29 MHD B144853 30 2 17 1
(MHD) ( ) MHD 2
1 5 1.1................. 5 1.2 MHD...................... 6 1.3....................... 11 1.4 Sweet-Parkaer............ 13 1.5 Petschek............... 15 1.6............. 16 1.7......................... 17 2 18 2.1.......................... 18 2.2.......................... 19 2.3............... 21 3 25 4 36 4.1............. 36 4.2.......... 38 4.3.......................... 39 5 41 3
1 [1]................... 5 2 [2]................. 5 3 Sweet-Parker [11].......... 13 4 Petschek [13]............. 16 5 [9]................... 17 6................. 22 7 ( )... 22 8 y = tanh(x/a)( a = 2 )....... 23 9 η = 2.0 10 3 ν = 5.0 10 4 25 10 η = 2.0 10 3 ν = 1.0 10 3 26 11 η = 2.0 10 3 ν = 4.0 10 3 27 12 η = 2.0 10 3 0.. 28 13 η = 2.0 10 3 ν = 5.0 10 4 29 14 η = 2.0 10 3 ν = 1.0 10 3 30 15 η = 2.0 10 3 ν = 4.0 10 3 31 16 η = 2.0 10 3 0.. 32 17.................. 34 18......... 34 19......... 34 20 ηj z............... 35 21 A zmax.............. 35 22 η = 2.0 10 3 ν = 5.0 10 4, 1.0 10 3, 4.0 10 3 x = 0 v y y.............. 37 23 ν = 4.0 10 3 E zmax......... 38 4
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
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
[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
(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)
( 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 + 1 3 ( 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
(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
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
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
(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
{ 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
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
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
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
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
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 2 + 16u i 1 30u i + 16u i+1 u i+2 12 x (86) 19
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 + 1 2 k 1) (89) k 3 = t g(t n + t 2 1 2 1 2, un k 1 + k 2 ) (90) 2 2 k 4 = t g(t n + t, u n 1 2 + 1 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 1 + 2 1 2 (k 2 k 1 ) (95) k 3 = t g(t n + t 2, u 2) (96) q 1 = (2 2)k 2 + ( 2 + 3 )k 1 (97) 2 u 2, q 1, k 3 3) u 3 = u 2 + 2 + 1 2 (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
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 2.3 6 7 7 2 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
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
(x, y) = (0, 0) (68) (69) δ η η L y L y = 50 η = 2.0 10 3 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 4.0 10 3 L x Sweet-Parker L δ δ/l 10 3 5 2L x = 100 x 1000 y 10000 0.01 η 4.0 10 3 23
η = 2.0 10 3 ν = 5.0 10 4, 1.0 10 3, 4.0 10 3 η 0 24
3 x y ( ) ( ) 9 ν = 5.0 10 4 10 ν = 1.0 10 3 11 ν = 4.0 10 3 12 0 η = 2.0 10 3 time=200 9: η = 2.0 10 3 ν = 5.0 10 4 ( ) 25
10: η = 2.0 10 3 ν = 1.0 10 3 ( ) 26
11: η = 2.0 10 3 ν = 4.0 10 3 ( ) 27
12: η = 2.0 10 3 0 ( ) 28
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 = 2.0 10 3 ν = 5.0 10 4 ( ) 29
14: η = 2.0 10 3 ν = 1.0 10 3 ( ) 30
15: η = 2.0 10 3 ν = 4.0 10 3 ( ) 31
16: η = 2.0 10 3 0 ( ) 32
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 )} + 0.1 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
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 = 5.0 10!# ν = 1.0 10!" ν = 4.0 10!" 速度常時 0 18: ν = 5.0 10!# ν = 1.0 10!" ν = 4.0 10!" 速度常時 0 19: 34
ν = 5.0 10!# ν = 1.0 10!" ν = 4.0 10!" 速度常時 0 20: ηj z ν = 5.0 10!# ν = 1.0 10!" ν = 4.0 10!" 速度常時 0 21: A zmax 35
4 4.1 9 12 12( 0) 0 time=100 Sweet-Parker time=200 9 11 Sweet- Parker 10 13 16 12( 0) 0 9 11 time=200 9 11 11 time time=100 200 x = 0 v y y ν = 5.0 10 4 ν = 1.0 10 3 ν = 4.0 10 3 (η = 2.0 10 3 ) 22 (x, y) = (0, 0) +y ν = 5.0 10 4 ν = 4.0 10 3 ν = 4.0 10 3 ν = 1.0 10 3 time 36
ν = 5.0 10!# ν = 1.0 10!" ν = 4.0 10!" 22: η = 2.0 10 3 ν = 5.0 10 4, 1.0 10 3, 4.0 10 3 x = 0 v y y (time ) 37
4.2 18 21 E k E m y = 0 z E zmax z A zmax time=200 time=100 200 18 21 0 18 time=150 ν = 4.0 10 3 19 time=100 ν = 4.0 10 3 20 time=170 time=200 ν = 4.0 10 3 ( 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 = 4.0 10 3 E zmax ( E z ) 38
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
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 = 5.0 10 4, 1.0 10 3, 4.0 10 3 3 ν = 5.0 10 4, 4.0 10 3 ν = 1.0 10 3 4 5 time=200 (20) ν = 5.0 10 4 time=200 time L x 50 100 40
5 MHD 41
[1] NASA s Goddard Space Flight Center/S. Wiessinger [2] GEOTILE (http://www.isas.jaxa. 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 (http://rspa.royalsocietypublishing.org/content/472/ 2196/20160479.figures-only) Figure1 [7] Fast Magnetic Reconnection(https://farside.ph.utexas.edu/ teaching/plasma/lectures1/node78.html)figure 27 [8] (http://www-space. 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.509-520 [13] H.E.Petschek Magnetic Field Annihilation NASA Special Publication,p.425, 1964. [14] Kulsrud, Russell M. Intuitive approach to magnetic reconnection Physics of Plasmas, Vol.18, Issue 11, article id:111201 p.6 (2011) 42
[15] Shibata, Kazunari; Tanuma, Syuniti Plasmoid-inducedreconnection and fractal reconnection Earth, Planets and Space, Vol.53, p. 473-482.(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, 112102 (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, 105004(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, 100706 (2015); [19] Takashi Minoshima, Takahiro Miyoshi, and Shinsuke Imada Boosting magnetic reconnection by viscosity and thermal conduction Physics of Plasmas vol.23, 072122 (2016) 43
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