SOME OCEAN MODEL FUNDAMENTALS
63
very similar to the depth surfaces shown in Figure 11. These properties
greatly reduce difficulties of computing the horizontal pressure gradient
relative to terrain following sigma models discussed next. Additionally,
since z ∗ = z when η = 0, no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. This
behavior is in contrast to the case with sigma models, where pressure
gradient errors in the presence of nontrivial topographic variations can
generate spontaneous flow from a resting state. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in z ∗ models using the same techniques
as in z-models (see Chapters 13-16 of Griffies, 2004 for a discussion of
neutral physics in z-models).
The range over which z ∗ varies is time independent −H ≤ z ∗ ≤ 0.
Hence, all cells remain nonvanishing, so long as the surface height maintains η > −H. This is a minor constraint relative to that encountered
on the surface height when using s = z or s = z − η.
Because z ∗ has a time independent range, all grid cells have static
increments ds, and the sum of the vertical increments yields the time
independent ocean depth
k ds = H. The z ∗ coordinate is therefore
invisible to undulations of the free surface, since it moves along with the
free surface. This property means that no spurious vertical transport
is induced across surfaces of constant z ∗ by motion of external gravity
waves. Such spurious transport can be a problem in z-models, especially those with tidal forcing. Quite generally, the time independent
range for the z ∗ coordinate is a very convenient property that allows
for a nearly arbitrary vertical resolution even in the presence of large
amplitude fluctuations of the surface height.
Depth sigma coordinate.
The depth-sigma coordinate
σ = (z − η)/(H + η)
(111)
is the canonical terrain following coordinate. Figure 12 illustrates this
coordinate in a realistic model. The sigma coordinate has a long history
of use in coastal modelling. For reviews, see Greatbatch and Mellor,
1999 and Ezer et al., 2002. Models based on the sigma coordinate have
also been successfully extended to basinwide studies, as well as recent
global work by Diansky et al., 2002.
Just as for z ∗ = H σ, the range over which the sigma coordinate
varies is time independent and given by −1 ≤ σ ≤ 0. Hence, all cells
have static grid increments ds, and the sum of the vertical increments
yields unity
k ds = 1. So long as the surface height is not depressed
63
very similar to the depth surfaces shown in Figure 11. These properties
greatly reduce difficulties of computing the horizontal pressure gradient
relative to terrain following sigma models discussed next. Additionally,
since z ∗ = z when η = 0, no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. This
behavior is in contrast to the case with sigma models, where pressure
gradient errors in the presence of nontrivial topographic variations can
generate spontaneous flow from a resting state. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in z ∗ models using the same techniques
as in z-models (see Chapters 13-16 of Griffies, 2004 for a discussion of
neutral physics in z-models).
The range over which z ∗ varies is time independent −H ≤ z ∗ ≤ 0.
Hence, all cells remain nonvanishing, so long as the surface height maintains η > −H. This is a minor constraint relative to that encountered
on the surface height when using s = z or s = z − η.
Because z ∗ has a time independent range, all grid cells have static
increments ds, and the sum of the vertical increments yields the time
independent ocean depth
k ds = H. The z ∗ coordinate is therefore
invisible to undulations of the free surface, since it moves along with the
free surface. This property means that no spurious vertical transport
is induced across surfaces of constant z ∗ by motion of external gravity
waves. Such spurious transport can be a problem in z-models, especially those with tidal forcing. Quite generally, the time independent
range for the z ∗ coordinate is a very convenient property that allows
for a nearly arbitrary vertical resolution even in the presence of large
amplitude fluctuations of the surface height.
Depth sigma coordinate.
The depth-sigma coordinate
σ = (z − η)/(H + η)
(111)
is the canonical terrain following coordinate. Figure 12 illustrates this
coordinate in a realistic model. The sigma coordinate has a long history
of use in coastal modelling. For reviews, see Greatbatch and Mellor,
1999 and Ezer et al., 2002. Models based on the sigma coordinate have
also been successfully extended to basinwide studies, as well as recent
global work by Diansky et al., 2002.
Just as for z ∗ = H σ, the range over which the sigma coordinate
varies is time independent and given by −1 ≤ σ ≤ 0. Hence, all cells
have static grid increments ds, and the sum of the vertical increments
yields unity
k ds = 1. So long as the surface height is not depressed