Imbalanced Strong MHD Turbulence

We present a phenomenological model of imbalanced MHD turbulence in anincompressible magnetofluid. The steady-state cascades, of waves traveling inopposite directions along the mean magnetic field, carry unequal energy fluxesto small length scales, where they decay due to viscous and resistivedissipation. The inertial-range scalings are well-understood when both cascadesare weak. We study the case when both cascades are, in a sense, strong. Theinertial-range of this imbalanced cascade has the following properties: (i) theratio of the r.m.s. Elsasser amplitudes is independent of scale, and is equalto the ratio of the corresponding energy fluxes; (ii) in common with thebalanced strong cascade, the energy spectra of both Elsasser waves are of theanisotropic Kolmogorov form, with their parallel correlation lengths equal toeach other on all scales, and proportional to the two-thirds power of thetransverse correlation length; (iii) the equality of cascade time andwaveperiod (critical balance) that characterizes the strong balanced cascadedoes not apply to the Elsasser field with the larger amplitude. Instead, themore general criterion that always applies to both Elsasser fields is that thecascade time is equal to the correlation time of the straining imposed byoppositely-directed waves. Our results are particularly relevant for turbulencein the solar wind. Spacecraft measurements have established that, in theinertial range of solar wind turbulence, waves travelling away from the sunhave higher amplitudes than those travelling towards it. Result (i) allows usto infer the turbulent flux ratios from the amplitude ratios, thus providinginsight into the origin of the turbulence