The turbulent heating rate in strong magnetohydrodynamic turbulence with nonzero cross helicity

Different results for the cascade power ε in strong, incompressible magnetohydrodynamic turbulence with nonzero cross helicity appear in the literature. In this paper, we discuss the conditions under which these different results are valid. Our conclusions can be expressed in terms of the density ρ, the rms amplitudes z ⁺ and z ^- of Alfvénic fluctuations propagating parallel and antiparallel to the background magnetic field B ₀, and the correlation length (outer scale) measured perpendicular to B ₀, denoted L. We argue that if z ⁺ ≫ z ^- and if the z ^- fluctuations are sustained by the reflection of z ⁺ fluctuations in a strong background magnetic field, then ε ρ(z ⁺)² z ^-/L as in previous studies by Hossain, Matthaeus, Dmitruk, Lithwick, Goldreich, Sridhar, and others. On the other hand, if the minority wave type (z ^-) is sustained by some form of forcing that is uncorrelated with or only weakly correlated with the z ⁺ fluctuations, then ε can be much less than ρ(z ⁺)² z ^-/L, as in previous studies by Dobrowolny, Lazarian, Chandran, and others. The mechanism for generating the minority wave type strongly affects the cascade power because it controls the coherence time for interactions between oppositely directed wave packets at the outer scale.