Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces

There is considered the problem on the spectra of magnetohydrodynamic (MHD) waves with the frequencies of the order of the ion cyclotron frequency propagating almost along the toroidal magnetic field in the large-size tokamaks with a non-circular cross section of magnetic surfaces. In this case the...

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Дата:	2000
Автори:	D’yakov, V.Ye., Girka, I.A., Yegorenkov, V.D., Stepanov, K.N.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2000
Назва видання:	Вопросы атомной науки и техники
Теми:	Magnetic confinement
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/78499
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Цитувати:	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces / V.Ye. D’yakov, I.А. Girka, V.D. Yegorenkov, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 60-61. — Бібліогр.: 4 назв. — англ.

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spelling	irk-123456789-784992016-04-14T11:13:16Z Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces D’yakov, V.Ye. Girka, I.A. Yegorenkov, V.D. Stepanov, K.N. Magnetic confinement There is considered the problem on the spectra of magnetohydrodynamic (MHD) waves with the frequencies of the order of the ion cyclotron frequency propagating almost along the toroidal magnetic field in the large-size tokamaks with a non-circular cross section of magnetic surfaces. In this case the waves can propagate in the small vicinity of the extremum point for the square of the refractive index of the MHD wave travelling along the torus. Developing the dielectric permittivity tensor of the “cold” plasma in a Taylor series around this point enables one to separate the variables and to express the natural functions of the boundary problem for Maxwell’s equations through Hermite functions and to determine the natural frequencies of MHD waves. The solution obtained is a generalization of the previous result for arbitrary radial and azimuth numbers. On the ground of the perturbation theory the corrections to the natural functions and natural values are found which take into account the rotational transform. 2000 Article Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces / V.Ye. D’yakov, I.А. Girka, V.D. Yegorenkov, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 60-61. — Бібліогр.: 4 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78499 533.9 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	Magnetic confinement Magnetic confinement
spellingShingle	Magnetic confinement Magnetic confinement D’yakov, V.Ye. Girka, I.A. Yegorenkov, V.D. Stepanov, K.N. Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces Вопросы атомной науки и техники
description	There is considered the problem on the spectra of magnetohydrodynamic (MHD) waves with the frequencies of the order of the ion cyclotron frequency propagating almost along the toroidal magnetic field in the large-size tokamaks with a non-circular cross section of magnetic surfaces. In this case the waves can propagate in the small vicinity of the extremum point for the square of the refractive index of the MHD wave travelling along the torus. Developing the dielectric permittivity tensor of the “cold” plasma in a Taylor series around this point enables one to separate the variables and to express the natural functions of the boundary problem for Maxwell’s equations through Hermite functions and to determine the natural frequencies of MHD waves. The solution obtained is a generalization of the previous result for arbitrary radial and azimuth numbers. On the ground of the perturbation theory the corrections to the natural functions and natural values are found which take into account the rotational transform.
format	Article
author	D’yakov, V.Ye. Girka, I.A. Yegorenkov, V.D. Stepanov, K.N.
author_facet	D’yakov, V.Ye. Girka, I.A. Yegorenkov, V.D. Stepanov, K.N.
author_sort	D’yakov, V.Ye.
title	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
title_short	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
title_full	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
title_fullStr	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
title_full_unstemmed	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
title_sort	magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2000
topic_facet	Magnetic confinement
url	http://dspace.nbuv.gov.ua/handle/123456789/78499
