Dynamics of ions during development of parametric instability of langmuir waves

Nonlinear regimes of one-dimensional parametric instabilities of long-wave plasma waves are considered for the cases when the average field energy density is less (Zakharov’s model) or greater (Silin’s model) than the plasma thermal energy. The process of generation of short-wave plasma waves and pe...

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Datum:	2013
Hauptverfasser:	Belkin, E.V., Kirichok, A.V., Kuklin, V.M., Pryjmak, A.V., Zagorodny, A.G.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2013
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Нелинейные процессы в плазменных средах
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/112160
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Zitieren:	Dynamics of ions during development of parametric instability of langmuir waves / E.V. Belkin, A.V. Kirichok, V.M. Kuklin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 260-266. — Бібліогр.: 37 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1121602025-02-23T18:02:53Z Dynamics of ions during development of parametric instability of langmuir waves Динаміка іонів при розвитку параметричної нестійкості ленгмюрівських хвиль Динамика ионов при развитии параметрической неустойчивости ленгмюровских волн Belkin, E.V. Kirichok, A.V. Kuklin, V.M. Pryjmak, A.V. Zagorodny, A.G. Нелинейные процессы в плазменных средах Nonlinear regimes of one-dimensional parametric instabilities of long-wave plasma waves are considered for the cases when the average field energy density is less (Zakharov’s model) or greater (Silin’s model) than the plasma thermal energy. The process of generation of short-wave plasma waves and perturbations of ion density is found to be similar in both cases. It is shown that the ion energy after the instability is saturated proves to be of the order of the ratio of linear growth rate to the frequency in the case when the initial field energy exceeds the plasma thermal energy. In the opposite case of hot plasma, the ions acquire a part of initial field energy equal to the ratio of initial field energy to the plasma thermal energy. The trajectory crossing of ions in the vicinity of density cavities is a reason of instability quenching in both cases. Розглянуто нелінійні режими розвитку одновимірних параметричних нестійкостей довгохвильових ленгмюрівських хвиль у випадках, коли енергія поля менша (модель Захарова) і більша (модель Сіліна) за теплову енергію плазми. Процес генерації короткохвильового спектра плазмових хвиль і збурень іонної густини виявляється подібним в обох моделях опису параметричних нестійкостей. Показано, що енергія іонів при насиченні нестійкостей виявляється дорівнює за порядком відношенню лінійного інкремента до частоти у випадку, коли початкова енергія поля помітно перевищує теплову енергію плазми. В умовах гарячої плазми іонам передається частка енергії, що дорівнює половині відношення початкової енергії поля до теплової енергії плазми. Перетин траєкторій іонів поблизу каверн густини є причиною зриву нестійкості в обох випадках. Рассмотрены нелинейные режимы развития одномерных параметрических неустойчивостей длинноволновых ленгмюровских волн в случаях, когда энергия поля меньше (модель Захарова) и больше (модель Силина) тепловой энергии плазмы. Процесс генерации коротковолнового спектра плазменных волн и возмущений ионной плотности оказывается подобным в обеих моделях описания параметрических неустойчивостей. Показано, что энергия ионов при насыщении неустойчивостей оказывается порядка отношения линейного инкремента к частоте в случае, когда начальная энергия поля заметно превышает тепловую энергию плазмы. В условиях горячей плазмы ионам передается доля энергии, равная половине отношения начальной энергии поля к тепловой энергии плазмы. Пересечение траекторий ионов вблизи каверн плотности является причиной срыва неустойчивости в обоих случаях. 2013 Article Dynamics of ions during development of parametric instability of langmuir waves / E.V. Belkin, A.V. Kirichok, V.M. Kuklin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 260-266. — Бібліогр.: 37 назв. — англ. 1562-6016 PACS: 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/112160 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах
spellingShingle	Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Belkin, E.V. Kirichok, A.V. Kuklin, V.M. Pryjmak, A.V. Zagorodny, A.G. Dynamics of ions during development of parametric instability of langmuir waves Вопросы атомной науки и техники
description	Nonlinear regimes of one-dimensional parametric instabilities of long-wave plasma waves are considered for the cases when the average field energy density is less (Zakharov’s model) or greater (Silin’s model) than the plasma thermal energy. The process of generation of short-wave plasma waves and perturbations of ion density is found to be similar in both cases. It is shown that the ion energy after the instability is saturated proves to be of the order of the ratio of linear growth rate to the frequency in the case when the initial field energy exceeds the plasma thermal energy. In the opposite case of hot plasma, the ions acquire a part of initial field energy equal to the ratio of initial field energy to the plasma thermal energy. The trajectory crossing of ions in the vicinity of density cavities is a reason of instability quenching in both cases.
format	Article
author	Belkin, E.V. Kirichok, A.V. Kuklin, V.M. Pryjmak, A.V. Zagorodny, A.G.
author_facet	Belkin, E.V. Kirichok, A.V. Kuklin, V.M. Pryjmak, A.V. Zagorodny, A.G.
