Instability of zonal flows in a two-layer shallow water semi-geostrophic model

The parametric dependence of the instability of zonally symmetric basic flows is examined in a two-layer shallow water semi-geostrophic (TLSWSG) inodel on the f-plane. The relevant parameters are the Rossby number (Ro), domain aspect ratio (mu), and Burger number (B). The cut-off values of the Bur.-er and Richardson numbers (Ri) for stability are estimated for a constant shear basic flow based on the pseudo-energy and pseudo-momentum conservation equations. Unstable normal-mode growth rates are calculated for a wide range of parameters for a constant shear basic flow and a cosine-type basic flow. The results show that within the SG regime, a small Burger number tends to generate strong baroclinic instability for a constant sheai basic flow, but tends to suppress barotropic-baroclinic instability for a cosine-type basic flow. Increasing the Rossby number enhances both baroclinic and barotropic-baroclinic instability when B > 0.16 but reduces the instability of large-scale disturbances when B < 0.16. Strong anisotropy (large mu) leads to strong barotropic-baroclinic instability for a cosine-type basic flow, but tends to reduce baroclinic instability for a constant shear basic flow. It is also found that strong horizontal shear tends to suppress baroclinic instability.