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Effects of waves on coastal water dispersion in a small estuarine bay
M. T. Delpey,
1
F. Ardhuin,
2
P. Otheguy,
1
and A. Jouon
1
Received 27 September 2013; revised 2 December 2013; accepted 9 December 2013; published 7 January 2014.
[1] A three-dimensional wave-current model is used to investigate wave-induced
circulations in a small estuarine bay and its impact on freshwater exchanges with the inner
shelf, related to stratified river plume dispersion. Modeled salinity fields exhibit a lower
salinity surface layer due to river outflows, with typical depth of 1 m inside the bay. The
asymmetric wave forcing on the bay circulation, related to the local bathymetry,
significantly impacts the river plumes. It is found that the transport initiated in the surf zone
by the longshore current can oppose and thus reduce the primary outflow of freshwater
through the bay inlets. Using the model to examine a high river runoff event occurring
during a high-energy wave episode, waves are found to induce a 24 h delay in freshwater
evacuation. At the end of the runoff event, waves have reduced the freshwater flux to the
ocean by a factor 5, and the total freshwater volume inside the bay is increased by 40%.
According to the model, and for this event, the effect of the surf zone current on the bay
flushing is larger than that of the wind. The freshwater balance is sensitive to incident wave
conditions. Maximum freshwater retention is found for intermediate offshore wave heights
1m<H
s
<2 m. For higher-energy waves, the increase in the longshore current reduces the
retention, which is two times lower for H
s
54 m than for H
s
52m.
Citation: Delpey, M. T., F. Ardhuin, P. Otheguy, and A. Jouon (2014), Effects of waves on coastal water dispersion in a small
estuarine bay, J. Geophys. Res. Oceans,119, 70–86, doi:10.1002/2013JC009466.
1. Introduction
[2] Coastal waters are the receptacle for pollutants from
estuarine watersheds, especially fecal indicator bacteria
[Boehm et al., 2002; Reeves et al., 2004]. The related deg-
radation of the water quality increases human health risks
and leads to large economic impacts [Grant et al., 2005 ;
Given et al., 2006]. Terrestrial storm water runoff often
drains first onto rivers and tidal channels, finding its way
into the coastal zone with freshwater inflows. In that case,
nearshore concentrations of fecal indicator bacteria are
closely related to river plumes dispersion and freshwater
exchanges with the inner shelf [e.g., Ahn et al., 2005]. Estu-
arine bays and lagoons are particularly impacted by such
contaminations, as freshwater concentrates in these coastal
systems due to their partial separation from the ocean [e.g.,
Fiandrino et al., 2003; Pereira et al., 2009]. Depending on
the system morphology and the local wave climate, ocean
waves may strongly impact dispersion processes. Surf zone
transport and mixing affect the suspended matter along-
shore distribution and nearshore retention [Boehm et al.,
2005; Feddersen, 2007; Spydell et al., 2007, 2009; Reniers
et al., 2009]. Waves also affect exchanges between the
nearshore and the inner shelf [Lentz et al., 2008].
[3] The interaction of ocean waves and nearshore cur-
rents can be decomposed in several processes. The momen-
tum carried by waves across the ocean is released in the
surf zone and captured in intense currents and sea level
changes. Waves contribute to vertical mixing in deep or
shallow water as turbulent kinetic energy is enhanced at the
surface by wave breaking [Craig and Banner, 1994; Terray
et al., 1996; Feddersen, 2012a, 2012b]. The mean wave
momentum, or Stokes drift [Stokes, 1847], contributes to
the advection as a surface-intensified current which adds to
the mean current. Following the pioneering work of Lon-
guet-Higgins and Stewart [1962, 1964], modeling of surf
zone circulations has long been based on phase-averaged
and depth-averaged equations, notably due to their compu-
tational efficiency. However, it appeared that the vertical
shear of the mean flow can influence lateral mixing [Svend-
sen and Putrevu, 1994]. Moreover, in a vertically sheared
flow, depth-integrated velocities are not representative of
velocities advecting a tracer with nonuniform vertical dis-
tribution. In fact, the Stokes drift always exhibits a strong
vertical shear near the surface [Miche, 1944; Ardhuin
et al., 2008b]. In the nearshore, this leads to large vertical
variations of the total cross-shore current. Also, laboratory
and field observations suggested that rip currents might
vary from depth-uniform to depth-varying outside the
breakers, with higher velocities near the surface [e.g., Haas
and Svendsen, 2002; Reniers et al., 2009]. To capture the
transport of a tracer in the presence of waves, the vertical
structure of the flow may thus be needed.
1
Centre Rivages Pro Tech, Lyonnaise des Eaux, Bidart, France.
2
Laboratoire d’Oc
eanographie Spatiale, IFREMER, Plouzan
e, France.
Corresponding author: M. T. Delpey, Centre Rivages Pro Tech, Lyon-
naise des Eaux, Technopole Izarbel, 2, all
ee Th
eodore Monod, FR-64210
Bidart, France. (matthias.delpey@ponts.org)
©2013. American Geophysical Union. All Rights Reserved.
2169-9275/14/10.1002/2013JC009466
70
JOURNAL OF GEOPHYSICAL RESEARCH : OCEANS, VOL. 119, 70–86, doi :10.1002/2013JC009466, 2014
[4] During the last decade, wave-current three-dimen-
sional (3-D) approaches have been proposed to account for
the vertical shear of wave momentum and forcing [e.g.,
Groeneweg, 1999; Mellor, 2003 ; Mc Williams et al., 2004 ;
Ardhuin et al., 2008b; Aiki and Greatbatch, 2012]. Some
of these have been further adapted into existing ocean
numerical models and applied to reproduce observed circu-
lations [e.g., Uchiyama et al., 2009] and nearshore disper-
sion [e.g., Reniers et al., 2009]. The main distinctions
between the different formulations are (1) the use of an
Eulerian or Lagrangian point of view and (2) the separation
or not of the momentum into wave and mean flow contribu-
tions [Lane et al., 2007; Bennis et al., 2011]. For the first
aspect, truly Eulerian approach requires a mathematical
extrapolation of the velocity profile from the trough level
to the mean sea level in order to define the mean flow in
the crest-to-trough region [Mc Williams et al., 2004]. This
can be avoided in the Lagrangian framework, as the
particle-following average is well defined in the whole
water column [Andrews and McIntyre, 1978]. The second
difference is the distinction of wave and current momentum
first introduced in depth-averaged equations by Garrett
[1976]. This distinction has the main benefit of avoiding a
common turbulent parameterization for both waves and
current, as the Stokes drift is not mixed by turbulence
unlike the mean current. It also turns out that the equation
for the mean flow momentum only requires wave-induced
forcing terms obtained from traditional two-dimensional
wave models, whereas forcing for the full momentum
requires more complex wave models, accurate to first order
in parameters like the bottom slope [Ardhuin et al., 2008a].
Taking advantages of both Lagrangian framework and
momentum separation, Ardhuin et al. [2008b] (hereinafter
A08) proposed a 3-D wave-current approach based on the
Generalized Lagrangian Mean (GLM) theory introduced
by Andrews and McIntyre [1978]. An asymptotic formula-
tion of the exact GLM-equations is provided to the second
order in wave nonlinearities. The obtained set of equations
is nondivergent, thanks to a transformation of the vertical
coordinate. The A08 formulation is consistent with Mc Wil-
liams et al. [2004] to the considered order of approximation
in the limit of a weak mean-current vertical shear [Ardhuin
et al., 2008b]. Bennis et al. [2011] confirmed the ability of
A08 formulation to provide the vertical structure of the
flow by numerical modeling of an adiabatic configuration.
Michaud et al. [2012] further showed the consistency of
the solution obtained from A08 formulation with results
from Haas and Warner [2009] and Uchiyama et al. [2010]
in the case of obliquely incident waves breaking on a plane
beach. Model results were further applied to a realistic con-
figuration on the French Mediterranean coast.
[5] Following these works, the A08 formulation shall be
used in the present study. The corresponding set of equa-
tions was implemented in the 3-D primitive equations
model MOHID Water [Martins et al., 2001; Braunschweig
et al., 2004]. This model is used to investigate dispersion
and mixing mechanisms in the bay of Saint Jean de Luz-
Ciboure under the combined influence of winds, tide and
waves. The bay is a small semienclosed estuary, located in
the high-energy wave environment of the southeastern Bay
of Biscay (Figure 1). As a sheltered bay where the currents
are generally weak compared to the neighboring exposed
nearshore areas, it may be impacted by runoff pollution
carried with river inflows. River plumes are characterized
by a remarkable stratification in the bay, with freshwater
concentrated in a thin surface layer.
