This paper describes the implementation of a coupling between a three-dimensional ocean
general circulation model (NEMO) and a wave model (WW3) to represent
the interactions of upper-oceanic flow dynamics with surface waves. The focus is on the impact
of such coupling on upper-ocean properties (temperature and currents) and mixed layer depth (MLD)
at global eddying scales. A generic coupling interface has been developed, and the NEMO
governing equations and boundary conditions have been adapted to include wave-induced terms following
the approach of and . In particular, the contributions of Stokes–Coriolis,
vortex, and surface pressure forces have been implemented on top of the necessary modifications of the
tracer–continuity equation and turbulent closure scheme (a one-equation turbulent kinetic energy – TKE – closure here). To assess the new
developments, we perform a set of sensitivity experiments with a global oceanic configuration at 1/4∘
resolution coupled with a wave model configured at 1/2∘ resolution. Numerical simulations show a global
increase in wind stress due to the interaction with waves (via the Charnock coefficient), particularly at high
latitudes, resulting in increased surface currents. The modifications brought to the TKE closure scheme and
the inclusion of a parameterization for Langmuir turbulence lead to a significant increase in the mixing, thus
helping to deepen the MLD. This deepening is mainly located in the Southern Hemisphere and results in
reduced sea surface currents and temperatures.
Introduction
An accurate representation of ocean surface waves has long been recognized as essential
for a wide range of applications from marine meteorology to ocean and coastal
engineering. Waves also play an important role in the short-term forecasting of extratropical
and tropical cyclones by regulating sea surface roughness .
More recently, the impact of waves on oceanic circulation at the global scale has triggered
interest from the research and operational community e.g.,.
In particular, surface waves are important for an accurate representation of air–sea
interactions, and their effect on fluxes of mass, momentum, and energy through the
wavy boundary layer must be taken into account in ocean–atmosphere coupled models.
For example, the momentum flux through the
air–sea interface has traditionally been parameterized
using near-surface winds (typically at 10 m) and the
atmospheric surface layer stability .
The physics of the coupling depend on the kinematics and dynamics of the wave field.
This includes a wide range of processes from wind–wave growth, nonlinear wave–wave interaction,
and wave–current interaction to wave dissipation. Such complex processes can only be adequately
represented by a wave model.
Besides affecting the air–sea fluxes, waves define the mixing in the oceanic surface boundary
layer (OSBL) via breaking and Langmuir turbulence. For example, showed
that Langmuir turbulence should be important over wide areas of the global ocean, more
particularly in the Southern Ocean. In this region, they show that the inclusion of the effect of
surface waves on the upper-ocean mixing during summertime allows for a reduction of systematic
biases in the OSBL depth. Indeed, their large eddy simulations (LESs) suggest that under certain
circumstances wave forcing can lead to large changes in the mixing profile throughout the OSBL
and in the entrainment flux at the base of the OSBL. They concluded that wave forcing is always
important when compared to buoyancy forcing, even in winter. Moreover,
and emphasized the fact that the Langmuir cell intensity strongly depends
on the alignment between the Stokes drift and wind direction. Langmuir turbulence is maximum
when wind and waves are aligned and becomes weaker as the misalignment becomes larger.
highlighted that ignoring the alignment of wind and waves (i.e., assuming
that wind and waves are systematically aligned) in the Langmuir cell parameterizations
leads to excessive mixing, particularly in winter.
Most previous studies of the impact of ocean–wave interactions at the global scale have
used an offline one-way coupling and included only parts of the wave-induced terms in
the oceanic model governing equations e.g.,. In this study,
the objective is to introduce a new online two-way-coupled ocean–wave modeling system
with great flexibility to be relevant for a large range of applications from climate modeling to
regional short-term process studies. This modeling system is based on the Nucleus for European
Modelling of the Ocean NEMO; as the oceanic compartment
and WAVEWATCH III® hereinafter WW3;
as the surface wave component. NEMO and WW3 are coupled using the
OASIS Model Coupling Toolkit OASIS-MCT;,
which is widely used in the climate and operational community. The various steps for our
implementation are the following: (i) the inclusion of all wave-induced terms in NEMO,
only neglecting the terms relevant for the surf zone, which is outside the scope here; (ii) modification of the NEMO subgrid-scale physics (including the bulk formulation) to
include wave effects and a parameterization for Langmuir turbulence; (iii) development of
the OASIS interface within NEMO and WW3 for the exchange
of data between the models; and (iv) a test of the implementation based on a realistic global
configuration at 1/4∘ for the ocean and 1/2∘ for the waves.
To go into the details of those different steps, the paper is organized as follows. The modifications
brought to the oceanic model primitive equations, their boundary conditions, and the subgrid-scale
physics to account for wave–ocean interactions are described in Sect. . This includes the
addition of the Stokes–Coriolis force, the vortex force, and the wave-induced pressure gradient.
In Sect. our modeling system coupling the NEMO oceanic model and the
WW3 wave model via the OASIS-MCT coupler is described in detail.
Numerical simulations are presented in Sect. using a global configuration at 1/4∘
for the oceanic model and 1/2∘ for the wave model. Using sensitivity runs, we assess those
global configurations with particular emphasis on the impact of wave–ocean interactions on mixed layer
depth, sea surface temperature and currents, turbulent kinetic energy (TKE) injection, and kinetic energy.
Finally, in Sect. , we summarize our findings and provide overall comments on the impact of
two-way ocean–wave coupling in global configurations at eddy-permitting resolution.
Inclusion of wave-induced terms in the oceanic model NEMO
In order to set the necessary notations, we start by introducing the classical primitive equations
solved by the NEMO ocean model. Note that between the two possible options to
formulate the momentum equations, namely the so-called “vector-invariant” and “flux” forms,
we present the first one here, which will be used for the numerical simulations in Sect. .
With uh=(u,v) the horizontal velocity vector, ω the dia-surface velocity
component, θ the potential temperature, ρ the density, ζ the relative vorticity,
ph the hydrostatic pressure, and ps the surface pressure,
the Reynolds-averaged equations (with ⋅ the averaging operator omitted here for simplicity) are as follows.
1∂tu=+(f+ζ)v-12∂x‖uh‖2-ωe3∂ku-1ρ0∂x(ps+ph)-(∂xz)(∂kph)e3-1e3∂ku′ω′+Fu2∂tv=-(f+ζ)u-12∂y‖uh‖2-ωe3∂kv-1ρ0∂y(ps+ph)-(∂yz)(∂kph)e3-1e3∂kv′ω′+Fv3∂t(e3θ)=-∂x(e3θu)-∂y(e3θv)-∂k(θω)-1e3∂kθ′ω′+Fθ4∂te3=-∂x(e3u)-∂y(e3v)-∂kω5∂kp=-ρge3
Here, k is a nondimensional vertical coordinate, the lateral derivatives ∂x and ∂y
have to be considered along the model coordinate, and e3 is the vertical scale factor given
by e3=∂kz, where z is the local depth and ρ is given by an equation of
state . The necessary boundary conditions include a kinematic surface and bottom
boundary condition, which can be expressed in terms of the vertical velocity w,
w(z=η)=∂tη+uz=η∂xη+vz=η∂yη+(E-P),w(z=-H)=-uz=-H∂xH-vz=-H∂yH,
with η the height of the sea surface and (E-P) the mass flux across the sea surface
due to precipitation and evaporation, a momentum surface boundary condition for the Reynolds stress vertical terms,
-u′ω′z=η=τuoceρ0,-v′ω′z=η=τvoceρ0,
with τoce=(τuoce,τvoce)
a wind stress vector that represents the part of the stress that drives the ocean,
and a dynamic boundary condition on the free surface leading to the continuity of pressure across the air–sea interface.
The kinematic boundary conditions () for w(z=η) and w(z=-H) translate into
ω(z=η)=(E-P) and ω(z=-H)=0. We do not explicitly include the boundary
conditions for the tracer equations here since they are unchanged from classical primitive equation models
in the presence of wave motions. As mentioned earlier, in Eqs. () to ()
prognostic variables have to be interpreted in an Eulerian mean sense even if the averaging operator is
not explicitly included.
