On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D
In this paper, we prove the nonlinear instability of a given vertical shear of velocity between two rigid plane for the 3-D inviscid, non-conducting Boussinesq equations with rotation. When the Rossby number is zero, this rotating inviscid Boussinesq system reduces to the nonlinear geostrophic limit model. For non-zero small Rossby numbers, we establish the nonlinear instability of the shear flow, which is consistent with that of the geostrophic limit model. The proof relies on constructing a precise approximate solution, which comprises a growing profile derived from the nonlinear geostrophic limit model and a higher-order asymptotic expansion with respect to the small Rossby number. Notice that the instabilities (growing modes) are driven by the physical boundaries.