7.2.3.3 Vertical coordinate systems
The choice of the vertical discretization is of particular importance for ocean circulation models. Currently, a variety of forms are used. Among these
are models featuring alternative treatments of
topography and the vertical coordinate. Examples
include vertical coordinate systems based upon
geopotential levels, isopycnal layers, and topography-following (‘sigma’) surfaces. While the first of
these approaches is still the most prevalent, all
three have by now been applied successfully to
large-scale circulation modelling (Willebrand et al.,
2001). Nevertheless, all three approaches have
inherent limitations, including poor convergence
upon vertical grid refinement over topography,
spurious pressure gradient errors, poor representation of vertical structure in shallow or weakly stratified regions, and/or excessive diapycnal diffusion.
For models expressed in geopotential coordinates, the form of the topography changes with
vertical resolution and placement of the grid boxes,
especially in regions with very gentle and very steep
topographic slopes. Some flexibility exists in choosing the levels, although a grid spacing that varies
according to a smooth analytic function has some
formal advantages (Treguier et al., 1996). However, it is usually not possible to find a single discretization that is simultaneously accurate for both
topography and stratification. Convergence of the
numerical solution under refinement of the vertical
grid is also very slow for stepwise approximations
to the underlying bathymetry (Haidvogel and
Beckmann, 1999). Spatial approximations that
remove the staircase representation – e.g. the
sigma coordinate method (see below) or the use of
‘shaved cells’ (Adcroft et al., 1997) – seem to
avoid this problem.
A geopotential coordinate model initialized
with a uniform barotropic (vertically unsheared)
flow field over a sloping bottom will, in the presence of finite horizontal viscosity, undergo an artificial energy transfer from the barotropic to the
baroclinic mode on the deep side of each topographic step. This spurious generation of vertical
shear is proportional to the prescribed eddy coefficient and the relative change in fluid depth over
one grid size.
Though free of the foregoing types of errors, a
problem of terrain-following coordinate systems is
the possibility of significant systematic errors in
the horizontal pressure gradients (Haney, 1991;
Beckmann and Haidvogel, 1993; Mellor et al.,
1994). These errors arise due to the splitting of the
pressure gradient term into an ‘along-coordinate
surface’ component and a ‘hydrostatic correction’.
Unfortunately, both terms are large, and cancellation of the hydrostatic resting pressure is not
exact, due to numerical truncation errors in the
two terms. The resulting pressure gradient errors
depend on the steepness of the topography, both
the horizontal and vertical resolution, and the
strength of the stratification. In addition, topographic smoothing is often necessary to ensure
stable and accurate simulations using realistic
bathymetry. Interestingly, the relative change in
fluid depth over a single grid step is also found to
be related to the robustness of the results from terrain-following models (Beckmann and Haidvogel,
1993).
Models using isopycnic coordinates have their
own unique limitations. Chief among these are
the need to handle vanishing layer thicknesses
and/or isopycnals intersecting topography, the difficulty in representing intrinsically non-isopycnic
processes that occur (e.g.) at the ocean surface,
and pressure gradient errors (similar to those that
arise with terrain-following models) when compressibility effects are considered. The first of these
can in principle be handled with specialized advection schemes, though this treatment is formally of
low order and hence of reduced accuracy. The latter errors can be substantially reduced (though not
fully eliminated) by various corrective methods
(e.g. Eden and Willebrand, 1999; Sun et al., 1999).
In the future, important gains are likely to be
made by further innovations in the area of vertical
coordinates. In particular, it is in principle feasible
to design new coordinate systems that merge the
best properties of the three coordinate treatments
currently in use. Such improved coordinates would
likely be nearly geopotential at the ocean surface
in order to provide uniform resolution of the
oceanic mixed layer, would be quasi-isopycnal in
the bulk of the oceanic interior in order to limit
spurious diapycnal mixing, and would more
closely follow bottom topography in regions of
strong slopes in order to more accurately incorporate the constraints of underlying bathymetric features. Several groups are currently exploring
approaches to such hybrid vertical coordinates,
and their experience with these new systems is
eagerly awaited.
