Some mathematical remarks concerning the localisation of planetary waves in a stochastic background flow

In this article we develop some mathematically rigorous ideas to explain the phenomenon of localisation of planetary waves in a stochastic background flow as presented in the physical companion paper. For this purpose the barotropic vorticity equation linearised around a zonal background wind and driven by a local source is transformed into a Sturm-Liouville problem with random potential function. We distinguish between two types of localising mechanisms. The first type is a background effect of localisation symmetrical with respect to the equator which is due to the nodes of the potential function (critical lines). The second is a more subtle effect and forces localisation around the source. It comes from the superposition of the source term with the Green's kernel expressed in terms of the eigenfunctions of the spectral resolution of the random Sturm-Liouville operators involved. On average, this effect is moderate for zero damping, and stronger for small non-zero damping.