Proceedings of the 19th IAHR-APD Congress 2014, Hanoi, Vietnam

ISBN 978604821338-1

ON TRIDIMENSIONAL RIP CURRENT MODELINGPATRICK MARCHESIELLO(1), RACHID BENSHILA(2), RAFAEL ALMAR(3), YUSUKE UCHIYAMA(4), JAMES MCWILLIAMS(5) &ALEXANDER SHCHEPETKIN(6)(1) Institute of Research for Development, Toulouse, France,

Patrick.Marchesiello@ird.fr(2) Centre National de la Recherche Scientifique, Toulouse, France

Rachid.Benshila@legos-obs-mip.fr(3) Institute of Research for Development, Toulouse, France,

Rafael.Almar@ird.fr(4) Kobe University, Kobe, Japan,uchiyama@harbor.kobe-u.ac.jp(5) University of California at Los Angeles, Los Angeles, California

jcm@atmos.ucla.edu(6) University of California at Los Angeles, Los Angeles, California,

alex@atmos.ucla.edu

ABSTRACT
Low-frequency variability generated by lateral shear instabilities of the coastal circulation may promote stirring andmixing of coastal waters. The question remains open to whether tridimensional transient processes are important for thegeneration of these instabilities and for surf-shelf exchanges in general. An innovative modeling system withtridimensional wave-current interactions (McWilliams et al., 2004) was designed to investigate transient nearshore
currents and interaction between nearshore and innershelf circulations. We present here some validation of rip current
modeling in the French coastal zone (Bruneau et al., 2011) using in-situ and remote video sensing. We then proceed toshow the benefits of 3D versus 2D modeling for the simulation of mean rip currents and their low-frequency variability. It
is concluded that tridimensional nearshore models may provide a valuable and cost-effective alternative to more usual 2D
approaches, which miss the vertical flow structure and its nonlinear interaction with the 2D flow.
Keywords: Rip; Video; Modeling; Tridimensional; Biscarrosse.
1.

INTRODUCTION

Rip currents are narrow, seaward currents that extend
from the inner surf zone out through the line of breaking
waves. Rip currents are usually long (100 m), narrow (10m) and intense jet flow (reaching 1–2 m/s). They appear
to span the entire water column in the shallow breaking
zone but they remain confined near the surface as theyflow past the breaking zone into deeper water, showing
strong vertical shear. Understanding and predicting thecomplex tridimensional dynamics of rip currents remain arelevant scientific challenge because they play a key roleon the beach and surf zone morphodynamics, on thedispersion of material across the surf zone and are a major
hazard to swimmers [see MacMahan et al., 2006, for areview].
Despite substantial evidence of tridimensional effects,
most modeling studies of rip currents are performed
using depth-integrated shallow water equations. Bruneau
et al. (2011) used this class of model (MARS coupled with
SWAN) to study the Aquitanian coast of France. In June2007 an intense 5-day field experiment was conducted at
the mesotidal‐macrotidal wave‐dominated Biscarrosse
Beach on a well‐developed bar and rip morphology.
Previous analysis of the field data exposed the main
characteristics of a tide‐modulated rip current driven bylow‐ to high‐energy shore‐normal waves (Bruneau etal., 2009a). The model was able to reproduce some of thecharacteristics of the rip currents but discrepancies withobservations at the rip neck were evidenced, particularlyin the cross-shore component.
In a parallel study, Bruneau et al. (2009b) showed
occurrence of rip current variability, referred to as Very
Low Frequency motions (VLF), in the rip neck where VLF
pulsations were most intense (reaching 1m/s on timescales of 10 to 30 minutes). The model was specificallytuned (reduced viscosity compared to the otherwise more
realistic control simulation) to develop shear instabilities
consistent with the analytical solution of Haller andDalrymple (2001).
In this paper, we extend on the work of Bruneau et al.
(2009b and 2011) to show that tridimensional wavecurrent interactions cannot be dismissed in the study of
rip current dynamics and that their expression is evident
both in their persistent structure and instability behavior.
2.

