Drift-kinetic equations in magnetized current-carrying plasmas

Drift-kinetic equations in magnetized current-carrying plasmas

Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untr...

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Дата:2023
Автори: Grishanov, N.I., Azarenkov, N.A.
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Мова:English
Опубліковано: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2023
Назва видання:Problems of Atomic Science and Technology
Теми:
Relativistic and nonrelativistic plasma electronics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/196168
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Цитувати:Drift-kinetic equations in magnetized current-carrying plasmas / N.I. Grishanov, N.A. Azarenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 27-32. — Бібліогр.: 13 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1961682023-12-11T13:48:44Z Drift-kinetic equations in magnetized current-carrying plasmas Grishanov, N.I. Azarenkov, N.A. Relativistic and nonrelativistic plasma electronics Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories. Кінетичні моделі замагніченої плазми зі струмом розроблені для вивчення впливу ефектів магнітного дрейфу на взаємодію хвиля-частинка у токамаках та циліндричних плазмових системах із гвинтовим магнітним полем. Отримано дрейфово-кінетичні рівняння для збурених функцій розподілу захоплених і пролітних частинок у двовимірній осесиметричній тороїдальній плазмі з урахуванням їх баунс-коливань і кінцевої ширини орбіт їхніх бананових траєкторій. 2023 Article Drift-kinetic equations in magnetized current-carrying plasmas / N.I. Grishanov, N.A. Azarenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 27-32. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 52.55.Fa, 52.50.Qt DOI: https://doi.org/10.46813/2023-146-027 http://dspace.nbuv.gov.ua/handle/123456789/196168 en Problems of Atomic Science and Technology 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 Relativistic and nonrelativistic plasma electronics
Relativistic and nonrelativistic plasma electronics
spellingShingle Relativistic and nonrelativistic plasma electronics
Relativistic and nonrelativistic plasma electronics
Grishanov, N.I.
Azarenkov, N.A.
Drift-kinetic equations in magnetized current-carrying plasmas
Problems of Atomic Science and Technology
description Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories.
format Article
author Grishanov, N.I.
Azarenkov, N.A.
author_facet Grishanov, N.I.
Azarenkov, N.A.
author_sort Grishanov, N.I.
title Drift-kinetic equations in magnetized current-carrying plasmas
title_short Drift-kinetic equations in magnetized current-carrying plasmas
title_full Drift-kinetic equations in magnetized current-carrying plasmas
title_fullStr Drift-kinetic equations in magnetized current-carrying plasmas
title_full_unstemmed Drift-kinetic equations in magnetized current-carrying plasmas
title_sort drift-kinetic equations in magnetized current-carrying plasmas
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate 2023
topic_facet Relativistic and nonrelativistic plasma electronics
url http://dspace.nbuv.gov.ua/handle/123456789/196168
citation_txt Drift-kinetic equations in magnetized current-carrying plasmas / N.I. Grishanov, N.A. Azarenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 27-32. — Бібліогр.: 13 назв. — англ.
