Linear baroclinic instability in extended regime geostrophic models

The linear wave and baroclinic instability properties of various geostrophi c models valid when the Rossby number is small are investigated. The models are the "L-1" dynamics, the "geostrophic potential vorticity" equations, a nd the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to de termine whether these models accurately portray linear baroclinic instabili ty properties in various geophysically relevant parameter regimes, in a hig hly idealized and limited set of cases. The L-1 and geostrophic potential v orticity models are properly balanced (devoid of inertio-gravity waves, exc ept possibly at solid boundaries), valid on the beta plane, and contain bot h quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the de formation and planetary scales of motion. The L-1 model also includes the " frontal geostrophic" equations as a third limit. In fact, the choice to inv estigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion. The models are cast in multilayer form, and the dispersion properties and e igenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the L-1 and geostrophic potential vorti city models have sensible linear stability properties with no artifactual i nstabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic and planetary geostrop hic parameter regimes. The growth rate of baroclinic instability in the pla netary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of th e planetary geostrophic equations diverges at high wavenumbers, but it is s hown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.