[6] The present work focuses on the effect of waves on
river plume dynamics and on water exchanges between the
bay and the inner shelf. Due to the limited field data set
available in the bay for now [Delpey, 2012] and the diffi-
culty to use point measurements for the study of complex
flows, this paper proposes an analysis based on wave-
current numerical modeling, as a first step in the investiga-
tion of the complex dynamics of the bay. An original
description of main dispersion processes under the effect of
wave-current interactions is provided in a realistic configu-
ration with strong 3-D features. Section 2 describes the
study site and field data. Wave and current models imple-
mentation is given in section 3. An analysis of waves trans-
formation and modeled wave-induced circulation are then
provided in section 4, and effects of waves on freshwater
dispersion are investigated. Conclusions are summarized in
section 5.
2. Study Area and Event of Interest
[7] The bay of Saint Jean de Luz-Ciboure is located in
the south of the French Atlantic Coast, 10 km northward
from the Spanish border (Figure 1). This region is exposed
to energetic swells coming mainly from North Atlantic
with direction W-NW. The offshore mean significant wave
height and peak period are 1.6 m and 9.6 s, respectively
[Abadie et al., 2005]. The study site is a shallow mesotidal
bay, with tidal range about 4.5 m at spring tides. The
bathymetry of the area is presented in Figure 1d. The bay is
approximately 2 km long by 1 km wide. The area is semi-
enclosed by breakwaters which delimit two inlets connect-
ing the bay with the inner shelf. These shallow inlets with
mean depth 13 and 8 m are, respectively, 250 and 350 m
wide. The small cross section in the inlets causes the accel-
eration of tide-induced currents and related mass exchanges
between the bay and the inner shelf during the tide cycle.
Ocean waves can penetrate in the bay, especially, through
the wider eastern inlet. In the northeastern part of the bay,
wave breaking is regularly observed over a shallow rocky
platform. During energetic events, a high level of wave
energy is dissipated in this area. The platform mean depth
is about 0.5 m above the Lowest Astronomical Tide (LAT)
level, so that a part of it is intertidal when the tidal range is
large. The platform surface is rocky and very irregular.
Ponctual measurements of rock formations height show
that it often reaches several tens of centimeters high (Figure
2). A steep slope connects the platform to the deeper part
of the bay, where the bottom is mixed sandy/rocky with
gentler slopes.
[8] The bay receives freshwater inflows from two small
rivers (Figure 1b). The annual mean flow is 5 m
3
s
21
for
the main river (Nivelle river) and of the order of 1 m
3
s
21
for the smaller river. During intense rain events, measure-
ments of bacterial loads (not shown) revealed that both riv-
ers can receive significant inputs of fecal indicator bacteria
in runoff from their watersheds. Such contamination pro-
cess is a common feature in many coastal urban watersheds
[e.g., Fiandrino et al., 2003; Reeves et al., 2004]. These
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
71
bacterial contaminant are then likely to be introduced into
the bay with river inflows. The following dispersion proc-
esses and the related impact on water quality are an impor-
tant issue for the touristic area of Saint Jean de Luz-
Ciboure.
[9] In this paper, the bay dynamics are examined during
the episode from 22 to 28 September 2010. Environmental
conditions encountered during this time interval are plotted
in Figure 3. It can be seen that the studied interval corre-
sponds to a river flood event occurring under high-energy
wave conditions (max. offshore H
s
2.5 m). Wind velocity
is also significant on 24 September (>10 m s
21
), with N to
NW direction. Tidal range varies from 2.5 to 3.3 m over
the time period. The bay response to the river outflow
under the effect of these different forcings will be investi-
gated. During the studied time interval, field data were col-
lected by three bottom mounted sensors deployed in the
framework of LOREA 2010 experiment (Littoral, Ocean,
Rivers in Euskadi Aquitain, http ://www.lorea.eu/). Sensor
locations are indicated in Figure 1d. In the present work,
recorded wave data will be used and compared with model-
ing results. A Nortek Acoustic Waves and Current profiler
(AWAC) was settled in the eastern inlet at 9 m depth under
the LAT level. It recorded wave data using a sampling fre-
quency f
s
52 Hz. Inside the bay, two Nortek Vectors (here-
inafter ADVN and ADVS) were deployed to measure
Figure 1. (a) SPOT image of the Atlantic coast on either side of the France-Spain border. SPOT4 : 20
m color composite image, 22 July 2008. (b) Aerial photograph of the study site. Google Earth image,
Data SIO, NOAA, U.S. Navy, 3 September 2006. (c) Photograph of the bay taken at high tide from the
eastern breakwater, with the inlets on the right and the rocky platform surf zone in the center. (d)
Bathymetry of the studied bay. Depth is positive downward, relative to the Lowest Astronomical Tide
(LAT) level of Saint Jean de Luz harbor. Mean water level is 2.48 m above the LAT level. White
squares: wave measurement locations. Red dot : Socoa tide gauge location.
30cm
Eastern breakwater
Figure 2. Photograph taken on the intertidal domain of
the rocky platform (northeast of the bay).
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
72
waves in the east part of the bay, close to the surf zone
(Figure 1d). The ADVN was deployed southward from the
platform, at depth 4 m, in order to provide information
about wave refraction and dissipation over the platform.
The ADVS was settled farther from the surf zone, also at 4
m depth. The ADVN and ADVS recorded wave data using
a sampling frequency f
s
54 and 8 Hz, respectively. Due to
a sensor dysfunction, wave data are only available from 25
to 27 September at the ADVS location.
[10] Some point measurements of current velocity and
salinity profile were also collected as part of LOREA 2010
deployment. However, given the studied configuration com-
plexity, the present work will not seek for an evaluation of
the model ability to reproduce the few available flow meas-
urements, but rather focus on a description of dispersion
mechanisms at the scale of the whole bay, based on main
processes simulated by the wave-current model. A detailed
description of LOREA 2010 experiment and of preliminary
flow model-data comparison may be found in Delpey [2012].
3. Numerical Modeling
3.1. General Implementation
[11] The studied bay is partially exposed to energetic
wave conditions. It is also expected to involve significant
effects of the flow vertical structure, related to river plumes
dynamics in sheltered areas. To investigate freshwater dis-
persion in this configuration, a full 3-D wave-current model
is used, including a representation of the vertical shear of
wave momentum and forcing as well as wave-induced ver-
tical mixing. The present work is based on the coupling of
the flow model MOHID Water [Martins et al., 2001 ;
Braunschweig et al., 2004] and the spectral wave model
WAVEWATCHIII
V
R
(WWIII) [Tolman, 2008, 2009], using
the glm2z formulation proposed by Ardhuin et al. [2008b].
[12] To predict wave-current dynamics inside the bay, the
computation is carried out on three nested domains, repre-
sented in Figure 4. Offshore wave conditions are first propa-
gated up to the vicinity of the bay on domain G1. The
domain extends approximately 30 km off the bay. The corre-
sponding computational grid is regular with size 412 by 286
andmeshsteps100m.Wavesandcurrentarethencomputed
ontwonestedlevelswithrefinedgrids(G2andG3),from
approximately 20 m mean depth to the shoreline inside the
bay. The second level G2 aims at providing river inflows
forcing for the level G3. The flow is computed along the tidal
section of both rivers up to the bay. To allow a realistic mod-
eling of these very narrow sections with a limited grid size, a
variable mesh step is adopted in G2-grid, from 100 m steps
offshore to 20 m steps over the rivers upstream section. The
total number of grid points is 208 by 155. The solution of
level G2 is transmitted at open boundaries of the last level
G3 (in particular flow properties at river outlets). The domain
G3 covers the bay area, with an offshore boundary located 1
km off the bay inlets. Wave computation is carried out on a
246 by 260 regular grid with 10 m steps. Wave forcing terms
are then interpolated on a 123 by 130 regular grid with 20 m
mesh steps for the flow computation. In levels G2 and G3,
the flow model vertical discretization uses 10 sigma levels,
with finer resolution close to the surface and at the bottom.
[13] Bathymetry of level G1 was derived from data pro-
vided by the French Navy Hydrographic and Oceanographic
Service (SHOM). For levels G2 and G3, a compilation of
bathymetric and topographic surveys data was provided in
the framework of the LOREA project. Horizontal resolution
of these data ranges from 1 m (in the deeper parts of the
domain) to approximately 20 m (in most of the shallow areas
like the rocky platform, the intertidal domain, river sections).
3.2. Wave Model Implementation
[14] Wave transformation is simulated with the spectral
wave model WWIII in its version 4.04 [Tolman, 2008,
2009]. WWIII has been widely validated at global and
regional scales [e.g., Tolman, 2002; Ardhuin et al., 2008a,
2010], and more recently in nearshore areas [e.g., Filipot
Figure 3. (a) Mean water elevation gmeasured by Socoa
tide gauge inside the bay. (b) Wind velocity V
wind
and (c)
wind direction D
wind
measured by the Socoa meteorologi-
cal station (M
et
eo France). Hourly averages of offshore (d)
significant wave height H
s
, (e) peak period T
p
(full line)
and mean period T
m02
(dashed), and (f) peak direction D
p
(full line) and mean direction D
m
(dashed). Offshore wave
data were measured by the Donostia directional buoy,
located 30 km off the bay at (22.0126E, 43.56N). (g)
Upstream flow of the main (full line) and the secondary
(dashed) rivers.