citation_txt	Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces / V.Ye. D’yakov, I.А. Girka, V.D. Yegorenkov, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 60-61. — Бібліогр.: 4 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT dyakovvye magnetohydrodynamicwavespectrainlargetokamakswithnoncircularcrosssectionofmagneticsurfaces AT girkaia magnetohydrodynamicwavespectrainlargetokamakswithnoncircularcrosssectionofmagneticsurfaces AT yegorenkovvd magnetohydrodynamicwavespectrainlargetokamakswithnoncircularcrosssectionofmagneticsurfaces AT stepanovkn magnetohydrodynamicwavespectrainlargetokamakswithnoncircularcrosssectionofmagneticsurfaces
first_indexed	2025-07-06T02:34:41Z
last_indexed	2025-07-06T02:34:41Z
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fulltext	60 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 60-61 UDC 533.9 MAGNETOHYDRODYNAMIC WAVE SPECTRA IN LARGE TOKAMAKS WITH NON-CIRCULAR CROSS SECTION OF MAGNETIC SURFACES V.Ye.D’yakov* , I.А.Girka, V.D.Yegorenkov, K..N.Stepanov* * National Science Center “Kharkiv Institute of Physics and Technology”, Institute of Plasma Physics, Akademichna, 1, 61108, Kharkiv, Ukraine *V.N.Karazin Kharkiv National University, 4, Svobody Square, 61077, Kharkiv, Ukraine FAX: 35-26-64. E-mail: stepanov@ipp.kharkov.ua There is considered the problem on the spectra of magnetohydrodynamic (MHD) waves with the frequencies of the order of the ion cyclotron frequency propagating almost along the toroidal magnetic field in the large-size tokamaks with a non-circular cross section of magnetic surfaces. In this case the waves can propagate in the small vicinity of the extremum point for the square of the refractive index of the MHD wave travelling along the torus. Developing the dielectric permittivity tensor of the “cold” plasma in a Taylor series around this point enables one to separate the variables and to express the natural functions of the boundary problem for Maxwell’s equations through Hermite functions and to determine the natural frequencies of MHD waves. The solution obtained is a generalization of the previous result for arbitrary radial and azimuth numbers. On the ground of the perturbation theory the corrections to the natural functions and natural values are found which take into account the rotational transform. 1. Introduction The flows of fast ions formed under injection of fast neutral particles in a reactor-tokamak may provide for the induction-free current and plasma equilibrium. But the strongly nonequilibrium distribution of fast ions may lead to the excitation of unstable Alfven and fast magnetosonic waves when the current density exceeds the threshold value determined by the wave damping due to electron Cherenkov and ion cyclotron damping. The consideration of this problem for the ITER tokamak has shown that the threshold value is determined by the excitation of first radial modes in a simplified plasma model of a hollow cylinder. It has appeared that the threshold value is highly sensitive to the parameters of the ion beam and of the waves [1,2]. In order to solve the problem on the excitation of such waves by the beam of fast ions, it is necessary to know the natural frequencies of MHD waves for a more realistic model. This problem is solved in the present paper where the dispersion equation is obtained for the MHD waves propagating almost along the magnetic field in a tokamak with a non-circular cross section of magnetic surfaces. 2. Dispersion equation From the Maxwell’s equations for the plasma with small β for the MHD waves ( )tiim ωϕ −∝ exp where m is the toroidal mode number, with neglect of the longitudinal inertia of electrons ( )∞→3ε there follows the set of two equations for the left- and right- hand polarized components of the wave field ZR iEEE ±=± : =+      ∂ ∂− ∂ ∂+ ∂ ∂ ∂ ∂ +++ EkE RR i ZR R RR 2 2 21 −      ∂ ∂+ ∂∂ ∂+ ∂ ∂− ∂ ∂ ∂ ∂ E RR i ZR i ZR R RR 2 2 2 21 , =+      ∂ ∂+ ∂ ∂+ ∂ ∂ ∂ ∂ −−− EkE RR i ZR R RR 2 2 21 +      ∂ ∂− ∂∂ ∂− ∂ ∂− ∂ ∂ ∂ ∂ E RR i ZR i ZR R RR 2 2 2 21 (1) where ( )       −±=± 2 2 212 2 2 2 R m c k εεω , (2) ∑ − ≈ i Ci pi 22 2 1 ωω ω ε , ( )∑ − ≈ i CiCi pi 22 2 2 ωωω ωω ε , (3) R,ϕ and Z are the cylindrical coordinates, cmeB iCi ϕω = , ( )RRBB c0=ϕ is the toroidal magnetic field, Rc is the large radius of the toroidal chamber, ( )ZRpi ,ω is the ion Langmuir frequency, the subscript i denotes the summation over ion species. In the equations (1) we neglect the effect of a small poloidal field 22 ZRp BBB += compared with the toroidal field ϕB . As we consider the waves propagating almost along the torus, then within the region of wave localization 00 , ZZRR ≈≈ it is necessary to meet one of the approximate equalities ( ) ( )2 2 21 ω ωωω ω εε Rmc i CiCi pi ≈∑=± ! (4) For certainty we consider further the left-hand-polarized waves (Alfven waves) for which ( )221 ωεε