The hydrostatic primitive equations written in geopotential (z- or height) coordinates form the basis for today's most widely used numerical ocean models. In some applications, however, it may be advantageous to apply suitable transformations to the vertical coordinate. Alternatives include terrain-following () and density (isopycnal) coordinate systems. The former maps the total ocean depth to the interval [0,1], thus associating the lowermost coordinate level with the the ocean floor, with potential advantages in representing benthic processes. The second assumes a system of moving constant-density layers, and treats layer thickness as a prognostic quantity, with consequent advantages in the representation of thermohaline fronts.

In a generalized vertical coordinate, the unforced, inviscid primitive equations are (Bleck,1978)

Here, the generalized vertical coordinate Z can be chosen to be z for geopotential, for terrain-following, or for isopycnic coordinates, as discussed below.

In their continuous forms, all these systems are equivalent of course. Unfortunately, discretization for a numerical model creates truncation errors whose detailed form and behavior differ from one vertical coordinate system to another. Hence, each of these systems may be better suited for certain classes of problems than for others.

Over the past 10 years, a variety of ocean circulation models have been developed for both regional and basin-to-global-scale applications. For example, an international inventory of regional models of the coastal ocean, compiled by Haidvogel and Beckmann (1998), shows more than a dozen codes to be in wide use, including one or more representatives from each of the three vertical coordinate classes. Here, we briefly describe and compare one example from each model class, including a geopotential (MOM), a terrain-following (SPEM/SCRUM) and an isopycnic (MICOM) model.

All these models use *finite difference* arithmetic. The description of a fourth model (SEOM) is added to contrast the properties of a Galerkin-based *finite element* model. Note that all these models are constantly under development. Therefore, only the "main engines" are described below. Complete descriptions of the models in their most current versions can be obtained from the World Wide Web.

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