Ultrafast wave-particle energy transfer in the collapse of standing whistler waves

Efficient energy transfer from electromagnetic waves to ions has been demanded to control laboratory plasmas for various applications and could be useful to understand the nature of space and astrophysical plasmas. However, there exists the severe unsolved problem that most of the wave energy is converted quickly to electrons but not to ions. Here, an energy-to-ion conversion process in overdense plasmas associated with whistler waves is investigated by numerical simulations and a theoretical model. Whistler waves propagating along a magnetic field in space and laboratories often form standing waves by the collision of counter-propagating waves or through the reflection. We find that ions in standing whistler waves acquire a large amount of energy directly from the waves over a short time scale comparable to the wave oscillation period. The thermalized ion temperature increases in proportion to the square of the wave amplitude and becomes much higher than the electron temperature in a wide range of wave-plasma conditions. This efficient ion-heating mechanism applies to various plasma phenomena in space physics and fusion energy sciences.

Figure 1

Snapshots of the electric fields and ion phase diagram in the 1D PIC simulation of the interaction between counter whistler waves and plasmas. (a–d) Time evolution of the electric fields during counter irradiation of circularly polarized lights. The longitudinal and tangential fields ($E_x$ and $|{\mathit{E}}_{\perp }|$) are depicted by the black and red curves, respectively. Snapshot data are taken at (a) ${\omega }_{0}t=-4.71$, (b) ${\omega }_{0}t=73.8$, (c) ${\omega }_{0}t=152$, and (d) ${\omega }_{0}t=192$, where the origin of time $t=0$ is defined by the time at which the injected lights arrive at the target surface. The gray area represents the inside of the target layer. (e–h) Snapshots of the position-velocity ($x-v_{xi}$) phase diagram for ions taken at the same timing as in (a)–(d). The color denotes the particle number.

Figure 2

Energy conversion rate and energy spectra for ions and electrons in the fiducial run. (a) Time histories of the ion and electron energies normalized by the injected energy of the electromagnetic wave ${\mathcal{E}}_0$. The gray bar represents the pulse duration of the target irradiation, $0\le {\omega }_{0}t\le 70.6$. Arrows indicate the snapshot timing shown in Fig. 1. (b) Energy spectra for ions and electrons at the end of the calculation. The vertical axis is $\epsilon f_{\epsilon }\left(\epsilon \right)$, where $f_{\epsilon }\left(\epsilon \right)$ is the probability density function for the energy $\epsilon$, so that the peak value is related to the temperature as ${\epsilon }_{\mathrm{peak}}=(3/2)k_{B}T$ in the Maxwell-Boltzmann distribution. The ion spectrum is fitted well by a two-temperature model with $T_i=24$ and 2.9 keV, and the electron spectrum is almost identical to the thermal distribution of $T_e=9.6$ keV. Dotted curves show the fitted functions.

Figure 3

Detailed structures of a standing whistler wave and linearly growing velocity fluctuations of ions in the fiducial run. (a) Magnified view of the electric fields at the center of the target, $E_x$ (black curve) and $|{\mathit{E}}_{\perp }|$ (red curve), taken at ${\omega }_{0}t=152$, which is right after the formation of the standing whistler wave. The electromotive field ${({\mathit{v}}_e\times \mathit{B})}_x$ working on the electrons is also plotted by green dots. The blue curve denotes the electron density fluctuation $\delta n_e\equiv n_e-n_{e0}$. The length indicated corresponds to the wavelength of the whistler wave ${\lambda }_w={\lambda }_0/N$. (b) Amplitude growth of the ion velocity in a standing wave identified from four successive snapshots, at ${\omega }_{0}t=113$ (black curve), 152 (red curve), 192 (green curve), and 231 (blue curve). The indicated length scale is ${\lambda }_w/8$.

Figure 4

Dependence of the model prediction for the ion temperature on the initial parameters $a_0, B_{\mathrm{ext}}/B_c$, and $n_e/n_c$. (a) Predicted ion temperature given by Eq. (10) for cases of $n_{e0}/n_c=19.3$. The wavelength is assumed to be ${\lambda }_0=0.8\mu \mathrm{m}$. Overplotted colored circles indicate $T_i$ values obtained by 1D PIC simulations with given parameters $a_0$ and $B_{\mathrm{ext}}/B_c$. In these runs, a flat-top pulse shape is used with a sufficiently long duration, ${\tau }_0\gg {\tau }_{\mathrm{sat}}$. The valid range of the temperature prediction is within the area surrounded by three critical curves, which are obtained from the pressure-gradient condition [Eq. (14); solid curve] and the cyclotron resonance conditions [Eqs. (15) and (16); dashed and dot-dashed curves, respectively]. (b) The same diagram for cases of $n_{e0}/n_c=10$ and ${\lambda }_0=1$ cm.

Figure 5

Parameter dependence of the energy conversion rate for ions and electrons obtained by 1D PIC simulations. (a) Conversion rate for ions (black symbols) and electrons (red symbols) as a function of the wave amplitude $a_0$. The conversion rate is evaluated at ${\omega }_0{t}_{\mathrm{end}}=2.78\times {10}^3$. Filled and open circles denote the results with and without the Coulomb collision effects, respectively. The dotted vertical line indicates the fiducial parameter $a_0=2.65$. (b, c) Dependence on (b) the external field strength $B_{\mathrm{ext}}/B_c$ and (c) the plasma density of the target $n_{e0}/n_c$. The meanings of the symbols are the same as in (a). The initial parameters are the same as in the fiducial run except for $B_{\mathrm{ext}}$ (b) and $n_{e0}$ (c). The fiducial parameters are $B_{\mathrm{ext}}/B_c=7.47$ and $n_{e0}/n_c=19.3$.

Figure 6

The time evolution of the energy conversion rate for (a) ions and (b) electrons is depicted for various cases—$L_y/{\lambda }_0=3$ (red curve), $L_y/{\lambda }_0=1$ (green curve), and $L_y/{\lambda }_0=0.1$ (blue curve)—of the 2D runs. The initial parameters are the same as in the 1D fiducial run except for the target thickness, $L_x/{\lambda }_0=18.75$, and the pulse duration, ${\omega }_0{\tau }_0=35.3$. The dashed curve is the result of the 1D run, that is, $L_y=0$.

Figure 7

Energy spectrum $\epsilon f_{\epsilon }\left(\epsilon \right)$ of ions (black curves) and electrons (red curves) at the end of the calculation, ${\omega }_0{t}_{\mathrm{end}}=1.84\times {10}^3$, are plotted for the cases of $L_y/{\lambda }_0=3$ (solid curves) and $L_y/{\lambda }_0=0$ (dashed curves). These spectra are calculated from particles located in the standing wave region $|x/{\lambda }_0|\le 5$.

Figure 8

Spatial distributions of the energy density $u$ for (a) ions and (b) electrons in a 2D simulation. The energy density is measured in units of pascals. The initial parameters are the same as in the 1D fiducial run except for the target thickness, $L_x/{\lambda }_0=18.75$, and the pulse duration, ${\omega }_0{\tau }_0=35.3$. The box size in the $y$ direction is $L_y/{\lambda }_0=3$. These images are taken at ${\omega }_{0}t=329$.

Figure 9

Spatial distribution of the density $n$ for (a) ions and (b) electrons in the 2D run. The initial parameters and snapshot timing are the same as those in the 2D run shown in Fig. 9.