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Laws for third-order moments in homogeneous anisotropic incompressible magnetohydrodynamic turbulence

Published online by Cambridge University Press:  31 July 2008

J. J. PODESTA
J. J. PODESTA*
Affiliation:
Space Science Center, University of New Hampshire, Durham, NH 03824, USA
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Abstract

It is known that Kolmogorov's four-fifths law for statistically homogeneous and isotropic turbulence can be generalized to anisotropic turbulence. This fundamental result for homogeneous anisotropic turbulence says that in the inertial range the divergence of the vector third-order moment 〈|δv(r)|2δv(r)〉 is constant and is equal to -4ϵ, where ϵ is the dissipation rate of the turbulence. This law can be extended to incompressible magnetohydrodyamic (MHD) turbulence where statistical isotropy is often not valid due, for example, to the presence of a large-scale magnetic field. Laws for anisotropic incompressible MHD turbulence were first derived by Politano and Pouquet. In this paper, the laws for vector third-order moments in homogeneous non-isotropic incompressible MHD turbulence are derived by a technique due to Frisch that clarifies the relationship between the energy flux in Fourier space and the vector third-order moments in physical space. This derivation is different from the original derivation of Politano and Pouquet which is based on the Kármán–Howarth equation, and provides some new physical insights. Separate laws are derived for the cascades of energy, cross-helicity and magnetic-helicity, the three ideal invariants of incompressible MHD for flows in three dimensions. These laws are of fundamental importance in the theory of MHD turbulence where non-isotropic turbulence is much more prevalent than isotropic turbulence.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 609 , 25 August 2008 , pp. 171 - 194
Copyright
Copyright © Cambridge University Press 2008

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