TY - JOUR

T1 - Spectral properties of the linearization at the Burgers vortex in the high rotation limit

AU - Maekawa, Yasunori

N1 - Funding Information:
Y. Maekawa was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/12

Y1 - 2011/12

N2 - We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier-Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.

AB - We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier-Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.

KW - Burgers vortices

KW - Navier-Stokes equations

KW - vortex Reynolds number

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U2 - 10.1007/s00021-010-0048-4

DO - 10.1007/s00021-010-0048-4

M3 - Article

AN - SCOPUS:80855127898

VL - 13

SP - 515

EP - 532

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 4

ER -