Small scales and anisotropy in low Rm magnetohydrodynamic turbulence

A. Pothérat, T. Alboussière

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


In this paper, we derive estimates for size of the small scales and the attractor dimension in low Rm magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any n-dimensional volume of the phase space by the evolution operator associated to the Navier-Stokes equations. As explained in Constantin et al. [J. Fluid Mech. 150, 427 (1985)], the value of n for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the more precise case of a three-dimensional periodical domain, we are led to calculate the distribution of n modes which minimizes the total (viscous and Joule) dissipation. This set of modes turns out to exhibit most of the well known properties of magnetohydrodynamic turbulence, previously obtained by heuristic considerations such as the existence of the Joule cone under strong magnetic field. The sought estimates for the small scales and attractor dimension are then obtained under no physical assumption as functions of the Hartmann and the Reynolds numbers and match the Hartmann number dependency of heuristic results. A necessary condition for the flow to be tridimensional and anisotropic (as opposed to purely two-dimensional) is also built.

Original languageEnglish
Pages (from-to)3170-3180
Number of pages11
JournalPhysics of Fluids
Issue number10
Early online date5 Sept 2003
Publication statusPublished - Oct 2003
Externally publishedYes

Bibliographical note

© 2003 American Institute of Physics


  • Magnetohydrodynamics
  • Fluid mechanics
  • Navier Stokes equations
  • Cognitive science

ASJC Scopus subject areas

  • Computational Mechanics
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes


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