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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, C06021, doi:10.1029/2008JC005135, 2009

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ArticleWave-current interaction in nearshore shear
instability analyzed with a vortex force formalism
Yusuke Uchiyama,1 James C. McWilliams,1 and Juan M. Restrepo2
Received 25 September 2008; revised 9 March 2009; accepted 26 March 2009; published 24 June 2009.

[1] We examine wave-current interactions in littoral current shear instabilities above asingle-barred beach using a fully coupled wave and barotropic current model based on themultiscale asymptotic theory derived by McWilliams et al. (2004). This Eulerian waveaveraged model captures wave effects on currents (WEC) in a manner that leads to simple
interpretations. The dynamically conservative WEC are the vortex force and materialtransport by Stokes drift and the sea level adjustment by wave set-down and setup. In thesetting considered here there are also important current effects on the waves (CEW):
induction of a Doppler shift by currents and surface elevation variation in the wave
dispersion relation. Nonconservative effects, due to wave breaking and bottom drag, also
play a prominent role in generating and equilibrating the mean alongshore current,
consistent with prior studies. High bottom drag stabilizes the currents, while a dragreduction below a critical threshold value leads to shear instability with nearly periodic,
alongshore-fluctuating eddies. An even smaller bottom drag yields irregular eddy motions
with intermittent offshore eruption of vortex pairs from the meandering alongshore
current. Several alternative parameterizations of the bottom drag are contrasted here.
Including CEW in the model leads to a delay in the onset of the instability, a suppressionof fluctuations in cross-shore velocity and lateral Reynolds stress, and an enhancement of
the mean alongshore velocity. The WEC increase the Reynolds stress in the offshore
region, and the conservative vortex force and mean advection are comparable inmagnitude to the breaking acceleration and bottom drag. Conversely, the CEW reduce theReynolds stress and attenuate the breaking acceleration through refractive focusing by
current shear. Overall, the WEC enhance the instantaneous cross-shore momentum flux toinduce more energetic eddy motions and retard the mean alongshore current, while the
CEW stabilize the fluctuations and help maintain a strong mean current.
Citation: Uchiyama, Y., J. C. McWilliams, and J. M. Restrepo (2009), Wave-current interaction in nearshore shear instability
analyzed with a vortex force formalism, J. Geophys. Res., 114, C06021, doi:10.1029/2008JC005135.

1. Introduction[2] We investigate littoral currents induced by wave
breaking using a barotropic model with a barred beach
topography. Our focus is the horizontal shear instability of
the alongshore current, often called a ‘‘shear wave’’ for its
propagating behavior. Shear waves were first observed by
Oltman-Shay et al. [1989] during the SUPERDUCK field
experiment, and they have been modeled as an inflection
point, normal mode instability of a specified alongshore
current profile V(x) [cf. Bowen and Holman, 1989; Dodd etal., 1992, 1994, 2000; Reniers et al., 1997]. Here, x is abeach-normal coordinate. The instability is observed to bestronger on barred beaches than on planar beaches. A
1Institute of Geophysics and Planetary Physics, University ofCalifornia, Los Angeles, California, USA.
2
Departments of Mathematics and Physics, University of Arizona,
Tucson, Arizona, USA.

numerical study with time-dependent, rigid lid, nonlinear
shallow water equations and an initial V(x) shows that fully
developed shear waves alter the mean alongshore current
profile significantly and eventually limit the growth of theinstabilities [Allen et al., 1996; Slinn et al., 1998]. ÖzkanHaller and Kirby [1999] and Noyes et al. [2005] obtainbetter agreement with measurements of the growth rate andthe propagation speed of the resulting shear instability
waves with a breaking-generated current over a barred
topography and a refracting wave field. They employ
nonlinear shallow water equations with phase-averaged
surface gravity wave effects on currents (WEC), using
forcing terms represented by the divergence of the radiationstress tensor [Longuet-Higgins and Stewart, 1962; LonguetHiggins, 1970a, 1970b]. Özkan-Haller and Li [2003] (hereinafter referred to as OL03) couple a similar current model
to a WKB wave number refraction, action conservation
wave model with parameterized wave breaking [Thornton
and Guza, 1983]. Effects of the currents on the waves(CEW) by Doppler shifting in the dispersion relation are

Copyright 2009 by the American Geophysical Union.
0148-0227/09/2008JC005135$09.00

C06021

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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

included. Kennedy and Zhang [2008] perform a linear
stability analysis including both radiation stress and Doppler shifting. Phase-resolving models have also been utilized
in solving the flow, without any distinction between waves
and currents, to investigate vorticity transport in nearshore
circulations [e.g., Bühler and Jacobson, 2001; Chen et al.,
2003; Terrile et al., 2008].
[3] OL03 show that when the bottom drag has intermediate strength, leading to a marginally unstable shear flow,
the interaction of waves and currents results in a reductionin the offshore extent of the currents; a delay in the onset of
the instability; and an increase of the alongshore shear wave
propagation speed. Most of these effects are minimal in amore turbulent regime with weaker bottom drag, but theoffshore extent of the eddy mixing remains suppressed tosome degree. OL03 attribute these effects to refraction andenergy gain from the incident waves around offshoredirected flows associated with the shear instabilities.
Newberger and Allen [2007] extend OL03 using the samerefraction wave model and the three-dimensional, primitive
equation current model with depth-averaged WEC terms.
They qualitatively demonstrate that CEW tends to suppress
shear instability in a relatively high bottom friction regime.
[4] The primary purpose of this paper is to analyze how
the littoral current and its shear instability are determined by
the various WEC and CEW mechanisms. We will employ
an Eulerian, multiscale wave-current framework, based on
the asymptotic equations derived by McWilliams et al.
[2004] (hereinafter referred to as MRL04). This has a vortex
force formalism that cleanly separates conservative andnonconservative WEC mechanisms (unlike the radiationstress formalism). Lane et al. [2007] analyze how theradiation stress formalism [Longuet-Higgins and Stewart,
1964; Longuet-Higgins, 1970a, 1970b; Hasselmann, 1971]
relates to the vortex force formalism; also see Smith [2006].
Section 2 describes the wave-current interaction model that
includes nonconservative effects of wave breaking andbottom drag; section 3 presents the essential mean wave
and current structures. Since previous studies have shownthat bottom drag plays a critical role in nearshore shear
instability, a secondary purpose of this paper is to determine
how the shear wave evolution and pattern differ withdifferent parameterization formulas for the bottom drag(section 4). Section 5 analyzes the various wave-current
interactions in the nonlinear regime.

