Energy and momentum conservation upon reflection of a solitary pulse in a bounded magnetized plasma

When the nature of a magnetosonic pulse propagating in a bounded magnetized plasma slab is successively transformed from compression to rarefaction and vice versa upon reflection at a plasma-vacuum interface, both the energy and the longitudinal electromagnetic (EM) momentum of the plasma-pulse system are found to oscillate between two states. Simple analytical models and particle-in-cell simulations show that these oscillations are associated with EM radiation to and from the surrounding magnetized vacuum. For partial reflection supplemental losses in total pulse energy and mechanical momentum are identified and shown to follow, respectively, Fresnel's transmission coefficients for the energy and the magnetic perturbation. This mechanical momentum loss upon partial reflection is traced to the momentarily nonzero volume-integrated Lorentz force, which in turn supports that mechanical and EM momentum transfers are, respectively, associated with the magnetic and electric parts of the momentum flux density.

Figure 1

Sketch of the plasma slab-pulse system. The profiles depict the time evolution of the magnetic field disturbance $B_1$, which changes sign upon reflection at the plasma-vacuum interface. Both the vacuum and the plasma slab are permeated by a constant and homogeneous background magnetic field $B_0\widehat{\mathbf{z}}$. The amplitude of the transmitted pulse is exaggerated for clarity.

Figure 2

Profiles of solitary wave solutions for the analytical small-amplitude ${\mathrm{sech}}^2(s\sqrt{\epsilon }/2)$ magnetic perturbation given in Eqs. (4) (dashed line) and for the large amplitude numerically integrated solitary wave solution from Rau and Tajima [45] (solid line) for three pulse amplitudes $\epsilon =0.05,0.1$, and 0.5. All quantities are normalized by the peak value of the small-amplitude solution for the corresponding $\epsilon$. Differences become noticeable for $\epsilon \ge 0.1$. Background plasma parameters are identical to those used in the simulation and given in Table 3.

Figure 3

Map of the magnetic field perturbation $B_1$ corresponding at $t=0$ to a magneto-sonic soliton of amplitude $\epsilon B_0$ at $x=0$ in a finite width plasma slab. The horizontal dashed lines approximately indicate the start and finish of reflection events.

Figure 4

Electromagnetic field convention for incident, reflected, and transmitted pulses at the plasma-vacuum interface.

Figure 5

Map of the normalized longitudinal electric field $E_x/\left[{\epsilon }^{3/2}{\eta }^{-1}{v}_A{B}_0\right]$ (a), transverse electric field $E_y/\left[\epsilon v_A{B}_0\right]$ (b), magnetic perturbation $B_1/\left[\epsilon B_0\right]$ (c), transverse current $j_y/\left[e{n}_0{\epsilon }^{3/2}{\eta }^{-1}{v}_A\right]$ (d), and electron density $n_e/n_0$ (e) near the plasma-vacuum interface during the first reflection.

Figure 6

Time evolution of the EM energy of the pulse-slab system ${\mathcal{E}}_{sp}$ (a) and breakdown of the contribution of Joule losses and flux of the Poynting vector through the axial ends of the simulation domain (b). Joule losses plotted in dash-dotted dark blue in the lower panel are multiplied by 10 for clarity. The vertical dashed gray lines depict the start and finish of reflection events, consistent with those shown in Fig. 3.

Figure 7

Energy exchange for the slab-pulse system upon transformation from a rarefaction (compression) to a compression (rarefaction) pulse. ${\mathcal{E}}^0$ is the energy exchange corresponding to the change in pulse nature assuming perfect reflection. ${\mathcal{E}}^1\ll {\mathcal{E}}^0$ is the pulse energy loss due to the partial transmission of the magnetic field perturbation upon reflection.

Figure 8

Time evolution of the total pulse energy (a), the electromagnetic ${\mathcal{E}}_p$ and kinetic ${\mathcal{F}}_p$ components of the pulse energy (b), and the electron and ion internal energy ${\varepsilon }_I$ in the slab (c). Energies are normalized by the soliton EM energy ${\widehat{\mathcal{E}}}_p$ given in Table 2. The vertical dashed gray lines depict the start and finish of reflection events, consistent with those shown in Fig. 3.

Figure 9

Map of the electron temperature. A clear increase of temperature is observed in the wake of the pulse. The horizontal dashed lines approximately indicate the start and finish of reflection events.

Figure 10

Time evolution of the pulse energy content before the first reflection event. (a) Electron heating $\mathrm{\Delta }{\varepsilon }_{I_e}$ follows the nonzero ${j_y}^2/\sigma$ volumic resistive power losses in the pulse, as expected from Eq. (31), and both quantities can be reasonably matched using an electron-ion momentum transfer frequency ${\nu }_{ei}^{★}=2.6\times {10}^7{\mathrm{s}}^{-1}$. The analytic solution corresponds to Eq. (33). (b) The corresponding pulse energy loss $-\mathrm{\Delta }{\varepsilon }_{I_e}$ is observed to be evenly split between the mechanical and electromagnetic energy contents ${\mathcal{F}}_p$ and ${\mathcal{E}}_p$, respectively.

