Theory of incompressible magnetohydrodynamic turbulence with scale-dependent alignment and cross-helicity

A phenomenological anisotropic theory of MHD turbulence with nonvanishing cross-helicity is constructed based on Boldyrev's phenomenology and probabilities p and q for fluctuations δv⊥ and δb⊥ to be positively or negatively aligned. The positively aligned fluctuations occupy a fractional volume p and the negatively aligned fluctuations occupy a fractional volume q. Guided by observations suggesting that the normalized cross-helicity σ_c and the probabilities p and q are approximately scale invariant in the inertial range, a generalization of Boldyrev's theory is derived that depends on the three ratios w⁺/w^-, ε⁺/ε^-, and p/q. It is assumed that the cascade processes for positively and negatively aligned fluctuations are both in a state of critical balance and that the eddy geometries are scale invariant. The theory reduces to Boldyrev's original theory when σ_c = 0, ε⁺ = ε^-, and p = q and extends the theory of Perez and Boldyrev when σ_c ≠ 0. The theory is also an anisotropic generalization of the theory of Dobrowolny, Mangeney, and Veltri.