author_sort	Belkin, E.V.
title	Dynamics of ions during development of parametric instability of langmuir waves
title_short	Dynamics of ions during development of parametric instability of langmuir waves
title_full	Dynamics of ions during development of parametric instability of langmuir waves
title_fullStr	Dynamics of ions during development of parametric instability of langmuir waves
title_full_unstemmed	Dynamics of ions during development of parametric instability of langmuir waves
title_sort	dynamics of ions during development of parametric instability of langmuir waves
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2013
topic_facet	Нелинейные процессы в плазменных средах
url	https://nasplib.isofts.kiev.ua/handle/123456789/112160
citation_txt	Dynamics of ions during development of parametric instability of langmuir waves / E.V. Belkin, A.V. Kirichok, V.M. Kuklin, A.V. Pryjmak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2013. — № 4. — С. 260-266. — Бібліогр.: 37 назв. — англ.
series	Вопросы атомной науки и техники
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fulltext	ISSN 1562-6016. ВАНТ. 2013. №4(86) 260 DYNAMICS OF IONS DURING DEVELOPMENT OF PARAMETRIC INSTABILITY OF LANGMUIR WAVES E.V. Belkin, A.V. Kirichok, V.M. Kuklin, A.V. Pryjmak, A.G. Zagorodny** Kharkov National University, Institute for High Technologies, Kharkov, Ukraine; Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine E-mail: kuklinvm1@rambler.ru Nonlinear regimes of one-dimensional parametric instabilities of long-wave plasma waves are considered for the cases when the average field energy density is less (Zakharov’s model) or greater (Silin’s model) than the plasma thermal energy. The process of generation of short-wave plasma waves and perturbations of ion density is found to be similar in both cases. It is shown that the ion energy after the instability is saturated proves to be of the order of the ratio of linear growth rate to the frequency in the case when the initial field energy exceeds the plasma thermal energy. In the opposite case of hot plasma, the ions acquire a part of initial field energy equal to the ratio of initial field energy to the plasma thermal energy. The trajectory crossing of ions in the vicinity of density cavities is a rea- son of instability quenching in both cases. PACS: 52.35.Mw INTRODUCTION The interest in parametric instability of intense Langmuir waves, which can be easily excited in the plasma by various sources[1 - 9], was stipulated, in par- ticular, by the new possibilities in heating of electrons and ions in plasma. The correct methods for description of parametric instability of long-waveplasma waves was developed in the pioneering works of V.P. Silin [10] and V.E. Zakharov [11]. In early one-dimensional nu- merical experiments on parametric decay of plasma oscillations [12], the theoretical concepts were con- firmed [10] (see also [13, 14] and review [15]). How- ever, the greatest interest has been expressed by ex- perimenters in the mechanism of dissipation of wave energy discovered by V.E. Zakharov. The analytical studies, laboratory-based experiments and numerical simulations performed at an early stage of studying these phenomena have confirmed [16-18] the fact that in some cases a significant part of the pump field energy transfers during the instability development into the energy of short-wave Langmuir oscillations attended with bursts of fast particles [16 - 27]. In this paper, we attempt to compare the models of Silin and Zakharov by the example of one-dimensional description. The choice of one-dimensional approach, as was noted by J. Dawson [28], often keeps the main fea- tures of the processes, but simplify their description and leads to a fuller understanding of what the important phenomena are. The ion heating is of particular interest, so we use the kinetic description of ions in this work because of account of inertial effects can be significant at the nonlinear stage of the process [29]. 