Emission of electromagnetic waves as a stopping mechanism for nonlinear collisionless ionization waves in a high-$\beta$ regime

A high energy density plasma embedded in a neutral gas is able to launch an outward-propagating nonlinear electrostatic ionization wave that traps energetic electrons. The trapping maintains a strong sheath electric field, enabling rapid and long-lasting wave propagation aided by field ionization. Using 1D3V kinetic simulations, we examine the propagation of the ionization wave in the presence of a transverse MG-level magnetic field with the objective to identify qualitative changes in a regime where the initial thermal pressure of the plasma exceeds the pressure of the magnetic field ($\beta >1$). Our key finding is that the magnetic field stops the propagation by causing the energetic electrons sustaining the wave to lose their energy by emitting an electromagnetic wave. The emission is accompanied by the magnetic field expulsion from the plasma and an increased electron loss from the trapping wave structure. The described effect provides a mechanism mitigating rapid plasma expansion for those applications that involve an embedded plasma, such as high-flux neutron production from laser-irradiated deuterium gas jets.

Figure 1

Setup for 1D PIC simulations (a) and a schematic structure of this system with a propagating ionization wave (b). The blue color represents neutral hydrogen gas, the black color represents the original electrons initialized at $t=0$ (original electrons), and the red color represent the electrons generated through field ionization of the gas (generated electrons).

Figure 2

Sheath electric field $E_x$ and neutral gas density $n_{\mathrm{gas}}$ at the plasma edge expanding due to the ionization wave. The black curve is $E_x$ (left axis) and the orange curve is $n_{\mathrm{gas}}$ (right axis) normalized to the initial density $n_0$. The snapshot is taken at $t=2$ ps.

Figure 3

Plasma expansion into neutral gas (a, b) and into vacuum (c). (a) Time evolution of $E_x$. The black curve shows a representative trajectory of a trapped energetic electron from the original population. (b, c) Electron kinetic energy density ${\epsilon }_k$ normalized to its initial value ${\epsilon }_{k0}$. In (c), the black dashed lines are electron density contours ($n_e/n_0$) and the red dashed line is the ion front.

Figure 4

Ionization wave structure without the applied magnetic field at $t=2$ ps. (a) Electron phase space. The grey dash-dotted line denotes the average momentum of the original electrons trapped by the wave. (b) Longitudinal electric field $E_x$. (c) Electron density, normalized by the initial electron density $n_0$.

Figure 5

Structure of the sheath region in the ionization wave propagating through neutral gas with an applied magnetic field ($B_0=2$ MG). (a) Structure of electric and magnetic fields in the vicinity of the sheath region with thickness $l$. (b) Structure of a current layer with thickness $\mathrm{\Delta }x$ generated by the applied magnetic field. These snapshots are taken at $t=5$ ps.

Figure 6

Plasma expansion in neutral gas with an applied magnetic field of $B_0=2$ MG in $\left|x\right|>200\mu \mathrm{m}$. (a) Time evolution of $E_x$. The black curve shows a representative trajectory of a trapped energetic electron from the original population. (b) Electron kinetic energy density ${\epsilon }_k$ normalized to its initial value ${\epsilon }_{k0}$. The ionization wave stops at 9 ps.

Figure 7

Scan over the applied magnetic field strength $B_0$. Each marker represents a separate PIC simulation, with $l_{\mathrm{stop}}$ being the total distance traveled by the ionization wave in the region with the applied magnetic field and ${\beta }_0$ is the ratio of the initial plasma pressure to that of the applied magnetic field. For all the cases presented, the emission of the electromagnetic wave accounts for $\sim 70\%$ of the energy initially associated with the ionization wave. The dashed line is $l_{\mathrm{stop}}=\kappa {\beta }_{0}L$, where $\kappa =0.9$.

Figure 8

Schematic setup for calculating the energy losses by the ionization wave. The front and the back of the ionization wave are located at $x_{max}\left(t\right)$ and $x_{min}\left(t\right)$. The energy emitted by the electromagnetic wave is calculated by integrating the Poynting flux through the plane located at $x=x_b$.

Figure 9

Time evolution of the energy loss by the ionization wave associated with the electromagnetic wave emission (solid curve). The applied magnetic field is $B_0=2$ MG. The blue dashed (upper) curve is the total energy carried by the electromagnetic wave and the red dashed (lower) curve is the total energy of the expelled magnetic field [see Eq. (27) for details]. The orange dash-dotted line denotes the time when ionization wave stops. The ionization wave enters the magnetic field region at $t=t_0$. The energies are normalized to ${\mathcal{E}}_{\mathrm{init}}$, which is the energy in the ionization wave at $t=t_0$.