Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

Jacob Bedrossian, Michele Coti Zelati, Vlad Vicol

Research output: Contribution to journalArticlepeer-review

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 ∥ur(t) ∥ ≲ ⟨ t⟩ - 1 and ∥uθ(t) ∥ ≲ ⟨ t⟩ - 2 in the appropriate radially weighted L2 spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as t→ ∞, 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→ ∞, 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.

Original languageEnglish (US)
Article number4
JournalAnnals of PDE
Volume5
Issue number1
DOIs
StatePublished - Jun 1 2019

Keywords

  • Euler equations
  • Inviscid damping
  • Phase mixing
  • Scattering
  • Vortex axisymmetrization
  • Vorticity depletion

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Geometry and Topology
  • Mathematical Physics
  • Physics and Astronomy(all)

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