Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2D Euler equations

TL;DR

This paper proves inviscid damping and vortex axisymmetrization for the linearized 2D Euler equations around a monotone vorticity, revealing vorticity depletion and providing optimal decay rates for velocity fields.

Contribution

It establishes the first rigorous results on vorticity depletion and vortex axisymmetrization, with precise decay rates and dynamics in higher Sobolev spaces for the linearized 2D Euler equations.

Findings

Proves optimal decay rates for velocity fields in radially weighted spaces.

Shows vorticity weakly converges to radial symmetry over time.

Identifies vorticity depletion as a key non-local effect in vortex dynamics.

Abstract

Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the $\theta$-dependent radial and angular velocity fields with the optimal rates $\|u^r(t)\| \lesssim \langle t \rangle^{-1}$ and $\|u^\theta(t)\| \lesssim \langle t \rangle^{-2}$ in the appropriate radially weighted $L^2$ spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as $t \rightarrow \infty$, a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the $\theta$-dependent angular Fourier modes in the vorticity are ejected from the origin as $t \to \infty$, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. Our work appears to be the first to find vorticity depletion relevant for the dynamics of vortices.