series Problems of Atomic Science and Technology
work_keys_str_mv AT grishanovni driftkineticequationsinmagnetizedcurrentcarryingplasmas
AT azarenkovna driftkineticequationsinmagnetizedcurrentcarryingplasmas
first_indexed 2025-07-17T00:40:02Z
last_indexed 2025-07-17T00:40:02Z
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fulltext ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 27 https://doi.org/10.46813/2023-146-027 DRIFT-KINETIC EQUATIONS IN MAGNETIZED CURRENT-CARRYING PLASMAS N.I. Grishanov 1,2 , N.A. Azarenkov 1,3 1 V.N. Karazin Kharkiv National University, Kharkiv, Ukraine; 2 Ukrainian State University of Railway Transport, Kharkiv, Ukraine; 3 National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine E-mail: n.i.grishanov@gmail.com Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equa- tions are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories. PACS: 52.55.Fa, 52.50.Qt INTRODUCTION To study the influence of finite Larmor radius ef- fects on the resonant wave-particle interactions in mag- netized plasmas one should use the kinetic dielectric tensor accounting for the particle drifts from the mag- netic surfaces under their moving along the magnetic field lines. Corresponding kinetic wave theory [1, 2] should be based on the solution of the linearized Vlasov equations or the drift-kinetic equations [3 - 6] for the perturbed distribution functions of ions and electrons. Usually, the response of a collisionless plasma to global electromagnetic perturbations of an axisymmetric toroidal equilibrium is described by the perturbed distri- bution functions of charged particles expressed in terms of their linearized guiding center Littlejohn Lagrangian [4, 5], adopting a variational formulation for the guiding center motion and drift effects. In this work, the drift-kinetic equation is derived di- rectly from the Vlasov equation using the Fourier ex- pansion of the perturbed distribution functions of plas- ma particles over the polar angle (gyration angle) in velocity space for low-frequency wave processes in axisymmetric tokamaks with circular magnetic surfaces and large aspect ratio, up to first-order corrections in the magnetization parameters. 1. PLASMA MODEL To describe the stationary magnetic field 0H in an axisymmetric tokamak with a circular cross-section, we use the system of quasi-toroidal coordinates ( , , )r   , associated with cylindrical ( , , )z  as follows: 0 cosR r   ,   , sinz r   , (1) where 0R is the large radius of the torus, Fig. 1, deter- mined by the radius of the magnetic axis; r is the radius of a magnetic surface (magnetic surface equation: r const , 0 r а  , a is the small plasma radius);  is the poloidal angle, measured by small azimuth in the cross-section of the torus, 0 2   ;  is the toroidal angle measured along the major azimuth in the horizon- tal section of the torus, 0 2   . In an axisymmetric 2D tokamak, the plasma config- uration is homogeneous in  . As a result, the equilibri- um field 0H and other steady-state plasma-field param- eters do not depend on  . Moreover, for this reason, the single-mode harmonic approximation is valid for the perturbations, ~ exp( )in , where integer n is the toroi- dal mode number. In contrast to the usual notation, the angle  is measured from the outer side of the torus, shorting the formulas related to trapped particles. The components of the equilibrium magnetic field  0 0 0 0, ,rH H H H for the considered plasma model are determined from the conditions of the absence of magnetic charges 0 0 H , and have the form: 0 0rH  , 0 0 ( ) 1 cos H r H       , 0 0 ( ) 1 cos H r H       , (2) where 0/r R  is the inverse aspect ratio of a consid- ered toroidal magnetic surface r const ; and 0 ( )H r , 0 ( )H r are the amplitude values of poloidal and toroi- dal magnetic fields there (at / 2  ). Fig. 1. Cylindrical ( , , )z  and quasi-toroidal ( , , )r   coordinates, describing an axisymmetric tokamak with circular magnetic surfaces In this case, the unit vector along the 0H field does not depend on the poloidal angle and has projections:   00 0 2 2 2 2 0 0 0 0 0 0, , 0, , HH h h H H H H                     H h H . (3) When describing the plasma particle distribution functions in velocity space ( , , )F t r v , it is convenient to use the orthogonal normal, binormal, and parallel velocity components: 28 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 1 2 3 ||cos sin             v n b h n b h , (4) where n is the unit vector normal (radial) to a magnetic surface, b h×n is the binormal to n and h, Fig. 1; || and  are the parallel and perpendicular velocity components, respectively;  is the gyration angle (or the polar angle in velocity space). Using the small perturbation method, the plasma particle distribution functions can be found as ( , , ) ( , ) ( , , )F t F f t   r v r v r v , (5) where ( , )F r v and ( , , )f t r v are the steady-state and the perturbed distribution functions of ions and/or elec- trons (i,e), respectively, under the condition: ( , , ) ( , )f t F r v r v . The steady-state functions ( , )F r v must take into account the presence of a stationary equi- librium current in tokamaks, 0 0 0j j    j e e , dia- magnetic currents and be self-consistent with the con- fining magnetic field, 0 0 0H H    H e e . In the gen- eral case, ( , )F r v can be defined in the velocity space ||( , , )   by the Fourier expansion i || ||( , ) ( , , , , , ) ( , , , , )F F r F r e                    r v .(6) The harmonics 0 ||( , , , , )F r     allow us to define the main contribution of plasma particles to the field- aligned stationary current (parallel to 0H ), , 0|| 0 || 0 || || 0 2 ( . ) e i j q d F d                     j h . (7) Whereas the harmonics 1 ||( , , , , )F r      are neces- sary to describe the diamagnetic currents, connected with the Larmor radius gyration of plasma particles along the helical magnetic field lines and the gradients of their density and temperature. Further, we consider the simplest pressureless plas- ma model of a 2D tokamak, where the equilibrium current must be force-free, i.e., the current density must be parallel to the magnetic field: 0 0j H or 0 0 0 j H . Describing such plasma models, it is assumed that the steady-state current is created by electrons having the velocity 0 0e  , whereas 0 0i  for heavy ions. In this case, according to Ampere’s law,  0 0|| 0 0 4 4 ej n e c c       H h , (8) where 2 0 0 04 e H c n e      . (9) Here 0en is the electron density, e is the elementary charge, and the magnetic field parameter 2 is equal to 2 0 2 0( cos ) h rR hd r dr h R r             , (10) h and h are the poloidal and toroidal projections of unit vector h along a helical H0-field line, Eq. (3). The steady-state distribution functions for such 2D current-carrying toroidal plasma can be done by the following harmonics, satisfying the Vlasov equation: 2 2 || 0 0 ||0 0 3/2 3 2 2 ( ) exp 1 2M T T T n F F                                  , (11) 2 2 || 0 || 00 1 1 0 02 2 0 0 2 2 cos , T T h hF F F i i F i F r r R                                 || 00 1 1 02 0 0 2 sin T hF F F F r R                        , (12) where 2 2 0 0 0 1 cos 1 cos H Hq M c                 , 2 02 T T M      (13) are the cyclotron (Larmor) frequency and the squared thermal velocity of species i,e plasma particles with the mass M , charge q , and temperature 0T  . Further, we assume that the drift-current velocity is much less than the thermal velocity, 0 T   . 2. DRIFT-KINETIC EQUATION The linearized Vlasov equation [7, 8] for the per- turbed plasma particle distribution functions ||( , , , , , , )f t r      in the considered tokamak plasma model can be rewritten in the explicit form as || 0 0 3 2 ( cos ) hf f h h R f h t r h r R r                               || || 0 0 sin ˆ 2( cos ) cos h hh f f Vf R r r R r                    (14) || 0 0 sin ˆsin cos cos h hf h f Vf r R r R r                       2 2 2 2 2 2 || || 0 0 cos cos ( cos ) cos h h h h f r r R r R r                                    2 2 2 || || || 0 0 cos sin ˆ ˆcos cos cos h h hf f Vf Vf r r R r R r                              ||0 0 0 sinsin2 ˆ 2 ( cos ) cos hh h R f h h Vf r h r R r R r                                  0 || 0 0 cos2 sin ˆ 2 cos ( cos ) h h h R f Vf h h R r r h r R r                                    ||2 3 1 2 1 || ||3 1 2 1 2 