−2 −1.95 −1.9 −1.85 −1.8 −1.75 −1.7 −1.65 −1.6 −1.55 −1.5
43.3
43.35
43.4
43.45
43.5
43.55
Lon
g
itude (°)
Latitude (°)
Flow
measurement
G3 G2
G1
Meteo
station
Donostia
buoy Anglet
buoy
Tide
gauge
Figure 4. Coastline (gray line), computational domains
(black dashed line), and data locations (black points).
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
73
and Ardhuin, 2012; Michaud et al., 2012 ; Ardhuin et al.,
2012]. The model solves the spectral balance equation for
the wave action density N, with source terms S
in
,S
nl
,S
oc
,
and S
bot
for wind input, nonlinear four-waves interactions,
wave breaking dissipation and bottom friction dissipation,
respectively. The term S
oc
is considered as the sum of two
contributions S
ds
and S
db
, respectively, for whitecapping
and depth-induced breaking in very shallow water. Triad
interactions and bottom scattering effects are not taken into
account in the present study.
[15] The wind-wave generation and dissipation (terms
S
in
and S
ds
) are parameterized according to Bidlot et al.
[2005]. Nonlinear four-waves interactions are modeled
using the Discrete Interaction Approximation (DIA) as pro-
posed by Hasselmann et al. [1985]. In shallow areas, like
the bay studied here, wave energy dissipations by bottom
friction and depth-induced breaking become very impor-
tant, and S
bot
and S
db
are much larger than the whitecapping
dissipation term S
ds
. Parameterization of the bottom friction
dissipation term S
bot
uses the linear JONSWAP formulation
[Hasselmann et al., 1973], given by
Sbot ðkÞ52C2pf
gsinh ðkDÞ
2
EðkÞ;(1)
where kis the wave number vector, fis the frequency, gis
the gravity acceleration, Dis the water column height, E5
2pfN is the wave energy density spectrum, and Cis a con-
stant friction coefficient. Depth-induced breaking dissipa-
tion is parameterized according to Battjes and Janssen
[1978]. Wave heights are limited by the threshold height
Hmax 5cD, with a constant breaker parameter c. The dissi-
pation term S
db
due to depth-induced breaking is given by
Sdb ðkÞ520:25Qbfm
H2
max
Etot
EðkÞ;(2)
where Q
b
is the breaking probability of the random sea
state, f
m
is the mean frequency and Etot 5ðk
EðkÞdk. For the
present application, values of parameters Cand cwere
adjusted based on wave model-data comparison, as further
discussed in section 4.1.
[16] Wave spectra are discretized over 25 frequencies
exponentially spaced from 0.041 to 0.41 Hz so that two
successive frequencies f
i
and f
i11
are related by fi1151:1fi,
and 36 directions with a constant 10directional resolution.
Offshore wave conditions are obtained from hourly direc-
tional spectra provided by the Donostia buoy (depth 450
m), located 30 km off the bay at (22.0126E, 43.56N)
(Figure 4). On the three computational levels G1 to G3,
surface elevation is prescribed from the 10 min record of
the Socoa tide gauge and wind forcing is given by the
hourly measurement of the Socoa meteorological station
(see Figure 4). The current retroaction on waves is not
taken into account in this study.
3.3. Flow Model Implementation
[17] The present work is based on the code MOHID, a 3-
D baroclinic, incompressible (Boussinesq), hydrostatic, and
free-surface ocean model. MOHID uses a finite volume
method to discretize governing equations in a structured C-
grid and a semi-implicit (ADI) temporal algorithm. Equa-
tions are numerically solved by the model with a procedure
equivalent to a generic vertical coordinate [Martins et al.,
2001], allowing for different types of vertical discretiza-
tion, like the sigma coordinate used for the present study.
The numerical procedure allows a representation of the
wetting and drying of intertidal regions. MOHID system is
coupled to the General Ocean Turbulence Model (GOTM)
[Burchard and Bolding, 2001] for the vertical turbulent clo-
sure. Detailed description of the code implementation can
be found in Montero [1999] and Martins et al. [2001]. The
MOHID hydrodynamic model has been used successfully
in complex ocean and coastal applications [e.g., Martins
et al., 2001; Coelho et al., 2002; Leit~
ao et al., 2005; Mal-
hadas et al., 2009] and compared well with several state-
of-the-art ocean models in the Bay of Biscay [Riflet et al.,
2010].
[18] The model used in the present study, further referred
to as ‘‘MOHID-GLM,’’ is a new MOHID version modified
to allow a full 3-D modeling of the mean flow including the
effects of waves, with all other elements retained from the
original MOHID code. The new code version solves the
glm2z asymptotic formulation of the GLM wave-current
equations for the quasi-Eulerian momentum, as proposed
by Ardhuin et al. [2008b] and further adapted by Bennis
et al. [2011]. The quasi-Eulerian velocity field, noted
^
u5ð^
u1;^
u2;^
wÞ, is defined as [Jenkins, 1989]
ð^
u1;^
u2;^
wÞ5ðuL
1;uL
2;wLÞ2ðuS
1;uS
2;wSÞ;(3)
where uL5ðuL
1;uL
2;wLÞis the Lagrangian mean velocity
field and uS5ðuS
1;uS
2;wSÞis the Stokes drift velocity field.
The vertical coordinate change used by Ardhuin et al.
[2008b] corrects the vertical coordinate so that the glm2z
equations are nondivergent. The reader is referred to
Appendix A for a presentation of the complete set of equa-
tions solved by MOHID-GLM, and to Delpey [2012] for a
detailed description of the code implementation.
[19] MOHID-GLM is coupled with a K2model [Rodi,
1980] to determine the vertical eddy viscosity and diffusiv-
ity coefficients in the momentum and tracer conservation
equations. To account for the wave-enhanced vertical mix-
ing at the surface [Agrawal et al., 1992; Terray et al.,
1996; Feddersen, 2012a, 2012b], a wave-induced surface
TKE flux is introduced in the surface boundary conditions
of the K2scheme, and an enhanced surface roughness
length z
0,s
is used [Craig and Banner, 1994]. As proposed
by Terray et al. [1996, 2000], z
0,s
is considered propor-
tional to the total significant wave height H
S
, i.e.,
z0;s5a0HS, with a
0
a constant. Here the value a
0
50.6, pro-
posed by Soloviev and Lukas [2003], was adopted. For the
horizontal mixing, a constant horizontal eddy viscosity K
H
was used and set to K
H
51.0 m
2
s
21
(see Appendix A).
[20] The importance of bottom friction for wave-induced
surf zone currents has been demonstrated by several
authors, especially in the longshore direction [e.g., Lon-
guet-Higgins, 1970a, 1970b; Thornton and Guza, 1986]. In
the present work, the specification of the bottom roughness
length z
0,b
also appeared critical to the determination of
wave-induced currents in the surf zone. This parameter is
spatialized in order to account for the variable nature of the
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
74
bottom in the studied site: for relatively smooth sandy
areas, z
0,b
was set to 0.001 m, and for very rough rocky
areas, a maximum value z
0,b
50.1 m was used. This latter
value of z
0,b
is large compared to more often cited values
for sandy bottoms, which range usually from 0.001 to 0.01
m [e.g., Weir et al., 2011]. Frictional dissipation over rocky
seabeds has been less studied [Nunes and Pawlak, 2008].
In the present configuration, the value of z
0,b
in the rocky
area was adjusted to meet a qualitative consistency of
model results with preliminary flow measurements avail-
able in the bay (not shown). The adopted schematic param-
eterization of z
0,b
may be seen as a first attempt to account
for the unusually large roughness of the rocky platform
(Figure 2). However, as a tunable parameter, the selected
high z
0,b
value may also compensate for other unknown
sources of errors. As part of further work, a quantitative
analysis of the roughness elements distribution over the
rocky area would be required to improve the representation
of this feature.
[21] For the flow computation on domain G2, surface
elevations are prescribed at the offshore boundary from the
10 min measurements of the Socoa tide gauge. The flow of
each river is prescribed from in situ measurements at an
upstream location, where the influence of tidal oscillations
on the flow is negligible (see Figure 4). The flow is then
computed along the tidal section of both rivers, which is
approximately 6 km long for the main river and 3.5 km
long for the secondary river. At the most refined level G3,
open boundary conditions are imposed by the solution of
level G2. Both domains are forced by a homogeneous wind
field, given by the hourly record of the Socoa meteorologi-
cal station (Figure 4). Finally, wave related terms required
for the flow calculation are computed by WWIII from the
wave field and transmitted to MOHID-GLM every 30 min.
