A weak turbulence theory for incompressible magnetohydrodynamics

TLDR

The study derives a weak turbulence formalism for incompressible magnetohydrodynamics. The formalism is based on three‑wave interactions that yield kinetic equations for energy and helicity spectral densities, conserve energy in planes normal to the mean magnetic field, and are benchmarked against numerical simulations. Numerical and analytical solutions show that the Elsässer energy spectra scale as E±∼k⊥^n± with n+ + n− = −4, and the spectral constants depend on the velocity–magnetic field correlation.

Abstract

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 B 0 ê ∥ . Numerically and analytically, we find energy spectra E ± ∼ k n ± ⊥ , 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.