A numerical study of the nonlinear cascade of energy in magnetohydrodynamic turbulence

Power spectra of solar wind magnetic field and velocity fluctuations more closely resemble those of turbulent fluids (spectral index of -5/3) than they do predictions for magnetofluid turbulence (a -3/2 index). Furthermore, the amount the solar wind is heated by turbulence is uncertain. To aid in the study of both of these issues, we report numerically derived energy cascade rates in magnetohydrodynamic (MHD) turbulence and compare them with predictions of MHD turbulence phenomenologies. Either of the commonly predicted spectral indices of 5/3 and 3/2 are consistent with the simulations. Explicit calculation of inertial range energy cascade rates in the simulations show that for unequal levels of fluctuations propagating parallel and antiparallel to the magnetic field, the majority species always cascades faster than does the minority species, and the cascade rates are in better agreement with a Kolmogoroff-like MHD turbulence phenomenology than with a generalized Kraichnan phenomenology even in situations where the fluctuations are much smaller than the mean magnetic field. The ``Kolmogoroff constant'' for MHD turbulence for small normalized cross helicity is roughly 6.7 in two dimensions and 3.6 for one calculation in three dimensions. For large normalized cross helicity, however, none of the existing models can account for the numerical results, although the Kolmogoroff-like case still works somewhat better than the Kraichnan-like. In particular, the applied magnetic field has much less influence than expected, and Alfvénicity is more important than predicted. These results imply the need for better phenomenological models to make clear predictions about the solar wind.