2. Wave-Current Interaction Model

negligible effect on shear instabilities [Özkan-Haller andKirby, 1999]. The cross-shore and alongshore lengths of thedomain are set to 512 m (about four times the width of thesurf zone) and 768 m (about six to eight wavelengths ofshear waves in the marginally unstable regime). The
computational grid is uniform with a spacing of 2 m.
[6] In this region the waves have a typical period of tensof seconds. The spectra are somewhat narrow-banded, andtheir dynamics is weakly nonlinear outside the breaking
zone. The temporal scales of interest for the alongshore
currents and shear waves are hundreds of seconds or more,
longer than that of the primary wave oscillations. This
motivates a wave-averaged theoretical framework. We neglect the Coriolis force as dynamically relevant only on
longer time scales.
[7] The currents are assumed to be barotropic, and they
impart a Doppler shift to the wave frequency. The waveaveraged water column depth is time-varying because of sea
level changes from the dynamic pressure of the currents(including the so-called ‘‘wave setup’’ caused by nonconservative wave effect through wave breaking) as well as thequasi-static response (see MRL04) that we may alternatively call ‘‘wave set-down’’ where waves are weaker. The
Doppler shift and depth variation are the CEW elements.
The dynamically conservative elements in the WEC are thisquasi-static response; the vortex force that combines thetotal flow vorticity and the Stokes drift velocity; and thematerial advection by the Stokes drift. The nonconservative
element in the WEC is parameterized wave breaking. The
computational time step size is around Dt = 0.1 s, consistent
with the CFL criterion for the current and barotropic gravity
wave speeds with our fine grid spacing.
2.1. Currents[8] MRL04 derive a multiscale asymptotic model for thephase-averaged, conservative dynamical effects of surface
gravity waves on currents and infragravity waves withslowly varying space and time scales using a ‘‘vortex force’’formalism [e.g., Craik and Leibovich, 1976; McWilliams etal., 1997; McWilliams and Restrepo, 1999]. The dynamical
equation for infragravity waves driven by primary shortwaves derived in MRL04 is implemented in a numericalcoastal ocean model to investigate generation and propagation of infragravity wave in the deep ocean in the context ofseismic hum excitation [Uchiyama and McWilliams, 2008].
To extend this, assuming barotropic currents, the equations
for the horizontal current and dynamic sea level height z
are[5] We choose a west coast configuration with (x, y)
positive to the (east, north), and x = 0 as the shoreline; x= x > 0 is the offshore distance. To facilitate comparisonswith previous studies, we consider a simple, alongshore
uniform topography, inspired by the field campaigns atDuck, NC. The offshore resting ocean depth profile ishihðx Þ ¼ 0:015x þ 2:08 tanh½0:017x  2:5 exp 0:001ðx  100Þ2
ð1Þ

(Figure 1a), with an added constraint of h  0.1 m. The
swash zone is excluded since runup is reported to have a

C06021@z@^z
þ r Hu ¼   r USt ;
@t
@t

ð2Þ

@u
þ u ru þ grz  J ¼ B  D þ Rr2 u;@tð3Þwhere g is gravitational acceleration; u = (u, v) is the depthaveraged Eulerian current velocity; c = @v/@x  @u/@y is
its vorticity; H = h + z c is the local water column depth; z c =z +^z is the composite sea level elevation; ^z is the quasistatic response due to the conservative WEC specified

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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 1. (a) The single-barred bottom topography h(x) from (1) with an offshore bar crest at x = 100 m.
(b) Breaking wave dissipation function b(x) from (16). (c) RMS wave height Hrms(x) and wave angle
q(x) and (d) Stokes velocity USt/H from (11). The offshore wave parameters are Hrms = 1.0 m, peak period
Tp = 10 s, and peak direction qm = 10°.

in the subsequent subsection; USt is Stokes transport (i.e.,
vertically integrated Stokes drift velocity uSt); and
J ¼ Jx ; Jy  ^z

1 St
U c
H

ð5Þ

where b: wave energy dissipation rate due to depth-induced
breaking, k: wave number vector, and s: the intrinsic wave
frequency. Lateral diffusion with eddy viscosity R = 0.1 m2s1 is used throughout the present study to stabilize thenumerical solutions in the eddying regime. The bottom dragforce is
 tb
D ¼ Dx ; Dy ¼
;
rH

ð6Þ

with r the fluid density and t b the bottom stress (chosen as
one of several alternatives; section 4).
[9] The bottom drag D plays a crucial role in theappearance and maintenance of alongshore currents. We
consider three different parameterization models for thebottom stress t b. The first model is a linear drag law:t b ¼ rmu;"

ð4Þis the vortex force. t is time; x = (x, y) are horizontal
coordinates; and the vertical coordinate z is aligned withgravity. Note that the relation of the present formalism to theconventionally used radiation stress formalism is given by
Smith [2006]. The right side of the momentum equation
contains the parameterized nonconservative processes.
Acceleration due to wave breaking isb k
B ¼ Bx ; By ¼:
rHs

with a constant linear drag coefficient m [m s1]. The
second model is a combined wave and current dragproposed by Soulsby [1995]:

ð7Þ

jt w jt b ¼ t c 1:0 þ 1:2
jt w j þ jt c j

tc ¼ r3:2 #;

