# American Institute of Mathematical Sciences

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November  2010, 27(4): 1327-1351. doi: 10.3934/dcds.2010.27.1327

## On universal relations in 2-D turbulence

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

Received  February 2010 Revised  March 2010 Published  March 2010

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.
Citation: Nusret Balci, Ciprian Foias, M. S Jolly, Ricardo Rosa. On universal relations in 2-D turbulence. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1327-1351. doi: 10.3934/dcds.2010.27.1327
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