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

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-Prüss 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.

arXiv comment51 pages

References in corpus (5)

• Diffusion and mixing in fluid flow via the resolvent estimate
• Metastability of Kolmogorov flows and inviscid damping of shear flows
• The formulation of the Navier-Stokes equations on Riemannian manifolds
• The Euler and Navier-Stokes equations on the hyperbolic plane
• The incompressible Navier-Stokes equations on non-compact manifolds

Cited by in corpus (1)

• Nonlinear stability of the two-jet Kolmogorov type flow on the unit sphere under a perturbation with nondissipative part