2k1
juju; jt w j ¼ rfw juwb j2 ;lnðzm =zo Þ
2

ð8Þ

ð9Þ

where t c and t w are bottom stresses due to current andwaves [Pa]; k is the von Kármán constant; zm is a reference
depth above the bed (in the barotropic model we simply
define zm = H/2) and zo is bed roughness length [m]; fw is
the wave friction factor given by fw = 1.39(szo/juwb j)0.52[Soulsby, 1997]; and juwb j = sA/sinh kH is the wave orbital
velocity at the bottom. The third alternative is anothercombined wave-current, bottom drag model originallyproposed by Feddersen et al. [2000] based on data obtainedat Duck, NC: 1=3 " #1=2kajuj 2
2
ð1:16Þ þt b ¼ 0:015r
juwrms ju;Hjuwrms j

ð10Þ

pﬃﬃﬃ
where uwrms = uwb / 2 is the RMS wave orbital velocity
and ka = 0.0125 is the apparent roughness (m) [Ruessink etal., 2001]. Notice that Feddersen et al. [2000] only
parameterized the quadratic velocity moment for thealongshore momentum balance, while we crudely presume
here that the model is expandable to the two-dimensional
fields.
[10] All the governing equations are transformed tocurvilinear horizontal coordinates and implemented into a

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UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021barotropic version of Regional Ocean Modeling System(ROMS [e.g., Shchepetkin and McWilliams, 2005]) so that
we can make use of its features including capability ofparallel computing and the reflection-free open-boundary
schemes. All fields are y periodic. At the offshore edge,
radiation conditions [Flather, 1976] are applied with anudging toward z = 0, v = 0, and u = USt/H. At thenearshore boundary, there is a zero normal mass flux, u +uSt = 0; no tangential flow, v = 0; and the Neumann condition,dz/dx = 0. uSt  0 near the shoreline because of wave
dissipation by breaking.
2.2. Waves[11] In accordance with MRL04 for a monochromatic
surface wave field, the Stokes transport USt and quasi-static
response ^z are
USt ¼

A2 sk
A2 k
¼ Ak; ^z ¼ :
2k tanh kH
2 sinh 2kH

ð11Þ

k is the magnitude of k(x, t); A(x, t) is the wave amplitude;and A is the wave action. This expression is fornonbreaking, small wave slope waves that may not be truein the surf zone; however, we simply assume that (11) isapproximately applicable to the problem considered here.
Note that the same assumption is also implied for other
WEC models such as the radiation stress model. In thenearshore setting under consideration, the wave field is well
captured by this slowly varying envelope representation andexhibits very weak nonlinearity (OL03 and references
therein). So, A can be related to a field observable RMS
wave height Hrms and k to the spectrum peak wave number.
Notice that ^z is a quasi-static quantity, unrelated to any
currents. In terms of the wave action A = E/s, where E =2
2
1
1
2rgA = 8rgHrms, the ‘‘ray theory’’ evolution equations forthe wave field are


@A
b
þ r Acg ¼  ;
s
@t

ð12Þ

@k
ks
þ cg rk ¼ ðk rÞu rh;@tsinh 2kH

ð13Þ

s2 ¼ gk tanh kH;

w ¼ u k þ s;

ð14Þ

where w is the wave frequency. The wave group velocity is


s
2kH
cg ¼ u þ 2 1 þk:
2ksinh 2kH

ð15Þ

The parameterized, depth-integrated rate of wave energy
loss due to breaking b [Thornton and Guza, 1983] isb ¼

3 pﬃﬃﬃ
B3 s 7prg 4 r 5
H  0;
g H 2p rms
16

ð16Þ

C06021

A and k are specified, while at the nearshore boundary zerogradient, Neumann conditions are used.
2.3. Three Types of WEC/CEW Model Setup[12] To better characterize the dynamical roles played bywaves and currents in the alongshore current shear instability, we exploit the relatively clean separation of wave andcurrent effects in our formulation by defining three types ofmodel calculations:[13] 1. The WEC+CEW model uses full model computations (as in the preceding two subsections).
[14] 2. The WEC-only model uses full current modelcomputations but with no Doppler shift, i.e., u = 0 and
H = h in the wave model (while H = h + z c in the current
model).
[15] 3. The NW model excludes the conservative WEC
and CEW terms while retaining the background wavebreaking effects, namely,@zþ r Hu ¼ 0;
@t

ð17Þ

@u
þ u ru þ grz ¼ BNW  D þ Rr2 u;
@t

ð18Þ

NW
with BNW = (0, BNW= bky/rHs, where b, k, and sy ), Byare determined from steady state wave solutions without the
CEW. This NW model lacks Stokes drift and wave setup/
set-down, and it does not develop a cross-shore anti-Stokes
flow (see section 2.4). Notice that we neglect the crossin the NW model as Allen et al. [1996]
shore forcing BNW
xdid, whereas it could be easily incorporated giving a wave
setup that might make results slightly different.
[16] For large t b all three models yield steady, stable,
alongshore-uniform (i.e., one-dimensional (1-D)) solutions.
One-dimensional solutions also exist for any t b, and they
can be calculated by integrating the models without alongshore variation to steady state. In practice, this is achievedby restricting the number of the alongshore grid points totwo, instead of 384 points for the two-dimensional (2-D)
runs. To initialize the 2-D integrations, the 1-D steadysolutions are utilized for 2-D WEC+CEW and WEC-onlyruns. For 2-D NW runs, we calculate BNW
y (x) and v(x) fromthe 1-D steady WEC-only model. Then small perturbations
are imposed in the u and v fields.