4. Results
4.1. Wave Transformation
[22] Wave transformation inside the bay is examined
during the time interval from 22 to 28 September. Model
results are compared with field data from the Anglet buoy
(Figure 4) and from the three bottom mounted sensors
deployed in the bay at this time (see section 2).
[23] Figure 5 shows observed and simulated time series
of significant wave height H
s
and mean direction D
m
at the
AWAC, ADVN, and ADVS locations. Measurements from
the AWAC allow us to identify two different wave events,
for which frequency spectra are plotted in Figure 6 : (1)
low-energy waves from 22 to 24 September with H
s
1m,
T
p
10 s, and D
m
315and (2) a more energetic sea
state with maximum H
s
2 m and mean direction D
m
325from 24 to 28 September, which consisted in a very
long swell with T
p
20 s superimposed to a shorter wave
system with T
p
7 s. Inside the bay, a strong decrease of
0
1
2
Hs
AWAC (m)
0
2
4
ηAWAC
(m)
0
1
Hs
ADVN (m)
0
1
Hs
ADVS (m)
270
300
330
Dm
AWAC (°)
270
300
330
Dm
ADVN (°)
22 23 24 25 26 27 28
270
300
330
Da
y
of September 2010 (UT+0h)
Dm
ADVS (°)
Observations
Model (Γ = 0.07 m2.s−3)
Model (Γ = 0.5 m2.s−3)
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5. Observed and modeled time series of hourly
averaged wave height H
s
at the (a) AWAC, (b) ADVN, and
(c) ADVS location. Water level recorded by the AWAC is
also plotted (gray line) in Figure 5a. Observed and modeled
time series of hourly averaged mean direction D
m
at the (d)
AWAC, (e) ADVN, and (f) ADVS location. Modeled
results are shown for two different bottom friction coeffi-
cients: the default value C50.07 m
2
s
23
(red) and the
selected value C50.5 m
2
s
23
(blue).
0
0.5
1
1.5
2
Ef (m2/Hz)
Observations
Model (Γ=0.07 m2.s−3)
Model (Γ=0.5)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Frequency (Hz)
Ef (m2/Hz)
(a)
(b)
Figure 6. Hourly averaged frequency spectra E
f
at the
AWAC location from field data (black), model with the
default (red) and selected (blue) friction coefficient (a) dur-
ing the low-energy wave event (23 September at 16 :00)
and (b) during the high-energy wave event (25 September
at 00:00).
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
75
wave energy is observed (ADVN,S), related to important
modifications of waves when propagating in the shallow
eastern part of the bay. First, waves are refracted over the
S-N oriented isobaths of the rocky platform. It causes wave
direction to shift from incident values between 310and
330(AWAC) to directions ranging from 275to 300on
the ADVN (Figures 5d and 5e). Second, wave energy is
dissipated over the rocky platform, due to depth-induced
breaking and bottom friction. The ADVN time series also
shows that wave transformation over the platform is
strongly modulated by the water level. Measured H
s
at the
ADVN location are approximately two times higher at high
tide than at low tide. The direction shift due to refraction
appears larger at low tide by 10–20compared to high
tide.
[24] For the wave simulation, the breaking parameter c
and the wave friction coefficient Cwere adjusted to (c,
C)5(0.75, 0.5 m
2
s
23
) on refined levels based on compari-
son of WWIII results with wave data recorded by the
AWAC and the ADVN,S. Results obtained with selected
parameters compare reasonably with field data during the 6
days interval examined here, for both low and high-energy
conditions. Cis essentially a tuning parameter here, cor-
recting for all possible errors in the definition of the off-
shore boundary conditions, and necessary to produce
reasonable wave parameters in the bay. The reader is
referred to Appendix B for an additional discussion of this
aspect and of other wave model parameters and error
statistics.
4.2. Wave-Induced Circulation in the Bay
[25] Wave fields computed by the WWIII model are now
used to force the 3-D flow model calculation on the study
area. Results of the wave-current modeling are examined
during the time interval from 22 to 28 September 2010, in
order to illuminate the main features of the bay circulation
in response to the different nearshore forcings. Special
attention is given to the effects of the increase in incident
wave energy on 24 September, in order to further investi-
gate the related impact on freshwater dynamics in the bay
(next paragraphs). MOHID-GLM has been successfully
validated and compared to reference wave-current models
on simplified nearshore configurations [see Delpey, 2012],
so that its ability to capture nearshore processes can be
expected. The present work is a first step in the small-scale
application of such a 3-D wave-current model to study a
complex realistic circulation in a stratified environment,
under both low and high-energy wave conditions.
[26] Figure 7 shows the surface Lagrangian mean veloc-
ity field computed by the model at three different instants
of the examined time interval : during the low-energy wave
episode at low tide (Figure 7a), and during the high-energy
wave episode at high tide (Figure 7b) and at low tide (Fig-
ure 7c). As expected in the studied mesotidal environment,
model results show a significant role of tide in the bay cir-
culation. In particular, the small cross section of the inlets
and of both river mouths causes the acceleration of baro-
tropic tidal currents, leading to mass exchanges between
the rivers, the bay and the inner shelf. However, in addition
to the tidal circulation, the wave-current modeling also sug-
gests a remarkable role of waves in the circulation patterns
inside the bay. Due to the angle between incident wave
direction and the orientation of the rocky platform isobaths
in the northeastern part of the bay (Figures 1 and 5), wave
breaking over the platform generates a longshore current
with south to southeastward direction. This feature is in
Figure 7. Modeled surface Lagrangian mean velocity on
23 September 2010 at (a) 09:00 (low tide) and on 24 Sep-
tember 2010 at (b) 16:00 (high tide), and (c) 23 :00 (low
tide). Velocity fields are averaged over the top 20% of the
water column. The colorscale indicates the velocity modu-
lus. Isobaths are superimposed in gray every 1 m.
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
76
agreement with the relatively well known generation mech-
anism first exposed by Longuet-Higgins [1970a, 1970b].
The intensity of the wave-induced longshore current in the
surf zone is important compared to the slower circulation
in the rest of the bay. The existence of this surf zone current
was consistent with a pilot drifter experiment performed in
the bay in conditions very similar to 23 September shown
in Figure 7a. An alongshore current was visually observed
during the experiment with approximately the same loca-
tion and intensity as in the model.
[27] Surface velocity fields in Figures 7b and 7c illustrate
the longshore current dependence on the mean water level.
For a given incident wave height and direction, a decrease
in the mean water level results in a displacement of the
maximum wave dissipation by breaking toward the south-
western deeper part of the platform. As a consequence, the
location of the induced longshore current also moves south-
westward (Figure 7c). Conversely, an increase in the mean
water level induces a displacement of the longshore current
location toward the northeastern shallower part of the plat-
form (Figure 7b), which results in a strong tidal modulation
of the wave-induced surf zone current. Figures 7a and 7c
illustrate the correlation between the longshore current
intensity and the incident wave heights. Moreover, compar-
ison of Figure 7a and Figure 7c shows that for a given
water level, the location of the longshore current also
moves southwestward when incident wave heights
increase, as wave breaking occurs in the deeper western
part of the platform, bringing another contribution to the
variability of the eastern bay circulation.
4.3. Freshwater Dynamics
4.3.1. River Plume Dispersion in the Bay
[28] We focus on the time frame from 22 to 28 Septem-
ber 2010. Figure 8 gives salinity fields obtained from the 3-
D wave-current model at low tide during both the high-
energy wave episode and the river flood event. To illustrate
the effect of waves on river plume dispersion, a model sim-
ulation is also performed without taking wave forcing into
account. Results are given with (Figures 8c and 8d) and
without (Figures 8a and 8b) wave forcing for both surface
(Figures 8a and 8c) and bottom (Figures 8b and 8d) water
column layers.
[29] In both simulations, the presence of the river plumes
results in a thin surface layer with lower salinity and typical
depth between 1 and 2 m. This important stratification
Figure 8. Fields of modeled salinity at low tide during the high-energy wave episode (26 September at
11:00) (a, b) without and (c, d) with wave forcing activated. (a, c) Average over the top 20% of the water
column and (b, d) average over the lowest 20% of the water column.
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
77
produced by the 3-D model is consistent with the few salin-
ity profile measurements collected in the bay (not shown,
see Delpey [2012]). Salinity fields exhibit a significant
space-time variability along the tide cycle, related to vary-
ing river outflows and to the following freshwater disper-
sion and mixing under the effect of the circulation.
[30] Comparison of model results with and without wave
forcing suggests a significant effect of waves on the salinity
field. The freshwater plume from the main river, in the east-
ern half of the bay, is particularly affected by wave-induced
circulation (Figures 8a versus 8c). The surf zone longshore
current over the rocky platform tends to advect salt water
into the northeastern part of the bay and thus to push fresh-
waters southward. It results in an accumulation of freshwater
close to the main river mouth. Also, the vertical mixing over
the rocky platform is higher with waves, due to the TKE sur-
face flux induced by wave breaking. It contributes to
increase the surface salinities in the northeastern area by
mixing the surface freshwater with the underlying ocean
water. Without wave forcing, a larger northward and east-
ward spreading of the river plume is obtained in the surface
layer, with stronger stratification close to the rocky platform.
