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Dynamical stabilization of low frequency microinstabilities
Volume 27A, number 10
PHYSICS LETTERS
2. R.A. Ferrell, Phys. Rev. 111 (1958) 1214. 3. E.A. Stern, in Optical properties and electronic structure of metals and alloys, ed. F.Abeles (NorthHolland, Amsterdam, 1966) p. 396. 4. E.A. Stern and R. A. Ferrell, Phys. Rev. 120 (1960) 130. 5. H.J. Howe J r . a,~d R. H. Ritchie, Bull. Am. Phys. Soc. 10 (1965) 433. 6. G.E.Jones, L.S. Cram and E.T.Arakawa, Phys. Rev. 147 (1966) 515.
7 October 1968
7. L.S. Cram andE.T.Arakawa, Phys. Rev. 153 (1967) 455; J. C. Ashley, L.S. Cram and E. T. Arakawa, Phys. Rev. 160 (1967) 313. 8. R.E.Wflems andR.H.Ritchie, Phys. Rev. Letters 19 (1967) 1325. 9. P. Dobberstein, A. Hampe and G. Sauerbrey, 2nd Intern. Conf. on Vacuum ultraviolet radiation physics, Gatlinburg, Tennessee, May 1968. 10. U.Btirker and W. Steinmann, Phys. Rev. Letters 12 (1968) 143.
* * * * *
DYNAMICAL
STABILIZATION
OF
LOW
FREQUENCY
MICROINSTABILITIES
M. COTSAFTIS Association Euratom-Cea D$partement de la Physique du Plasma et de la Fusion ContrOl$e, Centre d'Etudes Nuclkaires, 92 Fontenay-aux-Roses, France Received 12 August 1968
Stability conditions for flute and universal microinstabilities are given when the magnetic field is periodically time dependant.
In c l a s s i c a l m e c h a n i c s , it i s p o s s i b l e to s t a b i l i z e an u n s t a b l e s t a t i o n a r y s y s t e m by allowing a p a r a m e t e r , say the potential, to v a r y in t i m e . In p l a s m a p h y s i c s , c a l c u l a t i o n s have mainly been done f o r M. H . D . p l a s m a s with o s c i l l a t i n g h.f. m a g n e t i c or e l e c t r i c f i e l d s , through the a v e r a g i n g method, showing s t a b i l i z a t i o n e f f e c t s . A p p r o a c h of the m i c r o i n s t a b i l i t i e s d y n a m i c a l s t a b i l i z a t i o n , though new, could be s t i m u l a t e d by the hope to l o w e r the l e v e l of the uptiU now u n c o n t r o l l e d fluctuations leading to a l a r g e diffusion of the p a r t i c l e s . Only the effect of an h.f. o s c i l l a t i n g f i el d has been p r e s e n t l y studied [1,2]. We s h al l g i v e h e r e and a n a l y z e the d i s p e r s i o n r e l a t i o n s for low f r e q u e n c y p e r t u r b a t i o n s in a t i m e p e r i o d i c m a g n e t i c f i e l d without r e s t r i c t i n g i t s f r e q u e n c y COo. In a slab infinite g e o m e t r y , let: B = B o ( t ) e z be the m a g n e t i c f i e l d * [1], g = g(t)" e x the g r a v i t y of a y - i n d e p e n d a n t s y s t e m ; without p o te n t ia l along z, t h e r e e x i s t two constants of the motion: P y = v~ + +~¢~2(t), P z = v z , with: ~2(t) = e B o ( t ) / m c . C o n f in e m en t along x is obtained when wo b el o n g s to a given set of i n t e r v a l s . A n a l y s i s cf the V l a s o