2.4. Mean Continuity and Momentum Balances[17] We now derive mean balance equations for thecurrents, averaging in time and alongshore coordinate(assuming y periodicity). We use a Reynolds decomposition; that is, for any f(x, y, t), f(x, y, t) = h f i + f 0, with h f 0i =0, and the averaging period is typically greater than an hour.
[18] In the presence of wave and current fluctuations, themean continuity balance can be integrated (2) in x from theshoreline to yield

Br = 0.8, and g = 0.4. Thus, the CEW occurs in (12) – (16)when u 6¼ 0 and H 6¼ h. At the offshore boundary, incident

4 of 15h H ihui þ hH 0 u0 i þ U St ¼ 0:

ð19Þ

C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 2. Nearshore steady 1-D (a) currents and (b) sea level with WEC (but not CEW) for the
topography and wave profiles in Figure 1. Also shown in Figure 2b is the quasi-static response ^z(x).
For steady waves and currents, this implies anti-Stokes
flow, hui = huSti. The mean momentum balance in thealongshore direction is
hui



@ hvi
@v0
þ uSt hci þ uSt0 c0
þ u0
@x
@x
DvE
þm
 By ¼ 0:
H

3. Steady Wave-Driven Current
ð20Þ

Here for didactic purposes we have used the linear bottom
drag model and ignored small lateral diffusion. In (20) hui iszero if the Stokes drift is zero. If the flow is steady, thebreaking and bottom drag terms balance:
DvEm¼ By :
H

ð21Þ

This force balance is the classical dynamical conception of alittoral current [Longuet-Higgins, 1970a, 1970b]. Instead, if
the Stokes drift is nonzero and we still assume there are no
current fluctuations, the mean advection cancels the meanvortex force because of the anti-Stokes flow; that is,
hui@x hvi ¼  uSt hci;

ð22Þ

where @ x denotes @/@x. So (21) applies in general for steadyalongshore currents, even though (22) was not originallyenvisioned as part of the classical conception. An example
for the steady state alongshore momentum balance
diagnosed with a 1-D littoral current solution is illustrated
later in Figure 3. For unsteady (unstable) currents andwaves, both the fluctuation advection (i.e., eddy Reynolds
stress divergence) and the fluctuation vortex force may
contribute as shown later. Similarly, the mean y momentum
balance for the NW model (18) is

 D
E
0
hvi
0 @v
 BNW
u
þm
¼ 0:
y
@x
H

ð23Þ

[19] Provided that there is no mean surface elevation(e.g., tide), wave setup z can be obtained from the meancross-shore momentum balance dominated by its pressure
gradient force and the breaking acceleration,
g@x hz i  hBx i:

By integrating (24) and including the conservative quasistatic response h^zi, the composite actual sea level hz ci isapproximately retrieved [e.g., Longuet-Higgins and Stewart,
1962; Bowen et al., 1968].

ð24Þ

3.1. Cross-Shore Profiles[20] To obtain a specific solution, we specify the incident
wave conditions typical of those measured at Duck, NC[Church and Thornton, 1993]; RMS wave height Hrms =1.0 m, peak period Tp = 10 s, and peak direction qm = 10°
away from the shore-normal direction. This generates
alongshore currents in the positive y direction for all cases.
We assume the waves are steady, ignore CEW for now(i.e., WEC-only model), and show cross-shore onedimensional steady state solutions.
[21] With shoaling barred topography (Figure 1a), wave
steepening and refraction occur (Figure 1c) as expected. The
RMS wave height Hrms gently amplify and steepen until
around x = 120 m where breaking begins slightly offshore
from the bar crest in accordance with breaking dissipation b(Figure 1b). The secondary breaking is also depicted near
the shoreline. The Stokes drift is strongly shoreward andmore weakly in the alongshore direction (Figure 1d) because of the small wave angle q (Figure 1c).
[22] The steady 1-D currents and sea level with the linear
bottom drag coefficient (m = 0.006 m s1) are shown in
Figure 2. The alongshore current v(x) has approximately theshape of the breaking acceleration B multiplied by H, asrequired by the balance (21). The cross-shore current u(x)z(x) manifests
is an anti-Stokes flow, uSt(x) (section 2.4). ^a weak wave set-down around the breaking region and amuch stronger onshore wave setup in z as Hrms drops
toward zero (Figure 2) [e.g., Bowen et al., 1968] asapproximated by (24). The result shown here is qualitatively
consistent with the previous works [e.g., Longuet-Higgins,1970a, 1970b; Thornton and Guza, 1986; Raubenheimer etal., 2001]
3.2. Alongshore Momentum Balance and Roles of
WEC and CEW[23] In the 1-D steady solution with the WEC+CEW
model with m = 0.006 m s1, the mean alongshore momentum balance for the same incident wave condition as insection 3.1 (Figure 3) shows that the forces are concentrated
in the two breaking zones, near the bar and near the

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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 3. Steady 1-D solutions for the WEC+CEW model with linear bottom drag near the criticalvalue for instability (m = 0.006 m s1): (a) terms in the y momentum balance (20) (excluding
contributions from fluctuations) and (b) differences between the advection and vortex force and between
the breaking acceleration and bottom drag terms. Notice the change of scale between the plots.
shoreline. In each zone the principal contributions are frombreaking acceleration, mean advection, mean vortex force,and bottom drag, and they are all comparable in magnitude.
Since this is a steady case, the two subbalance relations (21)and (22) hold separately, except for relatively small corrections due to lateral momentum diffusion. Breaking accelerates the mean alongshore current and bottom dragquenches it (note that the plotted terms are negative forces).
The vortex force acts to move the current toward theshoreline, and mean momentum advection by the antiStokes flow acts to push it further offshore.
[24] The 1-D solutions are further illustrated for models
with the WEC and either with or without the CEW(Figure 4). The wave height is rather similar to each other;
however, we see an enhanced wave height and stronger
anti-Stokes offshore flow, u < 0 inshore of the bar, becauseof the wave-steepening Doppler shift from the latter with
the CEW. OL03 did not show a similar comparison andmerely denoted that the CEW has little effect on theseresult. They assumed that the Doppler shift takes place on
Lagrangian current that implies zero cross-shore flow,whereas it occurs on Eulerian current (i.e., u 6¼ 0) in thepresent study. Hence, the alteration in Hrms, uSt and u in
OL03 is considered to be much less than the present result.
The wave setup effect in sea level z c further adds to the
CEW differences in Hrms and u; z c in the WEC-only
solution is very similar to the WEC+CEW solution (notshown), except for an increase of about 1 cm in the runnel(i.e., the trough behind the bar). This contrast between
WEC+CEW and WEC-only cases is essentially the sameas that will be found in the 2-D unstable solution shown
later in section 5. In the 1-D NW model u = 0, of course,and Hrms is the same as in the WEC-only model but is only
relevant as a basis for calculating BNW
y . The alongshore
current v is quite similar in 1-D solutions for the different
models (also not shown).

