012: The wind-and wave-driven
inner-shelf circulation. Annu. Rev. Mar. Sci., 4, 317–343,
https://doi.org/10.1146/annurev-marine-120709-142745.
——, 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, 2341–2357, https://doi.org/
10.1175/2008JPO3986.1.
Longuet-Higgins, M. S., 1953: Mass transport in water waves.
Philos. Trans. Roy. Soc. London, 245A, 535–581, https://doi.org/10.1098/rsta.1953.0006.
MacMahan, J. H., E. B. Thornton, and A. J. Reniers, 2006: Ripcurrent review. Coastal Eng., 53, 191–208, https://doi.org/
10.1016/j.coastaleng.2005.10.009.

1983

Marchesiello, P., R. Benshila, R. Almar, Y. Uchiyama, J. C.
McWilliams, and A. Shchepetkin, 2015: On tridimensional rip
current modeling. Ocean Modell., 96, 36–48, https://doi.org/
10.1016/j.ocemod.2015.07.003.
Masselink, G., and A. D. Short, 1993: The effect of tide range on
beach morphodynamics and morphology: A conceptual beachmodel. J. Coastal Res., 9, 785–800.
McWilliams, J. C., 2018: Surface wave effects on submesoscale
fronts and filaments. J. Fluid Mech., 843, 479–517, https://doi.org/10.1017/jfm.2018.158.
——, 2021: Oceanic frontogenesis. Annu. Rev. Mar. Sci., 13, 227–
253, https://doi.org/10.1146/annurev-marine-032320-120725.
——, J. M. Restrepo, and E. M. Lane, 2004: An asymptotic
theory for the interaction of waves and currents in coastal
waters. J. Fluid Mech., 511, 135–178, https://doi.org/
10.1017/S0022112004009358.
Nittrouer, C. A., and L. D. Wright, 1994: Transport of particles
across continental shelves. Rev. Geophys., 32, 85–113, https://doi.org/10.1029/93RG02603.
Pineda, J., 1994: Internal tidal bores in the nearshore: Warm-water
fronts, seaward gravity currents and the onshore transport ofneustonic larvae. J. Mar. Res., 52, 427–458, https://doi.org/
10.1357/0022240943077046.
Reniers, A., E. Thornton, T. Stanton, and J. Roelvink, 2004:
Vertical flow structure during sandy duck: Observations andmodeling. Coastal Eng., 51, 237–260, https://doi.org/10.1016/
j.coastaleng.2004.02.001.
Reniers, A. J., J. MacMahan, E. Thornton, T. Stanton, M. Henriquez,
J. Brown, J. Brown, and E. Gallagher, 2009: Surf zone surface
retention on a rip-channeled beach. J. Geophys. Res., 114,
C10010, https://doi.org/10.1029/2008JC005153.
Romero, L., D. Hypolite, and J. C. McWilliams, 2020: Submesoscalecurrent effects on surface waves. Ocean Model., 153, 101662,
https://doi.org/10.1016/j.ocemod.2020.101662.
Ruessink, B., J. Miles, F. Feddersen, R. Guza, and S. Elgar,
2001: Modeling the alongshore current on barred beaches.
J. Geophys. Res., 106, 22 451–22 463, https://doi.org/10.1029/
2000JC000766.
Shanks, A. L., 1983: Surface slicks associated with tidally forced
internal waves may transport pelagic larvae of benthic invertebrates and fishes shoreward. Mar. Ecol. Prog. Ser., 13, 311–
315, https://doi.org/10.3354/meps013311.
——, J. MacMahan, S. G. Morgan, A. J. M. Reniers, M. Jarvis,
J. Brown, A. Fujimura, and C. Griesemer, 2015: Transport oflarvae and detritus across the surf zone of a steep reflective
pocket beach. Mar. Ecol. Prog. Ser., 528, 71–86, https://doi.org/10.3354/meps11223.
Simpson, J., and J. Hunter, 1974: Fronts in the Irish sea. Nature,
250, 404–406, https://doi.org/10.1038/250404a0.
Soulsby, R., 1997: Dynamics of Marine Sands: A Manual for
Practical Applications. Thomas Telford, 249 pp.
Sullivan, P. P., and J. C. McWilliams, 2019: Langmuir turbulence andfilament frontogenesis in the oceanic surface boundary layer.
J. Fluid Mech., 879, 512–553, https://doi.org/10.1017/jfm.2019.655.
Suzuki, N., and B. Fox-Kemper, 2016: Understanding Stokes forcesin the wave-averaged equations. J. Geophys. Res. Oceans, 121,
3579–3596, https://doi.org/10.1002/2015JC011566.
——, ——, P. E. Hamlington, and L. P. Van Roekel, 2016: Surface
waves affect frontogenesis. J. Geophys. Res. Oceans, 121,
3597–3624, https://doi.org/10.1002/2015JC011563.
Trowbridge, J., and O. S. Madsen, 1984: Turbulent wave boundary
layers: 2. Second-order theory and mass transport. J. Geophys.
Res., 89, 7999–8007, https://doi.org/10.1029/JC089iC05p07999.

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1984

JOURNAL OF PHYSICAL OCEANOGRAPHY

Uchiyama, Y., J. C. McWilliams, and A. F. Shchepetkin, 2010:
Wave–current interaction in an oceanic circulation modelwith a vortex-force formalism: Application to the surfzone. Ocean Modell., 34, 16–35, https://doi.org/10.1016/
j.ocemod.2010.04.002.
——, ——, and C. Akan, 2017: Three-dimensional transient rip
currents: Bathymetric excitation of low-frequency intrinsic
variability. J. Geophys. Res. Oceans, 122, 5826–5849, https://doi.org/10.1002/2017JC013005.
Wang, P., and T. M. Özgökmen, 2018: Langmuir circulation with
explicit surface waves from moving-mesh modeling. Geophys.
Res. Lett., 45, 216–226, https://doi.org/10.1002/2017GL076009.

VOLUME 51——, J. McWilliams, Y. Uchiyama, M. Chekroun, and D. Yi, 2020:
Effects of wave streaming and wave variations on nearshore
wave-driven circulation. J. Phys. Oceanogr., 50, 3025–3041,
https://doi.org/10.1175/JPO-D-19-0304.1.
Weber, J. E. H., G. Broström, and Ø. Saetra, 2006: Eulerian versus
Lagrangian approaches to the wave-induced transport in theupper ocean. J. Phys. Oceanogr., 36, 2106–2118, https://doi.org/10.1175/JPO2951.1.
Xu, Z., and A. Bowen, 1994: Wave-and wind-driven flow inwater of finite depth. J. Phys. Oceanogr., 24, 1850–1866,
https://doi.org/10.1175/1520-0485(1994)024,1850:WAWDFI.
2.0.CO;2.

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