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1.1.2 Fixed Level, Isopycnal, Sigma-Coordinate and Semi-Spectral Models

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.