v - P o i s s o n s e t of equations f o r e l e c t r o s t a t i c p e r t u r b a t i o n s , shows that p e r f o r m ing the i n t e g r a l along the u n p e r t u r b e d o r b i t s n e e d s knowing t h em . H o w e v e r f o r flute and u n i v e r s a l m i c r o i n s t a b i l i t i e s due to t h e i r f r e q u e n c y domain, this difficulty can be avoided. U s e of a s y m p t o t i c t h e o r e m s , giving the u s u al r e s u l t when f~o = const., shows that only the v a l u e at the end point of the half past t r a j e c t o r y (t' = t) is n e e d e d f o r ions, w h e r e a s a s i m p l i f i e d t r a j e c t o r y along the f i e l d l i n es i s sufficient f o r the e l e c t r o n s . So, with the p e r t u r b e d e l e c t r o s t a t i c potenUal: ~/(r, t) ~ e x p i { w t + k y +~tz} ~o(x, t), one gets in the l o cal a p p r o x i m a t i o n the equations: 1
* A sheared field 8 = B o ( t ) ( e z + X e y / L s ) can also been used. 662
Volume 27A, number 10
(k2+~2)go =~21(t--~
PHYSICS LETTERS
l[
-1+
7October 1968
1
w + kvD(t) - o i
go +
(u)
+~t~ie(t~l l.go+[w_kvD(t) --~t] i~r.½ e-~jdv z exp[-~eV2] fexp[i~tVzT] go(x,t+T)d~-f for flute and u n i v e r s a l m i c r o i n s t a b i l i t i e s , r e s p e c t i v e l y , with ~Dj(I) the Debye length, VDj = VDj + - (kg/f~j)VDj(t) the d r i f t v e l o c i t y , Moi = M o { e x p i k [ y ( t + T ) - y ( t ) ] } the mean o v e r the s t a t i c ionic c y c l o t r o n
f r e q u e n c y . F o r both equations, a g e n e r a l solution: go(x, t) = ~ n don(x) exp [inwot ] can be t r i e d , leading to an infinite d e t e r m i n a n t . It i s known that t h e r e m u s t be a s t a b i l i t y domain, but it i s i m p o r t a n t h e r e to d e t e r m i n e what p a r t of t h e / e - s p e c t r u m can be s t a b i l i z e d . To p r o c e e d roughly, let us r e s t r i c t the d e t e r m i n a n t s to 3 × 3 o n e s * . A s s u m i n g T i = T e = T, M i = const., k D = const., kgX2 = 0, vD(t) --= VDo + ~ - - - + 1 VDe exp [iEwot ] + VD2E exp (2i~wot], w o > hVthe, one g e t s f o r u n i v e r s a l m i c r o i n s t a b i l i t i e s the d i s p e r s i o n r e l a t i o n : A3(U) :
U+--f(U) kVthe
V 2 - M 2 (I+k2OD+2OD. 2
: 2AU-B
:0
where: U = (2 -Mi)~o - k V D Mi, ~ = w / t ° o , VD" = VD" ~tooR - , ~ = kRo, R o r a d i u s of the p l a s m a f = to(to - k ~ D n ) and: a = i V D + l V D _ l , B = Mzfl , ~ - VD.lVD+ 2 + ~D+I~D_2 . W h en a - e-- 0 , one g e t s the u s u a I d i s p e r s i o n r e l a t i o n , giving an u n s t a b l e solution. It r e m a i n s u n s t a b l e when B = 0. When B ¢ 0, one has the s t a b i l i t y conditions: -
-
(l+k2VD+2VD-2)½ where: ~b+ = u n s t a b l e . So one has a l s o ~D = const.,
<
ilk3 1 +k2(OD+2OD_ 2 +OD+lOD_I )
< q~±kVDo
(C~)
