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1 29 MHD B

2 (MHD) ( ) MHD 2

3 MHD Sweet-Parkaer Petschek

4 1 [1] [2] Sweet-Parker [11] Petschek [13] [9] ( ) y = tanh(x/a)( a = 2 ) η = ν = η = ν = η = ν = η = η = ν = η = ν = η = ν = η = ηj z A zmax η = ν = , , x = 0 v y y ν = E zmax

1 1.1 序論磁気リコネクションの重要性磁気リコネクションとは反平行磁場の系において磁力線がつなぎ変わる現象でありつなぎ変わる前までに蓄積された磁気エネルギーがプラズマの運動エネルギーや熱エネルギーに変換されることが知られている近年巨大な太陽フレアによる世界規模での電波障害などの脅威が認識されてきたが

が短時間で急速にプラズマのエネルギーに変換され地球方向その反対方向へプラズマの塊が飛び出すこのプラズマの塊をプラズモイド (磁気島) といい一連の物理過程をサブストームという図 2: オーロラ発生のプロセス [2] 図 1: 太陽フレア発生の瞬間 [1] 上記の例のように磁気リコネクションは天体爆発現象のエネルギー源であり

5 1 1.1 序論磁気リコネクションの重要性磁気リコネクションとは反平行磁場の系において磁力線がつなぎ変わる現象でありつなぎ変わる前までに蓄積された磁気エネルギーがプラズマの運動エネルギーや熱エネルギーに変換されることが知られている近年巨大な太陽フレアによる世界規模での電波障害などの脅威が認識されてきたがその太陽フレアの原因も磁気リコネクションであると考えられている (図 1) 他にも地球磁気圏と太陽風 (太陽から地球へ向かってくるプラズマの流れ) の相互作用の一つであるオーロラ爆発の起きるプロセス (図 2) にも磁気リコネクションが関わっている太陽風のエネルギーが地球磁気圏の夜側へ集まっていきやがて溜め込まれたエネルギーが短時間で急速にプラズマのエネルギーに変換され地球方向その反対方向へプラズマの塊が飛び出すこのプラズマの塊をプラズモイド (磁気島) といい一連の物理過程をサブストームという図 2: オーロラ発生のプロセス [2] 図 1: 太陽フレア発生の瞬間 [1] 上記の例のように磁気リコネクションは天体爆発現象のエネルギー源でありこのメカニズムを明らかにすることでプラズマと磁場が普遍的に存在している宇宙の天体物理現象の理解につながることが期待されるしかし宇宙プラズマの観測データの磁気リコネクション率と磁気流体力学 (MHD) 理論を用いたシミュレーション結果のそれが大きく異なっており未だに高い磁気リコネクション率の原因を解明できていない本章では宇宙プラズマの MHD 方程式と磁場の誘導方程式の導出磁気リコネクション研究の歴史的背景を述べる 5

6 1.2 MHD MHD m i m e n i n e ρ mi ρ me V i V e p i p e ρ mi (r, t) [3] ρ mi (r, t) = m i n i (r, t) (1) x y z ( ) ( ) = ( ) { (ni V x ) + (n iv y ) + (n } iv z ) x y z = n i x y z (2) x y z n i (r, t) + {n i (r, t)v i (r, t)} = 0 (3) (1) (3) n e (r, t) + {n e (r, t)v e (r, t)} = 0 (4) (3) (4) (continuity equation) E B n i m i dv i dt n e m e dv e dt = p i + Zen i (E + V i B) R (5) = p e en e (E + V e B) + R (6) R ν ei R = n e m e (V e V i )ν ei (7) 6

7 [3] dv (r, t) dt = V (r, t) (5) (6) { } Vi n i m i + (V i )V i + {V (r, t) }V (r, t) (8) = p i +Zen i (E + V i B) R (9) { } Ve n e m e + (V e )V e = p e en e (E + V e B) + R (10) ρ m V ρ j (9) + (10) ρ m = n e m e + n i m i (11) V = n em e V e + n i m i V i ρ m (12) ρ = en e + Zen i (13) j = en e V e + Zen i V i (14) ρ m V + n em e (V e )V e + n i m i (V i )V i = (p e + p i ) + ρe + j B (15) A ( ) ( ) = 1836A m e m i 0 (16) ρ m V ( ρ m = n i m i 1 + m ) e Z n i m i (17) m i V = V i + m e m i Z(V e V i ) V i (18) 7

8 (15) ( ) = ρ m V + n im i { me V ρ m + n im i (V i )V i { } V ρ m + (V )V m i Z(V e )V e + (V i )V i } (19) p = p i + p e (15) { } V ρ m + (V )V = p + ρe + j B (20) [3] (18) (14) V e = V i n e e Zn i e j V j (21) en e en e (10) en e (21) m e m i p e + E + (V j ) B + R = 0 en e en e en e E + V B j B p e + R = 0 (22) en e en e en e (7) R = n e m e (V e V i )ν ei = n e e m eν ei n e e 2 ( en e)(v e V i ) = n e eηj (23) η def = m eν ei n e e 2 (24) (20) j B p e = ρ m dv (r, t) dt 8 + p i (25)

