Bernstein-Greene-Kruskal (BGK) Modes in a Three Dimensional Unmagnetized Plasma

Transcription

1 Bernstein-Greene-Kruskal (BGK) Modes in a Three Dimensional Unmagnetized Plasma C. S. Ng & A. Bhattacharjee Space Science Center University of New Hampshire Space Science Center Seminar, March 9, 2005

2 What is a BGK mode? Outline 1-D BGK mode BGK mode observations 3-D BGK mode theory Laboratory experiments Numerical simulations Space-based observations 3-D BGK mode for B = 0 Conclusion

3 What is a BGK mode? An exact nonlinear solution of the Vlasov-Poisson system of equations

4 What is a BGK mode? Vlasov equation df s dt = 0 v Collisionless EM forces x

5 What is a BGK mode? df s dt = f s t + v f s r + q s m s Vlasov equation E + v c B f s v = 0

6 What is a BGK mode? df s dt = f s t + v f s r + q s m s Vlasov equation Special solution: f (C 1,C 2,...) E + v c B f s v = 0 C 1, C 1, are constants of motion

7 What is a BGK mode? Vlasov equation -- electrostatic f s t + v f s r q s m s f s v = 0 E =

8 What is a BGK mode? Vlasov-Poisson equation f s t + v f s r q s m s f s v = 0 E = E = 2 = 4 = 4q s dvf s s

9 What is a BGK mode? Vlasov-Poisson equation -- linear solution f = f 0 + f 1 2 = 4 dvf 1 f 1 t + v f 1 r + e m e f 0 v = 0 Pasma oscillation: Tonks & Langmuir (1929), Vlasov (1938), Bohm & Gross (1949) Collisionless damping: Landau (1946):

10 What is a BGK mode? Vlasov-Poisson equation -- steady state nonlinear solution f t = v f r + f v = 0 2 = dvf 1 Normalized; uniform ion background (for simplicity) 1-D solution: Bernstein, Greene & Kruskal (1957) Traveling solution by change of reference frame: f (x v 0 t,v v 0 ) v v No Landau damping x x

11 1-D BGK mode v f(x,v) x Construction method [BGK 1957] + d(x) f (x,v) dx v = 0 1st equation can be solved by: d 2 (x) = dvf (x,v) 1 dx 2 f = f (w) w = v 2 /2(x) Integral-differential equation: d 2 (x) dx 2 = 2 (x ) dwf (w) 2[w +(x)] 1 Can be solved by given f(w) or (x).

12 1-D BGK mode Bi-polar electric field single-humped potential No total net charge [Chen 2002]

13 1-D BGK mode single-humped potential Electrons can either be trapped, or can pass through. Distribution functions for trapped and passing electrons can be different. To support a positive potential, trapped electrons are less dense -- electron holes. [Chen 2002]

14 1-D BGK mode Example -- [Chen 2002] (x) = 0 exp(x 2 /2 2 ) f = 2 -w /2 2 exp(w)/ 2, for w > 0 1 2ln 4w + exp(w) 0 2 [ 1 erf( -w)], for w < 0

15 1-D BGK mode Physical meaning Electron velocity increases in the center, so electron density decreases d 2 (x) dx 2 = 2 (x ) dwf (w) 2[w +(x)] 1 d 2 (x) dx 2 = 2 dwf (w) 1 since w = v 2 /2(x) v (x )

16 BGK mode observations Laboratory experiments

17 BGK mode observations Laboratory experiments Recent experiments, e.g., [Danielson et al., 2004]

18 BGK mode observations Numerical simulations

19 BGK mode observations Numerical simulations

20 BGK mode observations Numerical simulations

21 BGK mode observations Numerical simulations

22 BGK mode observations Space-based observations

23 BGK mode observations

24 BGK mode observations Space-based observations

25 BGK mode observations

26 BGK mode observations Space-based observations

27 BGK mode observations Width-amplitude relation

28 BGK mode observations Width-amplitude relation

29 BGK mode observations Three dimensional features

30 BGK mode observations Three dimensional features Traveling direction E bipolar unipolar E unipolar E

31 BGK mode observations Three dimensional features

32 BGK mode observations

33 3-D BGK mode theory Solution for infinite B

34 3-D BGK mode theory Solution for infinite B Basic assumption: electrons moving along B only back to 1-D problem Electric potential: Steady state Vlasov equation: Poisson-Vlasov equation: [Chen & Parks 2002]

