A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible MHD. 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 B-o(e) over cap (parallel to) Numerically and analytically, we find energy spectra E+/- similar to k(perpendicular to)(n+/-), such that n(+) + n(-) = -4, where E+/- are the spectra of the Elsasser variables z(+/-) = v +/- b in the two-dimensional case (k(parallel to) = 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.