Examination of the four-fifths law for longitudinal third-order moments in incompressible magnetohydrodynamic turbulence in a periodic box

The four-fifths law for third-order longitudinal moments is examined, using direct numerical simulation (DNS) data on three-dimensional (3D) forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is ${512}^3$. A generalized Kármán-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of $\mathit{r}$. Here, $\mathit{r}$ is a separation vector. The viscous, forcing, anisotropic and nonstationary terms in the generalized equation are quantified. It is found that the influence of the anisotropic terms on the four-fifths law is negligible at small scales, compared to that of the viscous term. However, the influence of the directional anisotropy, which is measured by the departure of the third-order moments in a particular direction of $\mathit{r}$ from the spherically averaged ones, on the four-fifths law is suggested to be substantial, at least in the case studied here.

Figure 1

Time evolution of ${\overline{E}}_u$, ${\overline{E}}_b$, and ${\overline{\epsilon }}_t$.

Figure 2

Kinetic and magnetic energy spectra, $E_u\left(k\right)$ and $E_b\left(k\right)$ vs nondimensional wave number $k{\eta }_{\mathrm{IK}}$. As a reference, the IK spectrum [33, 34] (i.e., $k^{-3/2}$) is plotted with the dotted line.

Figure 3

The $r/L_u$ dependence of the normalized forcing term $I_f/\left({\overline{\epsilon }}_{u}r\right)$ and viscous term $I_{\nu }/\left({\overline{\epsilon }}_{u}r\right)$ together with $-{\langle \langle {\left(\delta u_L\right)}^3\rangle \rangle }_r/\left({\overline{\epsilon }}_{u}r\right)$ and $6{\langle \langle b_L^2\delta u_L\rangle \rangle }_r/\left({\overline{\epsilon }}_{u}r\right)$. The gray solid line denotes $4/5$. As references, the power laws $r^2$ and $r^{-3/2}$ are plotted with the gray dotted and gray dashed lines, respectively.

Figure 4

Sum of normalized third-order terms, $\{-{\langle \langle {\left(\delta u_L\right)}^3\rangle \rangle }_r+6{\langle \langle b_L^2\delta u_L\rangle \rangle }_r\}/\left({\overline{\epsilon }}_{u}r\right)$, denoted by the black solid curve, vs $r/L_u$. The gray solid line denotes $4/5$.

Figure 5

Magnitudes of the normalized anisotropic terms, $|I_u|/\left({\overline{\epsilon }}_{u}r\right)$ and $|I_b|/\left({\overline{\epsilon }}_{u}r\right)$, and the magnitude of the normalized nonstationary term, $|I_t|/{\overline{\epsilon }}^{u}r$, vs $r/L_u$, compared with the viscous and forcing terms, $I_{\nu }/\left({\overline{\epsilon }}_{u}r\right)$ and $I_f/\left({\overline{\epsilon }}_{u}r\right)$.

Figure 6

(a) Normalized anisotropic measures $|{\tilde{\Delta }}_i^u\left(r\right)|$ $(i=1,2,3)$ and the average $|{\tilde{\Delta }}_{\mathrm{ave}}^u\left(r\right)|$ vs $r/L_u$. (b) Measures $|{\tilde{\Delta }}_i^b\left(r\right)|$ $(i=1,2,3)$ and $|{\tilde{\Delta }}_{\mathrm{ave}}^b\left(r\right)|$ vs $r/L_u$. The terms $-{\langle \langle {\left(\delta u_L\right)}^3\rangle \rangle }_r/\left({\overline{\epsilon }}_{u}r\right)$ and $6{\langle \langle b_L^2\delta u_L\rangle \rangle }_r/\left({\overline{\epsilon }}_{u}r\right)$ are plotted in (a) and (b), respectively.