Three-Dimensional Instability of Elliptical Flow

A theory is presented for Pierrehumbert's three-dimensional short-wave inviscid instability of the simple two-dimensional elliptical flow with velocity field $\mathbf{u}\mathrm{}(x,y,z)=\Omega (-Ey,E^{-1\mathrm{}}x,0)$. The fundamental modes, which are also exact solutions of the nonlinear equations, are plane waves whose wave vector rotates elliptically around the $z$ axis with period $\frac{2\pi }{\Omega }$. The growth rates are the exponents of a matrix Floquet problem, and agree with those calculated by Pierrehumbert.