Energy-Enstrophy Stability of beta-plane Kolmogorov Flow with Drag

TL;DR

This paper introduces the Energy-Enstrophy (EZ) stability method for 2D hydrodynamics, demonstrating it extends the stability analysis of beta-plane Kolmogorov flows with drag beyond traditional energy methods, with implications for understanding flow stability.

Contribution

The paper develops the EZ method tailored for 2D flows, providing a more comprehensive nonlinear stability analysis than existing energy methods, and identifies physically realistic disturbances affecting flow stability.

Findings

EZ method extends nonlinear stability region compared to energy method.

Most amplifying disturbances identified are more physically realistic.

Small gap between linear instability and nonlinear stability regions.

Abstract

We develop a new nonlinear stability method, the Energy-Enstrophy (EZ) method, that is specialized to two-dimensional hydrodynamics; the method is applied to a beta-plane flow driven by a sinusoidal body force, and retarded by drag with damping time-scale mu^{-1}. The standard energy method (Fukuta and Murakami, J. Phys. Soc. Japan, 64, 1995, pp 3725) shows that the laminar solution is monotonically and globally stable in a certain portion of the (mu,beta)-parameter space. The EZ method proves nonlinear stability in a larger portion of the (mu,beta)-parameter space. And 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 (mu,beta)-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.