11 - Numerical Methods for Solving Baroclinic Equations

Summary

Introduction

A baroclinic model of the atmosphere is one that does not invoke the assumptions of the barotropic model; i.e., a fluid whose density depends only upon pressure. Thus, this model is more general than a barotropic model; however, it is not fully general. Winds in the baroclinic model are still represented by geostrophic approximation. Geostrophic winds are essentially non-divergent. Because geostrophic winds are nondivergent, there exists a stream function Ψ such that

where ψ is stream function, is the horizontal wind and ζ is the vertical component of the relative vorticity. Moreover, as the geostrophic winds are non-divergent, they can be expressed as

However, most numerical weather prediction modelers employ baroclinic models that utilize primitive equations with hydrostatic approximation, but without quasi-geostrophic filtering. Quasi-geostrophic models were utilized for simpler problems where the main motivation was the understanding of atmospheric or oceanic dynamics. Historically, the height coordinate z was employed as the vertical coordinate. However, while utilizing primitive equations with hydrostatic approximation, it became apparent that employing pressure p as a vertical coordinate is more advantageous. The most commonly employed vertical coordinates are height z, pressure p, a normalized pressure coordinate σ, potential temperature θ, and some examples of hybrid coordinates (combination of the earlier mentioned coordinates). The most important requirement of the choice of vertical coordinates is that the vertical coordinate has to be a monotonic function of the height z.

Let any arbitrary variable ζ(x,y,z,t) be denoted as the vertical coordinate. It is of course assumed that z is a monotonic function of height z. In this section, the system of equations is derived for a generalised vertical coordinate ζ (x,y,z,t), where ζ is assumed to be related to the height z by a single-valued monotonic function. With transformation of the vertical coordinate z to ζ , a variable u(x,y,z,t) becomes u(x,y,ζ(x,y,z,t);. In the transformed coordinate, the horizontal coordinates remain the same. Let s represent x or y or t. Figure 11.1 shows a schematic diagram of the vertical coordinate transformation.