Three-dimensional modeling of a plasma in a strong azimuthal magnetic field

Three-dimensional magnetohydrodynamic simulations are able to model the generation of disk-shaped plasma, driven by laser ablation from a current-carrying rod in a pulsed-power machine producing azimuthal magnetic fields of $2-3$ MG. The plasma at such extreme conditions is unique in that the parameter space for the plasma $\beta$ and Hall parameter $\chi$ transition from below unity to greater than unity at different stages of the plasma generation. In simulations, the formation of the plasma disk in the azimuthal direction is driven by heat flux from the laser spot and depends on the set of transport coefficients used in simulations. The most recent set of transport coefficients leads to the formation of plasma ejecta at the back end of the rod, which qualitatively matches experiments. Specifically, the cross-gradient Nernst effect, which twists the magnetic field, is shown to have a large effect on the shape of the back-end ejecta. In the direction along the axis of the rod, there is propagation of perturbations from the disk as observed in experiments. In simulations, the period of temperature perturbations is in good agreement with experimental results. An instability due to coupling of heat flux and the magnetic field advection provides a possible explanation for perturbation growth along the axis of the rod, and the instability growth rate is consistent with experimental results.

Figure 1

Isovolume plot of electron temperature at different times showing formation of disk plasma.

Figure 2

Isovolume plot of electron density at 6 ns after the laser pulse in simulations using three different sets of coefficients

Figure 3

Schematic of magnetic-field advection by Nernst effect ${\beta }_{\wedge }$ (blue arrows) and cross-gradient Nernst effect ${\beta }_{\parallel }-{\beta }_{\perp }$ (white arrows).

Figure 4

Two-dimensional cross section of electron density at 6 ns after the laser pulse in simulations using (a) Epperlein-Haines and (b) Sadler-Walsh coefficients. The ejecta in the back-end structure is straighter and extends farther in simulations using Sadler-Walsh coefficients.

Figure 5

Electron temperature near the disk back end at 3 ns after the laser pulse in simulation with classical Nernst coefficient (a) and Nernst coefficient reduced to 0.2 of classical value (b). In the simulations with reduced Nernst coefficient, the back-end cavitation is observed instead of the back-end ejecta in (a).

Figure 6

Shadowgrams at 6 ns after the laser pulse for probing laser wavelengths of 532 and 266 nm from experiments [(a) and (b), respectively] from simulations using Sadler-Walsh coefficients [(c) and (d), respectively] and using Epperlein-Haines coefficients [(e) and (f), respectively].

Figure 7

Two-dimensional cross sections immediately after the laser pulse ends for the ratio of ion mean-free path to ion temperature scale length at $x=0$ (a) and $z=0$ (b) and for the ratio of ion Larmor radius to ion temperature scale length at $x=0$ (c) and $z=0$ (d). The red (light) contour denotes electron densities above $5\times {10}^{18}{\text{cm}}^{-3}$ and the black (dark) contour denotes Al plasma.

Figure 8

Isovolume plot of electron temperature at electron densities larger than critical density $n_c$ at 6 ns after the laser pulse. In the axial direction, temperature perturbations expand away from the laser spot.

Figure 9

Shadowgram from experiments with probing laser wavelength of 532 nm at 14 ns after the laser pulse showing perturbations along the rod.

Figure 10

Line outs of electron density, temperature, and magnetic field in the axial direction along the rod surface starting from the axial bottom of the disk at 4 ns after laser pulse. The period of perturbations is smaller closer to the disk.

Figure 11

Solutions for real and imaginary parts of frequency $\omega$ (GHz) using Davies-Wen coefficients (a) and (b), Epperlein-Haines coefficients (c) and (d), and Sadler-Walsh coefficients (e) and (f). The parameters used are $T_e=150$ eV, $n_e=n_c=1.1\times {10}^{21}{\text{cm}}^{-3}, B=200T$, and $l_b=l_t=100\mu \mathrm{m}$.

Figure 12

The growth rate as a function of electron temperature at various electron densities calculated with either Epperlein-Haines or Davies-Wen Coefficients. Parameters: $k_r=.05(1/\mu \mathrm{m}), k_r=0.01(1/\mu \mathrm{m}), l_b=l_t=100, \mu \mathrm{m}$, and $B=200T$.