The models of Bryan , Madala and Piacsek  and Killworth et al.  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  and Haidvogel et al.  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.  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.