Spontaneous transport barriers quench turbulent resistivity in two-dimensional magnetohydrodynamics

This Rapid Communication identifies the physical mechanism for the quench of turbulent resistivity in two-dimensional magnetohydrodynamics. Without an imposed, ordered magnetic field, a multiscale, blob-and-barrier structure of magnetic potential forms spontaneously. Magnetic energy is concentrated in thin, linear barriers, located at the interstices between blobs. The barriers quench the transport and kinematic decay of magnetic energy. The local transport bifurcation underlying barrier formation is linked to the inverse cascade of $\langle A^2\rangle$ and negative resistivity, which induce local bistability. For small-scale forcing, spontaneous layering of the magnetic potential occurs, with barriers located at the interstices between layers. This structure is effectively a magnetic staircase.

Figure 1

Time evolution of (a) magnetic energy $E_B$ and kinetic energy $E_K$; (b) $\langle A^2\rangle$ in run 1. The suppression stage is marked in orange, and the kinematic decay stage in green. The decay of $E_B$ is slow in the suppression stage, which is consistent with previous studies. The decay of $\langle A^2\rangle$ is also slow in the suppression stage, and is more smooth compared to $E_B$.

Figure 2

Row 1: $A$ field snapshots; row 2: $B^2$ field snapshots; row 3: PDF of $A$. Column (a): Run 1 at $t=10$ (suppression stage). The system exhibits a blob-and-barrier feature, and the PDF of $A$ is bimodal. Column (b): Run 1 at $t=17$ (kinematic decay stage). The distribution of the fields is trivial. Column (c): Run 2 at $t=10$. Two peaks still arise on the PDF of $A$ even though its initial condition is unimodal. Column (d): Run 3 at $t=4.5$. The system exhibits a staircase feature, and the PDF of $A$ has multiple peaks.

Figure 3

The time evolutions of PDF of $A$, and the values are in logarithm scale (base 10). (a) For run 1, the PDF is bimodal in the suppression stage, and $\mathrm{\Delta }A$ between the two peaks reduces in time. The PDF becomes unimodal in the kinematic decay stage. (b) For run 2, two peaks at $A\ne 0$ still arise spontaneously given a unimodal initial condition. (c) For run 3, with external forcing at a smaller scale, layering and coarsening can occur. See further explanations in the text.

Figure 4

Time evolution of (a) packing fraction $P$, and (b) barrier width $W$ in run 1.

Figure 5

A sketch showing the relationship between flux ${\mathrm{\Gamma }}_A$ and $\nabla A$. The total resistivity is ${\eta }_{\text{tot}}=\delta {\mathrm{\Gamma }}_A/\delta \nabla A$, and is composed of turbulent and collisional parts ${\eta }_{\text{tot}}={\eta }_T+\eta$. In the small $B$ limit, ${\eta }_{\text{tot}}\sim {\eta }_K$; in the large $B$ limit, the residual resistivity is ${\eta }_{\text{res}}\sim \frac{ul}{\mathrm{Rm}\frac{1}{{\mu }_0\rho }{\langle B\rangle }^2/\langle v^2\rangle +\mathrm{Rm}\frac{1}{{\mu }_0\rho }\frac{\langle A^2\rangle }{L_{\text{blob}}^2}/\langle v^2\rangle }+\eta$. The transition between the two limits is the transport bifurcation.