9 ( ρ 0 ) ( ) (22) (25) (22) E + V B = ηj (26) (ohm) [3] B = µ 0 j (27) E = B (28) B = 0 (29) (27) 0 dv ρ m = p + j B + ν { 2 V + 13 } dt ( V ) (30) E + V B = ηj (31) B = µ 0 j (32) E = B (33) B = 0 (34) (41) ν { 2 V ( V )} MHD (34) (32) (33) B = (V B) (ηj) = (V B) η j = (V B) η µ 0 ( B) (35) ( B) = ( B) 2 B (36) (35) B = (V B) + η µ 0 2 B (37) 9

10 (37) (31) (33) ρ m dv dt = p + 1 µ 0 {( B) B} (38) ( B) B = B( B) ( B)B + (B )B (B )B (39) ρ m dv dt = p + 1 µ 0 {(B )B (B )B} = (p + B2 2µ 0 ) + 1 µ 0 (B )B (40) ρ m dv dt = (p + B2 ) + 1 (B )B 2µ 0 µ 0 + ν { 2 V + 13 } ( V ) (41) B = (V B) + η µ 0 2 B (42) ρ m + (ρ m V ) = 0 (43) B = 0 (44) p + (V ) + γp( V ) = 0 (45) (45) γ [3] 10

11 1.3 (37) ( ) ( ) = V B η µ 0 ( 2 B) V B/L (B/L 2 )(η/µ 0 ) = µ 0V L η def = R m (46) R m V B L R m ( )τ R τ H R m = τ R τ H = µ 0L 2 /η L/V A (47) V A V ( ) R m 1) R m 1 0 (37) 2) R m 1 0 (37) B = η µ 0 2 B (48) B = (V B) (49) (V B) = V ( B) ( B)V + (B )V (V )B (50) B + B( V ) (B )V = 0 (51) 11

12 S ϕ B z n ϕ = B n S = B x y (52) S d( x) dt = d(x + x x) dt = V x (x + x) V x (x) = V x x (53) x y ( S) = ( V x x + V y ) S (54) y S ϕ ( ϕ) = (B S) = B ( S) S + B { } B = + B( V ) (B )V S z = 0 (55) R m η 0 ( ) [4] MHD (41) (34) (32) j = B µ 0 L (56) 12

13 (41) p ρ m V A T = B2 µ 0 L V A L T = B2 µ 0 ρ m VA 2 = B2 µ 0 ρ m B V A = (57) µ0 ρ m B 2 /2µ 0 ρ m ( ) 1.4 Sweet-Parkaer Sweet(1958)[11] Parker(1957)[12] Sweet-Parker +x +z +y x = 0 Sweet-Parker ( 3 ) ±z ( ) ±x ( ) 3: Sweet-Parker [11] x z y B = E (58) 13

14 { E = E y z, E z x E } x z, E y x = (0, 0, 0) (59) E y η η = 0 E + V B = ηj (60) (η = 0) E y = v out B z v in B x = const. v in B x + const. = v out B z (61) const. j B ( ) (60) B = µ 0 j (62) E = V B + η µ 0 B (63) j y [5] v in B x η µ 0 B x δ (64) 2δ [5] 2L Lv in = δv out (65) v in v in = η v out µ 0 L (66) 14

15 v out ( )v A M M = v in v out = M = v in v out (67) η µ 0 1 v A L = S 1/2 η (68) S 10 14, 10 7 Sweet-Parker [5] M (65) M = v in v out = δ L (69) L/δ 1.5 Petschek Sweet-Parker Petschek [13] Sweet-Parker L Sweet-Parker [5] 4 X ( ) v in L = v out δ (70) [8] Sweet-Parker Petschek 15

16 4: Petschek [13] Kulsrud(2001,2011)[14] Petschek 1.6 Sweet-Parker ( ) [5] Sweet-Parker 5 3 (X ) ( 5) Shibata,Tanuma(2001)[15] ( Bhattacharjee, et al., 2009[16], Samtaney, et al., 2009[17], Shibayama,et al., 2015[18] ) 16

17 5: [9] 1.7 Minoshima,Miyoshi,Imada(2016)[19] MHD (fully compressible visco-resistive MHD equations) L δ = L (v in ν )( v in ) (71) v out MHD ν Sweet-Parker Petschek MHD 17

