Low-frequency model of breathing oscillations in Hall discharges

A paradigm for Hall discharge modeling is presented whereby only the time scale of the lowest-frequency mode is explicitly resolved. The ability of such a low-frequency model to reproduce with excellent accuracy the breathing mode is demonstrated through comparisons with a fully time-dependent numerical model. Based on this formalism, an approximate linearized model is derived which essentially constitutes a one-dimensional generalization of the classical zero-dimensional predator-prey model. The model highlights the interaction of standing plasma waves with the transport of neutral species, which involves standing and convective waves of similar magnitude. It predicts a frequency which is in close agreement with the frequency of the small perturbation modes observed in simulations. Finally, it is shown that unstable modes are in general strongly nonlinear and characterized by frequencies obeying a scaling law different from that of linear modes.

Figure 1

Schematic of a typical Hall thruster.

Figure 2

Evolution of the discharge current $I$. Solutions for $U=180\text{V}$ and $B_{\text{max}}=22\text{mT}$ using (a) the fully time-dependent model and (b) the low-frequency model. Ion transit-time oscillations can be distinguished on subfigure (a), superimposed on the low-frequency signal.

Figure 3

Evolution of the density of neutrals $N$ in the conditions of Fig. 2. Solutions for (a) the fully time-dependent model and (b) the low-frequency model.

Figure 4

Evolution of the plasma density $n$ in the conditions of Fig. 2. Solutions for (a) the fully time-dependent model and (b) the low-frequency model. Note the presence of ion transit-time waves on subfigure (a).

Figure 5

Evolution of the temperature of electrons $T_e$ in the conditions of Fig. 3. Solutions for (a) the fully time-dependent model and (b) the low-frequency model. Note the presence of ion transit-time waves on subfigure (a).

Figure 6

Functions $\gamma$ and $\Psi$ determined numerically in the conditions of Figs. 2, 3, 4, 5. The oscillations of $\Psi$ in the neighborhood of the sonic point $\left(x\approx 1\text{cm}\right)$ are believed to constitute a numerical artifact.

Figure 7

Comparison between (a) the frequency of small perturbation modes of the low-frequency simulation and (b) the prediction from the linear approximation [Eq. (32)]. Linearly unstable modes $\left[\text{Im}\left(\omega \right)<0\right]$ are emphasized with black markers on subfigure (a).

Figure 8

Average values (solid lines) and standard deviations (vertical bars) of the discharge current for the periodic modes of the low-frequency simulation. The steady-state discharge current is plotted with a dashed line.

Figure 9

Comparison between (a) the frequency of periodic solutions of the low-frequency simulation and (b) the prediction of Eq. (41) for highly nonlinear modes [25].