Limits of applicability of the weakly relativistic approximation in the theory of plasma waves

Limits of applicability of the weakly relativistic approximation in the theory of plasma waves

The more general weakly-relativistic plasma dispersion functions are introduced and discussed in the frame of the fully relativistic approach for the theory of plasma waves.

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Datum:2018
Hauptverfasser: Pavlov, S.S., Castejón, F.
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Sprache:English
Veröffentlicht: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2018
Schriftenreihe:Вопросы атомной науки и техники
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Фундаментальная физика плазмы
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/148841
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Zitieren:Limits of applicability of the weakly relativistic approximation in the theory of plasma waves / S.S. Pavlov, F. Castejón // Вопросы атомной науки и техники. — 2018. — № 6. — С. 83-85. — Бібліогр.: 5 назв. — англ.

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spelling irk-123456789-1488412019-02-19T01:27:36Z Limits of applicability of the weakly relativistic approximation in the theory of plasma waves Pavlov, S.S. Castejón, F. Фундаментальная физика плазмы The more general weakly-relativistic plasma dispersion functions are introduced and discussed in the frame of the fully relativistic approach for the theory of plasma waves. Більш загальні слаборелятивістські функції вводяться і обговорюються в рамках точного повністю релятивістського підходу в теорії плазмових хвиль. Более общие слаборелятивистские функции вводятся и обсуждаются в рамках точного полностью релятивистского подхода в теории плазменных волн. 2018 Article Limits of applicability of the weakly relativistic approximation in the theory of plasma waves / S.S. Pavlov, F. Castejón // Вопросы атомной науки и техники. — 2018. — № 6. — С. 83-85. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.27.Ny http://dspace.nbuv.gov.ua/handle/123456789/148841 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 Фундаментальная физика плазмы
Фундаментальная физика плазмы
spellingShingle Фундаментальная физика плазмы
Фундаментальная физика плазмы
Pavlov, S.S.
Castejón, F.
Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
Вопросы атомной науки и техники
description The more general weakly-relativistic plasma dispersion functions are introduced and discussed in the frame of the fully relativistic approach for the theory of plasma waves.
format Article
author Pavlov, S.S.
Castejón, F.
author_facet Pavlov, S.S.
Castejón, F.
author_sort Pavlov, S.S.
title Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
title_short Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
title_full Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
title_fullStr Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
title_full_unstemmed Limits of applicability of the weakly relativistic approximation in the theory of plasma waves
title_sort limits of applicability of the weakly relativistic approximation in the theory of plasma waves
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate 2018
topic_facet Фундаментальная физика плазмы
url http://dspace.nbuv.gov.ua/handle/123456789/148841
citation_txt Limits of applicability of the weakly relativistic approximation in the theory of plasma waves / S.S. Pavlov, F. Castejón // Вопросы атомной науки и техники. — 2018. — № 6. — С. 83-85. — Бібліогр.: 5 назв. — англ.
series Вопросы атомной науки и техники
work_keys_str_mv AT pavlovss limitsofapplicabilityoftheweaklyrelativisticapproximationinthetheoryofplasmawaves
AT castejonf limitsofapplicabilityoftheweaklyrelativisticapproximationinthetheoryofplasmawaves
first_indexed 2025-07-12T20:25:28Z
last_indexed 2025-07-12T20:25:28Z
