Spatial scales and temporal scales in the theory of magnetohydrodynamic turbulence

Podesta, J. J.
January 2011
Physics of Plasmas;Jan2011, Vol. 18 Issue 1, p012906
Academic Journal
In the study of incompressible magnetohydrodynamic (MHD) turbulence the correlation lengths along the directions parallel and perpendicular to the local mean magnetic field may be defined by iso-energy surfaces derived from second order structure functions. The correlation time of the turbulence may be defined in a similar manner by means of a temporal second order structure function. Moreover, there is a natural correspondence between the lengthscales of fluctuations with a given energy and the timescale of fluctuations of the same energy so that each iso-energy surface is associated with a unique pair of parallel and perpendicular correlation lengths and a unique correlation time. In the case when the magnetic Prandtl number is unity, Prm≡ν/η=1, it is shown that the correlation time τ associated with an iso-energy contour of energy E is equal to the energy cascade time of the turbulence in the sense that E/τ∼[variant_greek_epsilon], where [variant_greek_epsilon] is the energy cascade rate. For balanced MHD turbulence, turbulence with vanishing cross-helicity, both the Goldreich and Sridhar theory and Boldyrev's theory are shown to have this same property. An attempt is made to generalize these ideas to imbalanced MHD turbulence, turbulence with nonvanishing cross-helicity. It is shown that if the normalized cross-helicity σc is a constant in the inertial range, independent of wavenumber, then one obtains a model in which the energy cascade times of the two Elsasser fields are equal throughout the inertial range, that is, τ+/τ-=1. This result appears to be contradicted by numerical simulations and, therefore, the generalization to imbalanced turbulence discussed here requires further development; this is an important unsolved problem.


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