Summer College on Plasma Physics August Mirror instability in space plasmas: solions and cnoidal waves

Transcription

1 05-41 Summer College on Plasma Physics 10-8 August 009 Mirror instability in space plasmas: solions and cnoidal waes Oleg A. Pokhotelo UIPE, RAS Russia

2 Mirror instability in space plasmas: solions and cnoidal waes O. A. Pokhotelo Institute of Physics of the Earth, Moscow, Russia

3 Co-authors: Ya. N. N. Istomin Lebede Physical Institute, Moscow, Russia Russia M. M. A. A. Balikhin ACSE, ACSE, Uniersity of of Sheffield, UK UK

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5 MI linear growth rate for the arbitrary distribution function Field line bending FLR effect Inerse response γ L 3 L 1 π D βk β 1 L=A-E- and D= W 0 mp k k β β = + F β ( ) W e qj μb0 Fj / W j = F / W j j res W W0 dw k ρ i Instabilty threshold

6 Notations - arbitrary elocity distribution function Instability threshold L 1 = A + β Anisotropy A ( q ) j μ B0 F j / W j p q F / W i i j j j 1 F j p = μ B j 0 p W p i i

7 Notations Bi-Maxwellean distribution A nisotropy A E T = T 1 (bi-m axw ell) ( T / T 1) T e T ( 1 + T e / T ) T hreshold L = ( T T ) T e ( 1 + e / ) T / 1 1 = 1 T T T T β Mirror force Electrostatic force

8 Mirror waes are found in: Mirror waes are found in: Ring-current plasma Earth s magnetosheath Planetary magnetosheaths Cometary comas Wake of Io Solar wind Laboratory plasmas (teta-pinches)

9 Solar-Terrestrial Interactions Solar-Terrestrial Interactions

10 CLUSTER MISSION CLUSTER MISSION

11 Cluster satellites Cluster satellites

12 Galileo

13 Vega

14 Themis 007/07/11 (a) B (nt) TH C B (nt) (b) TH C n i (cm 3 ) TH D B (nt) (c) TH D n i (cm 3 ) 01:56:00 01:56:30 01:57:00 01:57:30 01:58:00 01:58:30 01:59:00 01:59:30 0:00:00

15 THEMIS obserations

16 Perpendicular plasma pressure balance δ p B δ B 3 B δ B + ρ 0 z 0 i μ 0 μ 0 β β B 0δ B = 1 + z μ 0 z z = res + ad + hd δ δ δ δ p p p p

17 Flattening of the elocity distribution function

18 NL effects associated with trapped particles NL effects associated with trapped particles * / 3 / 1 * F is F and where lim = = + = T T T T res e n b F k d d t b m p π ν ν π δ ν 0 when response (Landau) linear standard the has one small is amplitude when arctan 1 i.e., When * 1/ 1 / / * lim >> = = << << res T res T T res T p k t b T T k p p b k d e t b k nm p b T δ ν π δ ν π π δ ν

19 Adiabatic part of the pressure peturbation 1/ ad 3p T 1 b δ p = 1/ k T T b t Contrary to the linear limit in the NL regim e the expansion p arameter ω /k is now replaced T * by ω /k = ω /k T b 1/

20 Quasi-Hydrodynamic part 1/ 15 T hd 1/ δp = Apb+ Ap b b 8 T Mirror force Effect of flattening of the ion distribution function

21 NL MI dispersion relation near the saturated state res Near saturate state δ p 0 and k β β 3 [ β A 1 (1 + ) k i 1/ 1/ T 1/ (3/) T β b b b 1/ T T b t 15 ] = 0 k T 8 k ρ

22 The model equation (in the dimensionless form) 1/ h 1 h h = ξ ξ τ

23 NL growth rate 1/ 1/4 1/4 T 1/ 1/ T b k T 3 γ = [ A 1 k NL β ρ 3 β 1/ β β 15 T (1 + ) b ] 8 T 1/ 1/ i

24 Maximum growth rate The maximum growth is attained at 15 T = 9 8 T k ρi β A 1 1/ β 15 T 1/ = b 7 8 T (1 ) + 1/ k ρi β A 1 β β b 1/ γ NL ω ci 1/ 3/ 1/4 3/4 b T 15 T 1/ βa 1 b β β 1/ 1/ T 8 T + = 93 β (1 )

25 General case 1 b 1 arctan b 1/ t b π λ 1/ b + = t ( 1/ ) L b b

26 MI growth rate (K=0.1, h=-0.005)

27 MI growth rate (K=0.5, h=0.05)

28 MI growth rate (K=0.8, h=-0.05)

29 Basic nonlinear equations δ p 1 3 ρ b p 4 (1 ) i + 1 b + 0 β + b β β β = 1 + s = μ μ p B fd d f f μ B f + = t s m s b 0

