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∥.
Numerically and analytically, we find energy spectra
E± ∼ kn±⊥,
such that n+ + n− = −4, where E±
are the spectra of the Elsässer variables z± = v ± b in the
two-dimensional case (k∥ = 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.