The Unstable Spectrum of the Surface Quasi-geostrophic Equation

We study the unstable spectrum of an equation that arises in geophysical fluid dynamics known as the surface quasi-geostrophic equation. In general the spectrum is the union of discrete eigenvalues and an essential spectrum. We demonstrate the existence of unstable eigenvalues in a particular example. We examine the spectra of the semigroup and the evolution operator. We exhibit the structure of these spectra for general flows and prove that a spectral mapping theorem holds. We observe that the spectral properties of the SQG equation are closely analogous to those of the two dimensional Euler equation .