Slaving Principles, Balanced Dynamics, and the Hydrostatic Boussinesq Equations

Abstract

The problem of finding higher-order generalizations of the quasigeostrophic and semigeostrophic models, that is, higher-order balanced models, is investigated systematically by using a slaving principle. It is shown that most existing balanced models are based on an underlying slave equation with varying master variables and that this slaving approach arises as an alternative to the secular conventional Rossby number expansions. While in the conventional expansions all variables are expanded in a power series in the Rossby number, the slaving approach suggests modified expansion or iteration procedures in which only the fast variables (associated with gravity modes) are expanded or iterated. The slow variables (associated with vortical modes) are now the master variables in the system; a well-known choice for large-scale extratropical flows in geophysical fluid dynamics is potential vorticity. To illustrate both the iteration and the modified expansion procedures, the slaving approach is applied to the rapidly rotating hydrostatic Boussinesq equations in a horizontally semi-infinite domain by using a Rossby number scaling. Higher-order generalizations of the quasigeostrophic equations result from modified expansions, and higher-order generalizations of the semigeostrophic equations result from nonlinear iterations. The validity of the latter for small Rossby numbers as well as small frontal parameters exemplifies the advantage nonlinear iterations have.