Suppression of Electron Thermal Conduction by Whistler Turbulence in a Sustained Thermal Gradient

The dynamics of weakly magnetized collisionless plasmas in the presence of an imposed temperature gradient along an ambient magnetic field is explored with particle-in-cell simulations and modeling. Two thermal reservoirs at different temperatures drive an electron heat flux that destabilizes off-angle whistler-type modes. The whistlers grow to large amplitude, $\delta B/B_0\simeq 1$, and resonantly scatter the electrons, significantly reducing the heat flux. Surprisingly, the resulting steady-state heat flux is largely independent of the thermal gradient. The rate of thermal conduction is instead controlled by the finite propagation speed of the whistlers, which act as mobile scattering centers that convect the thermal energy of the hot reservoir. The results are relevant to thermal transport in high-$\beta$ astrophysical plasmas such as hot accretion flows and the intracluster medium of galaxy clusters.

Figure 1

Two-dimensional plots from the largest simulation ($L_x=2L_0$) in a saturated state at $t=800{\mathrm{\Omega }}_{e0}^{-1}$. (a) Fluctuations in out-of-plane $B_z$. (b) Temperature $T_{exx}$ with four self-consistent particle trajectories overlaid. (c) Heat flux $q_{ex}$.

Figure 2

Temperature and heat flux profiles. (a) Line plots of $y$-averaged $T_{exx}$ for the run with $L_x=2L_0$ at times $t{\mathrm{\Omega }}_{e0}=0$ (black), 80 (blue), 200 (green), 400 (cyan), and 800 (red). (b) Line plots of $y$-averaged heat flux at times $t{\mathrm{\Omega }}_{e0}^{-1}=0$ (black), 30 (blue), 100 (green), 150 (cyan), 300 (purple), 450 (grey-blue), 600 (grey), and 800 (red). (c) Box-averaged $⟨q_{ex}(t){⟩}_{x,y}$ for the six simulations in Table 1. Inset: Linear fit to the steady-state heat flux $q_{ex,f}$ as a function of $1/{\beta }_{e0h}$.

Figure 3

Evidence for scattering of electrons by whistlers. (a) Trajectories of particles in $v_{\parallel }$,$v_{\perp }$ space showing significant deflection of pitch angle $\varphi ={\tan }^{-1}(v_{\perp }/v_{\parallel })$. (b) Plot of $⟨\mathbf{(}x(t)-x(t_0){\mathbf{)}}^2⟩$ with linear fit to the slope representing diffusion coefficient $D$.

Figure 4

Line plots of $x(t)$ for 150 particles in the $L_x=2L_0$ simulation indicating diffusive behavior.

Figure 5

(a) Spacetime plot ($t$ versus $x$) of whistler fluctuations propagating through the simulation with $L_x=L_0$, ${\beta }_{e0h}=64$ at $L_y/2$. At early times, the initial condition $f_0$ produces fluctuations that reverse direction and are overtaken by the whistlers, which then move to the right. The fluctuations move slowly compared to the thermal speed, and most do not reach the cold reservoir at $x=L_0$ by the end of the simulation at $t=800{\mathrm{\Omega }}_{e0}^{-1}$. (b) Fourier spectra, summed over $k_y$ and plotted as a function of $k_x$ for the simulations with $L_x=L_0$ and ${\beta }_{e0h}=32$, 64, and 128.