OCEAN MODELS
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scale models, the smallest spatial scale is often the width of the western
boundary current. When it is controlled by laplacian friction it is called
a Munk boundary layer. The condition that the grid scale δx be smaller
than the Munk layer width results in a minimum bound for viscosity
(Smith and McWilliams, 2003): ν > ν M ≈ βδx 3 . On the other hand,
viscosity cannot be arbitrarily large due to the stability constraint (similar to the CFL criterion for advection). This criterion is more severe
in ocean models that use explicit leap-frog time stepping schemes for
nonlinear advection, with the viscous terms lagged by one time step for
stability. For laplacian viscosity ν < δx 2 /8 δt. For a biharmonic operator the criterion is ν < δx 4 /64 δt (biharmonic coded as in the POP
model) or ν < δx 4 /128 δt (biharmonic coded as in the OPA model).
For the laplacian operator on coarse grids, the Munk layer constraint
implies very large viscosities: with δx=10 km at 45 ◦ N, ν M = 16 m 2 .s −1 ,
but with δx=100 km, ν M = 16000 m 2 .s −1 . It is impossible to represent equatorial dynamics with such a large viscosity, which is why
this constraint is not always taken into account. For example, in the
FOAM 1 ◦ model the viscosity is 5100 m 2 .s −1 . In that case, some level of
grid point noise usually develops near the western boundary. A second
strategy is to decrease the viscosity at the equator, while increasing the
meridional resolution there (parameterization of the ORCA2 model with
ν = 2000 m 2 .s −1 at the equator and ν = 40000 m 2 .s −1 at mid-latitudes,
Madec et al., 1998). Finally, Large et al. (2001) have proposed to make
the viscosity anisotropic, noting that the equatorial current are dominantly zonal while Munk boundary currents are predominantly meridional. This solution however does not prevent the apparition of numerical noise.
For the biharmonic operation the numerical stability criterion is often more stringent than the Munk layer constraint. With δx=10 km,
the latter gives ν > βδx 5 = 1.6 10 9 m 4 .s −1 . With a 1200 s time step,
numerical stability requires that ν < δx 4 /64 δt = 1.3 10 11 m 4 .s −1 . A decrease of the biharmonic coefficient with the grid spacing is often needed
in order to ensure stability on spatially variable grids, as in the case of
PSY2 (Table 2). Based on numerical experiments with a 1/12 ◦ isopycnic
model, Chassignet and Garraffo (personal communication) suggest that
the Gulf stream separation is improved using together a laplacian and
a bilaplacian operator. This has prompted the use of both operators in
FOAM (Table 2).
Smagorinsky (1963) has proposed to make the laplacian viscosity proportional to the deformation rate times the squared grid spacing δx 2 .
Such a parameterization can be physically motivated in three dimensional turbulence and is used in large eddy simulations. In ocean
99
scale models, the smallest spatial scale is often the width of the western
boundary current. When it is controlled by laplacian friction it is called
a Munk boundary layer. The condition that the grid scale δx be smaller
than the Munk layer width results in a minimum bound for viscosity
(Smith and McWilliams, 2003): ν > ν M ≈ βδx 3 . On the other hand,
viscosity cannot be arbitrarily large due to the stability constraint (similar to the CFL criterion for advection). This criterion is more severe
in ocean models that use explicit leap-frog time stepping schemes for
nonlinear advection, with the viscous terms lagged by one time step for
stability. For laplacian viscosity ν < δx 2 /8 δt. For a biharmonic operator the criterion is ν < δx 4 /64 δt (biharmonic coded as in the POP
model) or ν < δx 4 /128 δt (biharmonic coded as in the OPA model).
For the laplacian operator on coarse grids, the Munk layer constraint
implies very large viscosities: with δx=10 km at 45 ◦ N, ν M = 16 m 2 .s −1 ,
but with δx=100 km, ν M = 16000 m 2 .s −1 . It is impossible to represent equatorial dynamics with such a large viscosity, which is why
this constraint is not always taken into account. For example, in the
FOAM 1 ◦ model the viscosity is 5100 m 2 .s −1 . In that case, some level of
grid point noise usually develops near the western boundary. A second
strategy is to decrease the viscosity at the equator, while increasing the
meridional resolution there (parameterization of the ORCA2 model with
ν = 2000 m 2 .s −1 at the equator and ν = 40000 m 2 .s −1 at mid-latitudes,
Madec et al., 1998). Finally, Large et al. (2001) have proposed to make
the viscosity anisotropic, noting that the equatorial current are dominantly zonal while Munk boundary currents are predominantly meridional. This solution however does not prevent the apparition of numerical noise.
For the biharmonic operation the numerical stability criterion is often more stringent than the Munk layer constraint. With δx=10 km,
the latter gives ν > βδx 5 = 1.6 10 9 m 4 .s −1 . With a 1200 s time step,
numerical stability requires that ν < δx 4 /64 δt = 1.3 10 11 m 4 .s −1 . A decrease of the biharmonic coefficient with the grid spacing is often needed
in order to ensure stability on spatially variable grids, as in the case of
PSY2 (Table 2). Based on numerical experiments with a 1/12 ◦ isopycnic
model, Chassignet and Garraffo (personal communication) suggest that
the Gulf stream separation is improved using together a laplacian and
a bilaplacian operator. This has prompted the use of both operators in
FOAM (Table 2).
Smagorinsky (1963) has proposed to make the laplacian viscosity proportional to the deformation rate times the squared grid spacing δx 2 .
Such a parameterization can be physically motivated in three dimensional turbulence and is used in large eddy simulations. In ocean