The hydrostatic limit for stratified fluids

I will talk about the hydrostatic approximation of the 2d Euler-Boussinesq system, describing the evolution of an inviscid stratified fluid where the vertical length scale is much smaller than the horizontal one. Even though meaningful in oceanography, the justification of the hydrostatic limit in this context has remained an open problem.

I will discuss some recent results showing that some instability mechanisms may prevent this limit to hold. Along the way, I will also draw some connections with large-time issues for stratified fluids and with kinetic theory for plasmas.