Variational asymptotics for rotating shallow water near geostrophy: a transformational approach

Abstract

We introduce a unified variational framework in which the classical balance models for nearly geostrophic shallow water as well as several new models can be derived. Our approach is based on consistently truncating an asymptotic expansion of a near-identity transformation of the rotating shallow-water Lagrangian. Model reduction is achieved by imposing either degeneracy (for models in a semi-geostrophic scaling) or incompressibility (for models in a quasi-geostrophic scaling) with respect to the new coordinates.

At first order, we recover the classical semi-geostrophic and quasi-geostrophic equations, Salmon's $L_1$ and large-scale semi-geostrophic equations, as well as a one-parameter family of models that interpolate between the two. We identify one member of this family, different from previously known models, that promises better regularity – hence consistency with large-scale vortical motion – than all other first-order models. Moreover, we explicitly derive second-order models for all cases considered. While these second-order models involve nonlinear potential vorticity inversion and do not obviously share the good properties or their first-order counterparts, we offer an explicit survey of second-order models and point out several avenues for exploration.