# Some Mathematical Remarks Concerning the Localization of Planetary Waves in a Stochastic Background Flow

### Peter Imkeller, Adam Hugh Monahan, and Lionel Pandolfo

In this article we develop some mathematically rigorous ideas to explain the phenomena of localization of planetary waves in a stochastic background flow as presented in the physical companion paper. For this purpose the linearized equation for the zonally averaged vorticity subject to a zonal background wind and driven by a local source is transformed in to Sturm-Liouville problems with random potential function. There is a first background effect of localization symmetrical with respect to the equator which is due to the nodes of the potential function (* critical lines *). A second more subtle effect causes localization around the source, by 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 an average, this effect is moderate for zero damping, and stronger for small non-zero damping.

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