Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields

Properties and excitation of the vortical turbulence, excited in a cylindrical radially inhomogeneous plasma in crossed radial electric and longitudinal magnetic fields, are considered. The dispersion relation, which allows to determine the range of parameters for which the vortical turbulence is su...

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Published in:	Вопросы атомной науки и техники
Date:	2015
Main Authors:	Maslov, V.I., Yarovaya, I.P., Yegorov, A.M., Yuferov, V.B.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2015
Subjects:	Плазменная электроника
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/82113
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Cite this:	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields / V.I. Maslov, I.P. Yarovaya, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 131-133. — Бібліогр.: 6 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Maslov, V.I. Yarovaya, I.P. Yegorov, A.M. Yuferov, V.B.
author_facet	Maslov, V.I. Yarovaya, I.P. Yegorov, A.M. Yuferov, V.B.
citation_txt	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields / V.I. Maslov, I.P. Yarovaya, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 131-133. — Бібліогр.: 6 назв. — англ.
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container_title	Вопросы атомной науки и техники
description	Properties and excitation of the vortical turbulence, excited in a cylindrical radially inhomogeneous plasma in crossed radial electric and longitudinal magnetic fields, are considered. The dispersion relation, which allows to determine the range of parameters for which the vortical turbulence is suppressed, is derived from the general nonlinear equations for the vorticity. Рассматриваются свойства и возбуждение вихревой турбулентности, возбуждаемой в цилиндрической радиально-неоднородной плазме в скрещенных радиальном электрическом и продольном магнитном полях. Из общего нелинейного уравнения для завихренности получено дисперсионное уравнение, позволяющее определить диапазон параметров, для которых вихревая турбулентность подавляется. Розглядаються властивості і збудження вихрової турбулентності, яка збуджується в циліндричній радіально-неоднорідній плазмі в схрещених радіальному електричному і поздовжньому магнітному полях. Із загального нелінійного рівняння для завихренності отримано дисперсійне рівняння, що дозволяє визначити діапазон параметрів, для яких вихрова турбулентність пригнічується.
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fulltext	ISSN 1562-6016. ВАНТ. 2015. №1(95) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 131-133. 131 SUPPRESSION OF EXCITED VORTICAL TURBULENCE IN INHOMOGENEOUS PLASMA IN CROSSED RADIAL ELECTRICAL AND LONGITUDINAL MAGNETIC FIELDS V.I. Maslov, I.P. Yarovaya, A.M. Yegorov, V.B. Yuferov NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine E-mail: vmaslov@kipt.kharkov.ua Properties and excitation of the vortical turbulence, excited in a cylindrical radially inhomogeneous plasma in crossed radial electric and longitudinal magnetic fields, are considered. The dispersion relation, which allows to determine the range of parameters for which the vortical turbulence is suppressed, is derived from the general non- linear equations for the vorticity. PACS: 29.17.+w; 41.75.Lx INTRODUCTION It is well known from numerous numerical simulations (see, for example, [1]) and from experiments (see, for example [2]) that electron density nonuniformity in kind of discrete vortices are long- living structures. In experiments [2] a rapid re- organization of discrete electron density nonuniformity has been observed in the spatial distribution of vorticity in pure electron plasma when a discrete vortex has been immersed in an extended distribution of the background vorticity. In plasma lens [3-5] for ion beam focusing a small-scale turbulence has been excited in crossed radial electrical and longitudinal magnetic fields by unremovable gradient of external magnetic field. This turbulence is a distributed vorticity. In this paper the properties and excitation of similar vortical turbulence, excited in cylindrical radially inhomogeneous plasma in crossed radial electrical 0rE and longitudinal magnetic 0H fields [6], is investigated theoretically. From general nonlinear equation, presented in article [5], for vorticity the dispersion relation, which determines the range of experimental installation parameters, for which the vortical turbulence is damped. 