Linear Baroclinic Instability in Extended Regime Geostrophic Models

The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number is small are investigated. The models are the ‘‘L1’’ dynamics, the ‘‘geostrophic potential vorticity’’ equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes, in a highly idealized and limited set of cases. The L1 and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravity waves, except possibly at solid boundaries), valid on the b plane, and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the deformation and planetary scales of motion. The L1 model also includes the ‘‘frontal geostrophic’’ equations as a third limit. In fact, the choice to investigate 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 eigenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the L1 and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic and planetary geostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.