Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves

Some estimations of correctness of use “standard” solution of Vlasov equation for finite sistems are curried out.

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Бібліографічні деталі
Дата:	2007
Автори:	Kasilova, E.V., Khvesyuk, V.I.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
Назва видання:	Вопросы атомной науки и техники
Теми:	Basic plasma physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/110382
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves / E.V. Kasilova, V.I. Khvesyuk // Вопросы атомной науки и техники. — 2007. — № 1. — С. 61-62. — Бібліогр.: 1 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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fulltext	Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 61-62 61 ESTIMATIONS OF RESTRICTIONS ON USE “STANDARD” SOLUTIONS OF VLASOV EQUATIONS FOR DRIFT WAVES E.V. Kasilova, V.I. Khvesyuk Bauman Moscow State Technical University, 2nd Baumanskaya Str. 5, 105005, Moscow, e-mail: khves@power.bmstu.ru Some estimations of correctness of use “standard” solution of Vlasov equation for finite sistems are curried out. PACS: 52.35.-g In this work we analyse correctness of ordinary solution of Vlasov-equation for drift waves in infinite plasma for analysis of plasma instability in finite size. Such a problem occurs for example when processes in FRC-systems are investigated where instability evolves in finite size. The initial expression for perturbed ion density [1] is: ×−−= iBiB i Tk qin Tk qn ϕϕ δ 0 ∫ ∫ ∫ ∞ ∞− ∞ ⊥ −       Ω +× 0 0 0 y i iB k m Tkvf ω τ J ×      Ω ⊥kv2 0 (1) zzz dvdvdvki ⊥−× τω)](exp[ , (“standard” time integration is taking from ∞− to zero [1]). Results of integration is: ∫ − − −−− =− 0 )( )](exp[1)](exp[ τ ω ω ω zz zz zz vki vkidtvki .(2) And ∫ ∞ ⊥       − 0 2 exp iB i Tk vm J =      Ω ⊥⊥ ⊥⊥ dvvvk2 0 =       Ω −      Ω ⊥⊥ i iB i iB m Tkk m TkkI 2 2 2 2 0 exp . Problem comes to integration of zzz zz iB zi iB i dvvki vk Tk vm Tk m )])(exp( 1[ exp 2 2 τω ωπ ω −−− − −       − ∫ ∞ ∞− , when τ is constant. Physical meaning of this integral is that the first item in square brackets corresponds to plasma function, the second item is correction for finite time of integration. Let’s transform the last expression zz z z z z iB zi iB i z dvv k ik v k Tk vm Tk m k ]exp1[ exp 2 2             −−− −       − ∫ ∞ ∞− τ ω ωπ ω Introduce dimensionless variables: iTk mvtv B i zz 2 =⇒ , iB i zz Tk m k z k 2 ωω =⇒ , i iB zz m Tkkk 2 ττ ⇒ . The first integral in (2) is ∫ ∞ ∞− − − = dt tz tizW )exp()( 2 π . The second integral is ∫ ∞ ∞− − − − = dttzia tz tiazW )](exp[)exp(),('' 2 π . Standard approach is acceptable to system of finite size if ),('' azW << )(zW . Drift waves phase velocity must satisfy condition: tiv << zk ω << tev . tiv is the ion thermal velocity, tev is the electron thermal velocity: ,2 i iB ti m Tkv = e eB te m Tkv 2 = . Parameter z is a complex value iyxz += . Relation between ),('' zaW and )(zW mailto:khves@power.bmstu.ru In particular for yx = corresponding approximately to so-called iη mode ( ITG - ion temperature gradient mode) we obtain restriction on z : z >1 or yx ≈ > 2 1 . Relation between ),('' zaW and )(zW is presented in the Figure. As one can see condition ),('' azW << )(zW is satisfied when a >2. Relation between ),('' zaW and )(zW is presented in the Figure above. As one can see condition ),('' azW << )(zW is satisfied when a >2. Therefore “standard” solution is satisfied when z tiz Lvka λ π τ 2 == >2. REFERENCES 1. N.A. Krall, A.W. Trivelpiece. Principles of Plasma Physics. New York: “Mc Graw-Hill Co.”, 1973. “ ” . , . . “ ” . , . .
spelling	irk-123456789-1103822017-01-05T03:02:16Z Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves Kasilova, E.V. Khvesyuk, V.I. Basic plasma physics Some estimations of correctness of use “standard” solution of Vlasov equation for finite sistems are curried out. Виконано оцінки коректності використання результатів інтегрування рівняння Власова для систем кінцевих розмірів. Выполнены оценки корректности использования результатов интегрирования уравнения Власова для систем конечных размеров. 2007 Article Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves / E.V. Kasilova, V.I. Khvesyuk // Вопросы атомной науки и техники. — 2007. — № 1. — С. 61-62. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS: 52.35.-g http://dspace.nbuv.gov.ua/handle/123456789/110382 en Вопросы атомной науки и техники 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	Basic plasma physics Basic plasma physics
spellingShingle	Basic plasma physics Basic plasma physics Kasilova, E.V. Khvesyuk, V.I. Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves Вопросы атомной науки и техники
description	Some estimations of correctness of use “standard” solution of Vlasov equation for finite sistems are curried out.
format	Article
author	Kasilova, E.V. Khvesyuk, V.I.
author_facet	Kasilova, E.V. Khvesyuk, V.I.
author_sort	Kasilova, E.V.
title	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves
title_short	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves
title_full	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves
title_fullStr	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves
title_full_unstemmed	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves
title_sort	estimations of restrictions on use “standard” solutions of vlasov equations for drift waves
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2007
topic_facet	Basic plasma physics
url	http://dspace.nbuv.gov.ua/handle/123456789/110382
citation_txt	Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves / E.V. Kasilova, V.I. Khvesyuk // Вопросы атомной науки и техники. — 2007. — № 1. — С. 61-62. — Бібліогр.: 1 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT kasilovaev estimationsofrestrictionsonusestandardsolutionsofvlasovequationsfordriftwaves AT khvesyukvi estimationsofrestrictionsonusestandardsolutionsofvlasovequationsfordriftwaves
first_indexed	2025-07-08T00:33:20Z
last_indexed	2025-07-08T00:33:20Z
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Estimations of restrictions on use “standard” solutions of Vlasov equations for drift waves

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