Stochastic flux freezing and magnetic dynamo

Magnetic flux conservation in turbulent plasmas at high magnetic Reynolds numbers is argued neither to hold in the conventional sense nor to be entirely broken, but instead to be valid in a statistical sense associated to the “spontaneous stochasticity” of Lagrangian particle trajectories. The latter phenomenon is due to the explosive separation of particles undergoing turbulent Richardson diffusion, which leads to a breakdown of Laplacian determinism for classical dynamics. Empirical evidence is presented for spontaneous stochasticity, including numerical results. A Lagrangian path-integral approach is then exploited to establish stochastic flux freezing for resistive hydromagnetic equations and to argue, based on the properties of Richardson diffusion, that flux conservation must remain stochastic at infinite magnetic Reynolds number. An important application of these results is the kinematic, fluctuation dynamo in nonhelical, incompressible turbulence at magnetic Prandtl number (Pr${}_m$) equal to unity. Numerical results on the Lagrangian dynamo mechanisms by a stochastic particle method demonstrate a strong similarity between the Pr${}_m=1$ and 0 dynamos. Stochasticity of field-line motion is an essential ingredient of both. Finally, some consequences for nonlinear magnetohydrodynamic turbulence, dynamo, and reconnection are briefly considered.

Figure 1

Mean dispersion of particle pairs: (a) backward dispersion, (b) forward dispersion. The backward-in-time results are plotted against $t^{\prime }=t_f-t$ and all quantities are nondimensionalized with viscous units (see text). The $\mathrm{Pr}=1$ results are plotted with solid lines, $\mathrm{Pr}=0.1$ results with dashed lines. We color code the lines with blue for raw data (including error bars), green for short-time molecular diffusion, and red for long-time Richardson diffusion.

Figure 2

Probability densities of pair-separation distances: (a) backward dispersion, (b) forward dispersion. The quantities at different times are normalized by $L(t)=\sqrt{\langle r^2(t)\rangle }$ as shown. We plot densities for three different times in the $t^3$ scaling range, with blue for $t=22.37$, green for $t=33.57$, and red for $t=44.79$. In the backward case, these times correspond to $t^{\prime }=t_f-t.$ The solid black line gives Richardson’s analytical prediction for the density (see text).

Figure 3

Illustration of the stochastic Lundquist formula. Three stochastic Lagrangian trajectories running backward in time from a common point are shown in red, green, and blue. Starting field vectors, represented by correspondingly colored arrows, are transported along the trajectories, stretched, and rotated to the common final point. These are then averaged to give the resultant magnetic field at that point, indicated by the black arrow.

Figure 4

Illustration of the stochastic Alfvén theorem. Shown are three members (red, green, and blue) of the infinite ensemble of loops obtained by stochastic advection of a loop $C$ (black) at time $t$ backward in time to $t_0.$ The average of the magnetic flux through the ensemble of loops is equal to the magnetic flux through $C.$

Figure 5

Error estimation. (a) The ratio $\rho (t)$ of the norms of the antisymmetric and symmetric parts of the approximate Cauchy-Green tensor. (b) The eigenvalues ${\varphi }_i(t)$, $i=1,2,3$, of the approximate Cauchy-Green tensor, with red for largest, green for middle, and blue for smallest.

Figure 6

Mean magnetic energy. Plotted in blue is the mean magnetic energy (solid line) calculated from Eqs. (48) and (49), along with plus and minus the error (dotted line) estimated from Eqs. (50) and (51). The straight line in red shows the least-squares linear fit over the time interval $t=6$ to 16.

Figure 7

Local slope of backward dispersion. Plotted in blue is the local slope of the backward dispersion $\langle r^2(t)\rangle ,$ as defined in (52). Plotted in red is the horizontal line segment from $t=6$ to 25 for the Richardson value $\sigma =3.$

Figure 8

Line-vector correlations: (a) longitudinal, (b) transverse. The line-vector correlations $R_L(r,t)$ and $R_N(r,t),$ respectively, as calculated numerically from formula (54) at time $t=11.12,$ are plotted in blue with solid lines and the correlations plus and minus their estimated errors with dotted lines. Also plotted with red lines are analytical results [11] for these correlations in the Kazantsev-Kraichnan model at ${\mathrm{Pr}}_m=0$, normalized as described in the text.

Figure 9

Contributions to magnetic energy. Plotted in blue is the function $4\pi r^2{R}_L(r,t)$ and in cyan the function $4\pi r^2{R}_N(r,t),$ both at time $t=11.12.$ These represent the contributions to magnetic energy in (53) from pairs of line vectors at initial separations $r$ and initially parallel and perpendicular, respectively, to the separation vector $\mathbf{r}.$

Figure 10

Line correlations for large separations: (a) longitudinal, (b) transverse. Plotted in blue are the numerical results from (54) at time $t=11.12$ and in red the analytical results [11] from the Kazantsev-Kraichnan model at ${\mathrm{Pr}}_m=0$. Straight lines in the log-linear plot versus $r^{1/3}$ correspond to the large-$r$ asymptotics (56). Shown in green is the prediction of that formula with the growth rate $\gamma =0.158$ determined in Fig. 6.