Figure 11

Time evolution of normalized pulse field energy ${\mathcal{E}}_p$, pulse amplitude $\epsilon$, and pulse mechanical momentum $\mathcal{Q}{}_{sp}$ before the first reflection event. The decrease rates for ${\mathcal{E}}_p$ and $B_1$ are in relatively good agreement with the values derived from electron-ion friction and the soliton scaling (dashed and dash-dotted curves). On the contrary, $\mathcal{Q}{}_{sp}$ decreases much faster than the $-{\nu }_{ei}^{★}\epsilon /15t$ obtained based on the pulse amplitude decrease.

Figure 12

Time evolution of the mechanical momentum $\mathcal{Q}{}_{sp}$ (a), and pulse ${\mathcal{P}}_p$ (b), and slab-pulse ${\mathcal{P}}_{sp}$ (c) electromagnetic momentum. Momenta are normalized by their soliton scaling $\widehat{\mathcal{Q}}$, $|{\widehat{\mathcal{P}}}_p^{\pm }|$ and ${\widehat{\mathcal{P}}}_{sp}$ given in Table 2. The vertical dashed gray lines depict the start and finish of reflection events, consistent with those shown in Fig. 3.

Figure 13

Time-integrated EM longitudinal momentum flux through the boundaries $\int D_f\left({\mathcal{T}}_E\right)dt$ [electric contribution only; see Eq. (44)] and variation in pulse EM longitudinal momentum ${\mathcal{P}}_p$. The amplitude of the momentum loss (gain) through the simulation boundaries upon reflection is greater than the pulse EM longitudinal momentum loss (gain) upon first (second) reflection, suggesting a momentum exchange with the plasma. Quantities are normalized by the pulse EM momentum $|{\widehat{\mathcal{P}}}_p^{\pm }|$ from Table 2. The $y$ axis is cut to highlight the dynamics and differences between predictions and numerical results away from reflection events.

Figure 14

Momentum exchange upon transformation from a rarefaction (compression) to a compression (rarefaction) pulse at the plasma-vacuum interface. For the slab-pulse system, momentum fluxes to the surrounding vacuum region can be traced back to a loss by ${\mathcal{P}}^{1,B}$ in mechanical momentum $\mathcal{Q}{}_{sp}$ at each partial reflection, and a loss (a gain) by ${\mathcal{P}}^0$ in EM momentum ${\mathcal{P}}_p$ when reflecting a compression (rarefaction) pulse into a rarefaction (compression) pulse.

Figure 15

Time evolution of rate of change of mechanical longitudinal momentum $\mathcal{Q}{}_{sp}$ and volume-integrated Laplace force $j_y{B}_z$. The Laplace force is observed to approach the rate of change of $\mathcal{Q}{}_{sp}$ during the reflection events, that is, for $0.6\le t{\omega }_{ci}\le 1.2$ and $2.4\le t{\omega }_{ci}\le 3.2$.

Figure 16

Illustration of the interference of incident and reflected longitudinal electric fields. (a) Away from reflection the volume integral of $E_x$ in the plasma slab $-L_p/2\le x\le L_p/2$ (olive area) is zero. (b) During reflection the incident and reflected pulses interfere so that, for incident and reflected pulses of different amplitudes, this integral is no longer zero.

Figure 17

Time evolution of the dimensionless longitudinal and transverse ion velocity ${\upsilon }_x=v_{i_x}/\left(\epsilon v_A\right)$ and ${\upsilon }_y=v_{i_y}/\left(\epsilon v_A\right)$ in response to the soliton electromagnetic field ($E_x,E_y,B_z$) with [Eq. (A5)] and without [Eq. (A4)] considering the effect of the magnetic perturbation $\delta B$, and solving numerically Eq. (A1) using a Boris scheme (full). In all cases the ion is initialized with thermal velocity directed along $\widehat{\mathbf{x}}$ at $\overline{t}=-50$, that is, much before the soliton field is noticeable at the ion position. The soliton field peaks at the ion position at $t=0$. Conditions are those of the PIC simulations given in Table 3. Upper and lower $x$ axes are related through $2\eta \overline{t}/\sqrt{\epsilon }=t{\omega }_{ci}$.

Figure 18

Time evolution of the dimensionless longitudinal and transverse population averaged ion velocity $\langle {\upsilon }_x\rangle =\langle v_{i_x}\rangle /\left(\epsilon v_A\right)$ and $\langle {\upsilon }_y\rangle =\langle v_{i_y}\rangle \left(\epsilon v_A\right)$ in response to the soliton electromagnetic field ($E_x,E_y,B_z$) with [Eq. (A5)] and without [Eq. (A4)] considering the effect of the magnetic perturbation $\delta B$, and solving numerically Eq. (A1) using a Boris scheme (full). In all cases we consider ions initialized with randomly directed thermal velocity at $\overline{t}=-50$, that is, much before the soliton field is noticeable at the ion position. The soliton field peaks at the ion position at $t=0$. Conditions are those of the PIC simulations given in Table 3. Upper and lower $x$ axes are related through $2\eta \overline{t}/\sqrt{\epsilon }=t{\omega }_{ci}$.