finite compressibility of seawater from the primitive equations, although it removes sound waves
from the system. Second, the related Boussinesq
approximation ignores the small density variations
in the ocean except for those associated with
the coefficient of gravity. Both approximations
together have a relative error of approximately
2%, which at present is much smaller than the
errors arising from uncertainties in forcing and
parameterizations or from the numerical discretization. In the future, however, one can envision a situation where such an error may no longer
be negligible, in particular when very accurate sea
surface height observations become available. In
such a case, the anelastic approximation constitutes an alternative; this also eliminates sound
waves, but has a smaller relative error of only
0.2%. See Mellor and Ezer (1995) and Dukowicz
(1997) for recent explorations of non-Boussinesq
effects in large-scale ocean models.
Third, the shallow water approximation –
which consists of the hydrostatic approximation
with the additional neglect of the horizontal component of the Coriolis vector (traditional approximation) – is valid when the vertical scale of
motion H is much smaller then the lateral scale L;
its accuracy is approximately of order H/L. For
ocean climate models and most eddy-resolving
models, the shallow-water approximation is well
justified, and the more complete equations of
motion are unnecessary. For models with a very
high resolution of order 10 km where this error
may no longer be tolerable, Marshall et al.
(1997b) have introduced the quasihydrostatic
approximation, which keeps the full Coriolis vector and has an accuracy of (H/L)
2
. For some
process studies (e.g. of convection, three-dimensional turbulence, etc.), the shallow-water approximation is no longer valid, and non-hydrostatic
models have to be considered.
Specialized alternative forms of the governing
equations have also been proposed for purposes
of superior computational performance, for more
robust behaviour when applied using ‘open’ boundary conditions, and/or for more complete removal
of high-frequency variability. Examples include the
balance equations (McWilliams et al., 1990; Allen,
1991), various filtered systems (Browning et al.,
1990), and equations based on Lattice Boltzman
methods (Salmon, 1999). None of these alternate forms has received systematic large-scale
application, and more study is warranted in this
area.
Many of the models mentioned above use a
rigid-lid condition to approximate the surface
mass balance. This approximation essentially
filters out barotropic gravity waves and long
barotropic Rossby waves, and in addition introduces some subtleties in the freshwater budget
(see, e.g., Huang, 1993). The overall error of the
rigid-lid-approximation for the barotropic circulation is approximately given by the ratio of surface
mass flux (P–E, in m s
91
) to the vertical Ekman
pumping velocity, and has a magnitude of 1–3%
or smaller. Much larger errors can however occur
in marginal seas when the surface salinity varies
between oceanic and freshwater values. The problems can be avoided by the use of the correct
free-surface boundary condition, although this
necessitates specialized time-stepping treatment
to remain computationally competitive with rigid
formulations.
7.2.3.2 Sub-grid-scale parameterizations
Perhaps the most pressing concern, particularly for
global climate modelling on non-eddy-resolving
grids, is the improvement of parameterizations for
‘sub-grid-scale’ processes. For coupled climate
models with typical grid spacing of several hundred kilometres, these processes encompass all of
the energetic mesoscale variability, as well as many
sub-basin-scale thermodynamic processes such as
dense overflows, deep convection and diapycnal
mixing processes in boundary layers and in the
ocean interior. At present, a wide variety of closure approaches is used. While traditionally many
of them are largely ad hoc in form and motivation,
dynamically motivated parameterizations are
increasingly becoming available. Recent reviews of
sub-grid-scale closures can be found in Haidvogel
and Beckmann (1999) and Chassignet and Verron
(1998).
Sub-grid-scale transports are traditionally represented by Fickian diffusion and friction, following
the classical ideas of Prandtl and Kolmogorov.