METHODS
2.1 Video monitoring

A main limitation for understanding nearshore
processes is lack of appropriate observation, particularly intropical environments. Traditional in-situ measurement
techniques can provide high sampling rates and a direct
estimation of many parameters but with coarse spatial
resolution that often miss complex dynamical interactions.
1

Additionally, instruments must be deployed in highenergy and sometimes hazardous environments (wavebreaking, strong currents), endangering not only the
instruments, but also the personnel involved. As analternative, remote sensing techniques can provide
synoptic coverage over large areas with a wide range of
temporal and spatial resolutions.
The innovative video imaging technique (Holman andHaller, 2013) is particularly suited to coastal observation:
low cost and easy to deploy. It is also considered as a nondisruptive observation technique for nearshore research.
A video system was deployed at Biscarosse beach in 2007
and new inversion methods (Almar et al., 2009; see Fig. 1)
were used to provide a remote estimation of the completenearshore system, i.e., hydrodynamics (wave height,
period, celerity, direction, wavelength, water level,
current) and morphodynamics (shoreline, sub- and intertidal bathymetry) continuously and over long-term
periods and large areas (approximate 2x1 km).

Figure 1. Coastal video monitoring: description of applications;(a) 5-camera system for Biscarrosse (France); (b) Spatial-temporal
post-processed image showing wave dynamics. The image is
generated from the high frequency acquisition of a cross-shore
array of pixels. (c) Bathymetric inversion from wave dispersion.
(d) Video-estimated water level. (e) 2D field of wave-breaking
dissipation.

2.2 The model: ROMS
The objective of the paper is to assess the benefit oftridimensional coastal models. An innovative modelingapproach for 3D wave-current interactions (McWilliams etal., 2004) was implemented in the Regional Oceanic
Modeling System (ROMS; Shchepetkin and McWilliams,
2005) by Uchiyama et al. (2010) and used for idealized rip
current studies (Weir et al., 2011). An upgraded version ofthis implementation is proposed here based on the AGRIF
version of ROMS (Penven et al., 2006; Debreu et al., 2012).
It allows additional capabilities such as nesting andwetting/drying (Warner et al., 2013), the latter being
crucial to the meso- macro-tidal environment of Biscarosse
beach.
ROMS is a hydrostatic, incompressible, free-surface andterrain-following coordinate model with non-conservative
forcing, diffusion, and bottom drag. It uses baroclinic–
barotropic mode splitting, with explicit fast time-steppingand subsequent conservative averaging of barotropic
variables. The discretization is with high-order finite
differences that provide both accurate and cost-effective
solutions. Open boundary conditions use a combination ofcharacteristic methods (barotropic part) and radiation(baroclinic part) with Newtonian and viscous damping(Marchesiello et al., 2001).
The interaction of surface gravity waves and currents is
implemented in ROMS through vortex-force (VF)formalism. Eulerian wave-averaged current equations for
mass, momentum, and tracers are included based on an
asymptotic theory by McWilliams et al. (2004) plus nonconservative wave effects due to wave breaking,associated surface roller waves, bottom streaming, andwave-enhanced vertical mixing and bottom drag. The
nonlinear parameterization of Soulzby (1995) for waveenhanced bottom drag is particularly relevant to thepresent case with strong tidal flow. All parameterizations
are described in details in Uchiyama et al., 2010 (see also
Blaas et al., 2007 for bottom drag formulation).
The currents are coupled with a ray-theory spectrum-peak
propagation and refraction model (WKB model) that
includes the effect of currents on waves and dissipation
due to shoaling-induced wave breaking. The system isthus fully coupled within a unique executable code that iseasy to handle.
The advantage of the vortex-force over radiation-stressformalism is to cleanly decouple conservative and nonconservative effects on currents and, within theconservative effects, the vortex force and Bernouilli headcomponents.
The
non-conservative
components(acceleration/dissipation) are those that requireparametrization and are thus responsible for the largest
uncertainties in our model formulation. Clear
identification of these terms is thus of primary interest.
Another advantage of the VF formalism is numerical since
it requires taking one less derivative in space and thus
produces fewer numerical approximations (Weir et al.,2011).
The coupled system is applied to the nearshore surf zone
during the 2007 Biscarosse field measurement campaign(see Bathymetry in Figure 2). Offshore measurements of
tidal elevation, wave height and winds are used to force
the model for 5 days from the 13th to the 17th of June 2007.
The horizontal resolution is 10m; there is 20 vertical sigma
levels equally spaced. The horizontal viscosity coefficientis given by the flow- and resolution-dependent
Smagorinsky formulation (between 0 and 2 m2/s). The
baroclinic time step is 2 seconds and 30 barotropic substeps are performed every baroclinic step. The cost of 3D
computations is thus moderate and the full 3D model is
only marginally more expensive than the 2D part.

