Dynamic Alignment and Exact Scaling Laws in MHD Turbulence

The Kolmogorov theory of hydrodynamic turbulence yields an exact relation for the third-order longitudinal velocity structure function, namely 〈δv L(r)〉 = −4/52r, where δvL(r) = [v(x+r)−v(x)]· r/r and 2 is the rate of energy dissipation. One therefore expects the velocity scaling δv(r) ∝ r, which leads to the Kolmogorov energy spectrum E(k) ∝ k−5/3. In 1998, Politano and Pouquet found that in magnetohydrodynamic turbulence certain third-order structure functions scale linearly with r. This, in turn, suggests that the spectrum of MHD turbulence also has the Kolmogorov scaling. However, recent high-resolution direct numerical simulations suggest that the spectrum is E(k) ∝ k−3/2. Here we propose that this apparent contradiction is a manifestation of the phenomenon of scale-dependent dynamic alignment recently discovered in MHD turbulence in [12–14].