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Finite-Amplitude Wave-Activity Invariants and Nonlinear Stability Theorems for Shallow Water Semigeostrophic Dynamics

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  • 1 Department of Physics, University of Toronto, Toronto, Ontario, Canada
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Abstract

Motivated by the boundary contributions to the wave-activity invariants and stability theorems of a class of Salmon’s L1-like Hamiltonian balance models, Arnold’s method is applied in this work to derive finite-amplitude wave-activity invariants and corresponding stability theorems for shallow water semigeostrophic (SWSG) dynamics.

It is shown that the Jacobian term in the potential vorticity of the SWSG model affects the stability properties in two ways: it generates stability constraints in the interior, and it makes the stability condition of cyclonic shear of basic flow at boundaries inescapable even when Ripa’s “subsonic” condition is satisfied in the interior. The latter effect makes the stability properties of the SWSG model different from those of the L1-like Hamiltonian balance models for which the condition of cyclonic shear of basic flow on the boundaries is not necessary when Ripa’s “subsonic” condition is satisfied. The physical reason for this difference is given and the implications of the stability theorems are discussed.

* Current affiliation: ARMA, Atmospheric Environment Service, Downsview, Ontario, Canada.

Corresponding author address: Dr. Shuzhan Ren, ARMA, Atmospheric Environment Service, 4905 Dufferin Street, Downsview, ON M3H 5T4, Canada.

Email: shuzhan.ren@ec.gc.ca

Abstract

Motivated by the boundary contributions to the wave-activity invariants and stability theorems of a class of Salmon’s L1-like Hamiltonian balance models, Arnold’s method is applied in this work to derive finite-amplitude wave-activity invariants and corresponding stability theorems for shallow water semigeostrophic (SWSG) dynamics.

It is shown that the Jacobian term in the potential vorticity of the SWSG model affects the stability properties in two ways: it generates stability constraints in the interior, and it makes the stability condition of cyclonic shear of basic flow at boundaries inescapable even when Ripa’s “subsonic” condition is satisfied in the interior. The latter effect makes the stability properties of the SWSG model different from those of the L1-like Hamiltonian balance models for which the condition of cyclonic shear of basic flow on the boundaries is not necessary when Ripa’s “subsonic” condition is satisfied. The physical reason for this difference is given and the implications of the stability theorems are discussed.

* Current affiliation: ARMA, Atmospheric Environment Service, Downsview, Ontario, Canada.

Corresponding author address: Dr. Shuzhan Ren, ARMA, Atmospheric Environment Service, 4905 Dufferin Street, Downsview, ON M3H 5T4, Canada.

Email: shuzhan.ren@ec.gc.ca

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