Energy-enstrophy stability of -plane Kolmogorov flow with drag

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 -plane flow driven by a sinusoidal body force and retarded by drag with damping time scale −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 , -parameter space. The EZ method proves nonlinear stability in a larger portion of the , -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 , -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. © 2008 American Institute of Physics. DOI: 10.1063/1.2958321