1. GENERALIZED SILIN’S EQUATIONS When the intensity of external long-wave field is much greater than the temperature of electrons in plas- ma 2 0 0\| \| /4 eW E n Tπ= >> , it is reasonable to explore the approach presented by V.P. Silin [30] 0 , v v e v u E v t x m x α α α α α α α ∂ ∂ ∂ + − = − ∂ ∂ ∂ (1) 0 0 ( ) ,n n v n vu n t x x x α α α α α α α ∂ ∂ ∂ ∂ + + = − ∂ ∂ ∂ ∂ (2) 4 .E e n x β β β π∂ = ∂ ∑ (3) Let set the wavelength of the external electric field infinite 0 0 0 0 0 (\| \| exp{ } \| \| exp{ }) / 2. E i E i t i E i t i ω φ ω φ = − + − − − − (4) The particles oscillates under the action of this field with velocity 0 0 0( \| \| / ) cosu e mα α α ω= − Ε ⋅ Φ . Substituting into Eq.(1) the electric field obtained from Eq.(3) 04 ( ) /n in enE ie n n k nπ= − − , we find 0 0 0 0 4n n т n m m m v e i u ik n v e n t k n m ik m v v α α α α β β βα α α π − ∂ + ⋅ ⋅ + ⋅ = ∂ ⋅ = − ⋅ ⋅ ∑ ∑ (5) Let use the following variables exp{ sin }n n ne n iaα α α αν = ⋅ ⋅ − ⋅ Φ , (6) exp{ sin }n n nv iaα α αθ = ⋅ − ⋅ Φ , (7) 2 0 0 0/na ne k E mα α α ω= ⋅ , (8) where 0tω φΦ = + . After this, Eqs.(1)-(2) take the form 0 0 0 n n n m m m ik n e n ik n t α α α α α α ν θ ν θ− ∂ + ⋅ ⋅ = − ⋅ ⋅ ⋅ ∂ ∑ (9) 0 0 4 exp{ ( )sin } . n n n n n m m m e i i a a t k n m ik m α α β β α βα α α θ π ν θ θ− ∂ + ⋅ − Φ = ∂ ⋅ = − ⋅ ⋅ ∑ ∑ (10) It is clear that 2 0 0 0( / )in en e na a n ek E m aω− ≈ ⋅ = , where, 2 04 /pe ee n mω π= , 2 04 /i e n MπΩ = , im M≡ and 0nk nk= defines a set of wave numbers. Equations (9)-(10) for electrons becomes 0 0 0 en en en m em m ik n en ik n t ν θ ν θ− ∂ − ⋅ ⋅ = − ⋅ ⋅ ⋅ ∂ ∑ (11) 0 0 4 ( exp{ sin }) . en en in n e en m em m ei ia t k n m ik m θ π ν ν θ θ− ∂ − + ⋅ ⋅ Φ = ∂ ⋅ = − ⋅ ⋅∑ (12) Then, we use ISSN 1562-6016. ВАНТ. 2013. №4(86) 261 0 0 0 0 ( ) (0) (1) 0 2 2( 1) (2) ( 2) exp{ ) i ts en n n n s i t i t i t n n n u is t u u e u e u e u e ω ω ω ω ν ω − −− − = ⋅ ⋅ = + ⋅ + + ⋅ + ⋅ + ⋅ ∑ (13) 0 0 0 0 ( ) (0) (1) 0 2 2( 1) (2) ( 2) exp{ ) i ts en n n n s i t i t i t n n n v is t v v e v e v e v e ω ω ω ω θ ω − −− − = ⋅ ⋅ = + ⋅ + + ⋅ + ⋅ + ⋅ ∑ (14) and well known expansion exp{ sin } ( ) exp{ }m m ia J a im ∞ =−∞ ⋅ Φ = ⋅ Φ∑ , (15) where ( )mJ x is the Bessel function, and 0 0( ) ( )J x J x= − , 1 1 1( ) ( ) ( )J x J x J x−= − − = − , 2 2 2( ) ( ) ( )J x J x J x−= = − [31]. Below, we find the non-resonance terms for perturba- tion of density (0) (2) ( 2), ,n n nu u u − and velocity (0) (2) ( 2), ,n n nv v v − in the oscillating reference frame [32 - 34]: (0) ( 1) ( 1) ( 1) (1)0 0 (1) ( 1) ( 1) (1) 0 ( )[ ) 1 [ ] , n n m m n m m m n k v n m v v v v v vv v i x x ω ω + − − − − − − = − ⋅ − ⋅ = ∂ ∂ = − ∂ ∂ ∑ (16) 2 2 (0) (1) ( 1)0 0 2 (1) ( 1) 0 2 ( ) 4 ( ) [ ] , 4 e n in n n m m m e in n n k n m u J a v v e m J a v v e x ν π ν π − − − ⋅ = − ⋅ + ⋅ ⋅ = ∂ = − ⋅ − ∂ ∑ (17) ( 2) 2 ( 1) ( 1)0 0 2 0 0 0 ( 1) 2 ( 1)0 2 0 0 0 2 ( ) 3 2 1( ) [ ] , 3 i n in n n m m m i in n n k v J a e mv v k n en vJ a e v k n en i x φ φ ω ν ω ω ν ω ± ± ± ± ± − ± ± ± ± ± = ⋅ ⋅ ⋅ ± ∂ = ⋅ ⋅ ∂ ∑m m (18) 2 ( 2) 2 ( 1) ( 1)0 0 2 2 ( 1) 2 ( 1)0 2 2 1 ( ) 3 1 ( ) [ ] . 3 i n in n s n s spe i in n n pe k n enu J a e sv v en vJ a e v x x φ φ ν ω ν ω ± ± ± ± ± − ± ± ± ± ⋅ = ⋅ − ⋅ ∂ ∂ = ⋅ + ∂ ∂ ∑ (19) The obtained equations should be supplemented by equations for resonant values 2( 1) 1( 1) 0 0 2 (0) ( 1) ( 1) (0) 0 0 (0) ( 1) ( 1) (0) 0 0 ( 1) ( 2) ( 2) ( 1) 0 0 0 ( ) 2 { } 2 [ ] ( ) [ ] [ [ ], i in pe nn n n m m n m m m n m m n m m m n m m n m m m m i J a eu i i u t k n en m v v v v ik n i u v u v k n en mv v u v φν ω ω ω ω ω ±± ±± ± ± − − ± ± − − ± ± − − ⋅∂ ± Δ ∂ = ⋅ ⋅ + ⋅ − − ⋅ ⋅ ± ⋅ + ⋅ + + ⋅ ± ⋅ ∑ ∑ ∑ ∑m m m m (20) where 2 2 0 0( ) / 2peω ω ωΔ = − . Authors of [32 - 34] have used the following representation ( 1) ( 1) ( 1) 0 0 0 0/ / 4n n nu k n en v ik n Eω π± ± ±= ± ⋅ = ⋅ . In this case, gathering in the r.h.s. of Eq.