Modification of governing equations and boundary conditions
Asymptotic expansions of the wave effects based on Eulerian velocities or
Lagrangian mean equations lead to the same self-consistent set of equations
for weak vertical current shears. These are further applied and discussed by ,
, , and . The three-component Stokes drift vector is
us=(ũs,ṽs,ω̃s)
and is non-divergent at lowest order . The coupled wave–current
equations for the Eulerian mean velocity and tracers in a vector-invariant form (the equivalent flux
form is given in Appendix ) are as follows.
7∂tu=+(f+ζ)(v+ṽs)-12∂x‖uh‖2-(ω+ω̃s)e3∂ku-(∂xz)e3ũs∂ku+ṽs∂kv-∂x(ps+p̃J)ρ0-1ρ0∂x(ph+p̃Shear)-(∂xz)∂k(ph+p̃Shear)e3+1e3∂ku′ω′+Fu+F̃u8∂tv=-(f+ζ)(u+ũs)-12∂y‖uh‖2-(ω+ω̃s)e3∂kv-(∂yz)e3ũs∂ku+ṽs∂kv-∂y(ps+p̃J)ρ0-1ρ0∂y(ph+p̃Shear)-(∂yz)∂k(ph+p̃Shear)e3+1e3∂kv′ω′+Fv+F̃v9∂t(e3θ)=-∂x(e3θ(u+ũs)-∂y(e3θ(v+ṽs))-∂k(θ(ω+ω̃s))-1e3∂kθ′ω′+Fθ10∂te3=-∂x(e3(u+ũs))-∂y(e3(v+ṽs))-∂k(ω+ω̃s)11∂kph+p̃Shear=-ρge3+ρ0ũs∂ku+ṽs∂kv
Here, wave-induced terms are represented with tildes. The F̃u and F̃v terms
represent the sink–source of wave momentum due to breaking, bottom friction, and wave–turbulence interaction.
These terms will be neglected since they are expected to play a significant role only in the surf zone.
The other extra contributions to the momentum equations include the Stokes–Coriolis force WSt-Cor,
the vortex force WVF, and a wave-induced pressure WPrs:
WSt-Cor=fṽs-fũs0,WVF=ζṽs-ω̃se3∂ku-(∂xz)e3ũs∂ku+ṽs∂kv-ζũs-ω̃se3∂kv-(∂yz)e3ũs∂ku+ṽs∂kvũse3∂ku+ṽse3∂kv,WPrs=-1ρ0∂xp̃J+p̃Shear-(∂xz)∂k(p̃Shear)e3∂yp̃J+p̃Shear-(∂yz)∂k(p̃Shear)e31e3∂kp̃Shear,
where the terms involving horizontal derivatives of ω have been neglected in
WVF.
In WPrs, the p̃J
term corresponds to a depth-uniform wave-induced kinematic pressure term
In the notations of this term corresponds to p̃J=ρ0SJ.
, while p̃Shear
is a shear-induced three-dimensional pressure term
In the notations of this term corresponds to p̃Shear=ρ0SShear.
associated with the vertical component of the vortex force.
The vortex force contribution WVF can be further simplified by neglecting the
terms involving the vertical shear.
In particular, the vertical component of the vortex force is absorbed in a pressure term p̃Shearthat gives the Sshear term in the notations of. That particular term was neglected in
because of the generally weak vertical shears in the wave mixed layer. The effect of that term was also found
to be much weaker than p̃J in shallow coastal environments, except in the surf zone.
This assumption has the advantage of leaving the hydrostatic relation
(Eq. ) unchanged. Our implementation of wave-induced terms in NEMO is in line
with and corresponds to the simplified form of Eq. ():
WSt-Cor=fṽs-fũs0,WVF=ζṽs-ω̃se3∂ku-ζũs-ω̃se3∂kv0,WPrs=-1ρ0∂xp̃J∂yp̃J0.
Because of geostrophy, it is obvious that the addition of the Stokes–Coriolis force requires the effect of the
Stokes drift on the mass and tracer advection to be taken into account.
Regarding the joint modification of the tracers and continuity equations, it is clear that constancy
preservation is maintained (i.e., a constant tracer field should remain constant during the advective transport)
and that an additional wave-related forcing must be added to the barotropic mode. The NEMO barotropic mode
has been modified accordingly since the surface kinematic boundary condition (Eq. ) in terms
of vertical velocities w and associated w̃s now reads
w+w̃s=∂tη+(uz=η+ũsz=η)∂xη+(vz=η+ṽsz=η)∂yη+(E-P)
to express the fact that there is a source of mass at the surface that compensates for the convergence of the Stokes drift;
hence, the barotropic mode is
∂tη=-∂x(H+η)(u‾+ũ‾s)-∂y(H+η)(v‾+ṽ‾s)+P-E,∂tu‾=+fv‾-g∂xη-Cb,x(H+η)u‾+Gx‾+G̃x‾∂tv‾=-fu‾-g∂yη-Cb,y(H+η)v‾+Gy‾+G̃y‾,
where ϕ‾=1H+η∫-Hηϕdz, Cb=(Cb,x,Cb,y) represents the
bottom drag coefficients, and G=(Gx‾,Gx‾) is the usual NEMO forcing
term containing coupling terms from the baroclinic mode and slowly varying barotropic terms (including nonlinear
advective terms) held constant during the barotropic integration to gain efficiency. In Eq. (),
G̃x‾ and G̃y‾ contain the additional wave-induced barotropic forcing
terms corresponding to the vertical integral of the WSt-Cor
and WVF terms, which are also held constant during the barotropic integration.
A thorough analysis of the impact of the additional wave-induced terms on energy transfers within an oceanic model can
be found in . Note, however, that the study of is based on the Craik–Leibovich
equations, which are a special case of the more general wave-averaged primitive equations. Those sets of equations are
equivalent to each other only at lowest order in vertical shear.
Computation and discretization of Stokes drift velocity profile
Reconstructing the full Stokes drift profile us in the ocean circulation model would require obtaining the
surface spectra of the Stokes drift from the wave model. Instead, profiles are generally reconstructed
considering a few important parameters, including the Stokes drift surface value uhs(η)
and the norm of the Stokes volume transport ‖Ts‖. In ,
Stokes drift velocity profiles are derived under the deepwater approximation in the general form
uhs(z)=uhs(η)S(z,ke), with ke a depth-independent
spatial wavenumber chosen such that the norm of the depth-integrated Stokes transport (assuming an ocean
of infinite depth) is equal to ‖Ts‖. The functions SB14(z,ke) from
and SB16(z,ke) from for z∈[-H,η] are given by
SB14(z,ke)=e2ke(z-η)1-8ke(z-η),SB16(z,ke)=e2ke(z-η)-2keπ(η-z)erfc(2ke(η-z)).
with erfc the complementary error function. It can be easily shown that for an ocean of infinite depth,
the vertical integrals of those functions are respectively equal to 16ke for SB16 and
1.340898ke≈15.97ke for SB14. Standard computations of Stokes
drift in numerical models are done in a finite-difference sense; however, due to the fast decay of
uhs(z) with depth, a finite-volume approach seems more adequate in this case.
(uhs)k=uhs(η)(e3)k∫zk-1/2zk+1/2S(z,ke)dz=uhs(η)(e3)kI(zk+1/2,ke)-I(zk-1/2,ke)
Such a finite-volume interpretation of the Stokes drift velocity can also be found in and .
The SB16 function is more adapted for this kind of approach since the primitive
function only requires special functions available in the Fortran standard.
IB16(z,ke)=16kee2ke(z-η)+4ke(z-η)SB16(z,ke)
Since NEMO is discretized on an Arakawa C grid, the components of the Stokes drift
velocity must be evaluated at cell interfaces, and a simple average weighted by layer thicknesses is used:
ũi+1/2,j,ks=(e3)i,j,kui,j,ks+(e3)i+1,j,kui+1,j,ks2(e3)i+1/2,j,k,ṽi,j+1/2,ks=(e3)i,j,kvi,j,ks+(e3)i,j+1,kvi+1,j,ks2(e3)i,j+1/2,k.
Note that no explicit computation of the vertical component of the Stokes drift is necessary since
in Eqs. ()–() ω̃s only appears summed
with ω such that the relevant variable is ω+ω̃s as a whole.
