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.