Numerical measurements of the spectrum in magnetohydrodynamic turbulence

We report the results of an extensive set of direct numerical simulations of forced, incompressible, magnetohydrodynamic (MHD) turbulence with a strong guide field. The aim is to resolve the controversy regarding the power-law exponent ($\alpha$, say) of the field-perpendicular energy spectrum $E\left(k_{\perp }\right)\propto k_{\perp }^{\alpha }$. The two main theoretical predictions $\alpha =-3/2$ and $\alpha =-5/3$ have both received some support from differently designed numerical simulations. Our calculations have a resolution of ${512}^3$ mesh points, a strong guide field, and an anisotropic simulation domain and implement a broad range of large-scale forcing routines, including those previously reported in the literature. Our findings indicate that the spectrum of well-developed, strong incompressible MHD turbulence with a strong guide field is $E\left(k_{\perp }\right)\propto k_{\perp }^{-3/2}$.

Figure 1

The (compensated) field-perpendicular energy spectra $E\left(k_{\perp }\right)$: (a) case 1, (b) case 2, (c) cases 1a (solid curve) and 2a (dashed curve), and (d) case 3. The description of the runs is given in Table .

Figure 2

The alignment angle $\theta$ as a function of scale for each of the simulations. The straight line has a slope 0.25 corresponding to the energy spectrum $E\left(k_{\perp }\right)\propto k_{\perp }^{-3/2}$; see [13, 14, 18].