This quantity is diagnosed from the continuity equation (Eq. , where the temporal
evolution of vertical scale factors ∂te3 is given by the free-surface evolution when
a quasi-Eulerian vertical coordinate is used; e.g., z⋆ or terrain-following coordinates).
(a) Reconstructed zonal component of a Stokes drift profile for ‖Ts‖=0.4 m2 s-1, us(z=η)=0.1 m s-1,
and vs(z=η)=0 m s-1 for a 1 m resolution vertical grid using the function (black dots), the function (grey dots),
and the finite-volume function (black vertical lines).
(b) Their continuous counterparts.
As illustrated in Fig. , for the typical vertical resolution used in most global models the properties
of the discretized Stokes profiles can be very different from their continuous counterparts. Indeed, the
SB16(z,ke) function has been considered superior to SB14(z,ke) because the
vertical shear near the surface is expected to be better reproduced. However, in Fig. it is shown
that this is no longer the case at a discrete level since the discrete vertical gradients at 1 m of depth turn
out to be larger for SB14(z,ke) compared to SB16(z,ke). In this case, the fast variations
of SB16(z,ke) near the surface cannot be represented by the computational vertical grid.
A vertical resolution finer than the one currently used in most global ocean models near the surface
would be required to properly represent the Stokes drift shear.
Subgrid-scale physicsTurbulent kinetic energy prognostic equation and boundary conditions
Under the assumption of horizontal homogeneity generally retained in general circulation models,
the contribution from Stokes drift to the turbulent kinetic energy (TKE) prognostic equation arises
from the vortex force vertical term WVFz=ũs∂zu+ṽs∂zv
in the hydrostatic relation (Eq. ). Mimicking the way the TKE equation is usually
derived see, e.g., and using an averaging operator ⋅ satisfying
the “Reynolds properties”, we find that the turbulent fluctuations, defined as ϕ′=ϕ-ϕ,(ϕ=p,ρ,u,v),
associated with the WVFz term are
(WVFz)′=ũs∂zu′+ṽs∂zv′.
After multiplication by w′ and averaging, we obtain
w′(WVFz)′=ũs∂zu′w′+ṽs∂zv′w′-ũsu′∂zw′-ṽsv′∂zw′,
where the last two terms on the right-hand side cancel, with similar terms appearing when forming the equations
for u′∂tu′ and v′∂tv′ (see Eqs. A.7 and A.8 in ).
The extra terms associated with the Stokes drift in the horizontally homogeneous TKE equation are thus
us∂zu′w′ and vs∂zv′w′, which can be further rewritten as
ũs∂zu′w′=-u′w′∂zũs+∂zũsu′w′,ṽs∂zv′w′=-v′w′∂zṽs+∂zṽsv′w′.
The first term will modify the shear production term; it can also be derived by taking the Lagrangian mean of the
wave-resolved TKE equation . The second will enter the TKE transport term that
is usually parameterized as -Ke∂ze. The prognostic equation for the turbulent kinetic energy e in
NEMO under the assumption that Ke=Avm, with Avm the eddy viscosity, is thus
∂te=Avme32(∂ku)2+(∂kv)2+(∂ku)(∂kũs)+(∂kv)(∂kṽs)-AvtN2+1e3∂kAvme3∂ke-cϵe3/2lε2,
with Avt the turbulent diffusivity, N the local Brunt–Väisälä frequency,
lε a dissipative length scale, and cε a constant parameter (generally
such that cε≈1/2). Once the value of e is know, eddy
diffusivity and viscosity are given by
Avm=Cmlme,Avt=Avm/Prt,
with Prt the Prandtl number (see Sect. 10.1.3 in , for the detailed computation
of Prt), lm a mixing length scale, and Cm a constant.
In addition to the modification of the shear production term in the TKE equation, the wave will affect the
surface boundary condition for e, lm, and lε. The Dirichlet boundary condition
traditionally used in NEMO for the TKE variable is modified into a Neumann boundary condition,
Avme3∂kez=z1=-ρ0g∫02π∫0∞Socedωdθ=Φoce,
meaning that the injection of TKE at the surface is given by the dissipation of the wave field via the
wave–ocean Soce term, which is a sink term in the wave model energy balance equation
usually dominated by wave breaking, converted into an ocean turbulence source term. In practice,
this sum of Soce is obtained as a residual of the source term integration; hence, it also includes
unresolved fluxes of energy to the high-frequency tail of the wave model. Due to the placement at cell
interfaces of the TKE variable on the computational grid, the TKE flux is not applied at the free surface
but at the center of the topmost grid cell (i.e., at z=z1). This amounts to interpreting the half-grid cell
at the top as a constant flux layer, which is consistent with the surface layer Monin–Obukhov theory.
The length scales lm and lε are computed via two intermediate length scales lup
and ldwn, respectively estimating the maximum upward and downward displacement of a water
parcel with a given initial kinetic energy. lup and ldwn are first initialized to the length
scale proposed by : lup(z)=ldwn(z)=2e(z)/N2(z).
The resulting length scales are then limited not only by the distance to the surface and to the bottom but
also by the distance to a strongly stratified portion of the water column such as the thermocline. This
limitation amounts to controlling the vertical gradients of lup(z) and ldwn(z)
such that they are not larger that the variations of depth ∂kl⋅≤e3,l⋅=lup,ldwn
Then, the dissipative and mixing length scales are given by lm=lupldwn
and lε=minlup,ldwn. Following
(their Sect. 4.2.3), a boundary condition consistent with the Monin–Obukhov similarity theory for
the length scale ldwn (while lup necessitates only a bottom boundary condition) is
ldwn(z=η)=κ(Cmcε)1/4Cmz0,
with κ the von Karman constant and Cm and cε the constant parameters in the
TKE closure. The surface roughness length z0 can be directly estimated from the significant wave
height provided by the wave model as z0=1.6Hstheir Eq. 5, which provides
a proxy for the scale of the breaking waves. Note that in our study, no explicit parameterization of the mixing
induced by near-inertial waves has been added . As highlighted by , without
activating this ad hoc parameterization in the standard NEMO TKE scheme, the model
does not mix deeply enough. They also speculated that this ad hoc mixing could mask the effects of
wave-related mixing processes such as Langmuir turbulence. For this reason, it is not used in the
present simulations.
Langmuir turbulence parameterization
Langmuir mixing is parameterized following the approach of . This parameterization
takes the form of an additional source term PLC in the TKE equation ().
PLC is defined as
PLC=wLC3dLC,
where wLC represents the vertical velocity profile associated with Langmuir cells
and dLC their expected depth. Following , wLC and dLC are
given by
wLC=cLC‖u^LCs‖sin-πzdLC,if-z≤dLC0,otherwise,-∫-dLCηN2(z)zdz=‖u^LCs‖22,
where ‖u^LCs‖ is the portion of the surface Stokes drift contributing to
Langmuir cell intensity and cLC a constant parameter. In the absence of information
about the wave field it is generally assumed that ‖u^LCs‖∝‖τ‖.
As mentioned in the Introduction, and showed that
the intensity of Langmuir cells is largely influenced by the angle between the Stokes drift and the
wind direction. To reflect this dependency we account for this angle in our definition of
‖u^LCs‖ via
‖u^LCs‖=maxus(η)⋅eτ,0,
with eτ the unit vector in the wind stress direction.
The difference between the surface Stokes drift ‖us(η)‖ and
‖u^LCs‖ given by Eq. () is shown
in Fig. and compared to the usual parameterization of ‖us(η)‖ as
0.377‖τoce‖/ρ0 in the uncoupled case see.
The modulation of ‖u^LCs‖ depending on the wind stress orientation
significantly reduces the input of the surface Stokes drift contributing to Langmuir cell intensity, especially
in the Southern Ocean, while other regions are less affected.
Finally, a value for the parameter cLC must be chosen. Based on single-column
experiments detailed in Appendix ,
we find that parameter values in the range 0.15–0.3 provide satisfactory results
compared to the LESs of and will be considered for the numerical experiments
discussed later in Sect. .
Annual average of the surface Stokes drift module ‖us(η)‖ (m s-1)(a),
the portion of the Stokes drift aligned with the wind, as given in Eq. () (b),
and the surface Stokes drift as parameterized by 0.377‖τoce‖/ρ0
in the uncoupled case (c).
