Linear stability theorems for shallow water semi-geostrophic dynamics

Abstract

The linear stability of zonal basic flow in the shallow water semi-geostrophic (SWSG) model is examined with an emphasis on the contributions of lateral boundaries to the stability properties of the model. Two conservation equations for wave activities, analogous to the pseudomomentum and pseudoenergy densities in the shallow water quasi-geostrophic (SWQG) theory, are obtained. Two linear stability theorems are established based on the pseudoenergy conservation equation. The first theorem, corresponding to positive definite pseudoenergy, requires the gradient of basic state potential vorticity to be negatively correlated with basic flow in the interior, and the shear of basic flow to be cyclonic at lateral boundaries. The second theorem, corresponding to negative definite pseudoenergy, requires the gradient of basic state potential vorticity to be positively correlated with basic flow in the interior and the shear of basic flow to be anti-cyclonic at lateral boundaries, plus some additional conditions. In both theorems, Ripa's subsonic condition U ≤ √gH is found unnecessary, and the unnecessity is found to be associated with the dynamic structure of the model. The general properties of normal mode disturbances are discussed. Bounds on the phase speed and growth rate of unstable normal modes are estimated. The contributions of lateral boundaries to the instability of constant shear and cosine type basic flows are investigated numerically. It is shown that the contributions crucially depends on the value of the Rossby number. When stability criterion in the interior is violated, unstable growth rate can be significantly enhanced by the boundary contributions. The underlying physical interpretation is given.