Moreover, comparison of Figures 8a and 8c shows that a
higher freshwater outflow occurs through the bay inlets
when wave forcing is not taken into account.
[31] As illustrated by Figure 8, model results suggest a
significant modification of freshwater distribution inside
the bay due to the asymmetric forcing of the longshore cur-
rent on the bay circulation. Despite its restriction to the
confined and shallow surf zone area, the current induced by
wave breaking is much stronger than the slower tide- and
wind-induced circulations in the bay (especially at low
tide), so that the transport initiated in the surf zone affects
freshwater outflow through the bay inlets. This suggests
that the wave-induced circulation could also impact fresh-
water exchanges between the bay and the inner shelf. This
feature is now examined.
4.3.2. Freshwater Balance Under the Effect of Waves
[32] In the context of contamination processes related to
river outflows, it is interesting to examine how different
forcings, here tides, winds, and waves, affect the ability of
the bay to evacuate freshwaters. Despite the complexity of
the small-scale flow patterns inside the bay, the representa-
tion of the different processes by the wave-current 3-D
model can be used to draw tendencies in the local fresh-
water balance in response to a rainfall event. Here the com-
puted salinity is used to establish the balance of river
waters inside the bay during the time interval from 22 to 28
September. As mentioned previously, this interval corre-
sponds to a river flood event occurring under high-energy
wave conditions (see Figure 3). The total salt quantity in
the bay Qs;tot can be obtained by mixing a volume V
s
of
ocean water with salinity S
s
and a volume V
f
of freshwater,
so that Qs;tot 5VsSs. Introducing the total water volume in
the bay Vtot 5Vs1Vf, an equivalent freshwater volume
(EFV) can be estimated by
Vf5Vtot 2Qs;tot
Ss
:(4)
[33] The EFV is computed from model results based on
four simulations: (1) a wave-current simulation with all
forcings, (2) a model simulation with wave forcing
switched off, (3) a wave-current simulation with wind forc-
ing on the flow switched off, and (4) a wave-current simu-
lation without Stokes drift contribution, i.e., with u
S
50.
The EFV time series for the four simulations are plotted in
Figure 9, together with time series of the wind speed, the
incident wave height H
s
and the sum of the two river flows.
In Figure 9b, the instantaneous EFV (dashed) are superim-
posed to a residual EFV (full lines) estimated by applying a
24 h 50 min average sliding window. Finally, the time
derivative of this residual EFV is plotted in Figure 9c. The
water flux d(EFV)=dt gives the overall freshwater flow into
the bay, which equals the sum of freshwater fluxes through
the two inlets and through the two river outlets.
[34] The comparison of simulations 1 and 3 suggests that
the wind has little effects on the freshwater balance inside
the bay during the examined time period, although the
wind speed exceeds 10 m s
21
on 24 September. On the
contrary, the comparison of simulations 1 and 2 suggests a
significant effect of waves on the freshwater balance. The
computed EFV is higher in simulation 1 than in simulation
2 at the end of the river flood event on 28 September (Fig-
ure 9b). Following the increase in the river flow (on 24
September), results of simulation 2 show a clearly corre-
lated response of the bay to this freshwater input. An
increase of the EFV is first obtained, followed by a com-
pensating decrease starting less than 16 h after the peak of
the river flood. In simulation 1, the increase in the EFV is
slower but with a longer duration. The EFV decreasing
phase only starts at the end of the examined time interval,
i.e., more than 40 h after the peak of the river flood, when
incident wave height H
s
becomes lower than 1 m. Thus,
0
1
2
3
Hs (m)
0
5
10
15
Wind speed (m.s−1)
1
1.5
2
2.5
3
3.5
x 105
EFV (m3)
1) Full model
2) Model without waves
3) Model without wind
4) Full model with uS=0
22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 27 27.5 28
−1
−0.5
0
0.5
1
1.5
d(EFV)/dt (m3.s−1)
Day of September (UT+0h)
2
3
4
5
6
7
River flow (m3.s−1)
(a)
(b)
(c)
Figure 9. Time series of (a) the wave height H
S
measured
by the AWAC (black) and the wind speed (gray); (b) the
instantaneous (dashed) and the residual (full line) EFV ;
and (c) the time derivative of the residual EFV (colored)
and the sum of the two river flows (gray).
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
78
compared to simulation 2, a delay of approximately 24 h
is obtained for the EFV flux to become negative. At the
end of the river flood event, the EFV in simulation 1 is
about 40% higher than in simulation 2. The freshwater
outflow on 27 September is of the order of 0.1 m
3
s
21
with waves, which is still much lower than the 0.5 m
3
s
21
outflow obtained without waves.
[35] The obtained tendencies suggest that waves contrib-
ute to maintain freshwaters inside the bay. Although the
EFV increases faster without waves at the peak of the river
flood event, at the end of this event, a higher volume of
freshwater is retained inside the bay due to wave forcing.
Waves delay the evacuation of this volume and limit the
freshwater outflow until low-energy conditions are met. An
additional experiment performed in the wave case confirms
that the extra freshwater volume retained in the bay would
be subsequently evacuated from 28 to 30 September if very
low-energy waves were encountered during this time inter-
val (not shown). The Lagrangian transport by waves
(Stokes transport) could be a possible driving factor of
freshwater retention in the bay. However, a simulation with
Stokes transport turned off (Figure 9, light blue lines)
shows that the Stokes drift has little contribution in the
present configuration. This result suggests that the outflow
reduction is mainly due to the wave-induced longshore cur-
rent in the eastern bay surf zone. It generates a transport
oriented toward the inner bay, in opposition with tidal cur-
rents during ebb tide. This reduces the primary outflow of
freshwater through the eastern inlet (also illustrated in Fig-
ure 8), yielding a reduction of the bay flushing during the
examined time interval.
4.3.3. Freshwater Balance Sensitivity to Incident
Wave Characteristics and Longshore Current Velocity
[36] To investigate the variability of the bay flushing
with incident wave characteristics, the modeled bay
dynamics is now examined in a simplified configuration
under different offshore wave conditions. Simulations are
performed using a sinusoidal tidal forcing of the water level
with period 12 h 25 min (corresponding to the dominant
M2 component) and a constant 3 m tidal range. Wind forc-
ing is switched off and upstream river flows are constant,
set to 2.5 m
3
s
21
for the main river and 0 m
3
s
21
for the
secondary river. At the offshore boundary of the domain
G1 (Figure 4), constant wave conditions are prescribed by
a JONSWAP spectrum. First to investigate the effect of
incident wave energy, simulations are carried out using an
offshore peak period T
p
510 s, peak direction h
p
5310,
and significant wave height H
s
in {0.5, 1, 2, 3, and 4 m}.
Second to investigate the effect of incident wave direction,
simulations are carried out with offshore T
p
510 s, H
s
52
m, and h
p
in {280, 340}. Finally, a simulation is also per-
formed with wave forcing switched off. The flow model
computation is initialized without wave forcing. Waves are
then introduced at day 0 and the subsequent bay response is
examined. Figure 10 shows the surface Lagrangian mean
velocity field computed by the model for different offshore
wave conditions at the same time instant, during ebb tide of
day 2. In addition, Figure 11 represents the maximum
velocity V
max
of the longshore current over the simulated
time interval, as a function of the offshore wave height H
s
.
Finally, residual EFV obtained from the ensemble of simu-
lations are plotted in Figure 12.
Figure 10. Modeled surface Lagrangian mean velocity during ebb tide of day 2 for (a) (H
s
,h
p
)5(0.5
m, 310), (b) (H
s
,h
p
)5(2 m, 280), (c) (H
s
,h
p
)5(2 m, 310), (d) (H
s
,h
p
)5(2 m, 340), (e) (H
s
,
h
p
)5(3 m, 310), and (f) (H
s
,h
p
)5(4 m, 310). Velocity fields are averaged over the top 20% of the
water column.
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
79
[37] Figure 12b shows that simulation without waves
(black line) gives a nearly constant residual EFV during the
examined time interval. This results from the equilibrium
between tidal and river inflow forcings. Considering wave
cases with offshore H
s
51 m and H
s
52 m (Figure 12b,
red and light blue lines), a comparable EFV increase of
approximately 40% is obtained at day 2, consistent with
results from the realistic simulation exposed in the previous
section. With offshore H
s
50.5 m (dark blue line), the
residual EFV is lower than with H
s
between 1 and 2 m by
approximately 15%. Figures 10a and 11 show that this EFV
reduction is correlated with a reduction of the longshore
current intensity V
max
, due to lower incident wave energy
(for a given incident direction). Conversely, Figures 10e,
10f, and 11 illustrate the increase in V
max
with higher inci-
dent wave energy H
s
53 and 4 m compared to H
s
52m.