While the lateral coefficients and mixing lengths
are often chosen to be constant or to depend on
grid spacing, one may increasingly expect formulations that reflect their dependence on energy and
length scale of the sub-grid-scale motions, and/or
the rate of strain such as in the early work by
Smagorinsky (1963). In ocean models that do not
SECTION 7 INSIGHTS FOR THE FUTURE
550
from the system. Second, the related Boussinesq
approximation ignores the small density variations
in the ocean except for those associated with
the coefficient of gravity. Both approximations
together have a relative error of approximately
2%, which at present is much smaller than the
errors arising from uncertainties in forcing and
parameterizations or from the numerical discretization. In the future, however, one can envision a situation where such an error may no longer
be negligible, in particular when very accurate sea
surface height observations become available. In
such a case, the anelastic approximation constitutes an alternative; this also eliminates sound
waves, but has a smaller relative error of only
0.2%. See Mellor and Ezer (1995) and Dukowicz
(1997) for recent explorations of non-Boussinesq
effects in large-scale ocean models.
Third, the shallow water approximation –
which consists of the hydrostatic approximation
with the additional neglect of the horizontal component of the Coriolis vector (traditional approximation) – is valid when the vertical scale of
motion H is much smaller then the lateral scale L;
its accuracy is approximately of order H/L. For
ocean climate models and most eddy-resolving
models, the shallow-water approximation is well
justified, and the more complete equations of
motion are unnecessary. For models with a very
high resolution of order 10 km where this error
may no longer be tolerable, Marshall et al.
(1997b) have introduced the quasihydrostatic
approximation, which keeps the full Coriolis vector and has an accuracy of (H/L)
2
. For some
process studies (e.g. of convection, three-dimensional turbulence, etc.), the shallow-water approximation is no longer valid, and non-hydrostatic
models have to be considered.
Specialized alternative forms of the governing
equations have also been proposed for purposes
of superior computational performance, for more
robust behaviour when applied using ‘open’ boundary conditions, and/or for more complete removal
of high-frequency variability. Examples include the
balance equations (McWilliams et al., 1990; Allen,
1991), various filtered systems (Browning et al.,
1990), and equations based on Lattice Boltzman
methods (Salmon, 1999). None of these alternate forms has received systematic large-scale
application, and more study is warranted in this
area.
Many of the models mentioned above use a
rigid-lid condition to approximate the surface
mass balance. This approximation essentially
filters out barotropic gravity waves and long
barotropic Rossby waves, and in addition introduces some subtleties in the freshwater budget
(see, e.g., Huang, 1993). The overall error of the
rigid-lid-approximation for the barotropic circulation is approximately given by the ratio of surface
mass flux (P–E, in m s
91
) to the vertical Ekman
pumping velocity, and has a magnitude of 1–3%
or smaller. Much larger errors can however occur
in marginal seas when the surface salinity varies
between oceanic and freshwater values. The problems can be avoided by the use of the correct
free-surface boundary condition, although this
necessitates specialized time-stepping treatment
to remain computationally competitive with rigid
formulations.
7.2.3.2 Sub-grid-scale parameterizations
Perhaps the most pressing concern, particularly for
global climate modelling on non-eddy-resolving
grids, is the improvement of parameterizations for
‘sub-grid-scale’ processes. For coupled climate
models with typical grid spacing of several hundred kilometres, these processes encompass all of
the energetic mesoscale variability, as well as many
sub-basin-scale thermodynamic processes such as
dense overflows, deep convection and diapycnal
mixing processes in boundary layers and in the
ocean interior. At present, a wide variety of closure approaches is used. While traditionally many
of them are largely ad hoc in form and motivation,
dynamically motivated parameterizations are
increasingly becoming available. Recent reviews of
sub-grid-scale closures can be found in Haidvogel
and Beckmann (1999) and Chassignet and Verron
(1998).
Sub-grid-scale transports are traditionally represented by Fickian diffusion and friction, following
the classical ideas of Prandtl and Kolmogorov.
While the lateral coefficients and mixing lengths
are often chosen to be constant or to depend on
grid spacing, one may increasingly expect formulations that reflect their dependence on energy and
length scale of the sub-grid-scale motions, and/or
the rate of strain such as in the early work by
Smagorinsky (1963). In ocean models that do not
SECTION 7 INSIGHTS FOR THE FUTURE
550