4. Alternative Bottom Drag Formulations[25] Several parameterizations for the bottom drag havebeen proposed. Here we show how the alongshore shear
instability outcome qualitatively changes depending on themagnitude of the drag coefficient, but it has less dependence
on the particular parameterization form used. We use thefull wave-current model specified in section 2.
[26] We consider a total of eight cases. In all cases theincident wave conditions are the same as those in section 3.
The first six cases illustrate the dependence on the coefficient in the linear drag model (7), with values that range
between m = 0.002– 0.007 m s1. The two other cases are
the wave-enhanced drag models in (8) and (10). All the 2-D
simulations are initiated with the corresponding 1-D, alongshore uniform, steady wave and current fields with small
perturbations imposed in u and v. Notice that the results
shown below are essentially equivalent to those drawn by
Allen et al. [1996], Özkan-Haller and Kirby [1999], andFigure 4. Steady profiles in WEC+CEW (light line) and WEC-only (heavy line) 1-D solutions: (a) wave
height Hrms and (b) u. These cases have a linear drag with m = 0.006 m s1.
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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021(Figure 6). The averaged wave properties, hui and hzi areabout the same for all the cases (e.g., Figures 1 and 2) andthus not shown here. The alongshore flow hvi(x) has its
primarily maximum near the bar and a secondary maximum
in the nearshore breaking zone. It shows significant changes
with m, broadening as the bottom drag weakens. The
vorticity hci has a strongly decreasing magnitude withdecreasing drag since the alongshore current broadens.
These two profile changes are a consequence of increasing
lateral eddy momentum flux due to the shear instabilities,
consistent with the increasing TKE and Reynolds stressamplitudes as drag decreases. The Soulsby and Feddersendrag cases are less productive in TKE and hu0v0i than thelinear drag cases because their bottom drag locally intensifies where hvi and hHrmsi are both large near the breaking
point, suppressing fluctuation amplitude and momentum
flux.

5. Roles of Waves and Currents in Shear
Instability
Figure 5. Alongshore-averaged frequency spectra for thefluctuating component of alongshore velocity v0 at the bar
crest for eight different bottom drags in the WEC+CEW
model. Spectra are calculated from the last 4.2 h of theintegration.

OL03, while we use the different formalism and two waveenhanced bottom drag models (section 2) additionally.
4.1. Unstable Fluctuations[27] Figure 5 shows the frequency spectra of the fluctuating component of alongshore velocity v0 = v  hvi at thebar crest (x = 100 m) for the eight cases, a –h. The linear
drag model makes it apparent that a smaller drag leads to amuch broader range of unstable modes, particularly in thehigh-frequency band, as shown by Allen et al. [1996] andÖzkan-Haller and Kirby [1999]. We denote flows such as
those in cases a – c as ‘‘turbulent’’ and cases d – f as‘‘marginally unstable’’ (‘‘weakly nonlinear’’ regime in thework by Feddersen [1998] with harmonics at multiples ofthe peak frequency), distinguishing between them by thebreadth of the frequency spectrum. The marginal regime
corresponding to higher drag is typified by the cases withm = 0.005 to 0.007 m s1 and with Soulsby model (8).
These have narrow-banded spectra with distinct peaks ataround 2 – 3
10 3 Hz (periods of 330 – 500 s). The
turbulent regime occurs with low drag (i.e., m = 0.002 to
0.004 m s1) and with the Feddersen model (10). These
cases have more energetic, broad-banded spectra. The
behavior of v0 with combined wave and current drag incases g and h is qualitatively similar to those with thelinear drag. The Soulsby model has a behavior closest tothat of the linear drag case with m = 0.006 m s1, and theFeddersen model is closest to the cases with m = 0.003 or
0.004 m s1.
4.2. Mean Fields[28] Now consider the time- and alongshore-averaged
fields in the presence of the unstable fluctuations

5.1. Mean Fields[29] To second order in fluctuation amplitude, the 2-D
mean momentum balance is the same as for the 1-D model(Figure 3), but in cases even only somewhat past themarginally critical value of bottom drag the fluctuation
contributions become significant and are reflected in thealongshore current hvi and its differences due to the WEC
and CEW (Figure 7a). The peak hvi (and thus hci) values
are largest for the WEC+CEW model, intermediate in the
NW model, and least in the WEC-only model. The decreaseof the peak in hvi due to exclusion of the CEW is by about
10% and that due to exclusion of the WEC is by about 4%.
The intensity and effects of the unstable fluctuations,0and Reynold stresses, are greatest
measured in terms of urms
for WEC-only, then NW, then WEC+CEW solutions in theoffshore region near the bar. Thus, there is an inverse
correlation between eddy activity and mean flow there. In
particular, the CEW act to weaken the eddies, especially inthe cross-shore direction. Near the shoreline strong fluctuations arise only in the WEC+CEW model, although they
have almost no effect there on hvi since the Reynolds stressis small. Thus, the shoreline fluctuations are not generated
by a local shear instability, but rather arise from the CEW
modulation of the breaking acceleration (section 5.5).
[30] In the turbulent regime differences among the three
types of models are less pronounced than in the marginal
regime, although most of them are still evident (Figures 7d–7f). The WEC-only model has the least intense peaks in hvi
and the strongest Reynolds stress in the offshore breaking
region. The role of the CEW is still evident in weakening
the offshore eddy Reynolds stress, limiting the offshore
extent of the fluctuations in u0rms, and supporting strong
fluctuations near the shoreline.
[31] In summary for both regimes, the WEC-only modelhas the most intense eddy Reynolds stress in the breaking
region among the three cases. Incorporation of the CEW
causes the attenuation of Reynolds stress and the eddy
fluctuations, particularly for u except in the shoreline region
where it has the opposite effect. Thus, the peak hvi isweakest and the alongshore current width greatest in the