(20) the terms responsible for electronic non-linearity we rewrite this equation for short-wave perturbations as follows 2( 1) 1( 1) 0 0 2 ( 1) 2 ( 1)0 2 0 0 0 0 0 ( ) 2 { } 2 [ ( ) ( )] ( / ) . i pe nn n in iin m m n m m n m m J a eu i i u i t n u J a e u J a en m k n en I φ φ ω ω ν ω ω ν ω ±± ±± ± ±− ± − − ⋅∂ ± Δ ⋅ + ∂ ⋅ + = = ⋅ ⋅ ∑ m m m (21) Obviously, that the electronic non-linearity (r.h.s. of Eq. (21)) is equal to zero as well as in [35]. Than the Eq. (21) can be rewritten [32 - 34] as 2( 1) 1( 1) 0 ( 1) 20 2 0 ( 1) 0 ( ) 2 [ ( ) 2 ( )]. i pe nn n in iin m m n m m m n m J a eu i u i t n i u J a e en m u J a φ φ ω ν ω ω ν ±± ±± ±− ± − ± − ⋅∂ Δ ⋅ = ∂ ± ⋅ ⋅ + + ⋅ ∑ m m m (22) If the electric field will be presented in the form [35] 0 0 0 0 ( ) 0 (1) ( 1) 2 2(2) ( 2) exp{ ) 1 ( ) 2 , s n n n s i t i t n n i t i t n n E E is t E E e E e E e E e ω ω ω ω ω −− −− = ⋅ ⋅ = + + ⋅ + ⋅ + + ⋅ + ⋅ ∑ (23) than ( 1) ( 1) ( 1) 0/ 2 4 /n n nE E iu k nπ± ± ±→ = − , and the Eq.(22) takes the form ( 1) ( 1) 1 0 ( 1) 20 2 0 ( 1) 0 8 ( ) exp( ) 2 [ ( ) 2 ( )]. pe inn n n i in m m n m m m n m E i E J a i t k n i E J a e en E J a φ πω ν φ ω ν ± ± ± ± − ± − ± − ∂ Δ ⋅ ± = ∂ ± ⋅ ⋅ + + ⋅ ∑ m m m (24) Going to the representation of pumping wave 0E corresponding to fixed velocity of oscillations 0 0 0( / ) cosu e E mα α α ω= − ⋅ ⋅ Φ , we find 0 0E iE→ − and * 0 0E iE→ . The equation for 0E can be written as [32 - 34] , ( 1) 20 0 2 0 0 ( 1) 0 8 [ ( ) 2 ( )] i m i m m m m m E i u J a e t en k m u J a φνπ ω − − − + − ∂ = ⋅ ⋅ + ∂ + ⋅ ∑ (25) or expressing the perturbations of density through the components of electric field ( 1) 20 0 , 2 0 ( 1) 0 [ ( ) 2 ( )]. i i m m m m m m E E J a e t en E J a φω ν − − + ∂ = − ⋅ ⋅ + ∂ + ⋅ ∑ (26) The slowly varying in time electric field [30] 0 (1) ( 1) 1 1 2 (1) ( 1) 0 0 (1) (1) 2 ( 1) ( 1) 22 0 4( )( { exp{ sin } ) [ ( ) ( ) ] ( ) ( ) ( ) 1 [ ]}, n en n in i i n n n n n n m m m i in n m m n m m m iE ia k n u J a e u J a e n J a u u en n m m nJ a u u e u u e en m φ φ φ φ π ν ν − − − − − − − − − − = − 〈 ⋅ − ⋅ Φ 〉 + = = ⋅ + ⋅ ⋅ − ⋅ − − − − ⋅ ⋅ + ⋅ ∑ ∑ (27) can be represented in another form (1) ( 1) 1 1 (1) ( 1)0 0 0 (1) (1) 20 2 0 ( 1) ( 1) 2 1 [ ( ) ( ) ] 2 ( ) 16 ( ) ( )[ 16 ], i i n n n n n n n m m m in n m m m i n m m m E E J a e E J a e ink J a E E en ik J a n m E E e en E E e φ φ φ φ π π − − − − − − − − − − = ⋅ + ⋅ ⋅ − − ⋅ − − − ⋅ + + ⋅ ∑ ∑ (28) ISSN 1562-6016. ВАНТ. 2013. №4(86) 262 that permits description of ions by particle-in-cell me- thod. Their equations of motion can be written as fol- lows 2 02 exp{ }s n s n d x e E ik nx Mdt = ⋅∑ , (29) and the ion density can be defined as 0 0 / 0 0 0 0 0 / exp[ ( , )] . 2 k in s s k k n n ink x x t dx π ππ − = ⋅ − ⋅ ⋅∫ (30) Note that the use of particle-in-cell method for de- scription of ion dynamics allows, what is more, to im- prove the computational stability [29]. The use of Eqs. (9) - (10), where the right-hand sides can be neglected in view of their smallness, permits the hydro dynamical description of ions. The equation for ion density at this takes the form [32 - 34]: 2 2 2 2 0 22 (1) ( 1) 1 1 2 (1) ( 1) 0 0 (1) (1) 2 ( 1) ( 1) 22 0 2{ [1 ( ) ( )] 3 [ ( ) ( ) ] ( ) ( ) ( ) 1 [ ]}. in i in n n i i n n n n n n m m m i in n m m n m m m J a J a t u J a e u J a e n J a u u en n m m nJ a u u e u u e en m φ φ φ φ ν ν − − − − − − − − − − ∂ = −Ω − + + ∂ + ⋅ + ⋅ ⋅ − ⋅ − − − − ⋅ ⋅ + ⋅ ∑ ∑ (31) Or after neglecting small terms 2 (1) ( 1)0 1 12 2 2 (1) ( 1)0 02 0 2 (1) (1) 20 2 2 0 ( 1) ( 1) 2 [ ( ) ( ) ] 8 ( ) 64 ( ) ( )[ 64 ]}. i iin n n n n n n m m m in n m m m i n m m ik n E J a e E J a e t n k J a E E en nk J a n m E E e en E E e φ φ φ φ ν π π π − − − − − − − − − − ∂ = + ⋅ − ∂ + ⋅ ⋅ + + ⋅ − ⋅ + + ⋅ ∑ ∑ (32) One can verify that the complex conjugate of Eq.