1. EXCITATION OF VORTICES Hydrodynamic equations for electrons and Poisson equation are used      2 t e He th e eV V V e m ,V V n n          ,(1)  t e en n V 0   , 0rE  ,  e i4 e n n    . (2) Here He 0 eeH m c  is the electron cyclotron frequency; e, me are the electron charge and mass; V, ne and Vth are the velocity, density and thermal velocity of electrons in the plasma, ni is the ion density; Eor is the external radial electric field,  is the electric potential of vortical perturbation. From equations (1), (2) one can derive, neglecting  2 th e eV n n in (1), equations    t He e He e z zd n n V          , (3)  t z e zd V e m   ,  t td V     , ze rotV  . Also from equation (1) one can derive expression for transversal ( 0V H  ) electron velocity. V   He ze m e ,       1 1 He t z He ze ,V e , V V                2 He z He te m e , e m            , (4)  r0 He He2eE rm e m        . (5) From (2), (5) it approximately follows  2 pe He e 0en n     that the vortical motion begins, as soon as there appears a plasma density perturbation en . From (3) one can derive    t He e He e z zd n n V    . (6) Taking into account the effect of unmagnetized ions, moving in longitudinal direction, from (2) one can obtain, searching the following dependence of perturbations en ,  i zn exp i k z t      . e4 e n    ,   22 pi z i01 k V    , e 0e en n n  . (7) Here pi is the ion plasma frequency; Vi0 is the longitudinal velocity of the ion flow. At first let us consider instability development. From (3) we derive      2 t He e He e 0e z 0d n e m n ik         , 0 0V r   . (8) Vo is the electron drift velocity in crossed radial electrical and longitudinal magnetic field. From (4), (7), (8) we obtain equation for     2 2 pe He r He t 0              2 2 z pe 0ik       . (9) Assuming that the value  1 2 r pe HeA r    is approximately independent on radius and looking for a transverse structure with the help of the Bessel functions from (9) one can find the linear dispersion relation, which describes the development of the instability pe is the electron plasma frequency.     22 2 pi z i0 01 k V A k          22 2 2 z pe 0k k 0      . (10) 132 ISSN 1562-6016. ВАНТ. 2015. №1(95) For short magnetic coil    2 2 He He0r 1 Br R   at B<<1 we have    1 2 2 2 r pe He pe0 He0A r 2B R .        For plasma density, decreasing on radius    2 2 e 0en r n 1 r R  , we have    1 2 2 2 r pe He pe0 He0A r 2 R .        Let us take into account that the ions pass through the system of length L during i i0L V  and electrons are renovated during e. (10) can be presented as     22 2 pi z i0 i 0 e1 k V i A k i              22 2 2 z pe 0 ek k i 0        . (11) Let us mean quick perturbations those, which phase velocity approximately equals ph 0V V . For them from (11) we derive in approximation kz=0, = (o) +, << (o) and neglecting e , i  0 pi 0      ,   2 0 pe He 0e2 n n     , qi  ,  1 q pik 2 A    , 0e 0in n n   . (12) From (12) it follows  i e He pe 0em m n n     , (13) that for typical parameters of experiments the perturbations with 1  are excited at a large magnetic field and at small electron density. For slow perturbations it is fulfilled ph 0V V . We derive for them from (11) in approximation kz=0 and neglecting e , i the following expressions      1 3 4 3 2 2 s pi pe He 0e3 2 2 n n          , 2 0k A    , s sRe 3   . (14) Here s is the growth rate of slow perturbation excitation. 