However, a further increase in the residual EFV with V
max
is not obtained for H
s
>2 m. The wave-induced EFV at day
2 is lower by 10% for H
s
53 m (Figure 12b, green line)
and by 25% for H
s
54 m (purple line), compared to the
case with H
s
52m.
[38] With smaller differences, the same tendency is
obtained when examining the different incident wave direc-
tions with H
s
52 m. Figures 10b, 10d, and 11 show an
increase in V
max
with the incident wave direction. This fea-
ture relates to the dependence of the longshore current
velocity on the angle between incident waves and the S-N-
oriented isobaths of the rocky platform. A larger angle (i.e.,
norther incoming waves) generates a stronger longshore
current. However, examining Figure 12b shows that this
stronger current does not result in a higher EFV at the end
of the simulation: the EFV is slightly lower for N/NW inci-
dent direction (red circles) than for NW and W/NW direc-
tion (red line and red squares).
[39] To further investigate the relationship between V
max
and freshwater exchanges with the inner shelf, freshwater
fluxes through the bay inlets are now examined. Consider-
ing that the total water flux F
tot
through one inlet cross-
section results from the sum of a flux F
s
of ocean water
with salinity S
s
and a flux F
f
of freshwater, an equivalent
freshwater flux through the inlet cross section can be esti-
mated from the model by
Ff5Ftot
Ss2S
Ss
:(5)
Then integrating this flux over the simulated time interval,
the total freshwater volume (or EFV) exchanged through
the considered inlet is computed. The flux F
f
and the corre-
sponding EFV exchange are counted positively when enter-
ing the bay and negatively when directed seaward. Figure
13 gives the total EFV exchange during the simulated time
interval partitioned into each inlet (Figure 13b) and through
both inlets (Figure 13c, sum of EFV exchanges through
both inlets). In addition, Figure 13a shows the residual
EFV in the bay at day 2. All of these freshwater volumes
are plotted as a function of the maximum velocity V
max
of
the wave-induced longshore current over the rocky plat-
form. The value V
max
50 is used to represent the no wave
case.
[40] Figure 13a shows the increase in the freshwater vol-
ume retained inside the bay for V
max
lower than approxi-
mately 0.6 m s
21
. This corresponds to H
s
<2 m for NW
incident waves. In consistency with the mechanism sug-
gested previously, Figure 13b shows that under these wave
conditions, the wave-induced transport initiated in the surf
zone reduces the primary freshwater outflow through the
eastern inlet. At the same time, a larger EFV is evacuated
through the western inlet. However, Figure 13c suggests
that this western outflow increase does not compensate the
eastern outflow reduction, as the sum of the EFV exchanges
through both inlets increases (reminding that it is counted
negatively seaward). This results in freshwater retention
inside the bay, with an increased residual EFV in the bay at
day 2 of the simulation.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Incident Hs (m)
Vmax (m.s−1)
Figure 11. Maximum velocity V
max
of the wave-induced
longshore current over the simulated time interval for the
different offshore wave conditions. Black squares corre-
spond to conditions with h
p
5310, the black triangle and
circle correspond to h
p
5280and h
p
5340, respectively.
1
2
3
4
η (m)
0 0.25 0.5 0.75 1 1.25 1.5 1.75 22.25
2
2.5
3
3.5 x 10
5
EFV (m
3
)
Days
No waves
Hs=0.5m,θ
p
=310°
Hs=1m,θ
p
=310°
Hs=2m,θ
p
=310°
Hs=3m,θ=310°
Hs=4m,θ
p
=310°
Hs=2m,θ
p
=280°
Hs=2m,θ
p
=340°
(a)
(b)
Figure 12. Time series of (a) mean water level ginside
the bay and (b) residual EFV in the bay computed without
(black line) and with (colored lines and symbols) wave
forcing under different incident wave conditions.
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
80
[41] On the contrary, a further increase in incident wave
energy promotes a larger bay flushing, as illustrated by Fig-
ure 13a for V
max
>0.6 m s
21
. Figures 13b and 13c show
that for these larger V
max
values, the increase in the fresh-
water outflow through the western inlet becomes more
important than the outflow reduction in the eastern inlet.
Indeed, the location of the surf zone and the very strong
longshore current in the rocky area result in an even more
important westward advection of freshwater masses inside
the bay. Freshwater finally concentrates in the western part
of the bay, close to the western bay inlet. This enables a
higher freshwater outflow through the western inlet during
the tide cycle, which results in a reduction of the retention
effect of wave forcing.
[42] Model results thus suggest a major sensitivity of the
bay flushing to the incident wave energy and the corre-
sponding longshore current velocity. For typical incident
peak period T
p
510 s and peak direction h
p
5310, maxi-
mum freshwater retention in the bay is found for H
s
between 1 and 2 m, which corresponds to V
max
values
between 0.5 and 0.6 m s
21
. This corresponds also to the
range of wave conditions encountered during the episode
from 22 to 28 September 2010, examined in this paper.
5. Conclusions
[43] A 3-D wave-current numerical model was used to
investigate the effect of tides, winds, and waves on the
dynamics of a small estuarine bay. In the studied configura-
tion, a strong salinity stratification is associated with river
plumes, leading to a highly 3-D behavior. To account for
this feature, the flow model resolves the vertical structure
of wave momentum and forcing, as well as wave-induced
vertical mixing. Although additional efforts will be
required for a quantitative small-scale validation of mod-
eled flow patterns, the present work takes advantages of the
3-D model representation to propose an analysis of local
dispersion processes, as a first step in the study of the com-
plex bay dynamics.
[44] A time interval, combining both a river flood event
and an increase in incident wave energy, was examined to
illustrate possible effects of waves on the bay response to
freshwater inputs. Results of numerical modeling suggest a
significant impact of waves on the bay dynamics, larger
than wind effect during the examined time interval.
Obliquely incident waves refract and break over a rocky
platform inside the bay, resulting in a longshore current
strongly modulated by tide. During the examined energetic
wave event (H
s
2 m), the computed current intensity is
very large in the surf zone area, compared to the slower cir-
culation in the other deeper parts of the bay.
[45] Modeled salinity fields show a thin surface layer
with lower salinity, associated with river plumes spreading
inside the bay. The surface layer depth and profile shape
exhibit a large space-time variability along the tide cycle,
related to transport and mixing inside the bay. During the
examined high-energy wave episode, model results suggest
that the wave-induced longshore current reduces freshwater
outflow through the bay inlets. The asymmetric wave forc-
ing on the circulation results in a higher concentration of
freshwater from the main river in the southeastern part of
the bay. The model is further used to determine the related
effects on the local freshwater balance of the bay at the
scale of the river flood event. It is found that waves can
contribute to concentrate freshwater inside the bay. At the
end of the examined event, the estimated freshwater vol-
ume inside the bay is 40% higher and the freshwater out-
flow is still reduced by a factor 5 due to waves.
[46] The sensitivity of the wave-induced freshwater
retention to incident wave conditions was finally investi-
gated. The analysis of model results shows a strong
dependence of the bay flushing to incident wave energy
and to the corresponding longshore current velocity. Maxi-
mum freshwater volume inside the bay was found for inter-
mediate offshore wave heights 1 m <H
s
<2 m and
longshore current velocity V
max
0.6 m s
21
, whereas
freshwater retention may be reduced by a factor 2 for lower
(H
s
0.5 m, V
max
0.25 m s
21
) or higher (H
s
4m,
V
max
0.8 m s
21
) incident wave energy.
[47] The wave-induced reduction of the bay flushing sug-
gested by this modeling study may have a large impact on
the exchanges between the bay and the inner shelf. In par-
ticular, it may be a key mechanism in the dispersion of ter-
restrial contaminant carried by rivers in the near shore,
contributing to determine water quality inside the bay.
Appendix A: MOHID-GLM Implementation
[48] This appendix presents the 3-D wave-current set of
equations implemented in MOHID-GLM. The formulation
corresponds to Ardhuin et al. [2008b] as adapted by Bennis
et al. [2011], with some specificities related to the MOHID
model and to the studied area. A synopsis of MOHID-GLM
is given below. The reader is referred to Delpey [2012] for
a detailed description of the code implementation.
2
3
4
EFV (105 m3)
Residual bay
−10
0
10
EFV (105 m3)
Inlet E
Inlet W
0 0.2 0.4 0.6 0.8
−5
−4
−3
−2
EFV (105 m3)
Vmax (m.s−1)
Inlet E + W
(a)
(b)
(c)
Figure 13. (a) Residual EFV in the bay at the end of the
simulation (day 2), (b) total EFV exchange through the
eastern inlet (triangles) and through the western inlet
(circles), and (c) total EFV exchange through both inlets
over the simulated time interval (day 0–2), as a function of
the maximum velocity V
max
of the longshore current.