1, ~ - = 2 (1 -M~)/Mi,:~ = s g n B , which shows that only the p a r t ]e < kc of the k - s p e c t r u m i s if kc ~< 1, one can e x p e c t c o m p l e t e s t a b i l i t y by boundary conditions. F o r the o t h e r r o o t s , s t a b i l i t y u n d e r s i m i l a r conditions. F o r flute m i c r o i n s t a b i l i t y , with M i = const., T e / T i << 1, tog = k g / ~ i = togO + togl c ° S t o o t ' one g e t s A3(F ) =- F [ F 2 - ot - 2 too 2 - ~t, ~ 2 ~j - ~ottoglto ~o2 G2 = 0
w h e r e T i F / T e i s the r i g h t - h a n d - s i d e of the u s u a l s t a t i c d i s p e r s i o n r e l a t i o n , a = 1 + k2k 2 - Mi, ~ = to + + togo' and G = G(~) a n o t h e r function equal to 0 in the s t a t i c e a s e . A s i m p l e sufficient s t a b i l i t y condition is: T e / T i < tog
with tog = ~ g l / t o g o '
~ g + ~D 2 + k2~2(~g+~D)2
(CF)
~D = ~D 1/~TD0, showing that now only the p a r t k > kc i s u n s t a b l e . If kc ~> 1, which
connects the p r e v i o u s c a s e , one can expect to s t a b i l i z e the m o s t d a n g e r o u s flute m i c r o i n s t a b i l i t y . F o r the o t h e r r o o t s , one g e t s a weak condition independant of k. So the r e s u l t f r o m (CF)(C~) i s that flute m i c r o i n s t a b i l i t y and u n i v e r s a l m i c r o i n s t a b i l i t y a r e both s t a b i l i z e d by a p e r i o d i c m a g n e t i c field, the e n e r g y of the o s c i l l a t i n g p a r t of which being f u r t h e r m o r e s m a l l c o m p a r e d to the m e a n one, which i s of i n t e r e s t f o r e x p e r i m e n t s and a s t r o p h y s i c a l a p p l i c a t i o n s . * This is the first harmonic approximation.
RejCe~'ences 1. A.B.Mikhailovskii and V. P. Sidorov, Soy. Phys. Techn. Phys. 12 (1968) 1192. 2. Ya.B. Fainberg and V.D. Shapiro, Sov. Phys. JETP 25 (1967) 189. 663
PHYSICS LETTERS
2. R.A. Ferrell, Phys. Rev. 111 (1958) 1214. 3. E.A. Stern, in Optical properties and electronic structure of metals and alloys, ed. F.Abeles (NorthHolland, Amsterdam, 1966) p. 396. 4. E.A. Stern and R. A. Ferrell, Phys. Rev. 120 (1960) 130. 5. H.J. Howe J r . a,~d R. H. Ritchie, Bull. Am. Phys. Soc. 10 (1965) 433. 6. G.E.Jones, L.S. Cram and E.T.Arakawa, Phys. Rev. 147 (1966) 515.
7 October 1968
7. L.S. Cram andE.T.Arakawa, Phys. Rev. 153 (1967) 455; J. C. Ashley, L.S. Cram and E. T. Arakawa, Phys. Rev. 160 (1967) 313. 8. R.E.Wflems andR.H.Ritchie, Phys. Rev. Letters 19 (1967) 1325. 9. P. Dobberstein, A. Hampe and G. Sauerbrey, 2nd Intern. Conf. on Vacuum ultraviolet radiation physics, Gatlinburg, Tennessee, May 1968. 10. U.Btirker and W. Steinmann, Phys. Rev. Letters 12 (1968) 143.
* * * * *
DYNAMICAL
STABILIZATION
OF
LOW
FREQUENCY
MICROINSTABILITIES
M. COTSAFTIS Association Euratom-Cea D$partement de la Physique du Plasma et de la Fusion ContrOl$e, Centre d'Etudes Nuclkaires, 92 Fontenay-aux-Roses, France Received 12 August 1968
Stability conditions for flute and universal microinstabilities are given when the magnetic field is periodically time dependant.
