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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-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models 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 based on the pseudomomentum; Part 2 is based on the pseudoenergy.\ud \ud A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear 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 to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.\ud \ud The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.\ud \ud The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.
HOSKINSB,. J. 1982 The mathematical theory of frontogenesis. Ann. Rev. Fluid Mech. 14, 131-151.
HOSKINSB,. J., MCINTYREM,. E. & ROBERTSONA,. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Met. SOC.111, 877-946.
KUSHNERP,. J. 1993 Nonlinear stability and wave, mean-flow interaction in semi-geostrophic theory. In Ninth ConJ on Atmospheric and Oceanic Waves and Stability. Preprint volume, pp. 363-366. American Meteorological Society.
KUSHNERP,. J. 1995 A generalized Charney-Stern theorem for semi-geostrophic dynamics. Tellus (In press).
KUSHNERP,. J. & SHEPHERDT,. G. 1995 Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory. J. Fluid Mech. 290, 67-104 (referred to herein as Part 1).
MCINTYREM, . E. & SHEPHERDT,. G. 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527-565 (referred to herein as MS87).
MAGNUSDOTTGIR.,& SCHUBERWT,. H. 1990 The generalization of semi-geostrophic theory to the fl-plane. J. Atmos. Sci. 47, 1714-1720 (referred to herein as MSc90).
MAGNUSWTTIRG,. & SCHUBERWT,. H. 1991 Semigeostrophic theory on the hemisphere. J. Atmos. Sci. 48, 1449-1456.
ROULSTONEI., & NORBURYJ,. 1994 A Hamiltonian structure with contact geometry for the semigeostrophic equations. J. Fluid Mech. 272, 21 1-233.
SALMONR,. 1985 New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461-477.
SALMONR,. 1988 Semigeostrophic theory as a Dirac-bracket projection. J.FluidMech. 1%, 345-358.
SCHUBERTW,. H. & MAGNUSDOTTIRG, . 1994 Vorticity coordinates, transformed primitive equations and a canonical form for balance models. J. Atmos. Sci. 51, 3309-3319.
SHEPHERDT., G . 1989 Nonlinear saturation of baroclinic instability. Part I1 :Continuously stratified fluid. J. Atmos. Sci. 46, 888-907.
SHEPHERDT,. G . 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287-338.
SHUTTS,G. J. & CULLENM,. J. P. 1987 Parcel stability and its relation to semi-geostrophic theory. J. Atmos. Sci. 44, 1318-1330.
SWATERSG, . E. 1986 A nonlinear stability theorem for baroclinic quasigeostrophic flow. Phys. Flu& 29, 5-6.