2. SPATIAL STRUCTURE OF VORTICES Let's describe structure of a fast vortex, placed on radius rq, in a rest frame, rotated with angular rate ph ph qV r  . Let's consider a chain (on ) of interleaving vortices  bunches and vortices  cavities of electrons. Neglecting non-stationary and non-linear on  terms, we receive the following equation    He z r0 He zV e m e ,E e m e ,            , (15) describing quasistationary dynamics of electrons in fields of crossed fields and vortical perturbation. From (15) we receive expressions for radial and azimuth velocities of electrons  r HeV e m      , V=Vo+(e/meHe)r , Vo=-(e/meHe)Ero=( 2 pe/2He)(n/noe)r . (16) V can been presented as the sum of the phase velocity of the perturbation, Vph, and velocity of azimuth oscillations of electrons, V, in the field of the perturbation, V=Vph+V. As V=rd/dt, we present d/dt as d/dt=d1/dt+ph, here ph=(n/noe)( 2 pe/2He)r=rq. Then from (16) we obtain d1/dt=(e/me)[Ero(r)/rHe(r)-Ero(rq)/rqHe(rq)]+ +(e/rmeHe)r , dr/dt=-(e/meHer). (17) At small deviations r from rq, decomposing n(r)/He(r) on rr-rq and integrating (17), we obtain (r) 2 +4/rqHer[Ero(r)/rHe(r)]r=rq=const. (18) The vortex boundary separates the trapped electrons, formed the vortex and moving on closed trajectories, and untrapped electrons moving outside the boundary of the vortex and oscillating in its field. For vortex boundary we receive the following expression from the condition r=-o=rcl, r=[-4(+o)/rqHer[Ero(r)/rHe(r)]r=rq+(rcl) 2 ] 1/2 . (19) Here rcl is the radial width of the vortex-bunch of electrons. From (19) the radial size of the vortex-cavity of electrons follows rh=[-8o/rqHer[Ero(r)/rHe(r)]r=rq+(rcl) 2 ] 1/2 . (20) From the equation of electron motion and Poisson equation one can derive approximately expression for the vorticity ezrotV , which is characteristic of the vortical motion of electrons -2eEro/rmHe+( 2 pe/He)ne/neo. From here it follows that up to certain amplitude of vortices the structure of electron trajectories in the field of the chain on  of fast vortices in the rest frame, rotated with phVph/rq, looks one kind and for large amplitude another kind. For large amplitudes of fast vortices in the region of electron bunches the reverse flows are formed. The vortex-cavity is rotated in the rest frame, rotated with frequency phVph/rq, in the same direction as nonperturbed plasma. The vortex-bunch is rotated in the opposite direction of rotation of nonperturbed plasma at ne>nnoe-noi. One can see that the size of the vortex is inversely proportional to [-8/rqHer[Ero(r)/rHe(r)]r=rq] 1/2 and is proportional to  1/2 o. That is the size of the vortex essentially depends on a gradient of the magnetic and electrical fields. At small r[Ero(r)/rHe(r)]r=rq] 1/2 already at small perturbations of electron density the size of the vortex, rh, can reach rhR/2, R is the radius of the plasma. Eq. (17) can be integrated without decomposing n(r)/He(r) on rr-rq. For this purpose in approximation nn(r) we approximate He(r)= Ho(1+r 2 /R 2 ). Then, integrating (17), we obtain 2+enr 2 [1-Ho/2He(rq)-He(r)/2He(rq)]=const. (21) From the condition r=-o=rq+rcl and (21) we derive the expression, determining the boundary of the vortex - cavity of electrons, [r 2 -(rv+rcl) 2 ][1-Ho/He(rv)]- [r 4 -(rv+rcl) 4 ]Ho/2R 2 He(rv)+2(+o)/en=const. (22) From (22) and r=o=rq+rh we derive the expression, determining the radial width of the vortex - cavity of electrons, o4R 2 He(rv)/en]Ho= =(rh-rcl)(2rv+rh+rcl)[rv(rh+rcl)+(rh 2 +rcl 2 )/2]. (23) Let's consider the vortex with the small phase velocity Vph in comparison with drift velocity of ISSN 1562-6016. ВАНТ. 2015. №1(95) 133 electrons, Vph<<Vo. The spatial structure of electron trajectories in its field for small amplitudes of the vortex looks like as corrugated structure. It is determined by that in all system  has the identical sign, >0. In other words, radial electrical field created by the vortex is less, than external