While the parameterization was already implemented in NEMO, there are three major
novelties in our implementation: (i) the online coupled strategy allows us to use the
surface Stokes drift directly delivered by the wave model instead of the
original value empirically estimated from the wind speed (e.g., 1.6 % of the 10 m wind).
(ii) We only considered the component of the Stokes drift aligned with the wind, and
(iii) based on a series of single-column simulations (see Appendix )
the coefficient cLC evaluated as 0.15 by is set to a 0.3 value. Those
changes, together with the new surface boundary condition for the TKE equation, lead to a
deeper penetration of the TKE inside the mixed layer and, as shown in Sect. ,
greatly improved the MLD distribution.
Modeling system and coupling strategy
Our coupled model is based on the NEMO oceanic model, the WW3
wave model, and the OASIS library for data exchange and synchronization
between the two components.
Numerical models and coupling infrastructureThe ocean model: NEMO
NEMO is a state-of-the-art primitive-equation, split–explicit, free-surface oceanic model
whose equations are formulated both in the vector-invariant and flux forms (see Eq.
for the vector-invariant form). The equations are discretized using a generalized vertical coordinate
featuring, among others, the z⋆ coordinate with partial-step bathymetry and the σ coordinate,
as well as a mixture of both . For efficiency and accuracy in the representation of
external gravity wave propagation, model equations are split between a barotropic mode and a baroclinic
mode to allow the possibility to adopt specific numerical treatments in each mode. The NEMO
equations are spatially discretized on an Arakawa C grid in the horizontal and a Lorenz grid in the vertical,
and the time dimension is discretized using a leapfrog scheme with a modified Robert-Asselin filter to
damp the spurious numerical mode associated with leapfrog . For the current study
the NEMO equations have been modified to include wave effects as described in Eqs. ()
and (). Moreover, the modifications to the standard NEMO one-equation
TKE closure scheme are given in Sect. .
The wave model: WW3
The NEMO ocean model has been coupled to the WW3 wave model.
In numerical models, waves are generally described using several phase and amplitude parameters.
We provide only the details sufficient to understand the coupling of waves with the oceanic
model here, and an exhaustive description of WW3 is given by .
WW3 integrates the wave action equation with the spectral density
of wave action Nw(kw,θw) discretized in wavenumber kw
and wave propagation direction θw for the spectral space (the subscript w is used here to avoid confusion with previously introduced notations):
∂tNw+∂ϕϕ˙Nw+∂λλ˙Nw+∂kwk˙wNw+∂θwθ˙wNw=Sσ,
where λ is longitude, ϕ is latitude, and S is the net spectral source term that includes
the sum of the rate of change of the surface elevation variance due to interactions with the atmosphere via
wind–wave generation and swell dissipation (Satm), nonlinear wave–wave interaction (Snl),
and interaction with the upper ocean that is generally dominated by wave breaking (Soce).
Those parameterized source terms are important in wave–ocean coupling. Indeed, as shown earlier
in Eq. (), the Soce term is used to compute the TKE flux transmitted to the ocean,
and the Sin term enters the computation of the wave-supported stress. They are computed here
following .
In Eq. (), the dot variables correspond to a propagation speed given by the following.
19ϕ˙=cgcosθw+vz=ηR-120λ˙=cgsinθw+uz=ηRcosϕ-121θ˙w=cgsinθwtanϕR-1+sinθw∂ωw∂ϕ-cosθwcosϕ∂ωw∂λkwR-122k˙w=-∂σ∂Hkkw⋅∇D-k⋅∇uh(z=η)
Here, R is the Earth's radius, uh(z=η)=(uz=η,vz=η)
represents the surface currents provided by the ocean model, cg is the group velocity,
ωw the absolute radian frequency, and H the mean water depth.
Equation () is solved for each spectral component (kw,θw)
coupled by the advection and source terms. Equations ()–()
show how the oceanic currents affect the advection of the wave action density; there are also indirect
effects via the source term .
The coupler: OASIS-MCT
The practical coupling between NEMO and WW3 has been implemented using
the OASIS-MCT software primarily developed for use in
multicomponent climate models. This software provides the tools to couple various models at low
implementation and performance overhead. In particular, thanks to MCT ,
it includes the parallelization of the coupling communications and runtime grid interpolations. For
efficiency, interpolations are formulated in the form of a matrix–vector multiplication whereby the matrix
containing the mapping weights is computed offline once for all. In practice, after compiling
OASIS-MCT, the resulting library is linked to the component models so that they have
access to the specific interpolation and data exchange subroutines. Now that we have described the
different components involved in our coupled system, we go into the details of the nature of the data
exchanged between both models.
Oceanic surface momentum flux computation
Surface waves affect the momentum exchange between the ocean and the atmosphere in two different ways.
First, the modification of surface roughness acts on the incoming atmospheric momentum flux τatm.
Second, a part of the momentum flux from the atmosphere is consumed by the wave field and contributes to the growing
waves (the so-called wave-supported stress); conversely, the waves release momentum to the ocean when they break
and dissipate. This implies that the wind stress transferred to the oceanic model
(we call it τoce) is different from the atmospheric wind stress τatm.
These two coupling processes are taken into account in our coupled framework.
The 10 m wind u10atm is sent to the wave model, which internally
computes the dimensionless Charnock parameter αch characterizing the sea
surface roughness . This information is used by the
wave model to compute its own atmospheric wind stress τww3atm
assuming neutral stratification, i.e., τww3atm=ρaCDN(αch)‖u10atm‖u10atm with CDN a neutral drag coefficient, which
is function of the Charnock parameter. Then the wave model computes the momentum flux transferred to the
ocean τww3oce. Using the latest available values
of αch, τww3atm, τww3oce,
and u10atm, the oceanic model computes an atmospheric wind stress τatm
using its own bulk formulation, and the local value of the momentum flux going into the water column is diagnosed as
τoce=τatm-τww3atm-τww3oce,
where the τww3 quantities are interpolated from the wave grid to the oceanic grid.
In NEMO, the wind stress is computed using the IFS (Integrated Forecasting System: https://www.ecmwf.int/en/forecasts/documentation-and-support/changes-ecmwf-model/ifs-documentation, last access: 2 July 2020)
bulk formulation such as implemented in the AeroBulk (https://github.com/brodeau/aerobulk, last access: 2 July 2020) package
. In particular, the roughness length that enters the definition of the drag coefficient is
a function of the Charnock parameter αch,
z0=αchu⋆2g+αmνu⋆,
where αm=0.11, u⋆ is the friction velocity, and ν the air kinematic viscosity whose contribution
is significant only asymptotically at very low wind speed. Note that in the uncoupled case the default value of the Charnock
parameter is αch0=0.018. In our implementation, the momentum fluxes are computed using the absolute
wind u10atm at 10 m rather than the relative wind u10atm-uh(z=η).
Indeed, several recent studies have emphasized that the use of relative winds is relevant only when a full coupling with an
atmospheric model is available since in a forced mode it leads to an unrealistically large loss of oceanic eddy kinetic energy e.g.,.
This is not a limitation of our approach since a simple modification of a namelist parameter allows us to run with relative winds,
but this case is not investigated in the present study.
In our coupling strategy two different values of the atmospheric wind stress and the wave-to-ocean wind stress
are computed with two different bulk formulations. This strategy is not fully satisfactory since it breaks the momentum conservation.
However, it was necessary in practice since the WW3 results were very sensitive to the bulk formulation, and at the same
time it was not conceivable to use the WW3 bulk formulation to force the ocean model because the latter ignores the
effect of stratification in the atmospheric surface layer. Previous implementations in NEMO
e.g., assumed that the wave field only acts on the
norm of τatm and not on its orientation. Instead of Eq. (), the atmospheric
wind stress was corrected as follows.
τoce=τatmτww3oceτww3atm
However, this approach potentially leads to artificially large values of τoce when
τww3atm is small, and it does not take into account the slight change in
τoce direction induced by the waves.
Additional details about the practical implementation
In Table the different variables exchanged between the oceanic and wave models are given.