SECTION 7 INSIGHTS FOR THE FUTURE
552
The choice of the vertical discretization is of particular importance for ocean circulation models. Currently, a variety of forms are used. Among these
are models featuring alternative treatments of
topography and the vertical coordinate. Examples
include vertical coordinate systems based upon
geopotential levels, isopycnal layers, and topography-following (‘sigma’) surfaces. While the first of
these approaches is still the most prevalent, all
three have by now been applied successfully to
large-scale circulation modelling (Willebrand et al.,
2001). Nevertheless, all three approaches have
inherent limitations, including poor convergence
upon vertical grid refinement over topography,
spurious pressure gradient errors, poor representation of vertical structure in shallow or weakly stratified regions, and/or excessive diapycnal diffusion.
For models expressed in geopotential coordinates, the form of the topography changes with
vertical resolution and placement of the grid boxes,
especially in regions with very gentle and very steep
topographic slopes. Some flexibility exists in choosing the levels, although a grid spacing that varies
according to a smooth analytic function has some
formal advantages (Treguier et al., 1996). However, it is usually not possible to find a single discretization that is simultaneously accurate for both
topography and stratification. Convergence of the
numerical solution under refinement of the vertical
grid is also very slow for stepwise approximations
to the underlying bathymetry (Haidvogel and
Beckmann, 1999). Spatial approximations that
remove the staircase representation – e.g. the
sigma coordinate method (see below) or the use of
‘shaved cells’ (Adcroft et al., 1997) – seem to
avoid this problem.
A geopotential coordinate model initialized
with a uniform barotropic (vertically unsheared)
flow field over a sloping bottom will, in the presence of finite horizontal viscosity, undergo an artificial energy transfer from the barotropic to the
baroclinic mode on the deep side of each topographic step. This spurious generation of vertical
shear is proportional to the prescribed eddy coefficient and the relative change in fluid depth over
one grid size.
Though free of the foregoing types of errors, a
problem of terrain-following coordinate systems is
the possibility of significant systematic errors in
the horizontal pressure gradients (Haney, 1991;
Beckmann and Haidvogel, 1993; Mellor et al.,
1994). These errors arise due to the splitting of the
pressure gradient term into an ‘along-coordinate
surface’ component and a ‘hydrostatic correction’.
Unfortunately, both terms are large, and cancellation of the hydrostatic resting pressure is not
exact, due to numerical truncation errors in the
two terms. The resulting pressure gradient errors
depend on the steepness of the topography, both
the horizontal and vertical resolution, and the
strength of the stratification. In addition, topographic smoothing is often necessary to ensure
stable and accurate simulations using realistic
bathymetry. Interestingly, the relative change in
fluid depth over a single grid step is also found to
be related to the robustness of the results from terrain-following models (Beckmann and Haidvogel,
1993).
Models using isopycnic coordinates have their
own unique limitations. Chief among these are
the need to handle vanishing layer thicknesses
and/or isopycnals intersecting topography, the difficulty in representing intrinsically non-isopycnic
processes that occur (e.g.) at the ocean surface,
and pressure gradient errors (similar to those that
arise with terrain-following models) when compressibility effects are considered. The first of these
can in principle be handled with specialized advection schemes, though this treatment is formally of
low order and hence of reduced accuracy. The latter errors can be substantially reduced (though not
fully eliminated) by various corrective methods
(e.g. Eden and Willebrand, 1999; Sun et al., 1999).
In the future, important gains are likely to be
made by further innovations in the area of vertical
coordinates. In particular, it is in principle feasible
to design new coordinate systems that merge the
best properties of the three coordinate treatments
currently in use. Such improved coordinates would
likely be nearly geopotential at the ocean surface
in order to provide uniform resolution of the
oceanic mixed layer, would be quasi-isopycnal in
the bulk of the oceanic interior in order to limit
spurious diapycnal mixing, and would more
closely follow bottom topography in regions of
strong slopes in order to more accurately incorporate the constraints of underlying bathymetric features. Several groups are currently exploring
approaches to such hybrid vertical coordinates,
and their experience with these new systems is
eagerly awaited.
SECTION 7 INSIGHTS FOR THE FUTURE
552