electrical field, Erq<Ero. Then in all system the azimuth velocities of electrons have the identical sign and there are no reverse flows of electrons. There is no separatrix in slow vortex of small amplitude. For the description of spatial structure of electron trajectories we use (16). Using in them V=rd/dt and excluding , we obtain for vortex boundary r() r=[r 2 s+(o-)2/en] 1/2 . (24) In the case of small amplitudes (24) becomes rr-rs=(o-)/enrs . (25) From (24) we derive the radial size of the slow vortex rsr=-o-rs=[r 2 s+4o/en] 1/2 -rs . (26) In the case of small amplitudes (26) becomes rs2o/enrs . (27) Because on r=rv , n(r=rv)=0, then the electron moves on it with Vo without radial perturbations. At r>rv radial displacement is positive, and when r<rv  negative radial displacement of the electrons. At large amplitudes, ne>n (or Erv>Ero), in the region, where the electron cavities place, the characteristic of the vortical motion  obtains the inverse sign, <0. In other words, on the axis, connecting the vortex-cavity and the vortex-bunch, the inequality Erv>Ero is executed and there is an azimuth reverse flow of electrons. Then in some regions the electrons are rotated in the direction inverse to their rotation in crossed fields. The slow vortex is a dipole perturbation of electron density, disjointed on radius. At ne>n the structure of the slow vortex is similar to the structure of Rossby vortex. 3. SUPRESSION OF EXCITATION OF VORTICAL PERTURBATIONS IN CASE OF MAGNETIZED IONS Similarly (10) one can derive dispersion relation     22 2 pi 0o1 A k 0        , which demonstrates the suppression of the instability in the case of magnetized ions Hi 2 r ieE m R  . CONCLUSIONS Properties and excitation of the vortical turbulence, excited in a cylindrical radially inhomogeneous plasma in crossed radial electric and longitudinal magnetic fields, have been described. The dispersion relation, which allows to determine the range of parameters for which the vortical turbulence is suppressed, is derived from the general non-linear equations for the vorticity. REFERENCES 1. C.F. Driscoll, J.H. Malmberg, K.S. Fine, R.A. Smith, X.-P. Huang, R.W. Gould // Plasma Phys. Contr. Fus. Res. 1989, v. 3, p. 507. 2. Y. Kiwamoto, A. Mohri, K. Ito, A. Sanpei, T.Yuyama. 2D interaction of discrete electron vortices // Non-neutral plasma physics. Princeton, 1999, p. 99-105. 3. A. Goncharov, A. Dobrovolsky, A. Kotsarenko, A. Morozov, I. Protsenko // Physica Plasmy. 1994, v. 20, p. 499 (in Russian). 4. A. Goncharov, I. Litovko. Electron Vortexes in High- Current Plasma Lens // IEEE Trans. Plasma Sci. 1999, v. 27, p. 1073. 5. V.I. Maslov, A.A. Goncharov, I.N. Onishchenko, Self-Organization of Non-Linear Vortexes in Plasma Lens for Ion-Beam-Focusing in Crossed Radial Electrical and Longitudinal Magnetic Fields // Invited Paper. In Proceedings of the International Workshop "Collective phenomena in macroscopic systems" / G. Bertin, R. Pozzoli, M. Rome' and K.R. Sreenivasan / Eds. World Scientific, Singapore. 2007, p. 20-25. 6. V.B. Yuferov, A.S. Svichkar, S.V. Shariy, T.I. Tkachova, V.О. Ilichova, V.V. Katrechko, A.I. Shapoval, S.N. Khizhnyak. About Redistribution of Ion Streams in Imitation Experiments on Plasma Separation // Problems of Atomic Science and Technology. Ser. “Physics of Radiation Damage and Radiation Material”. 2013, № 5(87), p. 100-103. Article received 06.12.2014 ПОДАВЛЕНИЕ ВОЗБУЖДАЕМОЙ ВИХРЕВОЙ ТУРБУЛЕНТНОСТИ В НЕОДНОРОДНОЙ ПЛАЗМЕ В СКРЕЩЕННЫХ РАДИАЛЬНОМ ЭЛЕКТРИЧЕСКОМ И ПРОДОЛЬНОМ МАГНИТНОМ ПОЛЯХ В.И. Маслов, И.П. Яровая, А.М. Егоров, В.Б. Юферов Рассматриваются свойства и возбуждение вихревой турбулентности, возбуждаемой в цилиндрической радиально-неоднородной плазме в скрещенных радиальном электрическом и продольном магнитном полях. Из общего нелинейного уравнения для завихренности получено дисперсионное уравнение, позволяющее определить диапазон параметров, для которых вихревая турбулентность подавляется. ПРИДУШЕННЯ ЗБУДЖЕННЯ ВИХРОВОЇ ТУРБУЛЕНТНОСТІ В НЕОДНОРІДНІЙ ПЛАЗМІ В СХРЕЩЕНИХ РАДІАЛЬНОМУ ЕЛЕКТРИЧНОМУ І ПОЗДОВЖНЬОМУ МАГНІТНОМУ ПОЛЯХ В.І. Маслов, І.П. Ярова, О.М. Єгоров, В.Б. Юферов Розглядаються властивості і збудження вихрової турбулентності, яка збуджується в циліндричній радіально-неоднорідній плазмі в схрещених радіальному електричному і поздовжньому магнітному полях. Із загального нелінійного рівняння для завихренності отримано дисперсійне рівняння, що дозволяє визначити діапазон параметрів, для яких вихрова турбулентність пригнічується.