30 Reduction of the kinetic equation New ariables ξ = k. ξ = k s / ω t f. f sin ξ f + ξ = 0. t ξ τ ξ w here τ = (m /k μ B b ) If b=const then. κ = τ ξ + sin ξ = 1 / 0 const Trapped particles - κ > 1 U ntrapped particles - κ < 1

31 Particles with different magnetic moments can be decomposed into distinct populations i. μ>μ, where 4m γ / B k b. This is 1 0 the adiabatic case. ii. μ<μ, Linear approximation is applied. 1

32 Adiabatic inariants If b co n st Trapped particles 1 1 [ E ( κ ) (1 κ ) K ( κ )] ν = = τ U ntrapped particles E ( κ ) σ = = const κτ const

33 Distribution function Trapped particles 1 f ( ) = [ f 0 (4 σ / π k ) + f 0 ( 4 σ / π k )] Untrapped particles f ( ) = f ( ) = f (4 ν / π k ) Plasm a pressure μ 1 B p = p 0 + μ μ δ ( μ, ) B 0 0 B d f d Adiabatic contribution Nonadiabatic contribution

34 NL pressure ariation 1 / δ p b b T π γ = b + (1 ) Φ ( α ) p T k 0 w here α = μ B / T = 8 γ / k b and Φ 1 0 α α α = - e e - e α α T T

35 Nonlinear eolution The transition from the linear to NL regime arises when μ T / B. Then NL eolution has explosie behaior. 1 The explosie dependence is arrested and then the system gradually approaches the saturated state

36 Temporal eolution

37 Stationary state

38 Saturated state Φ and α δ p b = b + p α d b 3 3 = b + b dx 5 General solution - cnoidal wae 1/ 1 3 b=b + ( b3 b) cn [ x, κ] or KdV soliton 3/5 b=- cosh ( x / ) ( b b )

39 Total plasma pressure δ δ δ δ p total disp p = p + p δ p total i = 1 + δ p 0 3 ρ 4(1 + b ) 3 B 0 ρ i 4 μ (1 + b ) b δ p 1 b + b + p0 β total 0

40 Nonzero electron temperature effects δ p + δ p 1 3 ρ b i e i + 1 b + p 0 β 4 (1 + b) β β β = 1 + s p i = B μ fd μ d f f μ B b e Ψ f t s m s m s = b 0 δ p e eψ = nt 0 eexp Te

41 Nonlinear pendulum. f f sin( ξ ) f + ξ =. t ξ τ ξ m τ = e Ψ k μ B0 b 1 + bμ B 0 0 1/ Nonlinear functions Φ ( α) = 1 ( α + 1) e 1 Φ ( α ) = 1 e Quasi neutrality condition a α α T T Φ α Φ + Φ π γ 1/ T ( / ) 1( ) 1 ( ) e ( T 1( α) Te ( α) 1 = T 1 + ( Te / T ) Φ ( α) ( T + TeΦ( α)) k T 1 B n = n + B d μ d δ f B μ where πγ ( ) F δ f = ( μ B0b + eψ ) 1 k m 1/ B T π γ ( ) n = n0 1 b[ Φ 1( α ) + a Φ ( α )] 1 B0 T k T where 4γ m eψ α = a and = ab k T b T μ1 0 i 0 0 B p i = p + B μdμ δf( μ, ) d B bi-maxwellian distribution B T p p bp a 1/ 0 0 i = [ i i Φ 3( α) + Φ1( α)] 1 B0 T k T Here Φ α = α e αe e α α α 3( ) π γ

42 Linear approximation 1/ eψ T T ( ) e T T T+ Te π γ = a = Tb T Te+ T ( Te+ T) k T δ p T ( T / T 1) T e = b 1 p i T T(1 + Te / T) (1 + Te/ T ) + (1 + T / T ) T + b (1 + / ) 1/ e π Te T T k T γ

43 Saturated state δ p e = b + p i T b β + β e / 5 0 1/ β + β e / 3 = cosh x where x = r / ρ [9 / 0(1+ β + β / 5)] i T b W ith the growth of the electron temperature the depth of the magnetic hole aries from 60 to 75 percent of the original magnetic field e 1/

44 Variation of total pressure in the presence of NET δ p p a b bt = (1 + ) b+ [ Φ 0 3( α) + aφ1( α)] i T

45 Conclusions The main nonlinear mechanism responsible for mirror instability saturation is associated with modification of the shape of the background ion distribution function in the region of small parallel particle elocities. The MI initially produces a depression in the magnetic field that has an explosie behaior as it was predicted before. In the course of NL saturation the mirror mode eolution ends with formation of cnoidal waes or solitons.

46 Nonzero electron temperature (NET) effects The MI threshold is enhanced in the presence of NET effect The transition from the linear to nonlinear regime occurs for smaller amplitudes NET effect results in a weak decrease of the spatial dimensions of the holes and in the decrease of hole depth

47 Thank you!