Figure 2. Model domain and interpolated bathymetry. Theorange dot shows the measurement station S4 at the rip neckwhere model-data comparisons are presented.
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In the following, the model results are compared toobservations and tridimensional effects are investigatedwith emphasis on the vertical profile of cross-shore
currents.
3.

CALIBRATION AND VALIDATION
3.1 Calibration with video data

There is still no consensus on many parameterizations:wave-driven flow acceleration and turbulent mixing,
contribution of the breaking wave roller, bottom friction
and boundary layer streaming (e.g., Uchiyama et al., 2010).
In particular, the parameterization of breaking wave
dissipation is crucial to nearshore dynamics but isgenerally guided by scarce data.

neck (see Figure 2). The observation shows a strong
increase of wave height on the 15th of June due to offshore
forcing but strongly modulated by large tides (with rangeof close to 4 m) that shift back and forth the cross-shore
position of the breaking line. High-energy conditions from
June 15th lead to strong offshore-directed rip currents inexcess of 75 cm/s. The model is able to reproduce theevolution of both wave height and currents. The wave
model fit is quite remarkable during high-energy
conditions, validating our assumption that nearshore
waves in this experiment are well described by ray
equations driving a monochromatic wave with the rootmean-square wave height and frequency corresponding to
the spectral peak. There is an equally remarkable fit ofsimulated currents with data, in contrast with the 2D
simulations shown by Bruneau et al. (2011).

One of the novelties of our method lies in the tuning of
breaking wave dissipation (including roller dissipation) in
the wave model by direct comparison with 2D video
imagery (Figure 3). The wave model uses theparameterization of Church and Thornton (1993), which
has the particularly of producing more shoaling of theincoming wave before breaking and thus more intense
breaking than other choices (Weir et al., 2011). It relies on
two empirical constants: the breaking wave parameter γb(wave height-to-depth ratio) and Bb the percentage of the
wave face that is broken. Values of 0.3 and 1.3,
respectively, provide the best fit to data. As opposed toBruneau et al. (2011), there is no need here to add a timedependence on γb as the fit to data appears valid for bothlow- and high-energy conditions. The fraction αr of
breaking waves converted into rollers is taken as 0.5 andreveals only little sensitivity.

Figure 4. Model-data comparison at station S4 for wave height(top) and cross-shore currents (bottom).

4.

ANALYSIS AND SENSITIVITY
4.1 Comparing 2D and 3D solutions

Figure 3. Normalized wave-breaking dissipation from the video
data and from ROMS after calibration.


3.2 Validation with in-situ data
Figure 4 shows a model-data comparison for wave heightand cross-shore currents at station S4 located at the rip

To analyze the extent of errors related to the 2D shallow
water hypothesis, we now compare our standard solution
with a 2D version. In this case the barotropic mode isadvanced alone. Note that the barotropic (depthaveraged) flow of the 2D model is different from that of
the 3D model because in the latter it receives contribution
from the baroclinic flow, in particular the baroclinic
contribution to nonlinear terms (Reynolds stresses of typeuu  uu , where the overbar denotes the depth average).
For the bottom drag due to currents 1 , we use anequivalent formulation to the 3D case. Assuming that inshallow water the logarithmic layer extends all the way to
1 The total bottom drag law is a nonlinear combination ofcurrent and wave contributions according to Soulsby(1995). The bottom stress due to waves has no
dependence on the vertical grid.