In c l a s s i c a l m e c h a n i c s , it i s p o s s i b l e to s t a b i l i z e an u n s t a b l e s t a t i o n a r y s y s t e m by allowing a p a r a m e t e r , say the potential, to v a r y in t i m e . In p l a s m a p h y s i c s , c a l c u l a t i o n s have mainly been done f o r M. H . D . p l a s m a s with o s c i l l a t i n g h.f. m a g n e t i c or e l e c t r i c f i e l d s , through the a v e r a g i n g method, showing s t a b i l i z a t i o n e f f e c t s . A p p r o a c h of the m i c r o i n s t a b i l i t i e s d y n a m i c a l s t a b i l i z a t i o n , though new, could be s t i m u l a t e d by the hope to l o w e r the l e v e l of the uptiU now u n c o n t r o l l e d fluctuations leading to a l a r g e diffusion of the p a r t i c l e s . Only the effect of an h.f. o s c i l l a t i n g f i el d has been p r e s e n t l y studied [1,2]. We s h al l g i v e h e r e and a n a l y z e the d i s p e r s i o n r e l a t i o n s for low f r e q u e n c y p e r t u r b a t i o n s in a t i m e p e r i o d i c m a g n e t i c f i e l d without r e s t r i c t i n g i t s f r e q u e n c y COo. In a slab infinite g e o m e t r y , let: B = B o ( t ) e z be the m a g n e t i c f i e l d * [1], g = g(t)" e x the g r a v i t y of a y - i n d e p e n d a n t s y s t e m ; without p o te n t ia l along z, t h e r e e x i s t two constants of the motion: P y = v~ + +~¢~2(t), P z = v z , with: ~2(t) = e B o ( t ) / m c . C o n f in e m en t along x is obtained when wo b el o n g s to a given set of i n t e r v a l s . A n a l y s i s cf the V l a s o v - P o i s s o n s e t of equations f o r e l e c t r o s t a t i c p e r t u r b a t i o n s , shows that p e r f o r m ing the i n t e g r a l along the u n p e r t u r b e d o r b i t s n e e d s knowing t h em . H o w e v e r f o r flute and u n i v e r s a l m i c r o i n s t a b i l i t i e s due to t h e i r f r e q u e n c y domain, this difficulty can be avoided. U s e of a s y m p t o t i c t h e o r e m s , giving the u s u al r e s u l t when f~o = const., shows that only the v a l u e at the end point of the half past t r a j e c t o r y (t' = t) is n e e d e d f o r ions, w h e r e a s a s i m p l i f i e d t r a j e c t o r y along the f i e l d l i n es i s sufficient f o r the e l e c t r o n s . So, with the p e r t u r b e d e l e c t r o s t a t i c potenUal: ~/(r, t) ~ e x p i { w t + k y +~tz} ~o(x, t), one gets in the l o cal a p p r o x i m a t i o n the equations: 1
* A sheared field 8 = B o ( t ) ( e z + X e y / L s ) can also been used. 662
Volume 27A, number 10
(k2+~2)go =~21(t--~
PHYSICS LETTERS
l[
-1+
7October 1968
1
w + kvD(t) - o i
go +
(u)
+~t~ie(t~l l.go+[w_kvD(t) --~t] i~r.½ e-~jdv z exp[-~eV2] fexp[i~tVzT] go(x,t+T)d~-f for flute and u n i v e r s a l m i c r o i n s t a b i l i t i e s , r e s p e c t i v e l y , with ~Dj(I) the Debye length, VDj = VDj + - (kg/f~j)VDj(t) the d r i f t v e l o c i t y , Moi = M o { e x p i k [ y ( t + T ) - y ( t ) ] } the mean o v e r the s t a t i c ionic c y c l o t r o n