id	nasplib_isofts_kiev_ua-123456789-82113
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1562-6016
language	English
last_indexed	2025-11-27T10:10:43Z
publishDate	2015
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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spelling	Maslov, V.I. Yarovaya, I.P. Yegorov, A.M. Yuferov, V.B. 2015-05-25T09:43:06Z 2015-05-25T09:43:06Z 2015 Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields / V.I. Maslov, I.P. Yarovaya, A.M. Yegorov, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 131-133. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx https://nasplib.isofts.kiev.ua/handle/123456789/82113 Properties and excitation of the vortical turbulence, excited in a cylindrical radially inhomogeneous plasma in crossed radial electric and longitudinal magnetic fields, are considered. The dispersion relation, which allows to determine the range of parameters for which the vortical turbulence is suppressed, is derived from the general nonlinear equations for the vorticity. Рассматриваются свойства и возбуждение вихревой турбулентности, возбуждаемой в цилиндрической радиально-неоднородной плазме в скрещенных радиальном электрическом и продольном магнитном полях. Из общего нелинейного уравнения для завихренности получено дисперсионное уравнение, позволяющее определить диапазон параметров, для которых вихревая турбулентность подавляется. Розглядаються властивості і збудження вихрової турбулентності, яка збуджується в циліндричній радіально-неоднорідній плазмі в схрещених радіальному електричному і поздовжньому магнітному полях. Із загального нелінійного рівняння для завихренності отримано дисперсійне рівняння, що дозволяє визначити діапазон параметрів, для яких вихрова турбулентність пригнічується. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Плазменная электроника Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields Подавление возбуждаемой вихревой турбулентности в неоднородной плазме в скрещенных радиальном электрическом и продольном магнитном полях Придушення збудження вихрової турбулентності в неоднорідній плазмі в схрещених радіальному електричному і поздовжньому магнітному полях Article published earlier
spellingShingle	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields Maslov, V.I. Yarovaya, I.P. Yegorov, A.M. Yuferov, V.B. Плазменная электроника
title	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
title_alt	Подавление возбуждаемой вихревой турбулентности в неоднородной плазме в скрещенных радиальном электрическом и продольном магнитном полях Придушення збудження вихрової турбулентності в неоднорідній плазмі в схрещених радіальному електричному і поздовжньому магнітному полях
title_full	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
title_fullStr	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
title_full_unstemmed	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
title_short	Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
title_sort	suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields
topic	Плазменная электроника
topic_facet	Плазменная электроника
url	https://nasplib.isofts.kiev.ua/handle/123456789/82113
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Suppression of excited vortical turbulence in inhomogeneous plasma in crossed radial electrical and longitudinal magnetic fields

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