Rmc≈+ . In this case 22 −± <<kk and one can neglect the right-hand side of the first equation of the set (1) proportional to −E and obtain the following equation +      ∂ ∂ + ∂ ∂ +E RRR 1 2 2 02 2 2 =+      ∂ ∂− ∂ ∂ +++ EkE ZR i Z (5) mailto:stepanov@ipp.kharkov.ua 61 which is valid for an arbitrary 2-D nonuniformity across the plasma column ),(22 ZRkk ++ = . Just this equation (5) will form the ground for studying the spectra of natural MHD oscillations of the toroidal plasma column. Assuming that the quantity 2 +k approaches its maximum at the point 00 , ZZRR == , we develop the function 2 +k into the Taylor series in powers of small deviations 00 , ZZzRRr −=−= . Then we obtain +      ∂ ∂+ ∂ ∂ +E rRr 0 2 2 1 . (6) [ ] 0222 0 0 2 2 =−−−+      ∂ ∂− ∂ ∂ ++ EcrzbzarkE zR i z , where ( )00 22 0 ,ZRkk += and ( ) , , 2 1 2 0 00 22 R ZRka ∂ ∂ −= + ( ) , , 2 1 2 0 00 22 Z ZRkb ∂ ∂ −= + ( ) 00 00 22 , ZR ZRkc ∂∂ ∂ −= + . (7) For a symmetric location of the plasma column with respect to the plane Z=0 с=0. Then the variables r,z can be separated straightforward. To perform the separation of variables in the equation (6) with arbitrary location of the column, we will make, as in [3], the change of variables: 22 1 , 1 α α α α + −=′ + +=′ zrzzrr , (8) where 2,1αα = , 1 2 2,1 +    −±−= c ba c baα . (9) This change not only makes diagonal the expression 2 +k in (6), but also provides for the absence of mixed derivatives with respect to new variables. After this change we obtain for +E the equation (6) in which one should make the following substitutions: zzrr ′→′→ , , ( ) 001 RiR −→ α , ( ) 00 1 RiRi α+→− , 0,,,00 →→→′→ cqbpakk , where ( )( )ααα cbap +++= 221 , ( )( )ααα cbaq −++= 221 , (10) ( ) 2 0 22 0 1 kk α+=′ . The solution of the modified equation (6) may be found by separating the variables: ( ) ( ) ( )[ ]2/exp νηµξηξ +−∝+ nHHE " , (11) where nHH ," are the Hermite functions, rp ′= 4ξ , zq ′= 4η , 0 4 Rp i−= αµ , 0 4 1 Rp iαν += . (12) The frequencies of natural MHD oscillations are found from the dispersion equation ( ) ( )12122 0 +++=′ nqpk " . (13). 3. Analytic solution Consider the simplest solution of the equation (13) for a single ion species at Ciωω≤ . Then in the zeroth approximation we will have for the frequency of natural oscillations the formula ±−== ± Ci A R vm ω ωω 2 0 22 0 2 1 2 0 222 2 0 22 2 R vm R vm A Ci A +      ω (14), where ( ) ( ) ( )00000 ,, ZRRcZRv piCiA ωω= . To the order of magnitude we have Ciωω ~0 at 1~ 0R mc piω (15) In this case the condition 22 −± << kk holds if 1~ 0 >>Rmaak ppF , where pa is the characteristic distance over which the density varies, i.e., this condition takes place for tokamaks with large dimensions. According to (13), the correction to the frequency (14) taking into account the finite values of the transverse wavenumbers is ( ) ( )[ ] × +−+ − ≈∆ 2 00 2 0 0 2 0 1 1 2 1 αωωωω ωω ω ω Ci CiAv ( ) ( )[ ]1212 +++× nqp " . (16) Using the expressions (10), we get for the correction (16) the order-of-magnitude estimate [ ] 11212~ 0 0 <<+++∆ pma Rn" ω ω . (17) Note that for the high number modes with 1>>+ n" the dispersion equation (13) was obtained by this method in the report [3], and for the first modes with 1~,n" in the report [4]. Taking into account the poloidal magnetic field of the tokamak will lead to the correction to the MHD wave frequency that is less than the correction (16): ϕω δω B B ma R ZR p ,0 0 ~ . (18) The consideration of the right-hand-polarized wave with this technique will lead to similar results. The work was performed under support from the Fund for Fundamental Research of the Ministry of Education and Science of Ukraine, contract 2.4/700 References 1. V.D.Yegorenkov, K.N.Stepanov. 16th EPS Conference on Controlled Fusion and Plasma Physics, Venice, March 13- 17, 1989, Contr. Pap. V. 3, pp. 1207-1210. 2. V.D.Yegorenkov, A.R.Polevoy, K.N.Stepanov, S.F.Sharapov. 18th EPS Conf. on Controlled Fusion and Plasma Physics, Berlin, June 3-7, 1991, Contr. Pap. Pt.IV, V. 15c, pt. IV, pp. 33-36. 3. V.D.Yegorenkov, K.N.Stepanov. 1994 Int. Conf. on Plasma Physics, Foz do Iguacu, Brazil, October 31 – November 4, Contr. Pap. V. 2, pp. 187-190. 4. V.Ye. D’yakov, V.D.Yegorenkov, K.N.Stepanov. In coll. Physical Phenomena in Solids (To the 190-th anniversary of Kharkov University). Materials of the 2nd Conference (1-3 February 1995), Kharkov, 1995, p. 26 (In Russian). UDC 533.9 V.Ye.D’yakov , I.À.Girka, V.D.Yegorenkov, K..N.Stepanov* FAX: 35-26-64. E-mail: stepanov@ipp.kharkov.ua References

Magnetohydrodynamic wave spectra in large tokamaks with non-circular cross section of magnetic surfaces

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