On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an ‘‘unstable disk theorem’’ through a simple energy–Casimir argument @Holm et al., Phys. Rep. 123, 1–116 ~1985!#. The eigenvalues of the linear Hamiltonian system are of four types: real pairs (c ,2c), purely imaginary pairs (id ,2id), quadruples (6c6id), and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincaré-type difference equation, i.e., it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constant-coefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory. © 2000 American Institute of Physics. @S0022-2488~00!00702-7#