All variables are 2D variables, meaning that no 3D arrays are exchanged through the coupler. All 2D interpolations
are made through a distance-weighted bilinear interpolation. The time discretization steps Δtww3
for WW3 and Δtnemo for NEMO are generally different with
Δtww3>Δtnemo and chosen such that Δtww3=ntΔtnemo
(nt∈N,nt≥1). In this case, coupling fields from NEMO to WW3 are
averaged in time between two exchanges, while fields from WW3 to NEMO are sent every
Δtww3 steps and therefore updated every nt time steps in NEMO. If
Δtww3>Δtnemo, the coupler time step is set to Δtww3. Our current
implementation does not include an explicit coupling between waves and sea ice, while it is known that
waves lead to ice breakup, pancake ice formation, and associated enhancement of both freezing and melting;
in return, this wave dissipation in ice-covered water e.g., leads to ice drift. Such explicit
coupling is currently under development within the NEMO framework .
Variables exchanged between NEMO (O) and WW3 (W) via the OASIS-MCT coupler.
The 10 m wind u10atm is interpolated online by WW3 and does not go through the OASIS-MCT coupler.
VariableDescriptionUnitsuh(z=η)Oceanic surface currentsO→Wm s-1u10atm10 m winds from external datasetO→Wm s-1uhs(z=η)Sea surface Stokes driftW→Om s-1‖Ts‖Norm of the Stokes drift volume transportW→Om2 s-1ΦocTKE surface flux multiplied by ρ0W→OW m-2αchCharnock parameterW→O–τwww3Wave-supported stressW→ON m-2p̃JWave-induced pressureW→Om2 s-2HsSignificant wave heightW→OmGlobal 1/4∘ coupled wave–ocean simulationsExperimental setup and experimentsThe global coupled ORCA25 configuration
The wave hindcasts presented here are all based on the WW3 model in its version 6.02
configured with a single grid at 0.5∘ resolution in longitude and latitude.
A spectral grid with 24 directions and 31 frequencies is exponentially spaced over the interval [fmin,fmax]
with fmin=0.037 Hz and fmax=0.7 Hz.
A one-step monotonic third-order coupled space–time advection scheme (also called the ultimate quickest scheme)
is used with a specific procedure to alleviate the so-called garden sprinkler effect .
As suggested in , the dissipation induced by wave breaking is proportional to the local
saturation spectrum see also.
The wind input growth rate at high frequency is based on the formulation of with an
additional “sheltering” term to reduce the effective winds for the shorter waves .
For the computation of nonlinear wave–wave interactions, the discrete interaction
approximation of is used.
This last approximation is known to be inaccurate, but it is thought that the associated errors are usually
compensated for by a proper adjustment of the dissipation source term .
As mentioned earlier in Sect. , the model
was run with 10 m winds, without any air–sea stability correction. No wave measurements
were assimilated in the model, but the stand-alone wave model was developed based on spectral
buoy and synthetic aperture
radar (SAR) data and calibrated against altimeter data by adjusting the
wind–wave coupling parameter . The WW3 time step for
the global configurations is Δtww3=3600 s.
For the oceanic component, we use a global ORCA025 configuration at a 1/4∘ horizontal
resolution . The vertical grid is designed with 75 vertical z levels with vertical
spacing increasing with depth. Grid thickness is about 1 m near the surface and increases
with depth to reach 200 m at the bottom. Partial steps are used to represent the bathymetry.
The LIM3 sea ice model is used for the sea ice dynamics and thermodynamics .
The vertical mixing coefficients are obtained from the one-equation TKE scheme described in Sect. ,
and the convective processes are mimicked using an enhanced vertical diffusion parameterization that
increases vertical diffusivity to 10 m2 s-1 at which static instability occurs. Water density
is computed from temperature and salinity through the use of a polynomial formulation of the
nonlinear equation of state .
The vector-invariant form of momentum advection
uses for the vorticity and a specific formulation to control the Hollingsworth instability
. Momentum lateral viscosity is biharmonic and acts along geopotential surfaces. It is set
to a value of 1.5×1011 m4 s-1 at the Equator and varies proportionally to Δx3
away from the Equator. Advection of tracers is performed with a
flux-corrected transport (FCT) scheme , and lateral diffusion of tracers is harmonic and acts
along an iso-neutral surface. It is set to a value of 300 m2 s-1 at the Equator, which varies
proportionally to Δx. The bottom
friction is nonlinear and the lateral boundary condition is free-slip. In this setup, the baroclinic time step
is set to Δtnemo=900 s and a barotropic time step 30 times smaller. Compared to
the standard uncoupled ORCA025 configuration, the additional computational cost associated with WW3
and the exchanges through the coupler is about 20 %.
Atmospheric forcings
The atmospheric fields used to force both ocean and wave models are based on the ECMWF (European Centre for
Medium-Range Weather Forecasts) ERA-Interim reanalysis . Corrections have been applied to guarantee
that the ERA-Interim mean states for rainfall as well as shortwave and longwave radiative fluxes are consistent with satellite observations
from the Remote Sensing Systems (RSS) Passive Microwave Water Cycle (PMWC) product and
GEWEX SRB 3.1 data (https://gewex-srb.larc.nasa.gov/, last access: 2 July 2020).
Momentum and heat turbulent surface fluxes are computed using the IFS bulk formulation from the AeroBulk package
using air temperature and humidity at 2 m, mean sea level pressure, and 10 m winds.
Sensitivity experiments and objectives
Sensitivity experiments have been conducted to check the proper implementation of various
components of the present coupled modeling system. For the sake of clarity, our developments are split
into four components: (i) the modification of the wind stress by waves through the Charnock
parameter and the inclusion of wave-supported stress, (ii) the modifications of the NEMO
governing equations through the Stokes–Coriolis, vortex force, and wave-induced surface pressure terms,
(iii) the addition of a Langmuir turbulence parameterization, and (iv) the
modifications to the TKE scheme. As summarized in Table , sensitivity experiments
are designed in such a way to incrementally increase the level of complexity and test the effect
of each component.
The No_CPL experiment corresponds to the classical NEMO setup in which the wave effect
is parameterized through a wind-stress-dependent TKE surface boundary condition as suggested
by . In this approach, a Dirichlet surface boundary condition is used and expressed
as follows: e(z=η)=12(15.8αCB)2/3‖τatm‖ρ0
with αCB=100. Based on the results of we expect that in
the uncoupled case the nature of the boundary condition (i.e., Dirichlet vs. Neumann) does not
significantly impact numerical solutions
In the authors consider a
Dirichlet boundary condition such that e(z=η)=12(15.8αCB)2/3u⋆2
and an equivalent Neumann condition Ke∂zez=η=2αCBu⋆3.
The authors claim that numerical solutions using a Dirichlet condition instead of a Neumann condition are qualitatively similar.
.
The WS_CPL experiment is identical as No_CPL except that
the wave coupling is introduced within the wind stress computation, as described in Sect. . The
ST_CPL experiment is as WS_CPL except that all terms relative
to the Stokes drift described in Sect. are added
in NEMO. TKE_CPL corresponds to ST_CPL but with the modified TKE scheme described in Sect. . All_CPL (1 and 2)
experiments are like TKE_CPL but with
a fully modified TKE scheme including the Langmuir cell parameterization described in Sect. .
All those simulations have been performed for 2 years (2013–2014), during which 2013 is spin-up and
only 2014 is analyzed.
We considered 2 years sufficient to illustrate the fact that our developments were actually producing the
expected results. Integrating longer in time could also lead to drifts in the stratification independently from the
wave effects and could thus distort our interpretation.
In any case, it must be clear that the objective here is not to go through a thorough physical
analysis of coupled solutions but to check and validate our numerical developments.
The wave distribution being inhomogeneous on the globe, it is expected that with the wave-modified wind stress
parameterization the stress should follow the wave patterns more closely.
In Fig. , the seasonal average of the significant wave height and of the difference between the Charnock
coefficient computed by the wave model and the default constant value used in the uncoupled case (αch0=0.018) are shown.
As expected, the Charnock parameter tends to be stronger in the area where the waves are higher.
Generally an increase in the Charnock parameter is observed in the northern and southern basin, while there is a net
decrease in αch near the Equator. There is also a strong seasonality in the Northern Hemisphere, with
a reduction in summer and a strong increase in winter. The differences between αch and αch0
are very latitudinal with very few longitudinal variations.
(a, c) Seasonal averages of significant wave height (in meters) for January–February–March (JFM, panel a)
and July–August–September (JAS, panel c).
