Transverse instability magnetic field thresholds of electron phase-space holes

A detailed comparison is presented of analytical and particle-in-cell simulation investigation of the transverse instability, in two dimensions, of initially one-dimensional electron phase-space hole equilibria. Good quantitative agreement is found between the shift-mode analysis and the simulations for the magnetic field ($B$) threshold at which the instability becomes overstable (time oscillatory) and for the real and imaginary parts of the frequency. The simulation $B$ threshold for full stabilization exceeds the predictions of shift-mode analysis by 20–30%, because the mode becomes substantially narrower in spatial extent than a pure shift. This threshold shift is qualitatively explained by the kinematic mechanism of instability.

Figure 1

Example PIC simulation of a kink instability of an initially planar hole. Contours of potential $\varphi$ in two spatial dimensions: $z$ is the direction of trapping and the magnetic field, $y$ is the transverse direction. The perturbation consists of sinusoidal wave variation in the $y$ direction of a displacement in the $z$ direction. This case has a displacement that is a growing standing wave whose amplitude in (b) at later time $t=238$ is opposite in sign and greater in magnitude than that in (a) at $t=194$. ($\psi =0.16, \mathrm{\Omega }=0.2$.)

Figure 2

Contours in the complex $\omega$ plane of the real and imaginary part of the momentum balance $F_{\text{tot}}$, which is required to be zero, i.e., the solution (denoted $@$) is the intersection of the two zero contours. (a) Purely growing case; (b) no solution; (c),(d) overstable solution. Hole parameters $\psi =0.16, {\omega }_b=0.2$.

Figure 3

Summary of dispersion solutions for (a) $\psi =0.16$ and (b) $\psi =0.36$, and six $k$ values.

Figure 4

Relation between oscillation frequency ${\omega }_r$ and wave number $k$ for overstable modes. Points are derived from numerical evaluation of the dispersion relation. The curve is an empirical approximate fit to the points.

Figure 5

Examples of unstable potential modes in hole simulations with $\psi =0.49$. (a) $\mathrm{\Omega }=0.2, \ell =1$, purely growing. (b) $\mathrm{\Omega }=0.25, \ell =2$, oscillatory. (c) $\mathrm{\Omega }=0.45, \ell =6$, oscillatory. The left-hand section $0<t<20$ is replaced by the shift mode $z$ profile for comparison with the PIC results. The color bars at right indicate mode amplitudes $A_{\ell }$. Only times prior to the onset of strongly nonlinear behavior are displayed.

Figure 6

Simulation observations of unstable real (open points) and imaginary (filled points) frequencies as a function of magnetic field strength. Analytic predictions are shown by dashed lines separating the three $\mathrm{\Omega }$ regions and showing the growth rate of overstability. Single numerals, denoting the mode number $\ell$ are placed at the predicted real analytic frequency fit Eq. (7) for the dominant $k$ observed. When the real frequency is nonzero, they are joined by vertical lines to the corresponding frequency ${\omega }_r$ observed in the simulation, to show the degree of agreement.

Figure 7

Comparison of the real frequency dependence on $k$ observed in PIC simulations (points with the same coding as Fig. 6) with the fit (line) of Eq. (7): ${\omega }_r/\sqrt{\psi }\simeq 0.45{(k/\sqrt{\psi })}^{0.75}$.