(3.24) with the lower sign becomes (the dummy in- dex in sums can be inverted m m→− ) ( 1) , 1( 1)* 0 * (1)* 20 , 2 0 ( 1)* 0 4 ( ) [( ( ) 2 ( )] 0. pe i n n in n i i m n m m n m m m n J aE i E e t k n i E J a e en E J a φ φ πω ν ω ν − −−− − − − − − − − − ∂ − Δ − ⋅ − ∂ − ⋅ ⋅ + + ⋅ = ∑ (33) At the same time this equation for positive indexes can be written as (1) (1) 1 0 ( 1) 20 2 0 (1) 0 4 ( ) ]} [ ( ) 2 ( )] 0. pe in in n n i in m m n m m m n m E i E J a e t k n i E J a e en E J a φ φ πω ν ω ν ± − − − − ∂ − Δ − ⋅ − ∂ − ⋅ ⋅ + + ⋅ = ∑ (34) а) It is clear that for ( 1) (1)( )n nE E− − = and , ,( )i n i nν ν− = Eqs. (3.31) and (3.32) are identical. Just as, it is easy to verify that ( 1) (1)( )n nE E− −= and , ,( )i n i nν ν −= , i.e. the perturbations of ion charges pos- sess the symmetry , ,( )i n i nn n− = . At this, for correct description of the instability it is sufficiently to use the high-frequency components (1) nE , (1) nE− and (1) 0E , as well as perturbations of ion charge ,i nν for positive values of index n . Other variables can be expressed through them. So, we can stop using the superscript. In this case we can rewrite Eqs. (3.23), (3.28) 1 0 20 2 0 0 4 ( ) e [ ( ) 2 ( )] 0, pe in in n n i in m m n m m m n m E i E J a t k n i E J a e en E J a φ φ πω ν ω ν − − − − ∂ − Δ − ⋅ − ∂ − ⋅ + + ⋅ = ∑ (35) 2 2 2 2 0 22 0 1 2 2 0 02 0 2 20 2 2 0 * * 2 2{ [1 ( ) ( )] 3 ( )[ ] 8 ( ) 64 ( ) ( )[ 64 ]}. in i in n n i i n n n n n m m m in n m m m i m n m J a J a t ik n J a E e E e n k J a E E en nk J a n m E E e en E E e φ φ φ φ ν ν π π π − − − − − − − − ∂ = −Ω − + ∂ + ⋅ − ⋅ − + ⋅ ⋅ + + ⋅ − ⋅ ⋅ + + ⋅ ∑ ∑ (36) In addition, when the particle-in-cell method is used, one can use the motion equation (30) and expression for ion density (31)with slowly varying electric field 2 2 0 2 0 * 1 0 0 0 20 2 0 * 2 4 2( ) [1 ( ) ( )] 3 1 ( )[ ] 2 ( ) 16 ( ) ( )[ 16 ]. n in n n i i n n n n n m m m in n m m m i m n m iE J a J a k n J a E e E e ink J a E E en ik J a n m E E e en E E e φ φ φ φ π ν π π − − − − − − − − = − − + + + ⋅ − ⋅ − − ⋅ − − − ⋅ ⋅ + + ⋅ ∑ ∑ (37) These equations should be supplemented by the equ- ation for pumping field 0E * 20 0 , 2 0 0 [ ( ) 2 ( )]. i i m m m m m m E E J a e t en E J a φω ν − − ∂ = − ⋅ ⋅ + ∂ + ⋅ ∑ (38) Note that the values corresponding to subscripts with different signs are independent that results in spatial distortion of integral perturbations not only owing to variation of amplitudes but also because of spatial dis- placement of different components of the wave packet. b) In the case , , ,( )i n i n i nn n n− = = , i.e. when the pertur- bations of ion density don’t change their location, the high-frequency electric field also remains spatially sym- metrical (1) (1) ( 1) ( 1)( ) ( )n n n nE E E E− − − −= = = . Then, the vari- ables (1) nE and ,i nn are sufficient for description of the process, that is stipulated by strong relation between val- ues with different signs. The structure of the field and density in this case, such as in the case of Zakharov’s model (that will be presented later), represents the mo- tionless spatial formation which amplitude increases and half-width decreases, at least in some region. c) Growing in time perturbations of ion density of the type , ,( )i n i nn n− = − are not realized. ISSN 1562-6016. ВАНТ. 2013. №4(86) 263 2. ZAKHAROV'S EQUATIONS When 1na << and 1( ) / 2n nJ a a≈ , 0 ( ) 1nJ a ≈ , 2 2 ( ) / 8n nJ a a≈ , the equations (35) - (38) are identical to equations derived in [35] under condition [37] 2 0 0\| \| /4 eW E n Tπ= << within the detuning 2 2 2 2 2 2 2 0 0 0 0 0( ) / 2 ( ) / 2pe pe Tek n vω ω ω ω ω ω− → − + and replacement 0 0E iE→− and * * 0 0E iE→ . 