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fulltext ISSN 1562-6016. ВАНТ. 2018. №6(118) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2018, № 6. Series: Plasma Physics (118), p. 83-85. 83 LIMITS OF APPLICABILITY OF THE WEAKLY RELATIVISTIC APPROXIMATION IN THE THEORY OF PLASMA WAVES S.S. Pavlov1, F. Castejón2 1National Science Center “Kharkov Institute of Physics and Technology”, Institute of Plasma Physics, Kharkiv, Ukraine; 2Laboratorio Nacional de Fusión, CIEMAT, Madrid, Spain The more general weakly-relativistic plasma dispersion functions are introduced and discussed in the frame of the fully relativistic approach for the theory of plasma waves. PACS: 52.27.Ny INTRODUCTION Presently, the analytical and numerical investigation of the excitation, propagation and absorption of electron cyclotron waves in thermonuclear plasma is performed as a rule in the frame of the weakly relativistic approximation, when longitudinal (to the magnetic field line) spatial dispersion of plasma is expressed in the terms of the weakly relativistic plasma dispersion functions (PDFs)                 0 2 , 2/1 2/1 2/12/3 azu eauIu du a e azF ua n n nn , ( ...2,1,0 n ), (1) where n is the number of electron cyclotron harmonic, 22 IINa  , 2)/( TeVc , c is speed of light in the vacuum, TeV is the thermal velocity of electrons, IIN is the longitudinal refractive index of plasma, )(2/1 xIn is modified Bessel function of the half-integer order [1]. Applicability limits of such approximation is not clear enough, since they could be defined exactly only on the basis of somewhat limit transitions in the frame of the exact fully relativistic approach, when plasma dielectric tensor elements are expressed in the terms of the Cauchy type integrals, named by the exact relativistic or fully relativistic PDFs [2]: for the case 10  IIN         21 2 )2( )( ,, 2 23 n aa n aeK e azZ                  0 21 21 * 2 )/1)2/(()/1)2/(( azu dueI u n n uuauu    , (2) where )1/(1 2 IIN , )(2 xK is the MacDonald function of the 2nd order, )N11( 2 II a , )(21 xIn is modified Bessel function with half-integer index, in the contrary case 1IIN         2123 )( ,, 22 )2exp( nn aK azZ a                          zt dttaK ttatta n n   exp2 )2( 2 )2( 2 21 21 21 , ( ...2,1,0 n ), (3) where )(2/1 xKn is MacDonald function with half- integer index The main scope of present work is the definition of applicability limits of the weakly relativistic approximation from fully relativistic PDFs (2) and (3) on the base limit transition  . 1. THE WEAKLY RELATIVISTIC PDFs WITHOUT TAKING INTO ACCOUNT THE TRANSVERSE SPATIAL DISPERSION From comparison of the weakly relativistic PDFs (1) and fully relativistic ones (2) and taking into account the asymptotic relation )2(~)(2   eK when  [3] it is easy to see, that    azFazZ nn ,,, 2/323   when  and аа  . The first limit transition corresponds to the transition into the cold plasma approximation, the second one corresponds to the transition into the nearly perpendicular (to the magnetic field) wave propagation case ( 12 IIN ), when аNа  2/2 II and 1 . Here it is necessary specially to note that the weakly relativistic PDFs (1) and fully relativistic ones (2) are fair only for the case 1IIN . In the contrary case 1IIN the exact relativistic PDFs are expressed by the Cauchy integrals, defined at the real axis in the form (3). Performing in the definition (3) the limit transition  and making the change )1/(1  2 IIN we will obtain the weakly relativistic PDFs of the new form, corresponding to the condition 12 IIN [2].         2123 2 ))1/(2exp( )1/(1, nn a Na NazF  2 II2 II              zt dtKta NtNtaan n )1/()1/(2 exp21 21 2 II 2 II . (4) 84 ISSN 1562-6016. ВАНТ. 2018. №6(118) Now making in the definition (2) only the limit transition  we will obtain the weakly relativistic PDFs also of the new form, corresponding to the condition 12 IIN .          2123 )1( 2 )21/()2( , n Naa n aN e azZ II II                        0 * 1 1 21 2 221 21 2 zau dueNu N a INu u n n  II II II . (5) It is easy to see that the form (4) goes to the form (1) when 0IIN (perpendicular wave propagation) and аа  (almost perpendicular wave propagation). It is easy to see, that  2 IINaа  112 and consequently for а is true condition ааa 2  when 10  IIN . Methods of evaluation of PDFs (4) and (5) are given in [2]. 