Cascade models for magnetohydrodynamic turbulence

The complex cascade model by Gloaguen et al. [Physica D 17, 154 (1985)] for magnetohydrodynamic turbulence is studied. In the first part the Alfvén effect is neglected. Stationary solutions are discussed. For purely kinetic turbulence a phenomenon, coherent structures in k spaces, is observed. For general magnetohydrodynamic turbulence, energy spectra are, in general, close to ${\mathit{k}}^{\mathrm{-}5/3}$, and inertial-range statistics, in particular structure functions, are similar to those observed in hydrodynamic (fluid) turbulence. In the second part the Alfvén effect is included. For the simplest case of a constant large-scale field ${\mathit{B}}_0$, energy spectra are ${\mathit{k}}^{\mathrm{-}1.25}$, flatter than the expected Iroshnikov-Kraichnan law ${\mathit{k}}^{\mathrm{-}3/2}$, and the inertial-range statistics are exactly Gaussian. Hence, the model has to be refined. Assuming ${\mathit{B}}_0$ to be the actual fluctuating field, intermittency is reintroduced, and the spectrum is close to ${\mathit{k}}^{\mathrm{-}3/2}$, though it tends to become flatter for high k, such that a uniformly valid inertial-range power-law spectrum may not exist.