Kolmogorov Scaling for Total Energy and Cross Helicity in Magnetohydrodynamic Turbulence

Abstract

The problem of scaling in isotropic magnetohydrodynamic (MHD) turbulence has remained unresolved, with competing predictions of $k^{-5/3}$ (Kolmogorov) and $k^{-3/2}$ (Iroshnikov-Kraichnan) scalings. In this paper, we address this long-standing controversy using high-resolution numerical simulations on $8192^2$ and $1536^3$ grids. We show that the total energy and cross helicity spectra are closer to $k^{-5/3}$ than $k^{-3/2}$. The fluxes and structure functions of the total energy and cross helicity also demonstrate robust support for Kolmogorov scaling. The magnetic energy shows $k^{-5/3}$ spectrum, but the kinetic energy exhibits $k^{-3/2}$ spectrum; the latter spectrum is due to the energy transfers from the magnetic field to the velocity field.

Figures (5)

• Figure 1: Various energy fluxes of MHD turbulence. The definitions are given in Eqs. (\ref{['eq:flux_ug_ul']}-\ref{['eq:flux_u_bl']}).
• Figure 2: Numerical results of isotropic MHD turbulence ($B_0=0$) simulations on $8192^2$ (2D) and $1536^3$ (3D) grids for energy injection ratios $\epsilon^{H_C}_\mathrm{inj}/\epsilon^T_\mathrm{inj} = 0$ and 1/4. Panels (a–d) show the normalized energy spectra $E_T(k),H_c(k)\,k^\alpha$ with $\alpha = 5/3$ or $3/2$. Panels (e–h) display the corresponding energy fluxes $\Pi_{T,H_c}(k)$ that match with the respective injection rates (dashed lines). Panels (i–l) present the normalized energy spectra $E_u(k) k^\alpha$ and $E_b(k) k^\alpha$. Panels (m–p) show energy fluxes $\Pi^{u<}_{u>}(k)$ (blue), $\Pi^{b<}_{b>}(k)$ (red), $\Pi^{b<}_u(k)$ (green), $\Pi^{u<}_b(k)$ (orange), $\Pi^{u<}_{u>}(k)+\Pi^{u<}_{b}(k)$ (black chained), and $\Pi^{b<}_{b>}(k)+\Pi^{b<}_{u}(k)$ (green chained). Panels (q–t) depict $-S_3^{T,H_c}(l) \propto l$ for the total energy and cross helicity.
• Figure 3: Numerical results of isotropic MHD turbulence ($B_0=0$) simulations on $8192^2$ (2D) and $1536^3$ (3D) grids for energy injection ratio of $\epsilon^{H_C}_\mathrm{inj}/\epsilon^T_\mathrm{inj} = 1/3$. Refer to Fig. \ref{['fig:figure']} for the description of the plots.
• Figure 4: (a) Schematic diagrams of energy transfer function $T(k)$ in Hydrodynamic and MHD turbulence. In hydrodynamic turbulence $T_u(k) = d\Pi_u(k)/dk \approx 0$, implying constancy of energy flux and $k^{-5/3}$ energy spectrum. Similarly, for our MHD turbulence, $T_b(k) \approx 0$, leading to $E_b(k) \sim k^{-5/3}$ spectrum. But, $T_u(k) > 0$, leading to $E_u(k)$ shallower than $k^{-5/3}$. See Eqs. (\ref{['eq:T_b']},\ref{['eq:T_u']}). (b) For the 3D simulations on $1536^3$ grid with $\epsilon^{H_C}_\mathrm{inj}/\epsilon^T_\mathrm{inj} = 1/4$ (solid curves) and $0$ (dashed curves) (third and fourth columns of Fig. \ref{['fig:figure']}), the numerically computed $T_{b,u}(k)$. We observe that $T_b(k)$ is nearly 0 on an average, but $T_u(k)$ is strongly positive.
• Figure 5: Numerical results of MHD turbulence simulations on $8192^2$ (2D) grid for energy injection ratios $\epsilon^{H_C}_\mathrm{inj}/\epsilon^T_\mathrm{inj} = 0$ with $\mathbf{B_0} = 1$ (1st coloum) and $3$ (2nd coloum). Panels (a,b) show the normalized energy spectra $E_T(k)\,k^{a}$ with $a = 5/3$ or $3/2$. Panels (c–d) display the corresponding perpendicular energy spectra $E_{T_\perp}(k_\perp)\,k_\perp^{a}$ with $a = 5/3$ or $3/2$. Panels (e,f) shows total energy fluxe $\Pi_{T}(k)$ that match with the respective injection rates (dashed lines). Panels (g,h) present the normalized energy spectra $E_u(k) k^\alpha$ and $E_b(k) k^\alpha$.