Decay of magnetohydrodynamic turbulence from power-law initial conditions

We derive relations for the decay of the kinetic and magnetic energies and the growth of the Taylor and integral scales in unforced, incompressible, homogeneous, and isotropic three-dimensional magnetohydrodynamic (3DMHD) turbulence with power-law initial energy spectra. We also derive bounds for the decay of the cross and magnetic helicities. We then present results from systematic numerical studies of such decay both within the context of a MHD shell model and direct numerical simulations of 3DMHD. We show explicitly that our results about the power-law decay of the energies hold for times $t<t_{*}$, where $t_{*}$ is the time at which the integral scales become comparable to the system size. For $t<t_{*}$, our numerical results are consistent with those predicted by the principle of “permanence of large eddies.”

Figure 1

(a) Plot of ${\mathcal{E}}_b^s∕k_n$ vs $k_n^2{\tau }^s$ for shells $n=9$ and $n=12$ with $q=1$, illustrating the data collapse implied by the scaling form of Eq. (10). (b) The analog for $q=0$ illustrating the lack of data collapse.

Figure 2

(a) Log-log plots of the decay of the magnetic energy $E_b^s$ as a function of time ${\tau }^s$. The observed slope is $-1.000\pm 0.001$ for $q=1$ in agreement with Eq. (11). A nominal slope of $-0.66\pm 0.01$ can be fit to the curve for $q=0$ in the range ${\tau }^s=1.28\times {10}^{-2}$ and ${\tau }^s=7.70\times {10}^2$. (b) Log-log plots of the decay of the kinetic and magnetic energies ($E_v^s$ and $E_b^s$, respectively), without initial equipartition of energy $\left(E_v^{0,s}={10}^3{E}_b^{0,s}\right)$ for $q=0$.

Figure 3

Log-log plots of the growth of the magnetic integral scale $L_b^s$ (for $q=0$ and $1$) as a function of time with $E_v^{0,s}=E_b^{0,s}$. The observed slope is $0.500\pm 0.002$ for $q=1$ in agreement with the magnetic integral-scale analog of Eq. (12).

Figure 4

Log-log plots of the MHD shell-model magnetic-energy density ${\mathcal{E}}_b^s$ as a function of wave number (temporal sequence with ${\tau }^s$ values indicated in the legend) for (a) $q=0$ and (b) $q=1$.

Figure 5

Log-log plots of the magnetic energy $E_{\mathbf{b}}$ as a function of time for $q=0$ and 1 for 3DMHD from our ${80}^3$ DNS.

Figure 6

Log-log plots of the magnetic integral scale $L_{\mathbf{b}}$ as a function of time for $q=0$ and 1 for 3DMHD from our ${80}^3$ DNS.

Figure 7

Log-log plots of the magnetic-energy density ${\mathcal{E}}_{\mathbf{b}}$ as a function of wave number $k$ (temporal sequence in steps of $0.02\tau$ for $q=0$ and $0.05\tau$ for $q=1$) for 3DMHD from our ${80}^3$ DNS.