The
very first 3-D ocean models were multi-level, **z**-coordinate models
applied to highly idealized oceans rectangular in shape and
uniform in depth
(Figure 2(a)).
The real oceans have complicated
geometry and large changes in bottom depth, ranging from a very
few meters near the coast to a few kilometers in the deep basins.
It is generally difficult to resolve the water column equally well
and equally efficiently in both shallow and deep regions of a
basin simultaneously. Since the number of vertical levels in an
ocean model is constrained principally by economic considerations,
either the deep or the near-surface regions have to be sacrificed.
Thus for strong topographic variations, **z**-coordinate models are
inherently disadvantageous. That is why, while rigid lid three-dimensional
**z**-coordinate models have enjoyed wide-spread
application to deep ocean basins, applications to shallow coastal
regions or basins where shallow regions are not excluded are quite
rare. Since turbulent mixing plays an important role in the
circulation in the entire water column in shallow water, a better
simulation of both the upper and bottom mixed layers is possible
with a topographically conformal vertical coordinate system, the
so-called sigma coordinate system
(Figure 2(b)).

In a sigma-coordinate system,
the number of vertical levels in the water
column is the same everywhere in the domain irrespective of the
depth of the water column. This is achieved by transformation of
the governing equations from **z**-coordinate to sigma-coordinate in
the vertical:

where , is the depth of the water
column. Notice that unlike the **z**-coordinate system, where the
* layer* thicknesses are uniform in the horizontal, it is the
normalized thicknesses that are uniform in the sigma-coordinate
system, while the * layer* thicknesses vary widely from grid point to
grid point. Also no grid point is wasted in the vertical, unlike
the **z**-coordinate model, where high resolution in deeper regions of
an ocean basin means inevitable loss of those grid points in
shallower regions. Besides, the sigma coordinate enables the
bottom (benthic) boundary layer to be better resolved everywhere
in the domain. It is easy to see that adequate resolution of the
benthic layer everywhere in the domain can be achieved in
**z**-coordinates only with the use of a prohibitively large number of
levels in the vertical.