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

Jacob Bedrossian, Michele Coti Zelati, Vlad Vicol

TL;DR

The work rigorously analyzes the linearized 2D Euler equations around radially symmetric vortices, proving inviscid damping with optimal decay rates and demonstrating vortex axisymmetrization in a weighted L^2 framework. A key novelty is the vorticity depletion mechanism, where θ-dependent modes are ejected from the origin, yielding faster damping than passive scalar evolution and enabling precise control of higher-order regularity via an intricate Rayleigh-Green’s function analysis. The authors develop a comprehensive spectral and functional-analytic apparatus, including contour integral representations, decomposition into depletion-carrying and damping components, and a robust treatment of both homogeneous and inhomogeneous Rayleigh problems across k≥1, culminating in sharp decay and scattering results. These results provide a rigorous linear-theory foundation for understanding persistent coherent vortices and offer detailed insight into the long-time dynamics of 2D vortical flows relevant to turbulence and plasma-like damping phenomena.

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 $θ$-dependent radial and angular velocity fields with the optimal rates $\|u^r(t)\| \lesssim \langle t \rangle^{-1}$ and $\|u^θ(t)\| \lesssim \langle t \rangle^{-2}$ 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 $θ$-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.

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

TL;DR

The work rigorously analyzes the linearized 2D Euler equations around radially symmetric vortices, proving inviscid damping with optimal decay rates and demonstrating vortex axisymmetrization in a weighted L^2 framework. A key novelty is the vorticity depletion mechanism, where θ-dependent modes are ejected from the origin, yielding faster damping than passive scalar evolution and enabling precise control of higher-order regularity via an intricate Rayleigh-Green’s function analysis. The authors develop a comprehensive spectral and functional-analytic apparatus, including contour integral representations, decomposition into depletion-carrying and damping components, and a robust treatment of both homogeneous and inhomogeneous Rayleigh problems across k≥1, culminating in sharp decay and scattering results. These results provide a rigorous linear-theory foundation for understanding persistent coherent vortices and offer detailed insight into the long-time dynamics of 2D vortical flows relevant to turbulence and plasma-like damping phenomena.

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 -dependent radial and angular velocity fields with the optimal rates and in the appropriate radially weighted spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as , a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the -dependent angular Fourier modes in the vorticity are ejected from the origin as , 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.

Paper Structure

This paper contains 65 sections, 74 theorems, 760 equations, 1 figure.

Table of Contents

  1. Introduction and statements of results
  2. Background
  3. Statement of main results
  4. Preliminaries and outline of the proof
  5. Skew-symmetric structure, neutral modes, and contour integral representation
  6. Outline of the proof for TEXT
  7. Outline of the proof for TEXT
  8. Depletion trick and contour integral decomposition
  9. Vanishing for $z \notin I_\alpha$ as $\varepsilon \to 0$
  10. Green's function for the Rayleigh problem for $z \in I_\alpha$ as $\varepsilon \rightarrow 0$
  11. Homogeneous Rayleigh problem
  12. Inhomogeneous Rayleigh problem
  13. Representation formulas and boundedness for $(r\partial_r)^j f_1$ and $(r\partial_r)^j f_2$
  14. Iteration scheme and representation formulas for $\partial_G^j$ derivatives for $j \leq k-1$
  15. Extension to $\partial_G^k$ or $r_c \partial_{r_c} \partial_G^{j}$ with $j\leq k-1$
  16. ...and 50 more sections

Key Result

Theorem 1.1

Let $k\neq 0$, and assume V1-V3. For all $1 \gg \delta \gg \eta_0 > 0$ and any smooth $\omega^{in}_{k} \in L^2_{\beta} \cap L^2_{f,\delta}$ satisfying the orthogonality condition eq:ortho, the solution $\omega,\psi$ to eq:EU obeys the following inviscid damping estimates Furthermore, there exist $f_{k;1}(t,r)$ and $f_{k;2}(t,r)$ such that and the following vorticity depletion estimates hold for

Figures (1)

  • Figure 1: This figure summarizes the limiting procedure used to treat \ref{['eq:integrallimit6']}. Region III represents $I_\alpha$. The contribution from this region is further decomposed into $f_1^\varepsilon$ and $f_2^\varepsilon$ (see \ref{['eq:fIone']} and \ref{['eq:fItwo']} below) which converge to the decomposition in \ref{['ineq:VortDep']}. The limiting procedure is done by constructing the Green's function for ${\textsc{Ray}}_z$ and making analyses of the resulting integral operators (carried out in §\ref{['sec:HomRay']}--§\ref{['sec:repbd']} together with Appendix \ref{['sec:BndConv']}). In regions I and IV we apply energy estimates on ${\textsc{Ray}}_z$ to prove these contributions vanish (carried out in Appendix \ref{['app:vanishingout']}). Here $\delta > 0$ is traded to gain the flexibility to take $\alpha > 0$. In region II, we apply a compactness-contradiction argument with a second order comparison principle that shows these contributions also must vanish (carried out in Appendix \ref{['sec:Ybd4']}).

Theorems & Definitions (140)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6: Vortex axisymmetrization
  • Corollary 1.7: Scattering to passive scalar evolution
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 130 more