Spectrum of Magnetohydrodynamic Turbulence

We propose a phenomenological theory of strong incompressible magnetohydrodynamic turbulence in the presence of a strong large-scale external magnetic field. We argue that in the inertial range of scales, magnetic-field and velocity-field fluctuations tend to align the directions of their polarizations. However, the perfect alignment cannot be reached; it is precluded by the presence of a constant energy flux over scales. As a consequence, the directions of shear-Alfvén fluid and magnetic-field fluctuations at each scale $\lambda$ become effectively aligned within the angle ${\varphi }_{\lambda }\propto {\lambda }^{1/4}$, which leads to scale-dependent depletion of the nonlinear interaction and to the field-perpendicular energy spectrum $E(k_{\perp })\propto k_{\perp }^{-3/2}$. Our results may be universal, i.e., independent of the external magnetic field, since small-scale fluctuations locally experience a strong field produced by large-scale eddies.

Figure 1

Sketch of a turbulent eddy in the Goldreich-Sridhar picture. The large-scale magnetic field is in the vertical direction. The field-perpendicular dimensions of the eddy are the same, while its field-parallel scale is $l\propto {\lambda }^{2/3}$. As the turbulent cascade proceeds toward the smallest, dissipative scales, $\lambda \to 0$, the Goldreich-Sridhar eddy assumes the shape of a filament.

Figure 2

Anisotropic turbulent eddy in our picture. The large-scale magnetic field is in the vertical direction. The field-perpendicular dimensions of the eddy are $\lambda$ and $\xi \propto {\lambda }^{3/4}$, and the eddy size in the field-parallel direction is $l\propto {\lambda }^{1/2}$. As the energy cascade proceeds toward the smallest, dissipative scales, $\lambda \to 0$, the eddy assumes the shape of a current sheet.

Figure 3

Sketch of three-dimensional angular alignment of shear-Alfvén velocity and magnetic-field fluctuations. The alignment angles consistent with an energy cascade are given by $\theta \propto {\lambda }^{\alpha /(3+\alpha )}$ and $\tilde{\theta }\propto {\lambda }^{1/(3+\alpha )}$. The maximal alignment is achieved for $\alpha =1$ (see the text).