35 3-D BGK mode theory Solution for infinite B [Chen & Parks 2002]

36 3-D BGK mode theory Width-amplitude relation -- inequality [Chen & Parks 2002] for >>1, for << 1, 1/4 [Chen et al. 2005]

37 3-D BGK mode theory Extension to finite B? Solutions not yet found for finite B No solution for B = 0? [Chen 2002] Only true for f = f(w)

38 3-D BGK mode for B = 0 1-D: No solution if f only depends on w = d 2 (x) dx 2 = 2 dwf (w) 1 = 2 2[w +(x)] (x) 2-D: = 2 = 2 dwf (w) 1 - (x ) dwf (w) v D: = 2 = 4 For 2-D and 3-D, d d 0 v dwf (w)

39 3-D BGK mode for B = 0 solution dependent on angular momentum 3-D BGK solution for B = 0 not constructed since [BGK 1957]! For a spherical potential = (r) f = f (w,l) with l = v r is also a solution of the Vlasov equation Possible to support the electric field self-consistently --- satisfying the Vlasov-Poisson equation

40 3-D BGK mode for B = 0 f = g(r)(v r ) v 2 + r d dr Toy example Poisson equation 1 r d 2 (r ) dr 2 = g(r) for (r) = 0 exp(r 2 / 2 ) requires width-amplitude inequality 2 > similar to Chen s results - - -

41 3-D BGK mode for B = 0 still a particular choice: More general solution f (w,l) = 1 2 boundary condition as r : 0 ( ) 3/2 ew f 1 (l) f 1 1 f exp(v 2 /2)/( 2 ) 3/2 One possible choice: f 1 (v r) = 1 (1 h 0 )exp(v 2 r 2 / x 0 2 ) f 1 (0)= h 0 0 Slight preference against zero angular momentum

42 3-D BGK mode for B = 0 Poisson equation 1 r More general solution d 2 (r ) = e h(r)1 dr 2 h(r) = ( h 0 + 2r 2 2 / x ) ( r 2 2 / x ) 0 Boundary condition: (r ) 0 (r = 0) = 0 (r = 0) = 0 Existence of solution: solve 0 for give h 0 and x 0

43 3-D BGK mode for B = 0 Numerical solution (for 0 h 0 < 1) Electron hole (r ) / r 2 No total net charge Fig. 1 (a) Numerical solution (r) for the case with h 0 = 0.9 and x 0 = 1. (b) The same solution in log-log plot. The dashed line showing = x 0 2 (1 h0 )/2. (c) Radial electric field. (d) Normalized charge density 1 e h(r).

44 3-D BGK mode for B = 0 Numerical solution (for h 0 > 1) Negative potential, not possible in 1-D without ion dynamics Mathematically possible not sure how physical Fig. 2 (a) Numerical solution (r) for the case with h 0 = 1.1 and x 0 = 1. (b) The same solution in log-log plot. The dashed line showing = x 0 2 (1 h0 )/2. (c) Radial electric field. (d) Normalized charge density 1 e h(r).

45 3-D BGK mode for B = 0 Width-amplitude inequality Fig. 3 Plots of 0 vs. x 0 for var ious values of h 0, from 0.1 (top curve) to 0.9 (bottom curve) in an increment of 0.1. The dashed line is 0 = 8x 0 2 (1 h0 ) for h 0 = 0.1. [Chen et al. 2005]

46 3-D BGK mode for finite B Equation to be solved Cylindrically symmetric potential: = (,z) Vlasov solution: f = f (w,2v + B 2 ) Poisson eqn: z = d 3 v 2 f 2,2v + B 2 1 v 2 More difficult to solve, but may be more relevant to observations, may even be more stable.

47 3-D BGK mode Future works (i) dynamical accessibility: how can such a 3D BGK state be produced? (ii) instability: how unstable? can it be stabilized? (iii) collision: what happens if a weak collision is at work? (iv) solution with finite B (v) compare with observations

48 3-D BGK mode Conclusion Localized BGK solution does not exist in 2D/3D unmagnetized plasmas if the distribution function only depends on energy. Solutions possible in 3D if distribution function also depends on angular momentum. Examples of such 3D solutions constructed. Properties of these solutions analyzed, including the width-amplitude relation that is consistent with studies of other BGK solutions [e.g., Chen 2002]. Works are ongoing. We welcome comments and collaborations, especially from the observation side.