1. 1 S. CHABLE a F. ROGIER b a - ONERA, 2 av. Ed. Belin, 31055 TOULOUSE Cedex 4 b - MIP, Université Paul Sabatier, 118, route de Narbonne, 31062 TOULOUSE Cedex Numerical Investigation and Modeling of Stationary Plasma Thruster Low Frequency Oscillations

2. 2 Scheme of a SPT Magnetic field  Electrons confined in the channel Electric field  Ions accelerated

3. 3 è Advantages : thrusters lower and more accurate than chemical conventional systems (thrust of some mN). è Development since three decades by the Russians. è Development nowadays in U.SA. (M.I.T.), in Russia, in Japan and in France (LPMI, CPAT, CNES, ONERA, LPGP, SNECMA, Astrium...). è Research axes : - clarifications of the mechanisms responsible for the plasma conductivity - reduction of the divergence of the plume - reduction of the low frequency oscillations è Numerical simulations Context

4. 4 1. Introduction 2. Physical model 3. Study of the linear instability modes 4. A simplified model 5. Control of the instabilities 6. Conclusion Plan

5. 5 Bibliography Classification of the low frequency oscillations (Choueri) : è 1-20 kHz band : « loop » axial oscillations, sensitive to the magnetic field and the applied voltage, related to a predator prey model or a depletion of the neutrals - Garrigues and Bœuf (1998), Chesta et al. (2001), Litvak and Fisch (2001). è 5-25 kHz band : rotating spoke related to the ionization process – Janes and Lowder (1966). è 20-60 kHz band : azimuthal oscillations associated with the gradient of density – Esipchuck and Tilinin (1976). è 20-100 kHz : oscillations due to the the low ionization of the plasma – Popovic and Melchior (1968).

6. 6 Physical model : Modelization Assumptions : è Ions transport scale time. è è Strong electron-neutral elastic collision rate : Maxwellian fluid for the electrons. è Ballistic neutrals and ions. è Magnetic field not sensitive to the plasma. è Ions not sensitive to the magnetic field. è Monokinetic neutral distribution è Quasineutrality è Inclusion of an electron-wall collision frequency (Bœuf-Garrigues model)

7. 7 Physical model Avec et 1D model and neutral and ion distribution functions,, and are the neutral, ion and electron densities. is the electron mean energy. is the electric potential, the magnetic field. and the ionization rate, the elastic collision rate and the inelastic collision rate. the electron-wall collision frequency and the electron mobility.

8. 8 Linearization Remark : the non local terms don’t allow us to conclude to the dissipation of the perturbed solution. Friedrichs system

9. 9 Link linear – non linear Discretization : the problem is unstable if the positive real part of the eigenvalues is positive Discretized system : Observation : linear instability è The link between the ion Vlasov equation and the electric potential is the mechanism responsible for the instabilities.

10. 10 Link linear – non linear Link between the growth rate and the amplitude of the non linear oscillations. Amplitude of the non linear oscillations obtained from the transient model Growth rate obtained from the linear model. Variation function of

11. 11 Simplified model Assumptions : è Electron mean energy given and constant. è Neutral density given and constant. è Source term neglected. è Linear model : with :

12. 12 Theorical instability Theorem : The problem is bivalued. If is solution so is solution too. è Unstable problem è Relationship with Buneman instability è Control of the instabilities : topology of B, electron-wall collision frequency.

13. 13 Conclusion Treated points : è Spectral study of the linear instability with a stationary quasineutral hybrid model  relationship between the growth rate and the amplitude of the non linear model. è Simplified model to explain the instabilities  relationship with the Buneman instabilities. è Control of the instabilities by modifying the magnetic field of the electron-wall collision frequency. è Predictive model

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