Energy-enstrophy stability of β-plane Kolmogorov flow with drag

Abstract

We develop a nonlinear stability method, the energy-enstrophy (EZ) method, that is specialized to two-dimensional hydrodynamics and basic state flows consisting of a single Helmholtz eigenmode. The method is applied to a β-plane flow driven by a sinusoidal body force and retarded by drag with damping time scale μ−1. The standard energy method [H. Fukuta and Y. Murakami, J. Phys. Soc. Jpn. 64, 3725 (1995)] shows that the laminar solution is monotonically and globally stable in a certain portion of the (μ,β)-parameter space. The EZ method proves nonlinear stability in a larger portion of the (μ,β)-parameter space than does the energy method. Moreover, by penalizing high wavenumbers, the EZ method identifies a most strongly amplifying disturbance that is more physically realistic than that delivered by the energy method. Linear instability calculations are used to determine the region of the (μ,β)-parameter space where the flow is unstable to infinitesimal perturbations. There is only a small gap between the linearly unstable region and the nonlinearly stable region, and full numerical solutions show only small transient amplification in that gap.