3the surface, we can integrate the velocity profile to obtaina 2D drag coefficient of the form:

CD 



2



lnH /z0  1

[1]

2

where H is total depth, z0 is bottom roughness and κ is
Von Karman constant. This formulation ensures that the
2D bottom
 stress equals that of the 3D model for the samebarotropic flow (again assuming logarithmic velocity
profiles).
Figure 5 shows a comparison of the barotropic
components of ROMS in its shallow water and 3Dversions (ROMS2D and ROMS3D, respectively). It
appears that ROMS2D strongly underestimates the crossshore flow during high-energy conditions, especially therip currents (negative values at time 150-170). This result
is also very comparable to that of the 2D model used in
Bruneau et al. (2011). Interestingly, the barotropic mode of
the 3D model shows larger velocities implying asignificant role played by the baroclinic contribution to
barotropic nonlinear terms. This result is quite importantas it shows that the total flow cannot simply be addressedby linear combination of a depth-averaged model and a
vertical profile technique, a common approach inmorphological models (e.g., De Vriend and Stive, 1987).

Figure 5. Depth-averaged component of cross-shore velocity in
ROMS2D and ROMS3D compared with data. See Figure 4 fortotal model cross-shore currents.

The vertical structure of the 3D flow has several
dependencies. First, it is strongly affected by the profile
distribution function of breaking and roller acceleration.
The best fit to data is obtained with a surface-intensified
breaking force, consistent with Uchiyama et al. (2010). On
the other end, the wave-induced mixing scale is larger
than breaking acceleration scale and requires a smoothervertical distribution function (again as in Uchiyama et al.,2010). We also find that the choice of bottom friction
formulation is less sensitive than breaking acceleration for
representing the proper flow structure. Finally, thecurrents’ vertical structure is significantly shaped by thevertical component of vortex force.

same time enhances nearshore eddy energy. In particular,
rip currents contain energetic low-frequency oscillations,which does not require low-frequency wave forcing.
5.1 BackgroundHaller and Darlymple (2001) developed an analytic model
to study the linear spatial stability of a rip current2 andshow that low-frequency oscillations can be explained byshear-flow (vortical) instabilities. In absence of friction
and bottom slope, the results are consistent with those of
the classical Bickley jet 3 (Bickley, 1939). The fastest
growing sinuous mode of Haller and Darlymple (2001)
produces jet meandering with non-dimensionalwavenumber k*=0.639 and frequency σ*=0.255, related toactual wavenumber k0 and frequency σ0 by a length-scaleand vorticity scale respectively. The oscillations period
and wavelength are given by:2 l 0  U max
2 2 l
L0 
 *
k0
k

T0 

2



*

Here, Umax is the maximum jet velocity and l its half-width
length-scale. It means that the oscillation frequency of arip current is closely controlled by the magnitude of its
vorticity, aresult supported by laboratory experiments(Kennedy and Zhang, 2008). With 0.5 m/s and 20m
respectively, we get a time-scale of about 15 min (andwavelength of about 200m). According to this scaling, thepeak period for weaker and wider jets can increase toorder 100 min. In Haller and Darlymple (2001), viscosity,
bottom friction and bottom slope play a large role incontrolling vorticity (through offshore jet spreading) andthus the instability process. In general, friction has astrong stabilizing effect and would increase the period of
unstable oscillations. The bottom slope has a similar effect.
Though vortex stretching tends to reduce jet spreading (asopposed to bottom friction), continuity of the depthaveraged flow has a stronger effect in reducing the jet
speed. In the tridimensional problem, surface trapping
past the surf zone would be less sensitive to both vortex
stretching and continuity effects but with uncertain
outcome on the overall contribution of bottom slope to jet
vorticity.
Kennedy and Zhang (2008) developed a stability analysis
for a more complex configuration with wave-current
interactions that required a numerical solver (for the
eigenvalue problem). They mostly confirmed the resultsof Haller and Darlymple (2001) and clarified thecharacteristics of the two types of shear-flow instability:
sinuous and varicose. The former and dominant one isvorticity driven and occurs for strong rip currents (0.5m/s) and the second for weaker rip currents has its
strongest bulging signature shoreward of the bar. In
Kennedy and Zhang (2008), oscillatory periods of unstablejet modes for various conditions (particularly friction)varied between around 8 and 14 min. Their peak periods
are a smaller than in Haller and Darlymple (2001) because
2 Exponential growth of normal modes along the jet axis is