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C06021Figure 6. Time- and alongshore-averaged quantities in the WEC+CEW model with different dragparameterizations: (a) alongshore velocity hvi, (b) vorticity hci, (c) turbulent kinetic energy (TKE), and(d) off-diagonal component of Reynolds stress hu0v0i for eight different bottom drags (see legend in
Figure 6d). The time average is for the last hour in a 6-h integration.
WEC-only model because it has the most energetic eddy
momentum fluxes.
5.2. Alongshore Momentum Balance[32] Cross-shore profiles of the terms in the mean alongshore momentum balances (20) and (23) for the three model
formulations are shown in Figures 8 and 9 for 2-D unstable
solutions. As in the 1-D steady WEC+CEW solution(Figure 3), the primary balance occurs among the breaking
acceleration, bottom drag, mean advection, and mean vortex
force. The fluctuation advection (Reynolds stress divergence) has a moderate contribution to the momentum
budget in the marginal regime, and it even becomescomparable to the bottom drag as a retardant of breaking
acceleration in the turbulent regime. The lateral viscosityand fluctuating vortex force play minor roles in the mean
momentum budget, the former by design.
[33] The WEC+CEW model has the smallest breaking
momentum production within the bar region while the NW
model has the largest near the shoreline. The hByi terms for
the WEC-only and NW models are nearly identical, as
expected. Inclusion of CEW leads to a reduction in hByi
around the breaking point by 6% (marginal) or 15%(turbulent) and an enhancement near the shoreline by 10%(marginal) or 4% (turbulent) compared to the WEC-onlymodel. This is caused by changes in the local depth (H = h +z c for the WEC+CEW model, but H = h for WEC-only) in
the wave field (Hrms, k and b) since B is proportional toH5 and H7rms according to (16). Mean bottom drag is less inthe WEC-only model since hvi is less (Figure 7), but it isgreater in the NW model because of the lack of reduction of
H without wave setup/set-down, especially inshore of thebar (Figure 6). Relative to the NW model, inclusion of the
WEC diminishes the y component of the drag by 7.6% in
the marginal regime and 16.5% in the turbulent regime in
the vicinity of the breaking point.

[34] The mean advection and mean vortex force in the
WEC+CEW model are larger than those in the WEC-onlymodel in the marginal regimes, while essentially the same in
the turbulent regime. In the NW model hui remains small inthe unstable regime, even though it does not need to vanish
if the cross-shore eddy mass flux were significant; hence,
the mean advection and mean vortex force are negligible.
The fluctuation advection is the largest in the WEC-onlymodel around the breaking point in both regimes, and its
magnitude is weaker in the NW and WEC+CEW models;these model differences are bigger in the marginal regime
than in the turbulent regime. Relative to the NW model, theWEC increases the momentum mixing due to eddies aroundthe breaking point, while inclusion of CEW reduces theeddy mixing. The contribution of the fluctuating vortex
force huSt0c0i appears only with WEC+CEW; it is generally
much smaller than the mean vortex force although it isrelatively largest near the shoreline in the turbulent regime.
5.3. Fluctuation Structure[35] A comparison of time series from the three different
models shows the familiar transition between nearly periodic fluctuations and more complex ones (Figure 10; seealso Figure 5) in both the waves with CEW and currents ingeneral.
[36] In the marginal regime there is a delay in the onset ofthe shear instability with CEW compared to the models
without CEW because of the suppression for eddy momentum flux (Figures 7c and 8e). The delay due to CEW is also
evident in the turbulent regimes although instability develops much more quickly. This delay is qualitatively consistent with OL03 whose fluctuations took longer to develop
since they started from resting conditions. There is also a
longer oscillation period with CEW. In the marginal regime
with fixed m, the WEC+CEW model time series is nearly
periodic, while the NW and WEC-only models show

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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 7. Time- and alongshore-averaged quantities in (a – c) the marginal regime (m = 0.006 m s1)and (d – f) the turbulent regime (m = 0.003 m s1) for 2-D WEC+CEW, WEC-only, and NW models:0, and Figures 7c
Figures 7a and 7d show hvi, Figures 7b and 7e show fluctuating cross-shore velocity u rms
and 7f show Reynolds stress. The time average is for the last 2 h of the 6-h integrations.
evident aperiodicity, consistent with their relative orderingin the fluctuation intensity measures (Figure 7). In the
turbulent regime there are similar model differences mainly
evident as a broader range of frequencies in the NW andWEC-only models. Notice that v0(t) and H0rms(t) covary
almost in phase in the WEC+CEW models.
[37] Vorticity snapshots at t = 6 h give similar indicationsof the model differences, now expressed in the spatial
structure of the fluctuations (Figure 11). In the marginal
regime an alongshore meandering pattern is prominent in caround the breaking points at x = 120 m. The wavelength ofthe meanders is longer in the WEC+CEW model than theother two models, corresponding to the longer period inv0(t). In the turbulent regime longer alongshore spatial scales
emerge as current streaks alternating with coherent vortices
that roll up from the unstable meanders near the maximum
in hvi. The vortices are ejected as dipoles that carry the twosigned structure of hci with them. Offshore cyclonic vortices are closest to circular shapes in the NW model without
any wave effects, and they reach farthest offshore in the
WEC-only model as agents of its most effective cross-shore
eddy momentum flux. Near the shoreline (x < 30 m), only
the WEC+CEW model has strong eddies (consistent withthe statistical profiles in Figure 7). Even in the marginal
regime the shoreline eddies are remarkably asymmetricbetween interiorward cyclonic centers and shoreward anticyclonic meanders, reflecting the proximate flow barrier at
the shoreline. In the turbulent regime the strongest nearshore eddies are again cyclones, and they occur quite
intermittently in the alongshore direction.