(b, d) Seasonal average of the difference between the Charnock parameter as
computed by the wave model and the default value αch0=0.018 for JFM (b) and JAS (d).
To isolate the effect of the Charnock parameter we compare the results obtained
in the No_CPL and WS_CPL experiments. Those two experiments
show relatively similar sea surface temperature patterns, meaning that the modification
of the wind stress ‖τoce‖ between those two cases is primarily
due to the use of different Charnock parameters and the inclusion of the wave-supported stress.
(a) Drag coefficient (CD) as a function of the 10 m wind speed ‖u10atm‖ and
(b) wind stress norm ‖τoce‖ as a function of ‖u10atm‖
(black curves represent the mean value, while the vertical bars represent the standard deviation).
Figure a illustrates that the Charnock parameter mostly affects the drag
coefficient CD, and hence the surface wind stress, for large winds. The ocean–wave
coupling does not lead to appreciable differences in the drag coefficient CD for wind
speeds lower than 8 m s-1. On the contrary, since large values of the
Charnock parameter are observed for large wind speeds, the coupling significantly
increases the drag (as well as its variance) at high winds. Figure b
shows how the wind stress is modified by this increase in the drag coefficient jointly
with the wave-supported stress, which tends to decrease the wind stress magnitude (Fig. ).
At low wind speed the wind stress magnitude is not affected by the coupling with waves,
while for strong winds the increase in wind stress associated with the increased drag coefficient
is always larger than the decrease associated with the wave-supported stress. This latter effect
reduces the wind stress by no more than 2 %; for the characteristic scales of our study, this
correction is thus almost negligible. The wind stress changes due to the coupling with waves
seen in our simulations are very localized in time and space and it is thus difficult to conclude
on their overall effect on upper-ocean dynamics such as Ekman pumping and the surface currents.
Wind stress difference ‖τoce‖-‖τatm‖ (Nm-2) due to the
correction made for growing waves for the WS_CPL experiment as a function of the 10 m wind speed.
Wave impact on surface TKE injection
As described in Sect. , in the ocean–wave coupled case, the surface boundary
condition for the TKE equation is a Neumann condition whose value is directly given by the wave model,
unlike the uncoupled case in which a Dirichlet condition is imposed. We aim here to assess the impact
on the order of magnitude of the near-surface TKE.
Since the Neumann boundary condition is applied at the center of the topmost grid box
(i.e., approximately at 50 cm of depth), we compare in Fig. the TKE value
at 1 m of depth between the coupled (All_CPL2) and the uncoupled (No_CPL) case.
Positive values mean that near-surface TKE is larger in the coupled simulation.
Seasonal difference of 1 m depth turbulent kinetic energy (m2 s-2)
between the coupled case (All_CPL2) and the uncoupled case (No_CPL).
(a) January, February, and March (JFM); (b) July, August, and September (JAS).
It shows an almost homogeneous increase in the TKE (up to more than 100 %) in the extratropical areas.
While low seasonal variability in the extratropical areas is visible in Fig. , a spatial averaging by
hemisphere (Fig. ) highlights seasonal variability with a strong increase in both the near-surface TKE value and
the TKE difference between the two experiments during winter.
In Figs. and (and also in the remainder of the paper), the spatial averaging is made between
25 and 60∘ S in the Southern Hemisphere and between 25 and 60∘ N
in the Northern Hemisphere
to avoid any conflicts with sea ice and to remove the equatorial region from the comparison.
The increase in the surface TKE injection associated with waves is expected to contribute to
an overall increase in mixed layer depth provided that the mixing length diagnosed by the
turbulent closure scheme allows for the effective propagation of this additional TKE deeper in the mixed layer.
Spatially averaged turbulent kinetic energy (m2 s-2) at 1 m of depth
over (a) the Southern Hemisphere and (b) the Northern Hemisphere.
Wave impact on mixed layer depth
In this section, we evaluate the wave effect on vertical mixing using the mixed layer depth (MLD)
as a relevant metric.
Figure represents the seasonally averaged difference in MLD
between the coupled (All_CPL2) and the uncoupled (No_CPL) case
relative to the No_CPL case (i.e., (hMLDnoCPL-hMLDCPL)/hMLDnoCPL
with hMLD considered negative downward). It shows a significant deepening of the mixed layer
at high latitudes in the coupled case with only a very few localized mixed layer shallowing up to 60 %, mainly in the Southern
Hemisphere.
(a, b) Seasonally averaged MLD differences (All_CPL2-No_CPL) relative to the uncoupled simulation No_CPL. Red corresponds to a deeper MLD for All_CPL2.
To assess whether the overall deepening of the mixed layer is realistic, we make a comparison with available observations. Available observations for 2014 were extracted following an updated dataset from . The MLD has been computed as the depth at which the density is 3 % smaller that the density at 10 m as in .
Figure represents the spatially averaged MLD; the blue line is the spatially averaged MLD obtained from
ARGO floats (available during the same period) in both hemispheres. In the Northern Hemisphere (Fig. a),
there is only a slight improvement compared to data during winter and late summer when implementing the coupling
with waves. In the Southern Hemisphere (Fig. b) the situation is rather different.
Spatially averaged MLD for (a) the Northern Hemisphere and (b) the Southern Hemisphere.
From January to July, the deepening of the MLD induced by the wave coupling significantly reduces the bias
between the model and ARGO data. From July to December, results in the coupled case show an overestimation
of MLDs, which were already too deep in the uncoupled case, thereby increasing the bias between the data and model.
Since mesoscale activity makes direct comparisons to data unreliable for such a short period of time,
we compare the normalized distribution of MLD between the different simulations and available ARGO data.
Results are presented in Fig. for the year 2014 (panel a) and during summer only (panel b). In both cases
the improvement in the Northern Hemisphere is very modest. As far as the Southern Hemisphere is concerned the
coupling with waves leads to a significant improvement compared to the MLD derived from ARGO floats despite the fact
that there are still too many low MLD values in the range 50–100 m. In comparison with the uncoupled
case there is a more realistic spreading toward deeper mixed layer depths. More particularly in summer (Fig. b),
the probability density function (PDF) in the coupled case matches the one computed from ARGO data almost perfectly.
Despite the fact that we did not activate the ad hoc extra mixing induced by near-inertial waves ,
our implementation of the wave–ocean interaction leads to a significant deepening of the MLD in a realistic way.
Mixed layer depth probability density function for (a) the full 2014 year and (b) summer 2014.
To better understand which components of the wave–ocean coupling are responsible for this improvement,
the summer PDF in the Southern Hemisphere has been computed for each of the experiments described
in Table . Results are shown in Fig. . First of all, it can be seen that all the wave–ocean
interactions described in previous sections lead to an improvement in terms of mixed layer depth distribution
compared to the uncoupled case. Indeed, the modification of the wind stress by the wave field introduced in
WS_CPL increases both surface currents and near-surface TKE values, resulting in a slight deepening
of the MLD. Adding the Stokes-drift-related terms in the primitive equations contributes only modestly to the
deepening of the MLD, while most of the improvement results from the modified TKE scheme, with some slight improvement when the Langmuir parameterization is activated. It is somewhat reassuring to see that the better agreement with ARGO data
is obtained when all components of the coupling are activated.
Mixed layer depth probability density function in the Southern Hemisphere during summer months. The details of each experiment can be found in Table .
Wave impact on sea surface temperature
Since the near-surface mixing is strengthened by the coupling, we can expect an impact on sea surface temperature (SST).
Figure represents the time series of SST for each hemisphere. The Northern Hemisphere is characterized by a
warm bias during summer with a very slight improvement when coupling with waves.
In the Southern Hemisphere (Fig. b) the summer warm bias is reduced by half in the coupled simulation
and a slight warming occurs during the winter. While the summer surface cooling might be linked to the mixed layer deepening,
the winter warming might be rather linked to advection as observed by for the Baltic sea. It could also result from an increased heat content during summer, leading to higher SST during winter.
Time series of the spatially averaged sea surface temperature (∘C); (a) Northern Hemisphere and (b) Southern Hemisphere.