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
81
A1. Governing Equations
[49] The quasi-Eulerian velocity field, noted
^
u5ð^
u1;^
u2;^
wÞ, is defined as [Jenkins, 1989]
ð^u1;^u2;^wÞ5ðuL
1;uL
2;wLÞ2ðuS
1;uS
2;wSÞ;(A1)
where uL5ðuL
1;uL
2;wLÞis the Lagrangian mean velocity
field and uS5ðuS
1;uS
2;wSÞis the Stokes drift velocity field.
[50] The vertical coordinate change used by Ardhuin
et al. [2008b] corrects the vertical coordinate for the GLM-
induced vertical displacement, so that ^
uand uLare nondi-
vergent. The nondivergence of uLis used by the model in
its vertically integrated form, from the bottom depth h to
the mean surface elevation ^
g, which results in the following
equation for ^
g:
@^
g
@t
1@h^
uai1huS
ai
@xa
50;(A2)
where tis the time; fxa;a2½1;2g are the horizontal space
coordinates; zis the vertical coordinate; hð:Þi denotes
depth-integrated variables; and the summation convention
for repeated indices is used. It should be noted that the ver-
tical coordinate change used by Ardhuin et al. [2008b] cor-
rects the quasi-Eulerian free surface position z5^
gfor the
Stokes correction of g, so that it is equal to the local phase-
averaged free surface position z5g.
[51] The conservation equation implemented in
MOHID-GLM for the quasi-Eulerian momentum is given
in a flux-divergence form by
@^
ua
@t
1
@ð^
ub1uS
bÞ^
ua
hi
@xb
1@ð^
w1wSÞ^
ua
@z
52 1
q0
@pH
@xb
2@SJ
@xb
1uS
b
@^
ub
@xa
1@
@xb
KH
@^
ua
@xb
1@
@zKV
@^
ua
@z
;
(A3)
where q
0
is the water mean density; pHdenotes the mean
pressure, which is assumed to be hydrostatic ; S
J
is the
wave-induced pressure term ; K
H
,K
V
are the horizontal and
vertical eddy viscosities, respectively. This equation is
equivalent to equations (12) and (13) in Bennis et al.
[2011] because u
S
is nondivergent [Ardhuin et al., 2008b].
Here the momentum equation is implemented using the
Cartesian z-coordinate, as the numerical resolution is
always achieved by MOHID in the Cartesian space for any
type of vertical discretization (Cartesian or not), with a pro-
cedure equivalent to a generic vertical coordinate [see Mar-
tins et al., 2001]. In the present study, a sigma vertical
discretization is used (terrain-following).
[52] In equations (A2) and (A3), the wave related terms
S
J
and uS
aare given, respectively, by
SJ5ðk
gk EðkÞ
sinh 2kD dk;(A4)
and
uS
a5ðk
USS
aðkÞcosh ð2kz12khÞ
sinh 2ðkDÞdkif kD <6;(A5)
uS
a5ðk
USS
aðkÞexp 2kðz2gÞ½dkif kD 6;(A6)
where k5jjkjj;ris the wave intrinsic pulsation; and USS
a
ðkÞ5rkaEðkÞdenotes the spectrum of the surface Stokes
drift horizontal components. The vertical component of
the Stokes drift w
S
can be computed using the nondiver-
gence of the Stokes drift velocity field by
wSðzÞ52uS
að2hÞ@h
@xa
2ðz
2h
@uS
a
@xaðz0Þdz 0:(A7)
[53] Density evolution is computed from Sand Tthrough
a state relationship [UNESCO, 1981]. If Cdenotes Sor T,
the conservation of Cis expressed by
@C
@t
1@ð^
ua1uS
aÞC
@xa
1@ð^
w1wSÞC
@z
5@
@xa
KH
@C
@xa
1@
@zKC
@C
@z
1SC;
(A8)
where K
C
and S
C
are the vertical eddy diffusivity and the
source term associated with the tracer C, respectively. In
equation (A8), the horizontal eddy diffusivity for Cis equal
to the horizontal eddy viscosity, both noted K
H
.
A2. Boundary Conditions for the Quasi-Eulerian Flow
[54] Boundary conditions in MOHID-GLM allow to
account for different effects of waves on the quasi-Eulerian
flow. Kinematic surface and bottom boundary conditions
are, respectively, given by
@^
g
@t
1^
ua1uS
a
@^
g
@xa
5^
w1wSat z5^
gðx1;x2;tÞ;(A9)
^
ua1uS
a
@ð2hÞ
@xa
5^
w1wSat z52hðx1;x2Þ:(A10)
[55] Fluxes of momentum from wind and from wave
breaking are introduced in the equations through the
dynamic surface boundary condition:
KV
@^
ua
@z
5sa
a2saw
a1soc
aat z5^
gðx1;x2;tÞ;(A11)
where s
a
is the total momentum flux from the atmosphere
to the ocean (wind stress), computed from the local wind
speed using a quadratic friction law, as proposed by Large
and Pond [1981]; s
aw
is the momentum flux from the
atmosphere to waves (or wave-supported wind stress) ; and
s
oc
is the momentum flux from waves to the mean current
due to wave breaking. s
aw
and s
wo
are computed from spec-
tral wave energy source terms S
in
and S
oc
, respectively
(introduced in paragraph 3.2.), according to Ardhuin et al.
[2009, 2010].
[56] At the bottom, the combined wave and current stress
s
b
is computed according to Soulsby et al. [1995]. It results
in the following condition:
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
82
KV
@^
ua
@z
5sb
c;a111:2jjsb
wjj
jjsb
cjj1jjsb
wjj
3:2
"#
at z52hðx1;x2Þ;
(A12)
where the mean-current bottom stress sb
cand the wave-
induced bottom stress sb
ware given, respectively, by
sb
c5q0CDjj^
ujj^
uand sb
w51
2q0Fwjju0jju0;(A13)
with C
D
aChezy-type bottom drag coefficient ; F
w
a bottom
friction coefficient; and u0the wave orbital velocity. In
(A13) both ^
uand u0are evaluated at the top of the bottom
boundary layer (which is not resolved here). The bottom
drag coefficient C
D
is given from a bottom roughness
length z
0,b
by
CD5j
log ðz01z0;b
z0;bÞ
0
@1
A
2
;(A14)
where z0is the distance from the top of the bottom bound-
ary layer. In accordance with Soulsby et al. [1995], the fric-
tion coefficient F
w
is evaluated by
Fw51:39 jju0jj
rpz0;b
20:52
;(A15)
where rp52pfp, with f
p
the peak frequency.
[57] At open boundaries, a mixed radiation-relaxation
condition is used, transferring the methodology proposed
by Marchesiello et al. [2001] to the present smaller scale
application. A Flather radiation condition is applied for the
barotropic flow [Flather, 1976]. The original formulation is
modified to account now for the linearized barotropic equi-
librium of the quasi-Eulerian flow, instead of the total flow.
It results in the following condition at open boundaries:
^
g2^
gext
56 ffiffiffiffi
D
g
rh^
ui1huSi
n2h^
uext i1huSext i
n
;(A16)
where ^
gext ;^
uext ;uSext are the imposed values of ^
g;^
u;uSat
the open boundary; nis the vector normal to the open
boundary. In addition, a relaxation scheme is used for the
baroclinic flow. If /denotes the horizontal components of
the baroclinic velocity, the open boundary condition is
given by
@/
@t
52 1
srelax ð/2/ext Þ;(A17)
where s
relax
is a relaxation coefficient, which is set a to low
value in a nudging layer near the boundary and to a very
large value in the rest of the domain. Baroclinic modes are
not radiated at open boundaries, following arguments pre-
sented, for example, by Blayo and Debreu [2005] and
Leit~
ao et al. [2008]. A similar relaxation scheme is applied
to salinity and temperature fields.
[58] At land boundaries, a free slip condition is assumed
with zero depth-integrated total mass flux in direction nor-
mal to the boundary.
A3. Turbulent Closure
[59] In equations (A3) and (A8), a constant value is used
for the horizontal eddy viscosity K
H
. This simple parame-
terization is used here as a first approximation, and the
main features described in this paper were still obtained
when testing different values of K
H
(not shown). However,
the refinement of this parameterization, e.g., to account for
horizontal diffusion due to wave breaking in the surf zone
[e.g., Feddersen, 2007; Spydell et al., 2007, 2009], would
be an interesting aspect to be investigated as part of further
work.
