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A two-dimensional vortex condensate at high Reynolds number

Published online by Cambridge University Press:  09 January 2013

Basile Gallet  and
William R. Young
Basile Gallet*
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
William R. Young
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: basile.gallet@ens.fr
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Abstract

We investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

JFM classification

Instability: Instability Instability: Transition to turbulence Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 715 , 25 January 2013 , pp. 359 - 388
Copyright
©2013 Cambridge University Press

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