On methods for solving the oceanic equations of motion in generalized vertical coordinates

We note that there are essentially two methods of solving the hydrostatic primitive equations in general vertical coordinates: the quasi-Eulerian class of algorithms are typically used in quasi-stationary coordinates (e.g. height, pressure, or terrain following) coordinate systems; the quasi-Lagrangian class of algorithms are almost exclusively used in layered models and is the preferred paradigm in modern isopycnal models. These approaches are not easily juxtaposed. Thus, hybrid coordinate models that choose one method over the other may not necessarily obtain the particular qualities associated with the alternative method.We discuss the nature of the differences between the Lagrangian and Eulerian algorithms and suggest that each has its benefits. The arbitrary Lagrangian–Eulerian method (ALE) purports to address these differences but we find that it does not treat the vertical and horizontal dimensions symmetrically as is done in classical Eulerian models. This distinction is particularly evident with the non-hydrostatic equations, since there is explicitly no symmetry breaking in these equations. It appears that the Lagrangian algorithms can not be easily invoked in conjunction with the pressure method that is often used in non-hydrostatic models. We suggest that research is necessary to find a way to combine the two viewpoints if we are to develop models that are suitable for simulating the wide range of spatial and temporal scales that are important in the ocean.