WAVE-ACTIVITY CONSERVATION-LAWS AND STABILITY THEOREMS FOR SEMI-GEOSTROPHIC DYNAMICS .2. PSEUDOENERGY-BASED THEORY

This paper represents the second part of a study of semi-geostrophic ( SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophi c (QG) model, but is also a good prototype for balanced models that ar e more accurate than QC dynamics. The development of such balanced mod els is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was b ased on the pseudomomentum; Part 2 is based on the pseudoenergy.A pseu doenergy invariant is a conserved quantity, of second order in disturb ance amplitude relative to a prescribed steady basic state, which is r elated to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linea r stability theorem analogous to Arnol'd's 'first theorem'; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous t o their quasi-geostrophic forms, and reduce to those forms in the limi t of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to be ta-plane compressible flow by Magnusdottir and Schubert. Novel feature s particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability cr iteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stab le basic states. The interior semi-geostrophic dynamics has an underly ing Hamiltonian structure, which guarantees that symmetries in the sys tem correspond naturally to the system's invariants. This is an import ant motivation for the theoretical approach used in this study. The co nnection between symmetries and conservation laws is made explicit usi ng Noether's theorem applied to the Eulerian form of the Hamiltonian d escription of the interior dynamics.