Alternative Vertical Coordinates
The model domain is usually represented in terms of spatial coordinates
x,y,z, or for large enough scale, longitude, latitude and altitude
(depth). However, in most oceanic situations the horizontal extent of
the domain is much greater than the vertical extent (maximum depth),
and bathymetric gradients and the vertical-to-horizontal aspect ratio of the flow are small.
Under latter situations, we can make a hydrostatic approximation to the
equation of motion, replacing the vertical momentum balance with the
hydrostatic balance between vertical pressure gradient and gravity.
Two other forms of vertical coordinates apart from height/depth have
proven themselves useful in oceanographic modeling:
- A terrain-following (sigma)
coordinate system designates the vertical location in terms of
fraction of the local depth (sigma), rather than by the depth itself.
In this way, the bottom of the model is always at sigma=1.
- Pros: Bathymetry can be represented in a smooth manner in the
model; bottom boundary condition always applies at sigma=1, making
- Cons: Where the bathymetric gradient is steep, along-sigma
evaluation of gradients can induce leading-order error (problem now
mostly solved). Processes such as diffusion tend to happen along
constant-sigma surfaces, which is unphysical and corrections to which
can incur significant computation.
- Models: Princeton
Ocean Model (POM) is the most popular pure sigma-coordinate model
in oceanography. Rutgers
Regional Ocean Model System (ROMS) uses S-coordinate system, a
generalization of sigma coordinates that allow controlled resolution
near the ocean surface and bottom boundary.
- A density (isopycnic) coordinate
system uses density of water, which would monotonically increase
with depth under static stratification, as the vertical coordinate. In
this coordinate system, ocean surface and bottom are not coordinate
surfaces.
- Pros: Since isopycnic (along-density-surface) and diapycnic
(across-density-surface) processes tend to be physically distinct in
the ocean, these processes are represented in a natural manner. It is
particularly straightforward to build conservative properties into the
model (e.g. potential vorticity).
- Cons: Care must be taken to evaluate isopycnic pressure
gradient across ocean surface or bottom (largely a solved
problem). Strongly non-conservative regions such as mixed layer and
convective regions require special treatment. Not suitable for
situations where density range and variability is great, such as an
estuary.
- Models: Miami
Isopycnic Coordinate Model (MICOM) is a commonly used density
coordinate model. Other choices include Hallberg
Isopycnic Model (HIM, in C) and Parallel Oregon State University
Model (POSUM). The latter two are both developed by UW graduates.
An effort is
under way to combine the virtues of different coordinates systems into
a single model.