5.4. Alongshore Propagation[38] Frequency – alongshore wave number spectra areuseful to detect celerities associated with the propagating
shear waves (Figure 12). A 2-D FFT is applied to v0(y, t) at
the bar crest position for the three model formulations,following the technique introduced by Özkan-Haller andKirby [1999]. The spectra are shown with three dispersion
curves in each plot: the dispersion relation for lowest-modeedge waves [Ursell, 1952] (to exclude them as the explanation for the fluctuations); the mean alongshore current hvi(as a Doppler shift influence on the fluctuations); and thefitted shear wave celerity cv = fv/kvy estimated from thespectrum. In the marginal regime the largest variations
occur in a discrete peak around (fv = 0.0025– 0.003 Hz,
kvy = 0.06– 0.007 m1) and higher harmonics; this reflects thenearly periodic behavior seen in Figures 10 and 11. In the
turbulent regime the spectra are broad-banded (Figure 12)
but still exhibit a ridge that indicates a dominant celerity
value. cv is similar for all three model formulations, although
inclusion of CEW increases cv relative to the WEC-onlymodel in both regimes. In turn, the vortex force in WEC
decreases cv relative to the NW model (as also demonstratedby Lane and Restrepo [2007] in the context of shorefaceconnected bar instabilities). The influences of WEC andCEW on propagation are closely related to the magnitude ofhvi. The celerity in the marginal regime closely matches hvi
at this particular location near the peak in hvi (x), but locallyfitted cv values are nearly the same in the x direction for eachcase (not shown). This implies that the fluctuations move asa coherent mode controlled by a cross-shore-averaged hvi
over the effective on-offshore span of the fluctuations. The

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UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021

C06021Figure 8. Time- and alongshore-averaged y momentum balance (20) or (23) in the marginal regime (m =0.006 m s1) for 2-D WEC+CEW, WEC-only, and NW models: (a) hByi, (b) bottom drag hDyi,(c) lateral viscosity, (d) mean advection, (e) fluctuation advection, and (f) mean vortex force (andfluctuation vortex force only for the WEC+CEW model with a dash-dotted line).
same cross-shore uniformity of cv is seen in the turbulent
regime but to a lesser degree. On the contrary, Kirby et al.
[2003] and Noyes et al. [2005] showed that cv in the
turbulent regime occurs mainly at the local magnitude ofhvi. In principle, noncoherent behavior of the fully nonlinear
shear waves may be relevant to the cross-shore variability in
cv, whereas for the linear shear waves inflection point
instability excites a single dominant propagation. The weakhvi dependency of cv found here is still an open question,
though this contrast may be attributed to the strength of thebottom drag (our turbulent regime result appears to havemoderate coherent structure; see Figure 11), incident wave
condition, or other unknown factors.
5.5. WEC and CEW in Fluctuation Dynamics[39] The inclusion of WEC results in an alteration in themagnitude in hu0v0i on both sides of the peak hvi (x) inboth regimes (Figure 7). Further inclusion of CEW reduces
u0rms and hu0v0i around the breaking point and increases
the fluctuations near the shoreline. To further expose howthese effects arise, we Reynolds decompose the advection
and vortex force terms in (3) as follows:u ru ¼ hui rhui þ hui ru0 þ u0 rhui þ u0 ru0^zuSt c ¼ ^z ^z

uSt hci  ^z
St0 0

u c:

uSt c0  ^z

ð25Þ

uSt0 hci
ð26Þ

Terms that are linear in the fluctuations vanish when
averaged; alongshore derivatives of mean quantities vanish;both the vortex force terms and hui (= 0) terms are absent
in the NW model; and uSt0 terms are present only in the
WEC+CEW model.
[40] The RMS magnitudes of the fluctuating advection
terms in the cross-shore component of (25) are displayed in
Figure 13 for all three models in the marginal regime. With
the WEC and hui 6¼ 0, huiu0x and u0huix contribute substantially, whereas they vanish in the NW model by definition.
With CEW the dominant alongshore advection by hviu0y , aswell as the eddy Reynolds stress divergence contributions
u0u0x and v0u0y , are greatly reduced since u0rms is weakercompared to WEC-only (Figure 7b). Analogous fluctuating
vortex force terms for the WEC models (Figure 14) havesomewhat smaller amplitudes than the advection terms
except nearshore. The vortex force terms / vSt0 that arise
only in CEW models are smaller than hvStic0 in the marginal
regime offshore but comparable nearshore. In the turbulent
regime vSt0c0 is comparable offshore and even dominantnearshore where it is a principal source of u0 variability (butalso see Bx; see below).
[41] Next we focus on the spatial structure of the crossshore velocity fluctuation, Reynolds stress, WEC advective
tendency (huiu0x and u0huix), and a component of crossshore VF, hvSti c0 (Figure 15). Coinciding with the locationsof peak cross-shore flow in the shear waves, the Reynolds
stress and an advective term huiu0x change sign in the
cross-shore direction as does the other advective term,
u0huix, which is somewhat weaker; see also Figure 13.
This behavior is consistent with the enhanced positive andnegative Reynolds stress values on the offshore and inshore
sides of the peak in hvi with the WEC seen in Figure 7.

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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 9. The same as Figure 8 except in the turbulent regime (m = 0.003 m s1).
These advective tendency terms act to strongly accentuate u0
on the offshore side and more weakly deplete it on theinshore side. Meanwhile, the cross-shore vortex force term
hvSti c0 varies approximately in phase with u0 and further
acts to strengthen the instability. These WEC characterizations are also valid in the turbulent regime (not shown).
[42] To further isolate the influences of CEW, two wave
simulations with and without CEW are performed to evaluate terms in the momentum equation (3) while fixing thecurrent field to be that of the WEC-only model. Instantaneous u and simulation differences are plotted in Figure 16.
The waves are significantly modified by both anti-Stokes

mean flow and shear wave refraction. Ray focusing occurs
in offshore-directed flow regions and increases Hrms in theadjacent runnel; conversely, ray divergence occurs in shoreward flow regions and decreases the runnel Hrms. The netincrease of Hrms with CEW implies an energy transfer fromthe offshore-directed flows associated with the shear waves(as remarked by OL03). In addition, wave refraction also
occurs on the anti-Stokes current (Figure 4) with the present
formalism. These current effects are somewhat analogous to
shoaling for the incident waves. The modifications to Hrms
and q lead to meandering in the breaking acceleration B andin J. CEW enhances the shoreward Bx around the breaking

Figure 10. Time series (a and b) of fluctuating Hrms in the WEC+CEW model and (c and d) of v 0 in thethree models at the bar crest (x, y) = (100 m, 384 m). Marginal regimes (m = 0.006 m s1) are shownin Figures 10a and 10c, and turbulent regimes (m = 0.003 m s1) are shown in Figures 10b and 10d.
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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 11. Instantaneous vorticity c for WEC + CEW, WEC-only, and NW models at t = 6 h.
(a) Marginal (m = 0.006 m s1) and (b) turbulent (m = 0.003 m s1) regimes.
point off the bar crest and in the shoreline region by about
15% of their peak values to oppose the instantaneous uheaded offshore. Conversely, the change in DBx is negativein onshore flow regions. This extra breaking acceleration
opposes the shear wave cross-shore flow and acts tosuppress it, with the evident CEW consequences of delayed
instability onset (Figure 10) and reduced velocity fluctuation amplitude and Reynolds stress (Figure 7). The shoreline CEW consequences in Bx are even more striking withintense shoreward accelerations at the ray focal centers.

