Journal of Plasma Physics

A weak turbulence theory for incompressible magnetohydrodynamics

S. GALTIER a1a2, S. V. NAZARENKO a1, A. C. NEWELL a1 and A. POUQUET a2
a1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
a2 Observatoire de la Côte d'Azur, CNRS UMR 6529, BP 4229, 06304 Nice Cedex 4, France


We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê[parallel R: parallel]. Numerically and analytically, we find energy spectra E± [similar] kn±[bot bottom], such that n+ + n = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k[parallel R: parallel] = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

(Received June 21 1999)
(Revised October 25 1999)