2. 1 Non-neutral Plasma Shock HU Xiwei （胡希伟） 工 HU Xiwei （胡希伟） HE Yong （何勇） HE Yong （何勇） Hu Yemin （胡业民） Hu Yemin （胡业民） Huazhong University of Science and Techonology 2006.10.25 Zhejiang University, Hangzhou, China

3. 2 Outline Introduction Motivation One dimensional case with magnetic field Two dimensional case (with magnetic field produced by shock current) The electrostatic instabilities on the non-neutral shock front

4. 3 Introduction Introduction to non-neutral plasma shock

5. 4 Plasma shock Plasma shock can arise: when a fluid velocity is larger than the ion sound velocity (without magnetic field) or the Alfven velocity (with magnetic field). A steady shock (or shock front, shock profile) is the result of a balance between compressive and dissipative effects.

6. 5 Front, Upstream and Downstream Shock front (or shock profile): a steady wave- front, propagating at supersonic speed through the undisturbed fluid. Upstream ⑴ : the unshocked (undisturbed) fluid before the shock front Downstream ⑵ : the shocked fluid (disturbed) behind the front front

7. 6 The neutral plasma shock In most cases, we describe the plasma shock with the single (neutral) fluid or magnetohydrodynamic (MHD) equations. So the plasma shock is neutral essentially. In neutral plasma shock there may be plasma current and magnetic field, but there is not charge separation and electrostatic field.

8. 7 Electric field in the front The ion fluid elements will run ahead of the electron fluid elements due to its larger inertial. Δφ This causes a charge separation, then a electrostatic field E and a potential difference Δφ between the up- and down stream.

9. 8 Effects of electric field in front (I) The electric field E will hold back the ion fluid elements and draw the electron fluid elements. This effect is similar the role of dissipation in the formation of steady shock (shock front or profile). So, the E could play very important role in the formation of plasma shock profile, especially the collisionless (non-dissipative) shock profile.

10. 9 Effects of electric field in front (II) Δφ The potential difference Δφ will modify the jump (discontinuity) relations – the so called Hugoniot relations and the critical Mach number M c (the least velocity for shock emergency). The electric field will drive a new kind of electrostatic instability in the shock front as the role of density gradient in the Rayleigh- Taylor instability.

11. 10 Motivation In our simulation for the imploding shock in a single sonoluminescing bubble, we found that there is a extreme strong electrostatic field in the shock front. The electric field is caused by the charge separation. The charge separation is due to the difference of electron fluid and various ion fluid velocities.

12. 11 Profiles of the charge number density Δn, electric field E, and electric current density J striding on the shock front at the moment near the emergence of the maximum electric field.

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14. 13 Electric field in strong laser and in imploding shock Plasma produced by extreme-short pulse laser Imploding shock front in SBSL E (V/m)5×10 11 2×10 10 Δ t (s) 10 -12 10 -11 Δ r (m) 10 -6 10 -8 n e (cm -3 ) 10 21 10 22 Polarization ElectromagneticElectrostatic

15. 14 One dimensional case Plasma shock carries low current. The effects of magnetic field produced by the shock current can be neglected.

16. 15 Static coupled equations in a reference frame on the front where is the equivalent heat flow The double fluid equations and Poisson equation

17. 16 Profiles: ━ M 1 =1.6 ┉ M 1 =1.5

18. 17 Jump conditions across the shock Particle fluxes Momentum fluxes Energy fluxes Where

19. 18 Critical upstream Mach number When, from we can obtain the minimum upstream velocity for the non-neutral shock emergency Z1234 M 1 min 1.4141.73222.236

20. 19 Two dimensional case Cylindrical Non-Neutral Plasma Shock with Current, Electric and magnetic Field. The current and electric field is in axial (z) direction. The magnetic field produced by the shock current is in poloidal (θ) direction.

21. 20 Basic steady equations for shock front

22. 21 fluxes Jump conditions of particle, momentum in r and z direction, and energy fluxes

23. 22 Profiles of velocity and temperature

24. 23 Downstream profiles ━ 2D case, ┉ 1D case

25. 24 Shock current changes the

26. 25 Summary (I) The jump conditions across the shock will be changed that the particle flux and energy flux are no longer conservative. The profiles of plasma parameters in downstream are no longer as same as the profiles in upstream.

27. 26 Summary (II) The critical Mach number ( the minimum upstream velocity for the non-neutral shock emergency) is larger than its value (1.0) in neutral shock. When J 0 =0, The positive shock current (from up- to down-stream) will reduce the, and the negative current will increase the.

28. 27 Electrostatic instabilities Electrostatic instabilities in shock front (1D case)

29. 28 The linear instability analysis x -- the direction of the shock propagating, y – the direction perpendicular to shock propagating.

30. 29 A Example Dispersion relation about k x Disturbance is in the direction of x – the shock propagating direction

31. 30 labeled x = 0.1, 0.3, 0.5, 0.7, 0.9 as No.1, No.3, No.5, No.7, No.9 Profiles of equilibrium parameters

32. 31 The numerical results The high frequency ( ) approximation vs. low frequency ( ) approximation. With electric field (E≠ 0 ) vs. E = 0. With dissipation – viscous and friction -- (α≠0 ) vs. α=0. Adiabatic process approximation vs. diabatic (energy equation) case.

33. 32 The high frequency approximation

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37. 36 Adiabatic vs. Diabatic (E≠ 0)

38. 37 Summary The E can drive the electrostatic instability in the shock front in both parallel (k x ) and perpendicular (k y ) directions. The E and density gradient are the destabilizing factors and dissipation (the viscosity etc.) is the stabilizing factor. The increasing rate There is evidence of zero frequency instable mode, which is a kind of absolute instability.

39. 38 Prospect The electrostatic instability in the shock propagating direction is comparable with the Rayleigh-Taylor instability in neutral plasma shock. The electrostatic instability with zero frequency in the perpendicular shock direction is comparable with the Richtmyer-Meshkov instability in neutral plasma shock. These works are in process.

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