On the wave‐activity invariants and stability of the two‐layer shallow‐water semi‐geostrophic model

Abstract

AbstractAlthough the two‐layer shallow‐water semi‐geostrophic (TLSWSG) model has been proven to be an important tool for the investigation of the dynamics of anisotropic and baroclinic flow in frontal zones, its general instability mechanism, however, has not been well understood. In light of recent work on the stability of the one‐layer SWSG model, this issue is re‐examined in this paper by extending a body of well developed theorems in the two‐layer quasi‐geostrophic (QG) model to the TLSWSG model. The generalized linearized TLSWSG model equations with respect to a parallel basic state are presented first. Conservation equations for two small‐amplitude wave‐activity invariants analogous to pseudo‐momentum and pseudo‐energy densities in QG dynamics are derived and then used to examine the stability of a parallel basic state. It is found that the TLSWSG model possesses two kinds of stability conditions: Arnold's first stability condition in the interior, plus conditions on lateral boundaries, and Arnold's second stability condition. In contrast to previous work, Ripa's ‘subsonic’ condition \documentclass{article}\pagestyle{empty}\begin{document}$ \[\left({\left({U_{G1},U_{G2} } \right)\max < \sum\nolimits_{j = 1}^2 {\left({{1 \mathord{\left/ {\vphantom {1 {cj}}} \right. \kern-\nulldelimiterspace} {cj}}} \right)^{ - 1} } } \right) \] $\end{document}, where UG1 and UG2 are basic flows at two levels and Cj is the phase speed of gravity waves at the jth level) is found to be unnecessary, and the underlying physical reasons, associated with the coastal Kelvin waves in the model, are given. It is also found that the phase speed of unstable normal‐mode disturbances of slow motion satisfies Howard's semi‐circle theorem.The energy—Casimir method is applied to derive small‐ and finite‐amplitude wave‐activity invariants and corresponding stability criteria for non‐parallel basic states. It turns out that the stability criteria for non‐parallel basic states include not only those for parallel basic states but also conditions associated with the anisotropy of basic flow. The nonlinear stability analyses in this work help us to understand the failure of the energy—Casimir method in deriving finite‐amplitude stability criteria in the continuously stratified semi‐geostrophic model.