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von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

Published online by Cambridge University Press:  06 March 2012

Minping Wan
Affiliation:
Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
Sean Oughton
Affiliation:
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand
Sergio Servidio
Affiliation:
Dipartimento di Fisica, Universita’ della Calabria, I-87036 Cosenza, Italy
William H. Matthaeus*
Affiliation:
Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
*
Email address for correspondence: whm@udel.edu

Abstract

We argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

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Papers
Copyright
Copyright © Cambridge University Press 2012

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