On universal relations in 2-D turbulence

Nusret Balci; Ciprian Foias; M. S Jolly; Ricardo Rosa

doi:10.3934/dcds.2010.27.1327

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2010, Volume 27, Issue 4: 1327-1351. Doi: 10.3934/dcds.2010.27.1327

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On universal relations in 2-D turbulence

1.
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455
2.
Department of Mathematics, Texas A&M University, College Station, TX 77843
3.
Department of Mathematics, Indiana University, Bloomington, IN, 47405
4.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ 21941-909

Received: January 31, 2010

Revised: February 28, 2010

Published: November 2010

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Abstract

A rigorous study of universal laws of 2-D turbulence is presented for time independent forcing at all length scales. Conditions for energy and enstrophy cascades are derived, both for a general force, and for one with a large gap in its spectrum. It is shown in the gap case that either a direct cascade of enstrophy or an inverse cascade of energy must hold, provided the gap modes of the velocity has a nonzero ensemble average. Partial rigorous support for 2-D analogs of Kolmogorov's 3-D dissipation law, as well as the power law for the distribution of energy are given.

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Mathematics Subject Classification: 35Q30, 76F02.

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