Time-dependent Tonks-Langmuir model is unstable

We investigate a time-dependent extension of the Tonks-Langmuir model for a one-dimensional plasma discharge with collisionless kinetic ions and Boltzmann electrons. Ions are created uniformly throughout the volume and flow from the center of the discharge to the boundary wall due to a self-consistent, zero-order electric field. Solving this model using a particle-in-cell simulation, we observe coherent low-frequency, long-wavelength unstable ion waves which move toward the boundary with a speed below both the ion acoustic speed and the average ion velocity. The maximum amplitude of the wave potential fluctuations peaks at $\approx 0.09T_e$ near the wall, where $T_e$ is the electron temperature in electron volts. Using linear kinetic theory, we identify this instability as slow ion-acoustic wave modes which are destabilized by the zero-order electric field.

Figure 1

Schematic of the Tonks-Langmuir discharge model. Cold ions are created and then accelerated to the walls by the self-consistent electric field.

Figure 2

Steady-state space-time plots of (a) the dimensionless potential $\eta =-e\varphi /k{T}_e$ and (b) the potential fluctuations $\delta \eta \left(x,t\right)=\eta \left(x,t\right)-〈\eta 〉\left(x\right)$ computed by subtracting the time-averaged potential. The dashed lines in panel (b) show the ion acoustic speed $c_s$ [Eq. (11)]. Standard ion-acoustic waves are visible near the center of the discharge with the speed $c_s$, while the long-wavelength fluctuations move with a speed well below $c_s$.

Figure 3

Power spectral density (psd) vs distance from the center of the discharge and frequency normalized by the ion plasma frequency at the center of the discharge ${\omega }_{\mathrm{pi}}$. A spectral peak with nearly constant frequency which grows with distance from the center of the discharge is found, indicating an unstable acoustic mode.

Figure 4

Time-averaged ion velocity distribution function $f\left(x,v\right)$. (a) The average is taken over $13{\omega }_{\mathrm{pi}}^{-1}$ so the effect of the instability on ion trajectories is evident. (b) The average is taken over $128{\omega }_{\mathrm{pi}}^{-1}$, which corresponds roughly to one period of the instability. Note that spatial fluctuations average to nearly zero.

Figure 5

The speed of unstable acoustic waves vs position $x$. The average ion velocity $〈v〉/c_s$ is shown for comparison. Speeds are normalized by the ion acoustic speed. The speed of the unstable waves is less than the average ion velocity, indicating that the unstable mode is propagating in the $-x$ direction but is carried to the wall by the ion flow.

Figure 6

Root-mean-squared (rms) (blue) and maximum (red) of potential fluctuations. The dashed line gives a linear spatial growth rate of $0.0094/{\lambda }_D$.

Figure 7

Root-mean-squared (rms) potential fluctuations vs normalized position $x/L$ for system lengths $L=256{\lambda }_D$, $384{\lambda }_D$ and $512{\lambda }_D$. The spatial dependence of the fluctuations is effectively independent of $L$, indicating the fluctuations are a feature of the time-dependent TL model in the plasma approximation ${\lambda }_D/L\ll 1$.

Figure 8

Average dispersion relations in the laboratory frame found by taking the two-dimensional Fourier transform of the potential fluctuations $\delta \eta$ for regions $128{\lambda }_D\times 512{\omega }_{\mathrm{pi}}^{-1}$ vs distance from the center of the discharge. For example, in panel (b) the spatial domain extends from $x=128{\lambda }_D$ to $256{\lambda }_D$. The white dashed lines in panel (a) give the ion acoustic speed $c_s$ for a stationary plasma. A standard acoustic wave with the speed $c_s$ is visible together with slow acoustic modes with speeds less than $c_s$ in the laboratory frame which propagate in the $-x$ direction and are carried to the wall by the ion flow.

Figure 9

Roots of the linear dielectric function from Eq. (20) with $k{\lambda }_D=0.05$, $T_e/T_i=10$, and ion flow speed $u_i=0:\mathrm{Re}\left\{\widehat{\varepsilon }\right\}=0$ (red solid lines) and $\mathrm{Im}\left\{\widehat{\varepsilon }\right\}=0$ (blue dashed lines). Circles are the crossing points, which represent the wave modes. (a) No electric field ($E=0$) and (b) with the electric field $eE/(T_e/{\lambda }_D)=-0.001$. The electric field leads to a growing mode.

Figure 10

(a) Growth rate and (b) real angular frequency of the linear dispersion relation computed from Eq. (20). The common data in each curve was $T_e/T_i=10$ and ion flow speed $u_i=0$. The five chosen values of the electric field are indicated in the legend.

Figure 11

Angular frequency and growth rate (inset) vs wave number for the linear dispersion relation computed from Eq. (20) with $T_e/T_i=10$ and $eE/(T_e/{\lambda }_D)=-0.001$ at the five values of $u_i/c_s$ indicted in the legend. All five curves are shown for the growth rate, but are indistinguishable, demonstrating that the growth rate is independent of the average ion velocity.

Figure 12

Growth rate and angular frequency (inset) of the linear dispersion relation computed from Eq. (20) for $T_e/T_i=10$, $eE/(T_e/{\lambda }_D)=-0.001$, and $〈v〉/c_s=0$ for four distinct wave modes.

Figure 13

(a) Profiles of electric field (solid line), ion flow speed (dashed line), rms ion speed (dotted line) and density (dash-dotted line) from the steady-state TL model (Ref. [15]). (b) The computed maximum linear growth rate computed from Eq. (20) using the parameters from panel (a). (c) Growth rate and (d) real frequency of the instability at the four spatial locations indicated in the legend.