2 2 2 2 2 0 0 0 0 0 00 2 { } 0 2 pe Ten n in in m m m k n vE i E t i n E n E n ω ω ω ω − ≠ − +∂ − − ∂ − ⋅ + =∑ (39) 2 2 2 * * 0 0 02 0, { }. 16 in n n n m m m n n k n E E E E E E Mt π − − − ≠ ∂ = − + + ∂ ∑ (40) We also give the expression for slowly varying electric field with account of the pump wave 0 0 02 0, ( 4 ), n n n p n m m m n ik ne E E E E E m E E ω − − − ≠ = − + + + ∑ (41) that makes possible the description of ions with the use of particle-in-cell method and Eqs. (28) and (29). The amplitude of the pump wave 0E can be found from equation 0 0 , 0 0. 2 i m m m E i n E t n ω − ∂ − ⋅ = ∂ ∑ (42) 3. LINEAR THEORY We restrict our consideration to the most interesting case of the long-wave pumping. The dispersion equation for the high-temperature case in supersonic limit 2 2 2 2 2 0/in in sn n t k c n∂ ∂ >> follows from linear approxima- tion of Zakharov’s equations (2.32) and (2.33) with the use of representation 1 /E E t i− ∂ ∂ = Ω : 2 2 2{ } 0.A−Ω Ω − Δ + Δ ⋅ = (43) In Zakharov’s model, the normalized to the Lang- muir frequency correction / peδ ω= Ω , in general, should be written in the form 2 4 2 2 , 2 4 Bδ Δ Δ = ± + Δ (44) where 2 0 0 \| \|1 . 2 4 e e m E B M n Tπ = (45) Since the value 4 2 1/ 2 2( 4 )BΔ + Δ − Δ increases mono- tonically with Δ , having no a distinct maximum, the instability increment for small 2 BΔ << , 2 Bδ ≈ −Δ and 2\| \| Bδ < is equal 1/2 1/42 2 2 2 0 0 2 0 \| \|1Im \| \| . 2 42 Te e pe epe k n v E m n T M ω πω ⎛ ⎞ ⎛ ⎞ Ω = Ω ≈ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (46) For the case of large 2 BΔ >> , 2 BΩ ≈ − , it has the form 1/22 0 0 \| \|1Im \| \| . 2 4 e pe e E m n T M ω π ⎛ ⎞ Ω = Ω ≈ ⎜ ⎟ ⎝ ⎠ (47) This means that the increment increases with the wave-number of perturbations, reaching the maximum (47). In Silin's model, the growth rate normalized to the plasma frequency reaches the value 1/3 1/3 2/3 13 3 ( ) 2 2 e n mi iA J a M δ ⎛ ⎞= ± ⋅ = ± ⎜ ⎟ ⎝ ⎠ (48) for the detuning value 3 / 2AΔ = or which the same for ( )1/3 2/3 12 ( ) me nm M J aΔ = The perturbations with wave-number 0m mk k n= for which 1.84 mna = , the Bessel function has a maximum and the growth rate reaches the maximal value 1/3 max 0.44 em i M δ ⎛ ⎞= ± ⎜ ⎟ ⎝ ⎠ . (49) With the development of the instability, the pump wave amplitude decreases and the increment maximum moves to shorter wavelengths. It is significant that the values of the maximal growth rate of parametric instability increase with de- creasing of perturbation amplitudes. Moreover, Zak- harov’s model shows that decrease in the amplitude of the pump field results in decrease in the growth rates within the entire unstable region, while Silin’s model demonstrates that similar process shifts the maximum growth rate in the short-wave band, without reducing its value (49). Thus, the process of energy transfer to the short-wave part of the spectrum in the two models is largely determined by the linear mechanism of perturba- tion growth. 4. SIMULATION RESULTS Consider the case when ion density perturbations are spatially symmetric and 0nΦ = . In addition, there is no spatial shift between perturbations of different scales during instability development. It can be shown that the energy transfer from long-wave plasma wave to elec- trons and ions will be most effective in this case. For low initial amplitude values (low noise level), the main energy of the growing instability spectrum have been concentrated in short-wave region at the close of the linear stage of the process. The perturbations in this spectrum region have a maximum growth rate and have significantly kept ahead the neighbor modes