A Generalization of a Sigma Coordinate Ocean Model
71
and if viscosity is not high enough. Applying a computational stability analysis to
one - dimensional, advection-diffusion equation,
i+l-i-l = 2R-l(i+I- 2i+i_l)
(Bryan et al. 1975) derived the stability condition R < 2, where R = ulll/A is the
grid cell Reynolds number. Apparently, the step structure of the z - leve1 grid
excites this mode (R varies spatially but is ofthe order 2 for the high viscosity runs,
but is much larger for the low viscosity runs) whereas the O' grids do not. Bell
(1997) has produced a more complex analysis which identifies vorticity errors due
to step structure.
4.8 Other Grids
We have looked at and compared only two conventional grids and a third somewhat unconventional, generalized sigma, or s - coordinate grid. (It should be mentioned however, that in the particular application tested here, the superiority ofthe s
- coordinate system over the standard O' - coordinate system has not been demonstrated). However, the possible grid structures are infinite. For example, there is no
reason not to have a sigma-like grid in one portion of a model domain and a z -
level in another portion where, for example, the topography in a low resolution
model might be especially steep. Given the model structure described in section
4.5, it would be possible to create an isopycnal model by creating a single enabling
subroutine. The attendant problems of surface and bottom layer representation
seem daunting to us, however. Other possibilities are an adaptive vertical coordinate system to resolve the surface mixed layer or an underlying sharp spring and
summertime thermocline such as observed in the Great Lakes (Bedford and
Schwab 1998).
4.9 Summary
There are two main thrusts to this paper. The first thrust is the creation of a model
algorithm whose vertical coordinate system is quite general; this algorithm invites
future inventiveness with other grids beyond the preliminary tests presented in this
paper. The second thrust is a model intercomparison study of conventional z - level
and O' - coordinate vertical grids and one unconventional generalized sigma grid.
The intercomparisons are unique in that alI aspects of the models are identical
except for the vertical grid specification. One finding is that the sigma grids can
tolerate much smaller levels of horizontal viscosity and diffusivity. In shalIow
water, errors arise in the z - level grid which, depending on viscosity (which itself
may be considered a source of error), are of a different nature and obviously much
larger than errors due to the sigma pressure gradient error.
71
and if viscosity is not high enough. Applying a computational stability analysis to
one - dimensional, advection-diffusion equation,
i+l-i-l = 2R-l(i+I- 2i+i_l)
(Bryan et al. 1975) derived the stability condition R < 2, where R = ulll/A is the
grid cell Reynolds number. Apparently, the step structure of the z - leve1 grid
excites this mode (R varies spatially but is ofthe order 2 for the high viscosity runs,
but is much larger for the low viscosity runs) whereas the O' grids do not. Bell
(1997) has produced a more complex analysis which identifies vorticity errors due
to step structure.
4.8 Other Grids
We have looked at and compared only two conventional grids and a third somewhat unconventional, generalized sigma, or s - coordinate grid. (It should be mentioned however, that in the particular application tested here, the superiority ofthe s
- coordinate system over the standard O' - coordinate system has not been demonstrated). However, the possible grid structures are infinite. For example, there is no
reason not to have a sigma-like grid in one portion of a model domain and a z -
level in another portion where, for example, the topography in a low resolution
model might be especially steep. Given the model structure described in section
4.5, it would be possible to create an isopycnal model by creating a single enabling
subroutine. The attendant problems of surface and bottom layer representation
seem daunting to us, however. Other possibilities are an adaptive vertical coordinate system to resolve the surface mixed layer or an underlying sharp spring and
summertime thermocline such as observed in the Great Lakes (Bedford and
Schwab 1998).
4.9 Summary
There are two main thrusts to this paper. The first thrust is the creation of a model
algorithm whose vertical coordinate system is quite general; this algorithm invites
future inventiveness with other grids beyond the preliminary tests presented in this
paper. The second thrust is a model intercomparison study of conventional z - level
and O' - coordinate vertical grids and one unconventional generalized sigma grid.
The intercomparisons are unique in that alI aspects of the models are identical
except for the vertical grid specification. One finding is that the sigma grids can
tolerate much smaller levels of horizontal viscosity and diffusivity. In shalIow
water, errors arise in the z - level grid which, depending on viscosity (which itself
may be considered a source of error), are of a different nature and obviously much
larger than errors due to the sigma pressure gradient error.