[60] For the vertical turbulent closure, the MOHID sys-
tem is coupled to the General Ocean Turbulence Model
(GOTM) [Burchard and Bolding, 2001], a 1-D water col-
umn model proposing several turbulent closure schemes. In
the present study, a K2model [Rodi, 1980] is used. The
vertical eddy viscosity and diffusivity (K
V
,K
C
) are parame-
terized as ðKV;KCÞ5ðSV;SCÞðq2=2Þ221,whereq
2
/2 is
the TKE, is the TKE-dissipation rate, and S
V
,S
C
are
stability functions. The formulation proposed by Canuto
et al. [2001] is used for S
V
and S
C
. Equations for q
2
/2 and
are given by
@q2=2ðÞ
@t
5@
@zKV
@q2=2ðÞ
@z
1Ps1Pb2; (A18)
@
@t
5@
@z
KV
r
@
@z
12
q2ðc1Ps1c3Pb2c2Þ;(A19)
with P
s
and P
b
the TKE productions by vertical shear and
buoyancy, respectively, given by Ps5@ua
@z
@ua
@zand Pb52 g
q0
@q
@z;ris the Schmidt number for ; and c
1
,c
2
,c
3
are empir-
ical constants, which are prescribed according to Canuto
et al. [2001] for this study.
[61] Wave breaking can greatly affect vertical mixing as
it provides an important source of TKE near the surface
compared to P
s
or P
b
[Agrawal et al., 1992; Terray et al.,
1996; Feddersen, 2012a, 2012b]. Effects of wave breaking
on vertical turbulence are taken into account through the
surface boundary conditions for (A18) and (A19), which
were adapted from Craig and Banner [1994], Craig [1996]
by Burchard [2001]. The surface boundary conditions for
q
2
=2 and are, respectively, given by
KV
@q2=2ðÞ
@z
5Foc;(A20)
KV
r
@
@z
5KV
r
C3
0
3
2
C3
0
SV
Foc1jðq2=2Þ3
2
1
j2ðz01z0;sÞ2;(A21)
with F
oc
the surface TKE flux due to wave breaking ; C3
0a
constant; z0the vertical distance from the surface; and z
0,s
a surface roughness length. In addition to these boundary
conditions, Burchard [2001] established a parameterization
of the Schmidt number ras a linear function of ðPs1PbÞ=
in order to obtain a behavior of the mixing length l5C3
0
ðq2=2Þ3
221like l5jðz01z0;sÞnear the surface, as prescribed
by Craig and Banner [1994]. In the present work, F
oc
is
specified from the wave breaking dissipation term S
oc
,
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
83
computed by the wave model according to Janssen et al.
[2004]. Wave breaking also envolves an enhanced value of
z
0,s
[Craig and Banner, 1994], which is proportional to the
total significant wave height H
S
in the present application
[Terray et al., 1996, 2000]:
z0;s5a0HS;(A22)
with a
0
a constant. The studied bay shows areas exposed to
incident waves and areas sheltered by breakwaters. The
main exposed area corresponds to the surf zone in the
northeast of the bay, where the whole wave spectrum
(including swell) can be affected by depth-induced break-
ing. The rest of the domain is mostly sheltered by the
breakwaters, so that the sea state generally consists in a
small wind sea which is generated locally. The total H
S
should thus be representative of the wave spectrum part
which is affected by breaking in both exposed (depth-
induced breaking) and sheltered (whitecapping) areas. As a
consequence, z
0,s
is considered proportional to the total H
S
,
because it generally provides the scale of breaking waves
in the studied bay.
[62] In the present work, wave related terms H
S
,f
p
,S
J
,
U
SS
,s
aw
,s
oc
, and F
oc
are computed by WWIII from the
wave field and source terms, and then transmitted to
MOHID-GLM for the flow computation.
Appendix B: Discussion of Wave Model Results
and Parameterization
[63] To obtain a better agreement with field data, the
friction coefficient in the bay has been significantly
increased compared to more common values used over
sand beds [e.g., Hasselmann et al., 1973]. Because of the
present site complexity involving combined refraction,
wave breaking and frictional dissipation, it is difficult to
confirm the role of bottom friction in the observed wave
transformation. However, preliminary tests with a constant
Nikuradse roughness length of 10 cm give similar results.
This value is comparable to the 12 cm used for rock plat-
forms by Ardhuin and Roland [2013]. Such a high friction
level could be related to the unusually rough surface of the
rocky platform (Figure 2) and of other rocky formations
over the inner shelf offshore of the bay [Augris et al.,
1999]. The importance of wave frictional dissipation has
been emphasized by several authors in shallow areas with
very rough bottoms, in particular coral reefs [e.g., Lowe
et al., 2005; Cialone and Mckee Smith, 2007 ; Filipot and
Cheung, 2012]. For example, based on field measurements
on the reef flat of Kaneohe Bay, Hawaii, Lowe et al. [2005]
calculated the averaged value of the wave friction factor
f
w
50.28 60.04, and mentioned that this value is 30 times
larger than the typical value of 0.01 for flat sandy bottoms.
According to the model of Madsen et al. [1988], f
w
can be
related to Cby
C5fw
g
ffiffiffi
2
pUrms ;(B1)
where U
rms
is the root-mean-square wave orbital velocity at
the bottom. For typical values of U
rms
between 0.1 and 0.5
ms
21
, this leads to Cvalues between 0.2 and 1.0 m
2
s
23
.
This range of Cvalues also corresponds to the order of
magnitude obtained for another Hawaiian reef area by
Cialone and Mckee Smith [2007], who used a coefficient
c
f
5C/gbetween 0.05 and 0.12 m s
21
. However, specific
investigations would be required to confirm that the level
of frictional wave dissipation over the studied rocky area
can be comparable to that induced by coral reefs. This large
Cvalue may be necessary here to correct for all possible
errors in the definition of the offshore boundary conditions.
[64] Figures 5 and 6 show that the increase in Csignifi-
cantly improves model results, in particular inside the bay
at the ADVN and ADVS locations. Table B1 gives root-
mean-square errors (RMSE) between observed and mod-
eled wave height H
s
, mean period T
02
and mean direction
D
m
for both default and selected model parameterization.
Normalized root-mean-square errors (NRMSE) are also
given for H
s
and T
02
. The two error indicators are defined,
respectively, as
RMSE51
NX
N
i51
Si2Oi
ðÞ
2
"#
1=2
;NRMSE5RMSE
RMS ;(B2)
where ðOiÞi51::Nand ðSiÞi51::Nare the observed and simu-
lated values, respectively, and RMS denotes the root-mean-
square of ðOiÞi51::N. Error statistics are given for the three
sensors deployed in the bay and also for the Anglet wave
buoy, located in the northeast of domain G1 at depth 50 m
(Figure 4).
[65] Results obtained with selected parameters compare
reasonably with field data during the 6 days interval exam-
ined here, for both low and high-energy conditions. Wave
energy associated with the long swell is still overestimated
by the model during the high-energy episode (Figure 6b),
possibly due to local nonresolved bathymetric effects out-
side the bay. The computed wave transformation in the east
part of the bay reproduces the direction shift and wave dis-
sipation due to the rocky platform and the related tidal
modulation. The higher error level at the ADVN location
could be related to the horizontal resolution of the
Table B1. Errors Between Observed and Simulated Wave Bulk
Parameters at Anglet Buoy, AWAC, ADVN and ADVS Locations
From 22 to 28 September
a
Sensor Parameter
Model With Default CModel With Selected C
RMSE, NRMSE RMSE, NRMSE
Anglet Buoy H
S
13 cm, 7% 13 cm, 7%
T
02
0.6 s, 8% 0.6 s, 8%
D
m
33
AWAC H
S
20 cm, 16% 16 cm, 13%
T
02
1.5 s, 25% 1.3 s, 21%
D
m
44
ADVN H
S
36 cm, 65% 17 cm, 31%
T
02
2.3 s, 37% 1.8 s, 31%
D
m
88
ADVS H
S
19 cm, 50% 7 cm, 19%
T
02
2.4 s, 34% 2.0 s, 32%
D
m
77
a
Error statistics at the ADVS location are given for the shorter time
interval from 25 to 27 September (due to the sensor dysfunction).
DELPEY ET AL.: EFFECTS OF WAVES ON COASTAL DISPERSION
84
computational grid, which limits the representation of wave
refraction over the complex bathymetry of the bay. Indeed,
calculations carried out with a spectral refraction-
diffraction model generally produced larger spatial gra-
dients (not shown), while giving the same patterns of wave
heights at scales of 100 m and larger inside the bay. The
use of a constant breaking parameter cmay also be a limi-
tation [e.g., Bruneau et al., 2011].
[66]Acknowledgments. M.T.D. acknowledges the support of a
research grant from ANRT (CIFRE grant), and F.A. is supported by
IOWAGA and Field_AC projects. The present study was supported in part
by the LOREA project, led by the Conseil G
en
eral des Pyr
en
ees Atlan-
tiques (http: //www.lorea.eu/). LOREA 2010 field experiment was per-
formed in collaboration with AZTI-Tecnalia (Marine Research Division),
G.E.O Transfert and CASAGEC. Lagrangian drifter experiment was per-
formed in collaboration with the SHOM.
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