during the linear stage of the instability development. The range of instability is found to berelativelynarrow. Therefore, severallong-livedsmall-scaledensity perturbations may arise on the length of the pumping wave. These cavities arise during all linear stage of the process due to phase synchronization of high-frequency modes. The phe- nomenon of phase synchronization of growing modes at the linear stage of the process was mentioned in earlier works [33]. For large initial mode amplitude RF spec- trum(i.e. at high levels of Langmuir noise), the number of density cavities arising on the length of the pumping wave decreases to only one or two [29]. Herewith, the ISSN 1562-6016. ВАНТ. 2013. №4(86) 264 spectra of the high-frequency field perturbations and ion density perturbations broaden. We used the following parameters for simulation. The number of simulated ion particles 0 2.500...5000s S< ≤ = , N n N− < < , 50...100N = , 2 0 0 0(0) (0) / 0.06e pea ek E m ω= = , 0 / 2s sk xξ π= , tτ δ= , 0 0/ \| \| 0s sd d vτ τξ τ = == = , 2 0 \| \| (0) / 0.0001...0.0025n e peek n e m ω = , 1Δ = , 3/ 10em M −= and 0/ 0.1eW n T = for Zakharov’s model. The ions acquire kinetic energy in the potential wells of the secaverns. At the nonlinear stage of the instabil- ity, the trajectories of ions cross each other, the ion den- sity perturbations are smoothed out and their amplitude increases. Relationship between ionic perturbations an- dhigh-frequency field is weakened and the instability saturates. The amplitude of the pumping wave is stabi- lized after some oscillation at a rather low level (Fig. 1). Fig. 1. Evolution of amplitude of the pumping wave (a – Zakharov's model; b – Silin's model) Fig. 2. Evolution of the value 2( / )ss d dtξ∑ (a – Zakharov's model; b – Silin's model) The main energy is now contained in the short-wave Langmuir spectrum. Some small part of the initial en- ergy is converted into kinetic energy of the ions (Fig. 2). The kinetic energy of ions positioned on a length of the pumping wave can be expressed through the esti- mated value of the sum of the squared dimensionless velocity ( )2 s s I d dξ τ= ∑ and the number of simulated particles S 2 2 2 0 0 2 0 00 42 1 2 2 2 sdx Mn n M I k dt kk S π δπ π⎛ ⎞⋅ 〈 〉 = ⋅⎜ ⎟ ⎝ ⎠ , (50) where 2( )sdx dt〈 〉 is the ensemble average. The ratio of the ion kinetic energy to the initial energy of intense long-wavelength Langmuir waves can be written as 2 2 0 0 0 0 2 2 2 0 0 2 2[ ] /{(\| \| /4 ) } 4 , s kin dx n M E k dt k I M W ma S π ππ π δ ⎛ ⎞〈 〉 ⋅ =⎜ ⎟ ⎝ ⎠ Ε = = (51) where kinΕ is the density of the ion kinetic energy, 2 0 0\| \| /4W E π= is the initial energy density of long- wavelength Langmuir waves. When the ion density per- turbations are spatially symmetric and 0nΦ = , the si- mulation shows that the maximum possible value 21.2 10sI −⋅ for 5000S = and 14,5 10sI −⋅ for 2.500S = are reached during the instability develop- ment for Silin’s and Zakharov’s models correspond- ingly. The ratio of time scales for these two models is equal to 1/6 1/ 2 01.6( / ) ( / ) 0.16e em M W n T = . Taking this into account, it is easy to see the energy of ions are of the same order in both models. The ratio of ion kinetic energy to the initial energy of long-wave oscillations occurs equal to 3 1/3 0/ 3.6 10 ( / )kin eW M m−Ε ⋅ for Silin’s model and 2 0 0 0/ 1.2 10 /kin eW W n T−Ε ⋅ for Zakharov’s model. This means that in Silin’s model the ions derive a portion of field energy of the order of maxδ . This effect was pre- dicted in [19] and confirmed in [29]. A portion of trans- ferred energy in Zakharov’s model is of the order of 0 0/ eW n T . Fig. 3. Evolution of speed distribution function half-width for simulated particles (a – Zakharov's model; b – Silin's model) In Zakharov’s model only