1ˆcos 1ˆsin . e F F H F E E VF E H M c c H F F H F E VF E H c c c                                                           Here the differential operator || || ˆ f f Vf             (15) has been used in velocity space to shorten the kinetic equations; the index i,e of particle species is omitted in Eqs. (14), (15). The connection between the projec- tions of vector values  , , ,A E H v j in the quasi- toroidal coordinates ( , ,rA A A  ) and their projections ( 1 nA A , 2 bA A , 3 hA A ) on the orts of an orthogo- nal trihedron generated by a magnetic field 0H , i.e. into the unit vectors n, b, h (see Fig. 1), is given by the for- ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 29 mulas 1 rA A  A n , 2A h A h A      A b , 3A h A h A      A h . (16) The linearized Vlasov equation, Eq. (14), is suitable for studying a wide class of electrodynamic problems in 2D tokamaks with circular magnetic surfaces, provided that the particle Larmor radius /T   is the smallest among all the characteristic dimensions-scales of the problem in the direction perpendicular to the equilibri- um magnetic field, using the periodicity of distribution functions on the polar angle  in velocity space. In this case, as for other axisymmetric plasma models in the cylindrical current-carrying plasmas [9 - 11] or elongat- ed tokamaks [12, 13], the perturbed distribution func- tions can be expanded into the Fourier series in  : || , ||( , , , , , , ) ( , , , )exp( )f t r v v f r v v i t in i              . As a result, Eq. (14) can be reduced to the set of coupled equations for harmonics ||( , , , )f r v v  , 1 ||( , , , )lf r v v  and 2 ||( , , , )f r v v  : || 0 0 3 2 ( cos ) h h h R i f i f i h f r h r R r                          || || 0 0 sin ˆ 2( cos ) cos h ih nfh f Vf R r r R r                     1 1 1 1 02 2( cos ) h f f h n f f i r R r                    2 2 2 2 2 2 || || 0 0 cos1 cos 2 ( cos ) cos h h h h r r R r R r                               1 1( 1) ( 1)f f     || 1 1 0 sin ˆ( ) 2( cos ) ih V f f R r             2 ||1 1 1 1 ˆ 2 2 hf f V f f r r             (17)   2 2 || || 1 1 0 0 cos sin ˆ 2( cos ) 2 ( cos ) h ih V f f R r R r                   1 1 2 2 0 sin ˆ( 1) ( 1) ( ) 4( cos ) h f f V f f R r                 || 0 2 2 0 ( 2) ( 2) 4 ( cos ) h h h R i h f f r h r R r                           0 2 2 0 ˆ( ) 4 ( cos ) h h h R i h V f f r h r R r                           || 2 2 0 sin ( 2) ( 2) 4( cos ) h f f R r                       1 11 2 3 1 1 || || 2 1 1 1 1 13 2 1 1 1 || 1 2 1 1 ˆ 2 2 ( 1) ( 1) 2 ˆ 2 2 1 ( 1) ( 1) . 2 F Fe F E H E V F F M c i E H F F c F FH E H i F i i V F F c c E H F F c                                                                   As is well known, by the plasma particle distribution functions one can estimate, in the scope of kinetic wave theory, the perturbations of particle densities and current density components involved in Maxwell’s equations for the perturbed electromagnetic fields ( , )E H in a considered plasma model. However, we have no exact solution for Eq. (17) in the general case. It is necessary to apply the approximation methods using the small parameters (e.g., the smallness of the Larmor radius of plasma particles or magnetization parameters) and the restrictions on the wave frequencies . As for the magnetization parameters, all of them, as usual, are inversely proportional to the cyclotron fre- quency, ,|| / 1X     , where X characterize the spatial scales of the inhomogeneity of particle density 0ln( ) /n n r     , temperature 0ln( ) /T T r     , wave numbers kr, /k m r  , 0/k n R  , where m and n are the poloidal and toroidal eigenmode numbers, respectively. It should be noted that, in contrast to the case of a straight equilibrium magnetic field (H0=H0ez, where ||0x y zk k  k e e e ) in a current-carrying plasma confined by a helical magnetic field, the wave vector ||r bk k k  k n b h always has three compo- nents, where the parallel ||k and binormal bk projections of k are defined as || 0 h nh m k r R   and 0 b h m h n k r R    . (18) Moreover, evaluating the main contribution of plas- ma particles to the perturbed longitudinal ( 3j ) and transverse 1( j , 2 )j current density components, there is enough to find the harmonics 0,f  and 1,f  : 1 (1) ( 1)nj j j j     j n , 2 ( 1) (1)[ ]bj j i j j    j b . (19) 1 2, , ,.. 