resolve mesoscale eddies, the lateral sub-grid-scale
transport mainly results from the eddies. Following
the work of Gent and McWilliams (1990), there
are presently intense efforts to develop parameterization schemes based on eddy dynamics and the
physics of baroclinic instability. Typically, these
transport parameterizations are diffusive for layer
thickness (Gent et al., 1995) or potential vorticity
(e.g. Killworth, 1997), in addition to an isopycnal
diffusion of tracers. These parameterizations
should be globally dissipative in order to simulate
the instability process; locally, however, this is not
necessarily so, and there exist parameterizations
(e.g. through topographic stress) that describe the
generation of mean flow by a pre-existing eddy
field (cf. Eby and Holloway, 1994). The parameterization of eddy-induced momentum transports is
much less advanced. While globally it is perhaps
less critical, due to the overriding geostrophy constraint, in regions of strong currents there are
strong dynamical eddy–mean flow interactions that
are not represented in most models.
Vertical (diapycnal) mixing in the ocean is associated with energy conversions, and parameterizations are often based on energy arguments. In their
most simple form, these suggest formulating mixing coefficients in terms of stability frequency
and/or Richardson number. Double-diffusive mixing has been included in models only to a limited
extent, and can be important at least for the
regional water mass distribution (Zhang et al.,
1998b; Merryfield et al., 1999). It is now documented that diapycnal mixing is distributed rather
inhomogeneously, and is concentrated over regions
of rough topography (Polzin et al., 1997), a fact
that so far has not been properly accounted for in
models. Tides are among the likely causes for that
inhomogeneity (Munk and Wunsch, 1998), and the
parameterization of mixing might be facilitated in a
model that includes both the circulation and the
tides.
The surface mixed layer has received special
attention because of its important role in air–sea
exchange. The ocean modeller can now choose
from various Bulk models of the Kraus–Turner
type, models with specified exchange as Price–
Weller–Pinkel (Price et al., 1986), Pacanowski and
Philander (1981), or KPP (K-profile parameterization; Large et al., 1994), TKE (turbulent kinetic
energy) schemes (Gaspar et al., 1990), and various
levels of second-order closure schemes (Mellor and
Yamada, 1982). A systematic intercomparison of
the properties, behaviour and accuracy of these
parameterizations in the context of circulation
models is lacking, and it is not entirely clear how
one would go about doing so for air–sea coupling
on the decadal time scales of interest to climate
modelling.
The diapycnal mixing across isopycnals in the
ocean is very weak. In a model based on isopycnal
layers, dyapycnic mixing can be easily controlled.
For other classes of numerical models, in particular
for geopotential and terrain-following coordinate
models, weak diapycnal mixing may be difficult to
achieve, and may depend on the rotation of the
mixing tensors away from the native lateral coordinate directions. Horizontal and along-sigma mixing
has long been the default for these classes of models; however, algorithms for stable and conservative isopycnal rotation of diffusion tensors are now
available (Griffies et al., 1998).
Parameterizations are not only needed to represent unresolved physical processes, but also to
compensate for deficiencies of the numerical algorithm. Monotonicity and positive definiteness are
important characteristics of a tracer distribution,
but not necessarily a property of numerical algorithms. The choice of diffusive closure is therefore
closely allied with the form of the discrete horizontal advection operator (e.g. Shchepetkin and
McWilliams, 1998). Also, dissipative parameterizations are necessary to check numerical instabilities. These requirements may best be satisfied by
biharmonic or higher-order operators, with viscous coefficients chosen adaptively, e.g. such that
grid-Reynolds and/or grid-Peclet numbers are
O(1). For the credibility and acceptance of model
results it seems important to state explicitly
whether a parameterization is primarily physically
based or numerically required.
A strong test of a sub-grid-scale closure would
involve a high-resolution simulation (using the
‘complete’ equations of motion) that resolves
the processes under consideration, followed by a
demonstration that a coarse-resolution experiment
with the new parameterization gives quantitatively
identical net (i.e. large-scale and/or time-mean)
results; however, this sort of systematic testing is
not often performed. The role of such simulation
studies, including the application of large-eddy
simulations (e.g. Wang et al., 1998), will become
increasingly important in the future.