5. RIP CURRENT INSTABILITIES
Turbulence generated by the coastal circulation isprobably responsible for large mixing and transport incoastal waters (Brocchini et al., 2004). Weir et al. (2011)
show that the feedback effect of currents on waves
reduces the cross-shore extension of turbulence but in theconsidered with real frequency and complex wave
number.
3 The Bickley jet is a parallel flow with profile function
sech2=1/cosh2 about the centerline and has inflection
points (change of sign of vorticity gradients) satisfying
the Rayleigh criterion for shear-flow instability. It
produces two types of unstable modes: the sinuous andvaricose modes.

4they consider a different maximum vorticity scale taken at
the feeder zone. Interestingly, the coarsening of grid
resolution in these experiments has a similar effect tofriction, since jet vorticity is underestimated in this case as
well. Kennedy and Zhang (2008) also expose a wavecurrent-type instability at higher periods (beyond 30mn)
but only for small wave heights. Wave-current modes are
due to a feedback mechanism between the rip current andlongshore pressure gradients associated with flow-driven
wave height variations. They grow slowly and may
impose a slow pulsation to the rip current (Haas et al.,2003) while they develop downstream of the jet, much asa convective instability. However, a major effect of wavecurrent interaction (for all wave height) is to reduce theintensity of rip currents, their offshore extension andshear instability (Haas et al., 2003; Weir et al., 2011).
5.1 Biscarosse
Figure 6 shows a snapshot of rip current flow advecting apassive tracer. The general pattern shows a rip flow
biased towards the south and a strong cyclonicrecirculation cell, very similar to that of Haas et al. (2003).
It also suggests the presence of turbulence, i.e.,
meandering and offshore-developing instability with amushroom-shaped pattern detaching from the rip
recirculation cell. The maximum vorticity within the rip
current is larger than 0.05 s-1, which is on the order of thevorticity scaling used as example above.

Figure 6. Snapshot of surface flow (vectors) advecting a passive
tracer initially released in 1-m depth.

Figure 7. Observed time series of cross-shore velocity at station
S4 during the June 2007 campaign (starting on the 13 th at noon).
The blue curve is low-pass filtered at 1 hour and the green curveis high-pass filtered at 30 min.

In the model, the frequency of variability associated with
VLF motions is consistent with the data although theamplitude of oscillations is lower (Figure 7). It should be
noted that the time interval of model wave forcing is 20
min; therefore all scales below 40 min are intrinsic in themodel. Part of the missing energy may thus be related toforcing conditions. Another part may be due to theresolution of fine vorticity scales. Some experiments withincreased resolution (5m) or decreased viscosity(Smagorinsky coefficient reduced by half) have enhanced
VLF energy.
To better extract and analyze the intrinsic part ofvariability in the simulation, we designed a stationary
experiment using the low-energy conditions of June 2007as a persistent forcing. The result compares well with thespectrum of unstable normal modes given by Haller andDarlymple (2001; see their Figure 19). There is a range of
VLF oscillations between 10-30 min in agreement with thescaling given above from analytical estimations. However,a range of energetic but slower oscillations (30-50 min) is
also present that is either due to wave-current interactions
or nonlinear growth of unstable perturbations.
Interestingly, increasing model viscosity removes thehighest frequency motions but leaves out part of theoscillations around 50 min.
VLF oscillations are nearly absent from the 2D simulations.
In Figure 7, all periods of oscillations below about 50 min
are considerably weaker than in the 3D model and even
weaker than in the data. Consistently, Bruneau et al.
(2009b) failed to capture VLF frequencies in their standard
2D simulations and significant viscosity reduction was
required to generate instabilities in their case. We
interpret this result as evidence that the barotropic
component of 2D equations is too weak to produce muchcurrent shear that triggers instabilities.

