Asymmetric one-dimensional slow electron holes

Slow solitary positive-potential peaks sustained by trapped electron deficit in a plasma with asymmetric ion velocity distributions are in principle asymmetric, involving a potential change across the hole. It is shown theoretically how to construct such asymmetric electron holes, thus providing fully consistent solutions of the one-dimensional Vlasov-Poisson equation for a wide variety of prescribed background ion velocity distributions. Because of ion reflection forces experienced by the hole, there is generally only one discrete slow hole velocity that is in equilibrium. Moreover the equilibrium is unstable unless there is a local minimum in the ion velocity distribution, in which the hole velocity then resides. For stable equilibria with Maxwellian electrons, the potential drop across the hole is shown to be $\mathrm{\Delta }\varphi \simeq \frac{2}{9}{f}^{\prime \prime \prime }\frac{T_e}{e}{\left(\frac{e\psi }{m_i}\right)}^2$, where $\psi$ is the hole peak potential, $f^{\prime \prime \prime }$ is the third derivative of the background ion velocity distribution function at the hole velocity, and $T_e$ is the electron temperature. Potential asymmetry is small for holes of the amplitudes usually observed, $\psi \lesssim 0.5T_e/e$.

Figure 1

Schematic of a hypothetical asymmetric electron hole.

Figure 2

Illustration of ion distributions on either side of the potential hill, and their integral $P\left(v\right)$, which is the cumulative probability distribution multiplied by $n_i\left(\varphi \right)$ (equal to $n_i\left(\varphi \right)$ at $v=+\infty$).

Figure 3

Illustrative asymmetric electron hole parameters (a) $\varphi \left(x\right)$, which is ion potential energy; (b) $-\varphi \left(x\right)$, electron potential energy; (c) $n_i\left(x\right)$ ion density; (d) ‘Classical’ potential giving force. The vertical bars in (a) and (b) indicate ranges of particle energy that are passing, reflected, or trapped. The two points in (c) show the matching distant electron density. The nonzero value of ${\int }_{-\infty }^{+\infty }{n}_{i}d\varphi$ shows this hole is subject to nonzero ion force.

Figure 4

Distant velocity distribution of the ions ($f_{i\infty }$) and the composition of the forces exerted on the electrons and ions, as $v_h$ is varied.

Figure 5

Converged equilibrium shape of potential $\varphi \left(x\right)$, ion density $n_i\left(x\right)$, and ion force, when the hole required velocity $v_h=0.57$ for equilibrium has been discovered. For the same $\psi$ and $f_{i\infty }$ as before.

Figure 6

(a) Potentials on the two sides of the hole. Prescribed side dashed line, Poisson solution side solid. (b) Corresponding trapped electron distribution function at $x=0, \varphi =\psi$.

Figure 7

Stability parameters as a function of $v_h$, for a range of distributions obtained by scaling the Maxwellian components' shift velocity ($v_{\text{shift}}\propto$ 0, 0.33, 0.66, 1.0, 1.33, 1.66, 2.0).

Figure 8

A $3\times 3$ display of nine different cases $T_2=(0.2,0.6,1), n_2=(0.1,0.3,0.5)$ for which $v_s=0.75+\sqrt{T_2}$; here $T_e=T_0$. Each case has two adjacent subframe plots: the left-hand side is $f_{i\infty }\left(v\right)$ and the right-hand side represents scaled ion density perturbation as $[n_i\left(x\right)-1]/\psi$. All corresponding frames have the same axis ranges $-4\le v_{\infty }\le 4, -12\le x\le 12$. Densities are plotted for three different potential peak heights $\psi =(0.4,0.04,0.004)$ indicated by different colors, but only if a stable equilibrium hole velocity ($v_h$) exists. If it does, then a cross is plotted on the left-hand side subframe at that velocity and distribution height.

Figure 9

(a) Comparison of numerical (solid black) and algebraic (dashed red) estimates of potential asymmetry $\mathrm{\Delta }\varphi$, showing its scaling proportional to the square of the electron hole peak potential $\psi$. In panel (b) is shown the ion distribution comprising Maxwellians $T_1=1, n_1=0.7, v_1=1.197,T_2=0.2, n_2=0.3, v_2=-1.197$, and $T_e=1$, in scaled units. The potential profiles in panel (c) are from the ends of the scaling range.