BAROCLINIC INSTABILITY in the SOLAR TACHOCLINE. II. the EADY PROBLEM

We solve the nongeostrophic baroclinic instability problem for the tachocline for a continuous model with a constant vertical rotation gradient (the Eady problem), using power series generated by the Frobenius method. The results confirm and greatly extend those from a previous two-layer model. For effective gravity G independent of height, growth rates and ranges of unstable longitudinal wavenumbers m and latitudes increase with decreasing G. As with the two-layer model, the overshoot tachocline is much more unstable than the radiative tachocline. The e-folding growth times range from as short as 10 days to as long as several years, depending on latitude, G, and wavenumber. For a more realistic effective gravity that decreases linearly from the radiative interior to near zero at the top of the tachocline, we find that only m = 1, 2 modes are unstable, with growth rates somewhat larger than for constant G, with the same value as at the bottom of the tachocline. All results are the same whether we assume that the vertical velocity or the perturbation pressure is zero at the top of the layer; this is a direct consquence of not employing the geostrophic assumption for perturbations. We explain most of the properties of the instability in terms of the Rossby deformation radius. We discuss further improvements in the realism of the model, particularly adding toroidal fields that vary in height, and including latitudinal gradients of both rotation and toroidal fields.