18 2 2.1 MHD V + (V )V = j B + ν 2 V (72) B = (V B ηj) (73) j = B (74) MHD MHD p E m β = p E m (75) (72) 0 ν 2 V V = 0 (76) (72) ( ) (73) (31) (33) (74) (73) (74) B = (V B) + η 2 B (77) R m (72) η η 18

19 2.2 MHD (72) (73) (74) 4 f(x + k) x = k f(x + k) = f(x) + kf (1) (x) + k2 2! f (2) (x) + k3 3! f (3) (x) + k4 4! f (4) (x) + (78) f(x k) f(x + 2k) f(x 2k) f(x k) = f(x) kf (1) (x) + k2 2! f (2) (x) k3 3! f (3) (x) + k4 4! f (4) (x) + (79) f(x + 2k) = f(x) + 2kf (1) (x) + (2k)2 f (2) (x) 2! + (2k)3 3! f (3) (x) + (2k)4 f (4) (x) + (80) 4! f(x 2k) = f(x) 2kf (1) (x) + (2k)2 f (2) (x) 2! (2k)3 3! f(x + k) f(x k) 2k f(x + 2k) f(x 2k) 4k f (3) (x) + (2k)4 f (4) (x) + (81) 4! = f (1) (x) + k2 3! f (3) (82) = f (1) (x) + 4k2 3! f (3) (83) f (1) (x) = 1 {f(x 2k) 8f(x k) + 8f(x + k) f(x + 2k)} (84) 12k k = 1 f(x k) = u i k u x = u i+1 u i x = u i 2 8u i 1 + 8u i+1 u i+2 12 x (85) 2 u x 2 = u i u i 1 30u i + 16u i+1 u i+2 12 x (86) 19

20 Runge-Kutta-Gill [10] u n+1 = u n + t {k 1 + (2 2)k 2 + (2 + } 2)k 3 + k 4 6 du = 1 {k 1 + (2 2)k 2 + (2 + } 2)k 3 + k 4 (87) dt 6 k 1 = t g(t n, u n ) (88) k 2 = t g(t n + 1 2, un k 1) (89) k 3 = t g(t n + t , un k 1 + k 2 ) (90) 2 2 k 4 = t g(t n + t, u n k 2 + k 3 ) (91) 2 2 g(t, u) = du dt t n Runge-Kutta-Gill 1) k 1 = t g(t n, u n ) (92) u 1 = u 0 + k 1 (93) 2 k 2 = t g(t n + t 2, u 1) (94) 3 u 1, k 1, k 2 2) u 2 = u (k 2 k 1 ) (95) k 3 = t g(t n + t 2, u 2) (96) q 1 = (2 2)k 2 + ( )k 1 (97) 2 u 2, q 1, k 3 3) u 3 = u (k 3 q 1 ) (98) k 4 = t g(t n + t 2, u 3) (99) q 2 = (2 + 2)k 3 + ( 2 3 )q 1 (100) 2 20

21 u 3, q 2, k 4 4) u 3, q 2, k 4 3 u n+1 ( u 4 ) u 4 = u 3 + k 4 6 q 2 3 (101) (85) 4 u i+1 u i, u i 1, u i+2, u i 2 4 Runge-Kutta-Gill x-y z 7 x, y x y ( 7 ) A z A z A z = ln{exp(y) + exp( y)} ln{exp(y max ) + exp( y max )} (102) B B x B y 2 z 0 B = A = ( A z y, A z x, B z) = (tanh(y), 0, B z ) (103) B z B z = 1 cosh(y) (104) 0 B z (x, y) = (0, 0) z 21

22 6: 7: ( ) B z A B V A z 0.1 exp ( x 2 y 2) 6 L x = 50 L y = 10 Sweet-Parker ( ) 22

23 (x, y) = (0, 0) (68) (69) δ η η L y L y = 50 η = L y B B = (tanh(y), 0, 1 cosh(y) ) (105) B x a y = tanh(x/a) 8: y = tanh(x/a)( a = 2 ) 8 a a = 1 η L x Sweet-Parker L δ δ/l L x = 100 x 1000 y η

24 η = ν = , , η 0 24

25 3 x y ( ) ( ) 9 ν = ν = ν = η = time=200 9: η = ν = ( ) 25

26 10: η = ν = ( ) 26

27 11: η = ν = ( ) 27

28 12: η = ( ) 28

29 x y ( ) ( ) z Ω z Ω z = ( V ) z = v y x v x y = v j 2 8v j 1 + 8v j+1 v j+2 12 x v i 2 8v i 1 + 8v i+1 v i+2 12 y (106) 13: η = ν = ( ) 29

