Numerical Demonstration of Kolmogorov Scaling in Magnetohydrodynamic Turbulence

TL;DR

This study addresses whether Kolmogorov or Iroshnikov-Kraichnan scaling governs isotropic magnetohydrodynamic turbulence. Using high-resolution 2D simulations up to $8192^2$ and 3D simulations up to $1024^3$, driven in a low-wavenumber band to steady states, the authors analyze energy spectra $E^\pm(k)$, inertial-range fluxes $\Pi^\pm(k)$, third-order structure functions $S_3^\pm(l)$, and intermittency exponents. The results favor Kolmogorov scaling: $E^\pm(k) \sim k^{-5/3}$, constant $\Pi^\pm(k)$ in the inertial range, and $S_3^\pm(l) \propto l$ with $\zeta_3^\pm \approx 1$, along with intermittency exponents aligning with Kolmogorov predictions; imbalanced flows show $\Pi^+\!>\Pi^-$ and $E^+/E^-$ scaling consistent with Kolmogorov-type energy transfer and unequal Kolmogorov constants $K^\pm$. These findings provide a robust numerical resolution to the IK-vs-Kolmogorov debate for isotropic MHD and have implications for modeling solar, stellar, and planetary dynamo- and wind-related processes.

Abstract

The two leading models of isotropic magnetohydrodynamic (MHD) turbulence have competing predictions: $k^{-5/3}$ (Kolmogorov) and $k^{-3/2}$ (Iroshnikov-Kraichnan) scalings. This paper identifies the valid MHD turbulence model using high-resolution numerical and diagnostics-structure functions, intermittency exponents, and energy spectra and fluxes of imbalance MHD. The energy spectra of our forced MHD simulations on $8192^2$, $4096^2$, $1024^3$, and $512^3$ support Kolmogorov's k^{-5/3} spectrum over Iroshnikov-Kraichnan's k^{-3/2} spectrum, but the difference in the spectral exponents is small. However, the numerically computed third-order structure functions and intermittency exponents support Kolmogorov scaling in both two and three dimensions. Also, the energy fluxes of the imbalance MHD follow the predictions of Kolmogorov scaling. These results would help in better modelling of solar wind, solar corona, and dynamos.

Figures (11)

• Figure 1: For MHD turbulence simulations on $8192^2$ and $1024^3$ grids, time series of (a,c) $E^\pm$ and (b, d) $\epsilon^\pm$. For ${\bf z}^+$ and ${\bf z}^-$, blue and red curves are for 2D, respectively, and black and green curves are for 3D, respectively. Solid, chained, and dashed curves represent $\epsilon_{\mathrm{inj}}^+/\epsilon_{\mathrm{inj}}^-=1$, 3, 5, respectively.
• Figure 2: For the 2D run on a $8192^2$ grid with $\epsilon_{\mathrm{inj}}^+/\epsilon_{\mathrm{inj}}^-=1$, (a) density plot of vorticity and vector plot of the velocity field; (b) density plot of current density $\nabla \times {\bf b}$ and vector plot of the magnetic field. (c) For the 3D run on a $512^3$ grid with $\epsilon_{\mathrm{inj}}^+/\epsilon_{\mathrm{inj}}^-=1$, the ring spectrum $E(k,\theta)$ for the total energy.
• Figure 3: Plots of the normalized energy spectra $E^\pm(k) k^\alpha$ with $\alpha =5/3$ and 3/2 on (a,b,c) $8192^2$, (d,e,f) $4096^2$, and (g,h,i) $1024^3$ grids. For 2D, the blue and red curves represent ${\bf z}^+$ and ${\bf z}^-$ fields, respectively. The black and green colors represent the respective curves for 3D.
• Figure 4: Plots of the energy fluxes $\Pi^\pm(k)$ on (a,b,c) $8192^2$, (d,e,f) $4096^2$, and (g,h,i) $1024^3$ grids. For 2D, the blue and red curves represent ${\bf z}^+$ and ${\bf z}^-$ fields, respectively. The black and green colors represent the respective curves for 3D.
• Figure 5: Plot of $E^+(k)/E^-(k)$ vs. $\Pi^+(k)/\Pi^-(k)$, with $k$ in the inertial range. The blue circles and purple squares are for $8192^2$ and $1024^3$ grids, respectively. The red and green lines represent $(\Pi^+(k)/\Pi^-(k))^2$ and $(\Pi^+(k)/\Pi^-(k))$, respectively. Light black lines represent the best-fit curves with slopes of $1.67$ and $1.5$.