The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Abstract

Abstract. 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 .

The Essential Spectrum of Advective Equations

“…So, the essential spectrum of the 2D Euler and SQG equations is a solid band (annulus) symmetric with respect to the imaginary axis. This result was obtained previously by Latushkin, Friedlander and the author in [41,40,13] via an explicit construction of approximate eigenfunctions for each point in the band.…”

“…These are the solid annulus and vertical strip, respectively, for any m = 0. The same spectral picture has been found for the SQG equation in [13]. What these equations have in common is that their b-cocycles have trivial dynamical spectrum Σ 0 = {0}.…”

“…Theorem 3.1. Suppose A ∈ L 0 is a pseudodifferential operator with semiclassical symbol a(x, ξ) so that decomposition (13) holds. Let…”

“…Subsequently, Shvydkoy and Vishik [42] have shown that, in fact, for any given Lyapunov exponent λ of the amplitude equation, the circle of radius e tλ contains a point of the essential spectrum. The geometric optics method has been applied to many other nondissipative equations of ideal hydrodynamics, such as Boussinesq approximation [14], SQG [13], Euler in vorticity form [23,24]. Equations with Coriolis forcing were treated in [16,43].…”

Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics

“…In this case we have trivially Σ B = {0} and there is no unstable continuous spectrum (see [15] for more information).…”