Linear stability and enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

TL;DR

This paper proves the linear stability and enhanced dissipation of a specific two-jet flow on the sphere, showing exponential decay of perturbations and the nonexistence of nonzero eigenvalues in the linearized operator.

Contribution

It establishes the linear stability of the two-jet Kolmogorov flow on the sphere for all viscosities and demonstrates enhanced dissipation through spectral analysis of the linearized operator.

Findings

Exponential decay of solutions to the linearized equation.

Nonexistence of nonzero eigenvalues of the perturbation operator.

Enhanced dissipation as viscosity tends to zero.

Abstract

We consider the Navier-Stokes equations on the two-dimensional unit sphere and study the linear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. We prove the linear stability of the two-jet Kolmogorov type flow for an arbitrary viscosity coefficient by showing the exponential decay of a solution to the linearized equation towards an equilibrium which grows as the viscosity coefficient tends to zero. The main result of this paper is the nonexistence of nonzero eigenvalues of the perturbation operator appearing in the linearized equation. By making use of the mixing property of the perturbation operator which is expressed by a recurrence relation for the spherical harmonics, we show that the perturbation operator does not have not only nonreal but also nonzero real eigenvalues. As an application of this result, we get the enhanced dissipation for the two-jet Kolmogorov type flow in the sense that a solution to the linearized equation rescaled in time decays arbitrarily fast as the viscosity coefficient tends to zero.