On hydrostatic flows in isentropic coordinates

Abstract

The hydrostatic primitive equations of motion which have been used in large-scale weather prediction and climate modelling over the last few decades are analysed with variational methods in an isentropic Eulerian framework. The use of material isentropic coordinates for the Eulerian hydrostatic equations is known to have distinct conceptual advantages since fluid motion is, under inviscid and statically stable circumstances, confined to take place on quasi-horizontal isentropic surfaces. First, an Eulerian isentropic Hamilton's principle, expressed in terms of fluid parcel variables, is therefore derived by transformation of a Lagrangian Hamilton's principle to an Eulerian one. This Eulerian principle explicitly describes the boundary dynamics of the time-dependent domain in terms of advection of boundary isentropes sB; these are the values the isentropes have at their intersection with the (lower) boundary. A partial Legendre transform for only the interior variables yields an Eulerian ‘action’ principle. Secondly, Noether's theorem is used to derive energy and potential vorticity conservation from the Eulerian Hamilton's principle. Thirdly, these conservation laws are used to derive a wave-activity invariant which is second-order in terms of small-amplitude disturbances relative to a resting or moving basic state. Linear stability criteria are derived but only for resting basic states. In mid-latitudes a time- scale separation between gravity and vortical modes occurs. Finally, this time-scale separation suggests that conservative geostrophic and ageostrophic approximations can be made to the Eulerian action principle for hydrostatic flows. Approximations to Eulerian variational principles may be more advantageous than approximations to Lagrangian ones because non-dimensionalization and scaling tend to be based on Eulerian estimates of the characteristic scales involved. These approximations to the stratified hydrostatic formulation extend previous approximations to the shallow- water equations. An explicit variational derivation is given of an isentropic version of Hoskins & Bretherton's model for atmospheric fronts.