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 ≈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.