2. ACCOUNT OF THE TRANSVERSE SPATIAL DISPERSION One-dimensional integral forms for fully relativistic plasma dielectric tensor, taking into account the transverse spatial dispersion and longitudinal one for the case 12 IIN , were given in [4]        1 1 22 11 )(Im 2 n Kua n Jduez n   , (6)          nn Kua n JJeduzi n 2 1 1 12 )(Im   ,            222 1 1 22 )(Im 1 nn Kua n JJeduz  ,        22 1 1 13 )(Im n Kua n JeuduzK n   ,          nn Kua n JJeuduzKi 2 1 1 23 )(Im 1   ,  ,)(2Im 22 1 1 22 33       n Kua n JeduuzK  )/1(2 )(  za Keez z   , 2)(2 2 0       K ep         ,  /)1(2 2uK  , . ),,(Im1 ),,(Re      zt dtta Pza ij ijij    After the weakly relativistic limit transition  in the integral forms (6) one will have functions, generating elements of plasma dielectric tensor for the weakly relativistic case with taking exactly into account the transverse and longitudinal spatial dispersion of plasma and suitable for numerical evaluations for arbitrary ECRF waves in the weakly relativistic plasma. The alternative case 1IIN was studied in [5], where were obtained plasma dielectric tensor elements for fully relativistic plasma with taking into account the transverse spatial dispersion and longitudinal one         1 2 11 2 2 Im n n JdueK uKan   , (7)            nn n JJeduKi uKan 2 1 12Im   ,          22 1 22 21 Im n n JeduK uKa   ,    , Im 2 1 13 2 )/1( 2            n n Je duK uKa zauK n         , Im 2 )/1( 2 1 23            nn n JJe duKi uKa zauK       , 2Im 2 1 33 2 2 )/1(            n n Je duK uKa zauK  . ),,(Im1 ),,(Re      zt dtta Pza ij ijij    After the same weakly relativistic transition  in formulae (7), one will obtain the functions, generating plasma dielectric tensor elements in the weakly relativistic case, taking into account the perpendicular and longitudinal dispersion of plasma and suitable for numerical investigation of fast and slow plasma EC waves in the thermonuclear plasma. CONCLUSIONS The next main conclusions can be extracted from this work: 1. In the frame of fully relativistic approach by means of limit transition  were derived the two sets of the weakly relativistic PDFs. ISSN 1562-6016. ВАНТ. 2018. №6(118) 85 2. One PDFs generalize Shkarofsky functions from quasi-perpendicular case into much more wide case 10  IIN . 3. Another PDFs give the weakly relativistic PDFs for the case 1IIN . 4. Method evaluating the weakly relativistic plasma dielectric tensor elements for arbitrary wave numbers or with taking into account transverse and longitudinal spatial dispersion of plasma is given as well. REFERENCES 1. M. Brambilla. Kinetic theory of plasma waves, homogeneous plasmas. Oxford University Press, 1998. 2. F. Castejon, S.S. Pavlov // Physics of Plasmas. 2006, v. 13, p. 072105. 3. Handbook of mathematical functions / Edited by M. Abramowitz and A. Stegun, 1964. 4. S.S. Pavlov. Exact relativistic maxwellian magnetized plasma dielectric tensor evaluation for arbitrary wave numbers // Problems of Atomic Science and Technology. Series “Plasma Physics” (22). 2016, № 6(106), p. 92. 5. S.S. Pavlov, F. Castejon. Fully relativistic approach to the theory of plasma waves / The work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programe 2014-2018 under grant agreement № 633053. This work has been also funded by the Spanish Ministerio de Economia y Competitividad under Project ENE2014-52174-P. Article received 10.10.2018 . ПРЕДЕЛЫ ПРИМЕНИМОСТИ СЛАБОРЕЛЯТИВИСТСКОГО ПРИБЛИЖЕНИЯ В ТЕОРИИ ПЛАЗМЕННЫХ ВОЛН С.С. Павлов, F. Castejón Более общие слаборелятивистские функции вводятся и обсуждаются в рамках точного полностью релятивистского подхода в теории плазменных волн. МЕЖІ ЗАСТОСУВАННЯ СЛАБОРЕЛЯТИВІСТСЬКОГО НАБЛИЖЕННЯ В ТЕОРІЇ ПЛАЗМОВИХ ХВИЛЬ С.С. Павлов, F. Castejón Більш загальні слаборелятивістські функції вводяться і обговорюються в рамках точного повністю релятивістського підходу в теорії плазмових хвиль.

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