30 14: η = ν = ( ) 30

31 15: η = ν = ( ) 31

32 16: η = ( ) 32

33 E k E m y = 0 z E zmax y = 0 z A zmax E zmax (74) ηj z = η( B y x B x y ) (107) (31) V B 0( V B ) y = 0 ηj z E zmax A zmax A z = ln{exp(y) + exp( y)} ln{exp(y max ) + exp( y max )} exp ( x 2 y 2) (108) y = 0 A zmax A zmax (103) B x = A z y (109) ψ 17 B x ψ = B x y = Ly A z (L y ) = 0 y A z y dy = A z (L y ) A z (0) (110) A z = ψ (111) 33

34 面積 ΔS を通過する磁場 B の束直線 Δy を通過する B! の束 17: ν = !# ν = !" ν = !" 速度常時 0 18: ν = !# ν = !" ν = !" 速度常時 0 19: 34

35 ν = !# ν = !" ν = !" 速度常時 0 20: ηj z ν = !# ν = !" ν = !" 速度常時 0 21: A zmax 35

36 ( 0) 0 time=100 Sweet-Parker time= Sweet- Parker ( 0) time= time time= x = 0 v y y ν = ν = ν = (η = ) 22 (x, y) = (0, 0) +y ν = ν = ν = ν = time 36

37 ν = !# ν = !" ν = !" 22: η = ν = , , x = 0 v y y (time ) 37

38 E k E m y = 0 z E zmax z A zmax time=200 time= time=150 ν = time=100 ν = time=170 time=200 ν = ( 23) E!"#$ はじめから存在する電流シート分割されて生じた電流シート実際に描かれたグラフ time 23: ν = E zmax ( E z ) 38

39 21 A zmax (111) A zmax = ψ (112) A zmax 4.3 p η j η j 2 MHD V + (V )V = j B + ν 2 V (113) B = (V B ηj) (114) j = B (115) (113) V { } V V + (V )V = V (j B) + V (ν 2 V ) (116) (v2 ) = V {V ( V )} V + V (j B) 2 ] + ν [ 2 ( v2 2 ) V 2 V 2 + ν[ {( V )V (V )V }] (117) (114) (115) B = (V B) + η 2 B (118) 39

40 B B B = B { (V B)} + B η 2 B (119) { } (B2 2 ) = B 2 V (B V )B V (j B) + ηj 2 + η (j B) (120) (117) (120) ν = , , ν = , ν = time=200 (20) ν = time=200 time L x

41 5 MHD 41

42 [1] NASA s Goddard Space Flight Center/S. Wiessinger [2] GEOTILE ( jp/j/forefront/2010/miyashita/02.shtml) [3] (2004) [4] (2001) MHD [5] (2015) [6] PROCEEDINGS OF THE ROYAL SOCIETY A MATHE- MATICAL,PHYSICAL AND ENGINEERING SCIENCES ( 2196/ figures-only) Figure1 [7] Fast Magnetic Reconnection( teaching/plasma/lectures1/node78.html)figure 27 [8] ( eps.s.u-tokyo.ac.jp/group/yokoyama-lab/thesis/2010ug_ matsui.pdf) [9] K. Shibata and S. Tanuma, Earth Planets Space 53, 473 (2001). [10] K s (Runge-Kutta-Gill method) (http: //kapapa.web.fc2.com/kadai26.htm) [11] P. A. Sweet The Neutral Point Theory of Solar Flares Electromagnetic Phenomena in Cosmical Physics (Cambridge University Press, 1958), Vol. 6, p. 123 [12] E.N.Parker Sweet s mechanism for merging magnetic fields in conducting fluids Journal of geophysical reserch Vol.62, p [13] H.E.Petschek Magnetic Field Annihilation NASA Special Publication,p.425, [14] Kulsrud, Russell M. Intuitive approach to magnetic reconnection Physics of Plasmas, Vol.18, Issue 11, article id: p.6 (2011) 42

43 [15] Shibata, Kazunari; Tanuma, Syuniti Plasmoid-inducedreconnection and fractal reconnection Earth, Planets and Space, Vol.53, p (2001) [16] A. Bhattacharjee1, Yi-Min Huang1, H. Yang2, and B. Rogers2 Fast reconnection in high-lundquist-number plasmas due to the plasmoid Instability Physics of Plasmas vol.16, (2009) [17] R. Samtaney, N. F. Loureiro, D. A. Uzdensky, A. A. Schekochihin, and S. C. Cowley Formation of Plasmoid Chains in Magnetic Reconnection Phys. Rev. Lett. vol.103, (2009) [18] Takuya Shibayama, Kanya Kusano, Takahiro Miyoshi, Takashi Nakabou, and Grigory Vekstein Fast magnetic reconnection supported by sporadic small-scale Petschek-type shocks Physics of Plasmas vol.22, (2015); [19] Takashi Minoshima, Takahiro Miyoshi, and Shinsuke Imada Boosting magnetic reconnection by viscosity and thermal conduction Physics of Plasmas vol.23, (2016) 43

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n 003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........