7.2 Numerical Ocean Circulation Modelling: Present and Future
551
Willebrand and Haidvogel
transport mainly results from the eddies. Following
the work of Gent and McWilliams (1990), there
are presently intense efforts to develop parameterization schemes based on eddy dynamics and the
physics of baroclinic instability. Typically, these
transport parameterizations are diffusive for layer
thickness (Gent et al., 1995) or potential vorticity
(e.g. Killworth, 1997), in addition to an isopycnal
diffusion of tracers. These parameterizations
should be globally dissipative in order to simulate
the instability process; locally, however, this is not
necessarily so, and there exist parameterizations
(e.g. through topographic stress) that describe the
generation of mean flow by a pre-existing eddy
field (cf. Eby and Holloway, 1994). The parameterization of eddy-induced momentum transports is
much less advanced. While globally it is perhaps
less critical, due to the overriding geostrophy constraint, in regions of strong currents there are
strong dynamical eddy–mean flow interactions that
are not represented in most models.
Vertical (diapycnal) mixing in the ocean is associated with energy conversions, and parameterizations are often based on energy arguments. In their
most simple form, these suggest formulating mixing coefficients in terms of stability frequency
and/or Richardson number. Double-diffusive mixing has been included in models only to a limited
extent, and can be important at least for the
regional water mass distribution (Zhang et al.,
1998b; Merryfield et al., 1999). It is now documented that diapycnal mixing is distributed rather
inhomogeneously, and is concentrated over regions
of rough topography (Polzin et al., 1997), a fact
that so far has not been properly accounted for in
models. Tides are among the likely causes for that
inhomogeneity (Munk and Wunsch, 1998), and the
parameterization of mixing might be facilitated in a
model that includes both the circulation and the
tides.
The surface mixed layer has received special
attention because of its important role in air–sea
exchange. The ocean modeller can now choose
from various Bulk models of the Kraus–Turner
type, models with specified exchange as Price–
Weller–Pinkel (Price et al., 1986), Pacanowski and
Philander (1981), or KPP (K-profile parameterization; Large et al., 1994), TKE (turbulent kinetic
energy) schemes (Gaspar et al., 1990), and various
levels of second-order closure schemes (Mellor and
Yamada, 1982). A systematic intercomparison of
the properties, behaviour and accuracy of these
parameterizations in the context of circulation
models is lacking, and it is not entirely clear how
one would go about doing so for air–sea coupling
on the decadal time scales of interest to climate
modelling.
The diapycnal mixing across isopycnals in the
ocean is very weak. In a model based on isopycnal
layers, dyapycnic mixing can be easily controlled.
For other classes of numerical models, in particular
for geopotential and terrain-following coordinate
models, weak diapycnal mixing may be difficult to
achieve, and may depend on the rotation of the
mixing tensors away from the native lateral coordinate directions. Horizontal and along-sigma mixing
has long been the default for these classes of models; however, algorithms for stable and conservative isopycnal rotation of diffusion tensors are now
available (Griffies et al., 1998).
Parameterizations are not only needed to represent unresolved physical processes, but also to
compensate for deficiencies of the numerical algorithm. Monotonicity and positive definiteness are
important characteristics of a tracer distribution,
but not necessarily a property of numerical algorithms. The choice of diffusive closure is therefore
closely allied with the form of the discrete horizontal advection operator (e.g. Shchepetkin and
McWilliams, 1998). Also, dissipative parameterizations are necessary to check numerical instabilities. These requirements may best be satisfied by
biharmonic or higher-order operators, with viscous coefficients chosen adaptively, e.g. such that
grid-Reynolds and/or grid-Peclet numbers are
O(1). For the credibility and acceptance of model
results it seems important to state explicitly
whether a parameterization is primarily physically
based or numerically required.
A strong test of a sub-grid-scale closure would
involve a high-resolution simulation (using the
‘complete’ equations of motion) that resolves
the processes under consideration, followed by a
demonstration that a coarse-resolution experiment
with the new parameterization gives quantitatively
identical net (i.e. large-scale and/or time-mean)
results; however, this sort of systematic testing is
not often performed. The role of such simulation
studies, including the application of large-eddy
simulations (e.g. Wang et al., 1998), will become
increasingly important in the future.
7.2 Numerical Ocean Circulation Modelling: Present and Future
551
Willebrand and Haidvogel