Kinetics of 2D Boussinesq: Beyond the realm of classical wave turbulence

The 2D Boussinesq equation is a non-isotropic, non-canonical Hamiltonian partial differential equation. This turns its kinetic description into an interesting and non-trivial problem since the classical wave turbulence approach is almost irrelevant. Similarly to the 3D Boussinesq equation - the theoretical derivation of the energy spectrum remains an open problem. I will discuss recent results which include the formal derivation of the kinetic equation and its turbulent solutions. In these the fact that the 2D Boussinesq has two quadratic invariants plays an essential role.