3 || || 0, 0 2 e i i hj j q d f d                   j h , 1 2, , ,.. 2 ( ) || , 0 e i i j q d f d                , 1  . In our previous papers [9 - 12] we have solved the Vlasov equations for harmonics 0,f  and 1,f  in the simplest case, i.e., in the zeroth-order over the magneti- zation parameters, neglecting the drift effects propor- tional to Larmor radius /T   . In this case, the har- monics 0,f  and 1,f  become independent of each oth- er, satisfying first-order differential equations with three partial derivates with respect to  and || : || 4 cos 1 cos 2 1 cos c h hf nq i f i f i r                            0 sin ˆ 2 (1 cos ) hr q f Vf Q q r R              , 0, 1  , (20) where q is the tokamak safety factor, see Eq. (22), 0/r R  , and Q terms for the equilibrium (Maxwel- lian) distribution functions are equal to 30 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 0 3 || 0 M q Q E F T  ,  1 1 2 0 M q Q E iE F T    , (21) 2 2 ||0 2 1.5 2 exp ( ) M T T n F               , 0 rh q R h    . (22) Eqs. (20)-(22) are suitable to study the wave-particle interaction accounting for the Cherenkov, cyclotron, and bounce resonances for both the trapped and untrapped particles in 2D axisymmetric tokamaks. In this paper, we derive the drift-kinetic equation for 0,f  in the first order over the magnetization parameters for the low-frequency perturbations, i   . After substituting 0 in Eq. (17), the equation for f0 can be rewritten in the form || || 00 0 0 0 0 sin ˆ cos 2( cos ) h ih nff h i f Vf r R r R r                        1 1 1 1 02 2( cos ) h f f h n f f i r R r                  (23) 2 2 2 2 2 2 || || 1 1 0 0 cos1 cos ( ) 2 ( cos ) cos h h h h f f r r R r R r                                || 1 1 0 sin ˆ( ) 2( cos ) ih V f f R r       2 ||1 1 1 1 ( ) ˆ( ) 2 2 hf f V f f r r           2 2 || || 1 1 1 1 0 0 cos sin ˆ( ) ( ) 2( cos ) 2 ( cos ) h ih V f f f f R r R r                   || 2 2 0 sin ˆ( ) 4( cos ) 2 hh h V f f i h R r r h                        ||0 2 2 2 2 0 0 sin ( ) ( ) ( cos ) 2( cos ) hh R f f f f r R r R r                0 2 2 0 ˆ( ) 4 ( cos ) h h h R i h V f f r h r R r                                     1 10 1 2 3 1 1 || || 1 12 1 2 1 1 ||1 1 1 2 1 1 1 ˆ 2 2 1 2 2 ˆ . 2 2 F Fq F E H E V F F M c F FE E H F F i c H i i V F F E H F F c c                                                     The influence of drift effects on the plasma particle distribution functions is described by the harmonics 1 ||( , , , )f r v v  and 2 ||( , , , )f r v v  , connected with 0 ||( , , , )f r v v  in first-order magnetization parameters as 2 2 || ||0 1 1 0 0 0 0 0 2 01 02 ˆ ˆcos 2 , T h hf f f i i Vf i Vf r r R F qHqE i i F M Mc                             ||0 1 1 0 0 0 0 0 0 0 1 02 02 ˆsin 2 , T h hf h n f f i f Vf r R R F qHqE F M Mc                                 2 2 0 0 0 sin ˆ 4 h f f i Vf R      , (24) 0 2 2 0 0 ˆ 4 ( cos ) h h h R f f i h Vf r h r R r                         . As one can see the exact drift-kinetic equation for 0f , after substituting Eqs. (24) into Eq. (23), is compli- cated, having four partial derivatives in ||, , ,r v v  : 0 0 0 0 0 0 || || f r f V f f i f i nf t t t r t                                      1 2 3( , , )Q E E E , (25) where     2 || 2 2 2 2 || ||2 0 0 0 cos 2 2 2 2 h h h h t r r rR                    ,  2 2 2 22 |||| 0 0 0 0 0 2 cos 2 2 ( cos ) h h hh h t R r R r r R r                        2 2 || 0 0 0 cos 2 2 ( cos ) h R R r          ,  2 2 || 0 0 sin 2 2 hr t R          , (26) 2 || 0 0 0 sinsin 2( cos ) 2 ( cos ) h hV h t R r r R r                , 2 2 || 0 || 32 2 ( ) 3 2 2 M M n T T T q qF Q F E M M                           2 2 2 2 |||| 2 1 22 2 M T q h h E H F E c Mr              (27)   2 2 2 2 2 2 2 || 22 2 0 0 cos 2M M T T q F q F E h h E MR M r                        2 2 2 1 1 || 12 2 0 0 0 sin 2 cos M M T T qh hq F E ih nE F E MR M r R r                          . It should be noted that the right-hand side of Eq. (25) is written in Eq. (27) for the case when the in- fluence of the equilibrium current on the magnetic drift effects (proportional to   ) can be neglected. In con- trast with initial Vlasov equations, where the plasma particle distribution functions depend on the three ve- locity variables ( || , ,   ), the drift-kinetic equations are written for the particle distribution functions aver- aged over the gyrophase angle  in velocity space, depending only the || and   velocities relative to H0. As a result, the drift-kinetic equations are simpler and more convenient for solutions in the low-frequency range. 