Bruneau et al (2009b) noted VLF oscillations in Biscarossetime-series with periods of 10-30 min. This is aconsiderably longer period than most subharmonic wave
energy but is well within the range of possible shear
instability (Kennedy and Zhang, 2008). The problem iscomplicated by the fact that the frequency spectrum fromthis data shows energy at all frequencies due to the wide
range of conditions during the 5-day campaign (Figure 8).
However, low-pass filtering of the data indicatesoccurrence of VLF oscillations that are correlated with thepresence of rip currents (Figure 7). This observation isconsistent with mean-eddy conversion associated withflow instability rather than low-frequency wave forcing.

5the maximum vorticity cell center. In addition, thecirculation cell is much weaker in a simulation withincreased viscosity, where no instability and VLF
oscillations are present.

Figure 8. Frequency spectrum of cross-shore velocity at the ripneck (station S4) during the 5-day Biscarosse campaign of 2007.
2D and 3D models are compared with data.

In linear models, lateral or bottom friction increases the jetspreading of the classical planar jet solution and causes
the centerline velocity to decay more rapidly. We havenoted that the bottom slope acts in the same direction as
friction because depth-averaged flow continuity
overwhelms the effect of vortex stretching (Haller andDarlymple, 2001). Interestingly and as already predicted
by Arthur (1962), this balance should be affected bytridimensionality. After rip currents leave the surf zone
they tend to be confined to a layer of thickness equal tothe depth at the seaward edge of the surf zone. This
would have a double effect. On one hand, surface
confinement should reduce bottom interaction andassociated stretching effect. On the other hand, bottom
friction should also decrease in the process. More
importantly, a surface-intensified jet has more intense
centerline velocity than its depth-averaged flow on a
sloping beach (due to continuity). From our results,
nonlinear interactions between baroclinic and barotropic
components also appear to increase the barotropic flow.
Overall, the increase of shear production from surface
intensification appears to easily dominate over depthaverage dynamical balance.
To localize in space the instability process, we now
compute the mean-eddy conversion terms in thestationery experiment (KmKe; e.g., Marchesiello et al.,2003). Figure 8 shows KmKe together with vorticity andvelocity vectors. When conversion is positive, it indicates
the work of shear instability. In contrast, negative values
show locations of feedback to the mean flow. The
maximum instability is clearly in the feeder zone on eachside of the rip center (shown by positive and negative
vorticity maxima). The instability decreases offshore along
the jet axis as predicted by the sinuous instability modes.
However, a new strong maximum appears at the offshore
branch of the re-circulation cell. This branch ischaracterized by surface-trapped currents in relatively
deep water, where bottom friction is weaker than in therip channel. The offshore growth of disturbances is thus
less impaired by friction. It allows the offshore branch of
the cell to wander offshore and sometimes create an
anticyclonic circulation as in Figure 6 (and Figure 4 ofHaas et al., 2003). The cyclonic recirculation cell may
partly result from nonlinearities and KmKe is negative in

Figure 8. Mean-eddy conversion KmKe in the rip current systemfrom a persistent 8-hour simulation of the low-energy conditionsof June 2007. Vorticity contours and velocity vectors are overlaid.

There are large implications of instability problems in theejection of surf-zone material by the rip-current system.
The recirculation cell would promote retention whilepulsation and filament formation would do the opposite
and increase the dispersion out of the surf-zone.
6.

CONCLUSIONS

We have presented some validation of tridimensional rip
current modeling in the Aquitanian coastal zone using insitu and remote video sensing. We showed the benefits of
3D versus 2D modeling for the simulation of mean rip
currents and their low-frequency variability. We conclude
that tridimensional nearshore models may provide a
valuable and cost-effective alternative to more usual 2D
approaches, which miss the vertical flow structure and its
nonlinear interaction with the 2D flow.

ACKNOWLEDGMENTS
We appreciate financial support from the IRD ad CNRS.
We also thank the EPOC group for providing in-situ datafrom the June 2007 campaign.

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