Nonlocal Model for Electron Heat Flux and Self-generated Magnetic Field

TL;DR

The paper addresses the problem of jointly modeling nonlocal electron heat flux and self-generated magnetic fields in inertial confinement fusion scenarios. It develops a self-consistent nonlocal framework that includes electric-field corrections, enabling simultaneous recovery of kinetic effects on heat conduction and magnetic-field evolution at hydrodynamic scales. The authors provide analytic and numerical demonstrations of nonlocal corrections to heat flux $q$, Biermann source, and Nernst velocity $v_N$, and show that nonlocality can generate magnetic fields without density gradients while modifying Biermann generation and field advection in laser ablation. Validation against Fokker-Planck benchmarks and FLASH simulations suggests that electric-field self-consistency is essential for physical results and reveals significant impacts on magnetic-field distributions and potential hydrodynamic-instability dynamics in ICF.

Abstract

Coupling of electron heat conduction and magnetic field takes significant effects in inertial confinement fusion (ICF). As the nonlocal models for electron heat conduction have been developed for modeling kinetic effects on heat flux in hydrodynamic scale, modeling kinetic effects on magnetic field are still restricted to flux limiters instead of nonlocal corrections. We propose a new nonlocal model which can recover the kinetic effects for heat conduction and magnetic field in hydrodynamic scale simultaneously. We clarify the necessity of self-consistently considering the electric field corrections in nonlocal models to get reasonable physical quantities. Using the new nonlocal model, the nonlocal corrections of transport coefficients in magnetized plasma and the magnetic field generation without density gradients are systematically studied. We find nonlocal effects significantly change the magnetic field distribution in laser ablation, which potentially influences the hydrodynamic instabilities in ICF.

Figures (5)

• Figure 1: Classical transport coefficients from the Fokker-Planck equation. Results for $Z=\infty$ limit is rigorous, while artificial BGK electron-electron collision operator fitted from Fokker-Planck simulations is used for finite Z condition. $\alpha_\perp, \alpha_\wedge$ are the electrical resistivity coefficients. $\beta_\perp, \beta_\wedge$ are the thermoelectric coefficients. $\kappa_\perp, \kappa_\wedge$ are the heat conduction coefficients. $\delta_\perp, \delta_\wedge, \gamma_\perp, \gamma_\wedge$ are the advection coefficients of magnetic field defined in literature 21PRL-Sadler.
• Figure 2: Normalized nonlocal electron heat flux, Biermann magnetic, and Nernst velocity with small temperature perturbation in unmagnetized plasma from our model, SNB model with $\mathbf{f}_1^M$ and $\mathbf{g}_1^M$ source terms respectively. The circles are simulation results from Fokker-Planck code IMPACTA04JCP-Kingham corresponding to $r=2$ in nonlocal model. Results from BGK and AWBS electron-electron collision operators are both shown.
• Figure 3: Normalized nonlocal results of heat flux $q_x$, $q_y$, Biermann magnetic field, Nernst velocity, and electric resistivity $\alpha_\perp, \alpha_\wedge'$ with small temperature perturbation and different magnetized parameters. Here $\alpha_\wedge'=\alpha_\wedge-3\sqrt{\pi}\Omega/4$ refers to the total coefficient of transverse electric field $E_y$ (including Hall effects) corresponding to current $j_x$.
• Figure 4: Results for magnetic field generation without density gradients. (a)Normalized temperature distribution and magnetic field generation with normalized temperature distribution $T=2+tanh[(x-192-16\cos(2\pi y/128))]$. (b)Normalized magnetic field generation with small temperature perturbation from Fokker-Planck simulations and our model. (c)The coefficient $f(k\lambda)$ for nonlocal magnetic field generation with small temperature perturbation in a wide range. (d)Magnetic field at 0.3 ns through FLASH simulations00-FLASH. The results are from only Biermann source and with nonlocal corrections.
• Figure 5: (a)Normalized electron heat flux with steep temperature gradient in unmagnetized plasma. $qSH$ refers to the classical Spitzer-Harm heat flux. $q-f1$ refers to the nonlocal heat flux without self-consistently considering electric field correction (i.e. the nonlocal heat flux in SNB model with $\mathbf{f}_1^M$ source term). $q-\Delta E$ refers to the heat flux corresponding to the return current from electric field correction. $q-With \Delta E$ refers to the total nonlocal heat flux in our model. (b)Normalized classical and nonlocal transverse electric field with steep temperature gradient with different magnetized parameters. The results with $\Omega=0.001$ is artificially enhanced by 10 times to be more clear. (c)Magnetic field at 1.0 ns in laser ablation from FLASH simulations. The left part is the results from classical Biermann source and the right part is with the nonlocal corrections. (d)Magnetic flux in half the space.