Figure 12. Frequency – alongshore wave number spectra for v0 at the bar crest (x = 100 m) in the (top)marginal and (bottom) turbulent regimes for WEC+CEW, WEC-only, and NW models. Thick solid lines
are the dispersion curves for the lowest edge mode (as estimated by Ursell [1952]), thin solid lines aretime- and alongshore-averaged hvi, and dashed lines are the inferred shear wave celerity cv.
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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

C06021Figure 13. Cross-shore profiles of the RMS Reynolds-decomposed advection terms in the cross-shore
momentum balances (3) and (18) for the marginal regime (m = 0.006 m s1).
[43] The CEW alterations to the wave field also influence
By in phase with the shear waves. DBy has positive andnegative values in phase with the q anomalies, both in theoffshore region and near the shoreline. hDByi is negativearound the bar crest and positive near the shoreline, consistent with Figures 8 and 9. This effects a net transfer ofalongshore momentum from the bar to the shore by diminishing the breaking near the bar that subsequently must
increase at the shore since the total wave energy must be
dissipated. Thus, both Bx and By are increased near the
shoreline, leading to the greatly enhanced fluctuations there
in the WEC+CEW model. Finally, CEW increases DJx by
about 40% and DJy by about 6% of their offshore maxima
in the WEC-only model; however, these vortex force terms
are much smaller than the breaking acceleration terms,
hence of lesser influence as agents of CEW on u0.

6. Concluding Remarks[44] Wave breaking acceleration is well understood to bea cause of littoral currents, with bottom drag providing the
primary equilibrating balance, even though the alongshore
vortex force and anti-Stokes momentum advection also contribute. In this paper we show that wave-current interactions
significantly influence the littoral shear instability phenomenon that mostly has been understood without these interactions. Some of our conclusions are implicit in the radiationstress and current refraction model of OL03, but here we givethem an explicit interpretation in our WEC+CEW model
formulation partly by separating B and J.
[45] The WEC enhance offshore shear instability and its
attendant eddy momentum flux, thereby weakening andspreading the mean littoral current hvi (x) near the offshore
bar. It does this partly through the advective influence ofa mean cross-shore anti-Stokes flow hui. The combined
WEC+CEW delay the onset of instability, suppresses crossshore velocity fluctuations, weakens breaking near the bar,and diminishes eddy momentum flux, leading to a stronger,
narrower hvi. The WEC + CEW also energizes the fluctuations in the nearshore region by refracting the gravity wave
rays around the shear wave meanders and modulating the
breaking near the shoreline.
[46] The onshore increase in cross-shore breaking acceleration is potentially even more important for offshore-directedflows associated with rip currents over an alongshore-varying
topography. This has been reported by Yu and Slinn [2003]
with a coupled wave-current modeling framework similar tothat given by OL03, where the CEW influence on breaking
effects significantly reduces the offshore extent of the rip
currents. This behavior is also being investigated with ourformalism and will be reported separately.
[47] We use the bottom drag parameterization to control
successive instability regime transitions to weakly unstable
and complex fluctuation behaviors (as also done previously).

Figure 14. Cross-shore profiles of the RMS Reynolds-decomposed vortex force terms in the crossshore momentum equation (3): (a) hvSti c0 for the WEC-only model in the marginal and turbulent regimes(m = 0.006 and 0.003 m s1) and hvSti c0, vSt0 hci, and vSt0 c0 for the WEC+CEW model in (b) themarginal and (c) turbulent regimes.
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C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISM

Figure 15. WEC-only model snapshots of instantaneous quantities at t = 6 h in the marginal regime(m = 0.006 m s1): (a) fluctuating cross-shore velocity u0; (b) Reynolds stress u0v0; (c) cross-shore
advection of u0, huiu0x; (d) cross-shore advection of hui, u0huix; and (e) cross-shore VF contribution
from c0, hvSti c0. Notice that the scales are not the same.

Figure 16. Instantaneous quantities in the marginal regime (m = 0.006 m s1) at t = 12 h. (a) The WEConly cross-shore velocity u and differences diagnostically calculated with and without CEW using thesame u field: (b) DHrms, (c) Dq, (d) DBx, (e) DBy, (f) DJx, and (g) DJy .

14 of 15

C06021

C06021UCHIYAMA ET AL.: WAVE-CURRENT INTERACTION WITH VORTEX FORCE FORMALISMThis control is cleanest with a decreasing coefficient mwith a linear current drag law. On the basis of the spectral
plots, qualitatively similar behaviors occur with the nonlinear, wave-current drag laws by Soulsby [1995] andFeddersen et al. [2000], that mimic high- and low-m
regimes, respectively.
[48] Acknowledgments. We appreciate the support from the National
Science Foundation (DMS-0723757) and Office of Naval Research(N00014-04-1-0166). We also appreciate the help of Emily Lane and Brad
Weir in code preparation and preliminary calculations. Thanks are also due
to two anonymous reviewers and James Kirby for their constructive
comments and suggestions.

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J. C. McWilliams and Y. Uchiyama, Institute of Geophysics andPlanetary Physics, University of California, Los Angeles, CA 900951565, USA. (uchiyama@atmos.ucla.edu)
J. M. Restrepo, Department of Mathematics, University of Arizona,
Tucson, AZ 85721, USA.

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