On instability of the Ekman spiral

Abstract

The aim of this note is to present a new approach to linear and nonlinear instability of the Ekman spiral, the famous stationary geostrophic solution of the 3D Navier-Stokes equations in a rotating frame. As former approaches to the Ekman boundary layer problem, our result is based on the numerical existence of an unstable wave perturbation for Reynolds numbers large enough derived by Lilly in [15]. By the fact that this unstable wave is tangentially nondecaying at infinity, however, standard approaches (e.g. by cut-off techniques) to instability in standard function spaces (e.g. $L^p$) remain a technical and intricate issue. In spite of this fact, we will present a rather short proof of linear and nonlinear instability of the Ekman spiral in $L^2$. The results are based on a recently developed general approach to rotating boundary layer problems, which relies on Fourier transformed vector Radon measures, cf.[11].