To better characterize the wave impact on the SST, we show in Fig. a
the difference in terms of annual mean between the No_CPL experiment and OSTIA analysis,
exhibiting a cold bias in the No_CPL simulation in equatorial and tropical regions and a warm bias in
the northern part of the Pacific Ocean. The coupling with
waves tends to diminish the cold bias (see Fig. b), especially in the Pacific Ocean, and
the warm bias in the North Pacific is significantly reduced.
(a) Annual average of the differences between No_CPL and OSTIA sea surface temperatures (∘C)
for the year 2014 (positive when the model is warmer).
(b) Annual average of the difference between All_CPL2 and No_CPL (positive when All_CPL2 is warmer).
As already noticed by the warming in the equatorial and tropical regions mainly results
from a lower wind stress caused by a value of the Charnock parameter lower than the value used in the
uncoupled case (see Fig. b, d). A consequence is a decrease in the drag coefficient, leading to
smaller turbulent exchange coefficients and reducing the heat flux.
As mentioned above, in extratropical regions, some warm bias tends to be partially reduced by the extra
mixing induced by the waves at high latitude and/or by the increased turbulent transfer coefficient.
The tendency of the wave coupling to improve the near-surface temperature distribution can also be verified
on a time–latitude Hovmöller diagram like the ones shown in Fig. . For instance, it can be seen that
the summer warm bias in the Northern Hemisphere (Fig. a) coincides well with the cooling induced
by the coupling with waves (Fig. b). Similarly we can also observe a warming in the tropical and equatorial
regions (Fig. b) corresponding to the cold bias seen in Fig. a. In the southern
extratropical region, a summer cooling is observed. It is induced by the wave coupling, whereas
Fig. a shows a slight warm bias. During winter we can observe a
warming in Fig. b north of 60∘ S, which again partially corresponds to a cold bias in Fig. a.
Hovmöller diagram of the longitudinally averaged sea surface temperature (∘C) differences between (a) No_CPL and OSTIA and (b) between All_CPL2 and No_CPL.
Time series of the spatially averaged surface kinetic energy (m2 s-2) for (a) the Northern Hemisphere and (b) the Southern Hemisphere.
Surface current and kinetic energy
The last aspect of our solutions we would like to evaluate is the impact of the surface waves
on surface currents and kinetic energy (KE). To do so, we show in Fig. time series of
the spatially averaged surface kinetic energy for both hemispheres. Whatever the hemisphere
there is a net decrease in surface KE (up to 20 % in the south) when a coupling with the waves
is included. This decrease in surface kinetic energy reflects a decrease in surface current magnitudes.
Indeed, as detailed in Fig. , which represents the vertical profile of the horizontal components
of the current in the oceanic surface boundary layer, the coupling with waves decreases both the surface
current magnitudes and the shear.
While currents from the WS_CPL are increased due to increased wind stress, the Stokes–Coriolis force
when included in momentum equations leads to a decrease in velocities in the whole boundary layer as
previously shown by (orange lines in Fig. ). The inclusion of vertical mixing
due to waves and Langmuir circulation attenuates the currents in the surface layer, resulting in further reduced
surface currents and stronger currents at the bottom of the boundary layer (purple lines in Fig. ).
This concludes our checking of the proper functioning of the coupling with waves as described in the present
paper.
Zonally averaged zonal (a) and meridional (b) currents (m s-1) between 60 and 25∘ S
as a function of depth (m) for the simulations described in Table .
Conclusions
In this paper we have described the implementation of an online coupling between the oceanic
model NEMO and the wave model WW3. The impact of such coupling
on the model solutions has been assessed from the oceanic point of view for a global configuration.
In particular, the following steps to set up the coupled model have been discussed in detail:
(i) the inclusion of all wave-induced terms in NEMO primitive equations, only
neglecting the terms relevant for the surf zone, which is outside the scope of the NEMO community;
(ii) modification of the subgrid-scale vertical physics (including the bulk formulation) to
include wave effects and a parameterization of Langmuir turbulence; (iii) development of a
coupling interface based on the OASIS-MCT software for the exchange of data between
the two models; and (iv) tests of our developments on a realistic global configuration
at 1/4∘ for the ocean coupled to a 1/2∘ resolution wave model. Compared to an
ocean-only simulation, the coupling with a wave model (with a resolution twice as coarse as the oceanic
model) leads to an additional computational cost of about 20 %.
Following and , in the weak vertical current shear limit,
the wave-induced terms implemented in NEMO include the Stokes–Coriolis force,
the vortex force, Stokes advection in tracer and continuity equations, and a wave-induced
surface pressure term. The prognostic equation for TKE also includes an additional forcing
term associated with the Stokes drift vertical shear and various modifications of its boundary
condition described in Sect. .
The development of a coupling infrastructure based on OASIS-MCT has several advantages
as it allows for an efficient data exchange (including the treatment of nonconformities between the
computational grids) but also for versatility in the inclusion of a wave model in existing ocean–atmosphere
or ocean-only models. At a practical level, the OASIS interface we have implemented in
NEMO is similar to other interfaces (e.g., toward atmospheric models) existing in the code,
which is important for maintenance and for further developments. It paves the way for a seamless and
more systematic inclusion of the coupling with waves for NEMO users. Unlike most previous studies of wave–ocean coupling using NEMO, we have shown that satisfactory
results can be obtained from the TKE vertical turbulent closure scheme without activating the ad hoc
parameterization for the mixing induced by near-inertial waves, surface waves, and swell (known as the ETAU parameterization).
This parameterization that allows users to empirically propagate the surface TKE at depth using a prescribed shape function
is a pragmatic way to cure the shallow mixed layer depths in the Southern Ocean found in simulations ignoring wave effects.
Previous studies of wave–ocean coupling by , , and have used the
ETAU parameterization in their setup. However, as suggested by , we can speculate that such a parameterization
could mask the impact of the wave coupling even though it turned out to be necessary to obtain realistic mixed layer depths.
We believe that our modification of the standard NEMO one-equation TKE scheme described in Sect.
is more physically justifiable than the ETAU parameterization and requires much less parameter tuning.
The numerical experiments based on the ORCA25 configuration discussed in Sect. were
meant to check that our developments were having the expected impact on numerical solutions. First, we confirmed
that using the Charnock parameter computed in the wave model instead of a constant value globally increases
the wind stress magnitude, particularly at middle and high latitudes, whereas accounting for the portion of the wind stress
consumed by the waves has a small impact (in our experiments it leads to a maximum 2 % decrease in the wind stress).
Second, using the mixed layer depth as an indicator to assess the amount of vertical mixing, the modifications brought
to the NEMO turbulence scheme (i.e., the new boundary condition for TKE and for the mixing length, the
addition of the Stokes shear in the TKE equation, and the modified parameterization for Langmuir cells)
lead to an important extra mixing contributing to a deepening of the surface mixed layer, particularly in the Southern
Hemisphere. When compared to ARGO data it shows a significant improvement during the summer, while during the winter
the extra wave-induced mixing deepens the already too deep mixed layer. Note that the parameterization
to account for the restratification induced by mixed layer instabilities during the winter
was not used in our experiments. This parameterization induces even more shallow summer mixed layer depths.
As far as the Northern Hemisphere is concerned, coupled results show an improvement when compared to ARGO
for winter with a deepening of the mixed layer, while in summer results are similar to the uncoupled case.
Since the comparison with ARGO data can be tricky due to the scarcity of data, we looked at the results
in terms of mixed layer depth (MLD) probability density functions. This allowed us to highlight the significant
improvement in MLD distribution when coupling with the waves. Furthermore, we noticed that all components
of the ocean–wave coupling act to deepen the mixed layer and therefore have a cumulative effect. However, the
main contributor is the fully modified TKE scheme including the Langmuir cell parameterization of , which is consistent with recent
results obtained by and using a K-profile parameterization (KPP) closure scheme.
Since the magnitude of the vertical mixing is increased by the coupling with waves we expect an impact on
sea surface temperature and currents. Indeed, the summer deepening of the mixed layer in the Southern
Hemisphere leads to colder sea surface temperatures, resulting in better agreement with the OSTIA SST analysis.
