Imbalanced Weak Magnetohydrodynamic Turbulence

Weak MHD turbulence consists of waves that propagate along magnetic field lines, in both directions. When two oppositely directed waves collide, they distort each other, without changing their respective energies. Each wave suffers many collisions before cascading; by contrast, in strong MHD turbulence, waves cascade on the same timescale at which they collide. "Imbalance" means that more energy is going in one direction than the other. In general, MHD turbulence is imbalanced. Yet imbalanced MHD cascades are not understood. For example, turbulence in the solar wind is observed to be imbalanced, so solar wind turbulence will not be understood until a theory of the imbalanced cascade is developed. We solve weak MHD turbulence that is imbalanced. Of crucial importance is that the energies going in both directions are forced to equalize at the dissipation scale. This "pinning" of the energy spectra was discovered by Grappin and coworkers. It affects the entire inertial range. Weak MHD turbulence is particularly interesting because perturbation theory is applicable. Hence, it can be described with a simple kinetic equation. Galtier and coworkers derived this kinetic equation. We present a simpler, more physical derivation, based on the picture of colliding wavepackets. In the process, we clarify the role of the zero-frequency mode. We also explain why Goldreich & Sridhar claimed that perturbation theory is inapplicable, and why this claim is wrong. (Our "weak" is equivalent to Goldreich & Sridhar's "intermediate.") We perform numerical simulations of the kinetic equation to verify our claims. We construct simplified model equations that illustrate the main effects. Finally, we show that a large magnetic Prandtl number does not have a significant effect, and that hyperviscosity leads to a pronounced bottleneck effect