On the cross-helicity dependence of the energy spectrum in magnetohydrodynamic turbulence

Podesta, J. J.
January 2011
Physics of Plasmas;Jan2011, Vol. 18 Issue 1, p012907
Academic Journal
Phenomenological theories of strong incompressible magnetohydrodynamic (MHD) turbulence derived by Goldreich and Sridhar (GS) in 1995 and by Boldyrev in 2006 are only applicable to turbulence with vanishing cross-helicity. In this study, these two theories are generalized to treat turbulence with nonvanishing cross-helicity in such a way that the relation (w+/w-)2=([variant_greek_epsilon]+/[variant_greek_epsilon]-)2 observed in numerical simulations is satisfied. The average energy (second order structure function) in the generalized GS theory is E(r⊥)=[lowercase_phi_synonym]1(σc)([variant_greek_epsilon]r⊥)2/3 and that in the generalized Boldyrev theory is E(r⊥)=[lowercase_phi_synonym]2(σc)(vA[variant_greek_epsilon]r⊥)1/2, where the function [lowercase_phi_synonym](σc) describes the dependence on the normalized cross-helicity σc. The form of the function [lowercase_phi_synonym](σc) is derived through a renormalization of the variable σc that yields a one parameter family of solutions. The theory derived by Lithwick, Goldreich, and Sridhar (LGS) in 2007 is a special case of the generalized GS theory derived here; however, other generalizations of the GS theory are obtained that have a different cross-helicity dependence than the LGS theory. This new class of solutions and similar generalizations of Boldyrev's theory are investigated to see how the energy cascade rate [variant_greek_epsilon] changes as a function of σc when the energy at a given scale is held fixed. The generalization of Boldyrev's theory derived here is applicable to homogeneous MHD turbulence in the solar wind, for example, and can be used to obtain the turbulent dissipation rate [variant_greek_epsilon] from measurements of the energy spectrum and the normalized cross-helicity.


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