More generally, although the global SST biases are not totally compensated for, they tend to be reduced when
considering the effect of waves (see Sect. ). The currents in the oceanic surface boundary layer
are reduced by the Stokes–Coriolis force which counteracts the Ekman current;. They are
also affected by the increased vertical mixing, which tends to reduce the surface currents (and thus the surface kinetic energy)
and strengthen the currents at the base of the surface boundary layer. The reduction of surface kinetic energy due
to the wave–ocean coupling in the global 1/4∘ resolution configuration is of the same order of magnitude
as the reduction observed when accounting for surface currents in the computation of the wind stress in a coupled
ocean–atmosphere model e.g.,. A fully coupled ocean–wave–atmosphere model would thus be
necessary to properly disentangle the different contributions at play impacting the oceanic surface kinetic energy.
Even if additional diagnostics on various configurations at different resolutions are still needed to exhaustively
evaluate the impact of each component of the ocean wave coupling, the results presented in the paper confirm
the robustness of our developments, and our implementation will serve as a starting point for the inclusion of
wave–current interactions in the forthcoming NEMO official release.
We can speculate that the ocean–wave coupled ORCA025 configuration might become a standard component
of future Coupled Model Intercomparison Project (CMIP) exercises.
We already mentioned as a perspective the addition of a coupling with an interactive atmospheric boundary layer
either via a full atmospheric model or a simplified boundary layer model e.g.,.
Furthermore, the gain of an online two-way coupling compared to a one-way coupling on the oceanic and
wave solution must be investigated in the future.
Indeed, the improvements of the quality of surface wave simulations associated with a coupling with large-scale oceanic currents
are well documented, particularly in the Agulhas current and in the Gulf Stream .
have also shown a strong impact of small-scale currents (10–100 km) on wave height variability at the same scales.
We can therefore expect improvements for both wave and ocean forecasts when the coupling is implemented in an operational context.
Flux-form wave-averaged momentum equations
In this Appendix we describe the necessary changes when a flux formulation for advective terms
in the momentum equations is preferred to the vector-invariant form presented in Eqs. ()
and (). For simplicity, we consider just the i component in horizontal curvilinear
coordinates and the z coordinate in the vertical (results will be extended to the j component and to a
generalized vertical coordinate). Consistently with the notations of , e1 and e2 are the horizontal scale factors.
We denote Avu the extra term needed to guarantee equivalence between the flux formulation and the vector-invariant form.
Avu is defined such that
∇⋅(usu)+Avu=-ζvs+wse3∂ku.
Since ∇⋅us=0, we have ∇⋅(usu)=us⋅∇u and thus
e1e2Avu=-vs∂i(e2v)-∂j(e1u)+e1e2e3ws∂ku-e2us∂iu+e1vs∂ju+e1e2e3ws∂ku=-vsv∂ie2-u∂je1+e2∂iv-e1∂ju-e2us∂iu-e1vs∂ju.
Hence,
Avu=-vse1e2v∂ie2-u∂je1︸Metric term on Stokes drift-use1∂iu+vse1∂iv︸Additional term.
The same computation for the j component leads to the following equations in generalized vertical coordinates.
1e3∂t(e3u)=-1e1e2∂i(e2(u+ũs)u)+∂j(e1(v+ṽs)u)+1e3∂k((ω+ω̃s)u)+f+1e1e2v∂ie2-u∂je1(v+ṽs)+ũse1∂iuz+ṽse1∂ivz-1ρ0e1∂i(ps+p̃J)-1ρ0e1∂iphz+1e3∂ku′ω′+Fu+F̃u1e3∂t(e3v)=-1e1e2∂i(e2(u+ũs)v)+∂j(e1(v+ṽs)v)+1e3∂k((ω+ω̃s)v)-f+1e1e2v∂ie2-u∂je1(u+ũs)+ũse2∂juz+ṽse2∂jvz-1ρ0e2∂j(ps+p̃J)-1ρ0e2∂jphz+1e3∂kv′ω′+Fv+F̃v
Here, ∂i•z and ∂j•z are derivatives along the z coordinate.
Sensitivity to the cLC parameter from single-column experiments
Single-column experiments based on have been performed to
study the behavior of the NEMO vertical closure with the Langmuir
cell parameterization of . In the experiments
the initial condition is given by
u(z,t)=v(z,t)=0,θ(z,t)=minT0-N02(z-5.0)αg,T0,
with α the thermal expansion coefficient in the equation of state defined as
ρ=-αρ0(T-T0) with ρ0=1024 kg m-3.
A zonal wind is imposed with u⋆=0.02 m s-1,
and the Stokes drift is given by
us=(us,0),us=2πaλ2gλ2πe-4πz/λ.
The various parameter values are
fcor=10-4s-1,hmax=120m,T0=16∘C,N02=10-5s-2,
with 96 vertical levels for the discretization and 16 h simulations.
We only consider the case with a=1 m and λ=40m, which gives a turbulent
Langmuir number of Lat≈0.32.
Numerical results are shown in Fig. (upper panels) and are consistent with the results of
with a deepening of the oceanic mixing length of about 10 m when Langmuir turbulence
is accounted for (see LES results in Fig. 3 in ). For CLC=0.15 in the parameterization,
the deepening is too weak, while for CLC=0.3 it is closer to the LES results.
Note that for those experiments, the value of dLC is almost identical to the mixed
layer depth. Figure (lower panels) illustrates the fact that for a stronger
stratification (i.e., with N02=2×10-4 s-2
instead of N02=10-5 s-2) the effect of Langmuir turbulence on mixed layer depth is negligible.
Indeed, in this case Langmuir cells do not provide enough mixing to erode the stratification.
Solution obtained for the single-column experiment after 16 h for different parameter values in
the Langmuir cell parameterization in the case N02=10-5 s-2 (upper panels) and N02=2×10-4 s-2 (lower panels).
Code and data availability
The changes to the NEMO code have been made on the standard NEMO
code (nemo_v3_6_STABLE). The code can be downloaded
from the NEMO website (http://www.nemo-ocean.eu/, last access: 11 July 2019, ).
The NEMO code modified to include wave–ocean coupling terms and the OASIS interface
is available in the Zenodo archive (10.5281/zenodo.3331463, ).
The WW3 code version 6.02 has been used without further modifications
and can be downloaded from the NOAA GitHub repository (https://github.com/NOAA-EMC/WW3, last access: 11 July 2019, ).
Our modifications of the OASIS interface in the WW3 code have already been
integrated in the official release. The OASIS3_MCT code is also freely available
(https://portal.enes.org/oasis/, last access: 11 July 2019, ).
The exact versions of the WW3 and OASIS3_MCT codes that were used
have also been made available in the Zenodo archive (10.5281/zenodo.3331463, )
The initial and forcing data for both the oceanic and wave model, analysis scripts,
namelists, and the data used to produce the figures are also available in the Zenodo archive.
Author contributions
XC prepared and carried out all the numerical experiments,
investigated the results, and wrote the paper with the help of all the coauthors.
GM, FL, RB, and XC made the changes in the
NEMO code to include the wave–ocean interactions. GS helped to
prepare the necessary datasets for the numerical experiments and analyze the model outputs. FA and JR helped to investigate the results and to formalize the
necessary wave-induced terms in both the primitive equations and the TKE closure.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank George Nurser and Oyvind Breivik, whose efforts helped to improve earlier versions of this paper, as well as Qiang Wang and Svenja Langer.
We also thank Knut Klingbeil, Patrick Marchesiello, and Patrick Marsaleix for useful discussions.
Xavier Couvelard, Florian Lemarié, and Jean-Luc Redelsperger
acknowledge support by Mercator-Ocean and the Copernicus Marine
Environment Monitoring Service (CMEMS) through contract
22-GLO-HR – Lot 2 (High-resolution ocean, waves, atmosphere interaction).
Numerical simulations were performed on Ifremer HPC facilities DATARMOR of
“P∘le de Calcul Intensif pour la Mer” (PCIM) (http://www.ifremer.fr/pcim, last access: 2 July 2020).
Mixed layer depth data were graciously provided by Clément de Boyer Montégut,
and SST data were downloaded from the CMEMS catalog. The authors also gratefully
thank Claude Talandier for help with NEMO, Mickaël Accensi for help with
WW3, and Eric Maisonnave and Laure Coquart for their help with OASIS3_MCT.
Review statement
This paper was edited by Qiang Wang and reviewed by A. J. George Nurser and Oyvind Breivik.
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