Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

TL;DR

This paper investigates the enhanced dissipation phenomenon for a specific two-jet flow on the sphere, demonstrating rapid decay rates of solutions in the small viscosity limit using advanced spectral analysis techniques.

Contribution

The authors extend pseudospectral and resolvent estimate methods to analyze the decay rates of linearized flows on the sphere, adapting techniques from planar flows to spherical geometry.

Findings

Solutions decay at rate O(e^{-\sqrt{ u} }) for small viscosity u

Derived resolvent estimates for the linearized operator on the sphere

Established rapid decay analogous to planar Kolmogorov flow

Abstract

We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Pr\"uss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate $O(e^{-\sqrt{\nu}\,t})$ when the viscosity coefficient $\nu$ is sufficiently small as in the case of the plane Kolmogorov flow.