Диссертация (1150622), страница 18
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L., Baumjohann W., and Nakamura R. A wavy twisted neutral sheet observed byCLUSTER // Geophys.Res.Let.,29(19), 1899, (2002) doi:10.1029/2002GL015544103Ïðèëîæåíèå ýòîì ðàçäåëå ïðèâåäåíû ðàñ÷¼òíûå ôîðìóëû äëÿ ïàðàìåòðîâ òîêîâîãî ñëîÿ ïðè êîëåáàíèÿõ, íå âîøåäøèå â ãëàâó 2.Ïàðàìåòðû äëÿ ñëó÷àÿ íà÷àëüíîãî âîçìóùåíèÿ ïî ãàóññèàíó, èçãèáíàÿ ìîäà (èíäåêñ ”+ ”ñîîòâåòñòâóåò ðåøåíèþ âûøå ñëîÿ (z ≥ z0 ), à èíäåêñ ”− ” ðåøåíèþ íèæå ñëîÿ (z ≤ −z0 )):ξy (t, y, z) ==vz =vy =Bz =By =Bx =ξz± (t, y, z) =ξy± (t, y, z) =vz± (t, y, z) =vy± (t, y, z) =Bx± (t, y, z) =By± (t, y, z) =Bz± (t, y, z) =∫Reλ(k) √ − k2−[πe 4 sin[λ(k)z]]ei[ω(k)t−ky] dk =2πik∫ ∞1λ(k) − k2√e 4 cos(ω(k)t)sin(ky)sin(λz)dkkπ 0∫ ∞k21ω(k)e− 4 sin(ω(k)t)cos(ky)cos(λz)dk,−√π 0∫ ∞1λ(k)ω(k) − k2−√e 4 sin(ω(k)t)sin(ky)sin(λz)dk.kπ 0∫ ∞k2b−√λ(k)e− 4 cos(ω(k)t)cos(ky)sin(λz)dk;π 0∫ ∞ 2bλ (k) − k2√e 4 cos(ω(k)t)sin(ky)cos(λz)dk;kπ 0∫ ∞k2ae− 4 cos(ω(k)t)cos(ky)cos(λz)dk.−√π 0∫ ∞k21√ek− 4 ∓kz cos[λ(k)]cos(ky)cos[ω(k)t]dk;π 0∫ ∞k21∓√ek− 4 ∓kz cos[λ(k)]cos(ky)sin[ω(k)t]dk;π 0∫ ∞k21−√ek− 2 ∓kz cos[λ(k)]ω(k)cos(ky)sin[ω(k)t]dk;π 0∫ ∞k21∓√ek− 4 ∓kz cos[λ(k)]ω(k)cos(ky)cos[ω(k)t]dk;π 0−aξz± ;∫ ∞k2b√kek− 4 ∓kz cos[λ(k)]cos(ky)sin[ω(k)t]dk;π 0∫ ∞k2bkek− 4 ∓kz cos[λ(k)]cos(ky)cos[ω(k)t]dk,∓√π 0(56)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(67)(68)sausage-ìîäà:ξzξy∫ ∞k21= √e− 4 cos[ω(k)t]cos(ky)sin[λ(k)z]dk;π 0∫ ∞1λ(k) − k2= −√e 4 cos[ω(k)t]sin(ky)cos[λ(k)z]dk;kπ 0104vz =vy =Bz =By =Bx =ξz± (t, y, z) =ξy± (t, y, z) =vz± (t, y, z) =vy± (t, y, z) =Bx± (t, y, z) =By± (t, y, z) =Bz± (t, y, z) =∫ ∞k21−√ω(k)e− 4 sin[ω(k)t]cos(ky)sin[λ(k)z]dk;π 0∫ ∞1λ(k)ω(k) − k2√e 4 sin[ω(k)t]sin(ky)cos[λ(k)z]dk;kπ 0∫ ∞k2b−√λ(k)e− 4 cos[ω(k)t]cos(ky)cos[λ(k)z]dk;π 0∫ ∞ 2bλ (k) − k2√e 4 cos[ω(k)t]sin(ky)sin[λ(k)z]dk;kπ 0−aξz ;∫ ∞k21±√ek− 4 ∓kz sin[λ(k)]cos(ky)cos[ω(k)t]dk;π 0∫ ∞k21−√ek− 4 ∓kz sin[λ(k)]cos(ky)sin[ω(k)t]dk;π 0∫ ∞k21∓√ek− 2 ∓kz sin[λ(k)]ω(k)cos(ky)sin[ω(k)t]dk;π 0∫ ∞k21−√ek− 4 ∓kz sin[λ(k)]ω(k)cos(ky)cos[ω(k)t]dk;π 0−aξz± ;∫ ∞k2bkek− 4 ∓kz sin[λ(k)]cos(ky)sin[ω(k)t]dk;±√π 0∫ ∞k2b−p √kek− 4 ∓kz sin[λ(k)]cos(ky)cos[ω(k)t]dk,π 0(69)Ïàðàìåòðû äëÿ ñëó÷àÿ íå÷åòíîãî íà÷àëüíîãî âîçìóùåíèÿ sin(y)Ce−a(k+1)22 y2, kink-ìîäà:(k−1)2ξz0 = e− 4 − e− 4 ;∫ ∞1ξz = √ξ 0 sin[ω(k)t]cos(ky)cos[λ(k)z]dk;π 0 z∫ ∞1λ(k) 0ξy = √ξ sin[ω(k)t]sin(ky)sin[λ(k)z]dk;k zπ 0∫ ∞1vz = √ω(k)ξz0 cos[ω(k)t]cos(ky)cos[λ(k)z]dk;π 0∫ ∞1λ(k)ω(k) 0vy = √ξz cos[ω(k)t]sin(ky)sin[λ(k)z]dk;kπ 0∫ ∞bBz = − √λ(k)ξz0 sin[ω(k)t]cos(ky)sin[λ(k)z]dk;π 0∫ ∞ 2bλ (k) 0By = √ξ sin[ω(k)t]sin(ky)cos[λ(k)z]dk;k zπ 0Bx = −aξz ;(70)Ïàðàìåòðû äëÿ ñëó÷àÿ íå÷åòíîãî íà÷àëüíîãî âîçìóùåíèÿ sin(y)Ce−a1052 y2, sausage-ìîäà:ξz =ξy =vz =vy =Bz =By =Bx =∫ ∞1√ξ 0 sin[ω(k)t]cos(ky)sin[λ(k)z]dk;π 0 z∫ ∞1λ(k) 0−√ξ sin[ω(k)t]sin(ky)cos[λ(k)z]dk;k zπ 0∫ ∞1√ω(k)ξz0 cos[ω(k)t]cos(ky)sin[λ(k)z]dk;π 0∫ ∞1λ(k)ω(k) 0−√ξz cos[ω(k)t]sin(ky)cos[λ(k)z]dk;kπ 0∫ ∞b√λ(k)ξz0 sin[ω(k)t]cos(ky)cos[λ(k)z]dk;π 0∫ ∞ 2bλ (k) 0√ξ sin[ω(k)t]sin(ky)sin[λ(k)z]dk;k zπ 0−aξz ;(71)106.
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