a fraction of a percent of the initial energy can be transferred to the ions during development of the long-wave parametric instability of plasma waves. Nevertheless, the stabilization of the instability and its saturation are mostly determined by the trapping of ions to the potential wells of caverns in both models. During the trapping, the mixing of ions and the destruction of cavities occur that evidenced by the sharp increase in the ion density. If the speed distribution of ions was Maxwellian, the half-width of such distribution (Fig. 3) would have been associated with the thermal velocity by the relation 1,18 Tv v= . ISSN 1562-6016. ВАНТ. 2013. №4(86) 265 Fig. 4. Distribution function of the value 2( / )sd dtξ (a – Zakharov's model; b – Silin's model) However, the simulation shows that the half-width reaches at the nonlinear stage of the process the value of 0,005v = for Zakharov’s model and 0,006v = for Si- lin’s model. In addition, if the speed distribution of ions was Maxwellian, the value of 2 20.72M TI S v S v= ⋅ ≈ ⋅ , will be of the order of 0,22 for Zaharov’s model and 30,73 10−⋅ for Silin’s model, while the simulation gives the value two times greater in Zakharov’s model and an order greater in Silin’s model. 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The large amplitude Langmuir waves and particle acceleration in plasma micro- wave discharge // Sov. Plasma Physics. 1986, v. 12, № 5, p. 552-565. 37. V.M. Kuklin. Similarity of 1D Parametric Instability description of Langmuir waves // Journal of Kharkоv National University. Physical series «Nu- clei, Particles, Fields». 2013, № 1041, іss. 2(58), p. 20-30. Article received 16.04.2013 ДИНАМИКА ИОНОВ ПРИ РАЗВИТИИ ПАРАМЕТРИЧЕСКОЙ НЕУСТОЙЧИВОСТИ ЛЕНГМЮРОВСКИХ ВОЛН Е.В. Белкин, А.В. Киричок, В.М. Куклин, А.В. Приймак, А.Г. Загородний Рассмотрены нелинейные режимы развития одномерных параметрических неустойчивостей длинновол- новых ленгмюровских волн в случаях, когда энергия поля меньше (модель Захарова) и больше (модель Си- лина) тепловой энергии плазмы. Процесс генерации коротковолнового спектра плазменных волн и возму- щений ионной плотности оказывается подобным в обеих моделях описания параметрических неустойчиво- стей. Показано, что энергия ионов при насыщении неустойчивостей оказывается порядка отношения линей- ного инкремента к частоте в случае, когда начальная энергия поля заметно превышает тепловую энергию плазмы. В условиях горячей плазмы ионам передается доля энергии, равная половине отношения начальной энергии поля к тепловой энергии плазмы. Пересечение траекторий ионов вблизи каверн плотности является причиной срыва неустойчивости в обоих случаях. ДИНАМІКА ІОНІВ ПРИ РОЗВИТКУ ПАРАМЕТРИЧНОЇ НЕСТІЙКОСТІ ЛЕНГМЮРІВСЬКИХ ХВИЛЬ Є.В. Бєлкін, О.В. Киричок, В.М. Куклін, О.В. Приймак, О.Г. Загородній Розглянуто нелінійні режими розвитку одновимірних параметричних нестійкостей довгохвильових лен- гмюрівських хвиль у випадках, коли енергія поля менша (модель Захарова) і більша (модель Сіліна) за теп- лову енергію плазми. Процес генерації короткохвильового спектра плазмових хвиль і збурень іонної густи- ни виявляється подібним в обох моделях опису параметричних нестійкостей. Показано, що енергія іонів при насиченні нестійкостей виявляється дорівнює за порядком відношенню лінійного інкремента до частоти у випадку, коли початкова енергія поля помітно перевищує теплову енергію плазми. В умовах гарячої плазми іонам передається частка енергії, що дорівнює половині відношення початкової енергії поля до теплової ене- ргії плазми. Перетин траєкторій іонів поблизу каверн густини є причиною зриву нестійкості в обох випад- ках.

Dynamics of ions during development of parametric instability of langmuir waves

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