The models of Bryan
[4],
Madala and Piacsek
[37]
and Killworth et al.
[36]
have all used fixed
levels in the vertical z-direction, with a variable spacing of the
depth levels to resolve the rapid changes in the surface layers.
On the other hand, Blumberg and Mellor
[3]
and Haidvogel et al.
[25]
have introduced a stretched coordinate usually referred to as
sigma (for its previous usage in meteorology), defined as
where D is the fluid depth. This leads to boundary surfaces that
can be put at 
and an automatic accommodation of the
depth contours. In addition Haidvogel et al.
[25]
have introduced a
semi-spectral representation of the vertical dimension (sigma
layout) in terms of Chebyshev polynomials (collocation method).
Both of these have employed the free-surface formulation. See
Figure 2
and Figure 3.
Figure 2: Vertical discretization in ocean models.
