Nonlinear instability for the Boussinesq equations with diabatic forcing

In this paper we prove the linear instability for the Boussinesq equations experiencing a diabatic forcing. To do so, we first prove the local well-posedness of the governing set of equations. An example of an unstable solution to the linearized set of equations is then constructed. Finally, we show that the unstable solution to the linearized problem is also unstable in the nonlinear sense.