f r e q u e n c y . F o r both equations, a g e n e r a l solution: go(x, t) = ~ n don(x) exp [inwot ] can be t r i e d , leading to an infinite d e t e r m i n a n t . It i s known that t h e r e m u s t be a s t a b i l i t y domain, but it i s i m p o r t a n t h e r e to d e t e r m i n e what p a r t of t h e / e - s p e c t r u m can be s t a b i l i z e d . To p r o c e e d roughly, let us r e s t r i c t the d e t e r m i n a n t s to 3 × 3 o n e s * . A s s u m i n g T i = T e = T, M i = const., k D = const., kgX2 = 0, vD(t) --= VDo + ~ - - - + 1 VDe exp [iEwot ] + VD2E exp (2i~wot], w o > hVthe, one g e t s f o r u n i v e r s a l m i c r o i n s t a b i l i t i e s the d i s p e r s i o n r e l a t i o n : A3(U) :
U+--f(U) kVthe
V 2 - M 2 (I+k2OD+2OD. 2
: 2AU-B
:0
where: U = (2 -Mi)~o - k V D Mi, ~ = w / t ° o , VD" = VD" ~tooR - , ~ = kRo, R o r a d i u s of the p l a s m a f = to(to - k ~ D n ) and: a = i V D + l V D _ l , B = Mzfl , ~ - VD.lVD+ 2 + ~D+I~D_2 . W h en a - e-- 0 , one g e t s the u s u a I d i s p e r s i o n r e l a t i o n , giving an u n s t a b l e solution. It r e m a i n s u n s t a b l e when B = 0. When B ¢ 0, one has the s t a b i l i t y conditions: -
-
(l+k2VD+2VD-2)½ where: ~b+ = u n s t a b l e . So one has a l s o ~D = const.,
<
ilk3 1 +k2(OD+2OD_ 2 +OD+lOD_I )
< q~±kVDo
(C~)
1, ~ - = 2 (1 -M~)/Mi,:~ = s g n B , which shows that only the p a r t ]e < kc of the k - s p e c t r u m i s if kc ~< 1, one can e x p e c t c o m p l e t e s t a b i l i t y by boundary conditions. F o r the o t h e r r o o t s , s t a b i l i t y u n d e r s i m i l a r conditions. F o r flute m i c r o i n s t a b i l i t y , with M i = const., T e / T i << 1, tog = k g / ~ i = togO + togl c ° S t o o t ' one g e t s A3(F ) =- F [ F 2 - ot - 2 too 2 - ~t, ~ 2 ~j - ~ottoglto ~o2 G2 = 0
w h e r e T i F / T e i s the r i g h t - h a n d - s i d e of the u s u a l s t a t i c d i s p e r s i o n r e l a t i o n , a = 1 + k2k 2 - Mi, ~ = to + + togo' and G = G(~) a n o t h e r function equal to 0 in the s t a t i c e a s e . A s i m p l e sufficient s t a b i l i t y condition is: T e / T i < tog
with tog = ~ g l / t o g o '
~ g + ~D 2 + k2~2(~g+~D)2
(CF)
~D = ~D 1/~TD0, showing that now only the p a r t k > kc i s u n s t a b l e . If kc ~> 1, which
connects the p r e v i o u s c a s e , one can expect to s t a b i l i z e the m o s t d a n g e r o u s flute m i c r o i n s t a b i l i t y . F o r the o t h e r r o o t s , one g e t s a weak condition independant of k. So the r e s u l t f r o m (CF)(C~) i s that flute m i c r o i n s t a b i l i t y and u n i v e r s a l m i c r o i n s t a b i l i t y a r e both s t a b i l i z e d by a p e r i o d i c m a g n e t i c field, the e n e r g y of the o s c i l l a t i n g p a r t of which being f u r t h e r m o r e s m a l l c o m p a r e d to the m e a n one, which i s of i n t e r e s t f o r e x p e r i m e n t s and a s t r o p h y s i c a l a p p l i c a t i o n s . * This is the first harmonic approximation.
RejCe~'ences 1. A.B.Mikhailovskii and V. P. Sidorov, Soy. Phys. Techn. Phys. 12 (1968) 1192. 2. Ya.B. Fainberg and V.D. Shapiro, Sov. Phys. JETP 25 (1967) 189. 663