3. TRAJECTORIES OF UNTRAPPED AND TRAPPED PARTICLES The number of partial derivatives in Eq. (25) can be reduced after introducing the new conventional varia- bles associated with the corresponding invariants of motion of charged particles in a considered plasma model. As usual, the conservation integrals (the motion invariants) should be connected with the particle energy ( 2 2 || const   ), magnetic moment ( 2 0/ H const  ), and, so-called, longitudinal invariant. According to Eqs. (26), in the zeroth approximation in Larmor radius corrections, we can introduce the new ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 31 variables  and  (nondimensional magnetic moment) instead of || and  as 2 2 ||    ,   2 2 2 || 1 cos            , (28) where 0/r R  is the inverse aspect ratio of a torus. Since the tokamak magnetic field H0 is nonuniform and has a minimum, all plasma particles should be sepa- rated into two groups of, so-called, untrapped and trapped particles. Such a separation [8, 12, 13] can be done by the inequalities for and  : 0 1    ,    untrapped particles, 1 1      , tt    trapped particles, analyzing the condition ||, 1- (1 cos ) 0s s       , where 1s distinguishing the positive and negative parallel velocity relative to H0. Here the stop (reflection, turning) points of trapped particles are defined as 1 arccost             . (29) However, in the first approximation in the Larmor ra- dius corrections, we should take into account that the toroidal drift of charged particles leads to the deflection of their trajectories from magnetic surfaces. By the char- acteristical equations for / t  and /r t  in Eqs. (26) we can define the radial coordinate of untrapped and trapped particles, moving along the H0-field lines, respec- tively, as , , ( , )u s u sr r r r   and , . ( , )t s t sr r r r   . Here r is the radius of the considered magnetic surface, 2 2 ||, , 0 ||, ( , ) 0,5 ( , )( ) ( , ) sin ( ) ( , ) s u s s r rq r r r d r r                   , (30) 2 2 ||, , 0 ||, ( , ) 0,5 ( , )( ) ( , ) sin ( ) ( , ) t s t s s r rq r r r d r r                   , (31) where, under 1  , ||, 0 ( , ) 1- 1 coss r r s R             , (32) 2 2 0 ( , ) 1 cos r r R            . (33) Projections of the typical guiding-center trajectories of untrapped and trapped particles on the transverse cross-section of the moderate magnetic surfaces in tokamaks, r=const (dashed circle lines, r = 0.7a), are plotted in Figs. 2 and 3, respectively. Fig. 2. The trajectories of the untrapped particles in an axisymmetric tokamak with circular magnetic surfaces As for positively charged untrapped particles (ions), moving along the H0-field lines, with s=+1, due to mag- netic drift they are shifted (red line) to a region of a weaker magnetic field (i.e., outward from the magnetic surface). While ions moving against a H0-field, with s=-1, drift to a region of a stronger magnetic field (i.e., inside the magnetic surface, green line). Both trajecto- ries in Fig. 2 are plotted for untrapped ions, starting at the inner part of the magnetic surface, r=const, under 0.18  , 0.5  and Larmor radius 0/ 0.04   cm. The main feature of the drift deflections of both the untrapped and trapped particles is that they are deter- mined by the particle Larmor rotation in the poloidal magnetic field 0H  (r), since 0 0/ q   , where 0 0 / ( )qH Mc   is the Larmor (cyclotron) frequency of charged particles in the 0H  -field, that depends sig- nificantly on r. After integration in Eq. (30): , 0 0 0 4 ( , ) 1 cos 1 1 cos 3 u s s r r r r R R                         0 0 4 1 1 1 r r R R                   . (34) As a result, the maximal deflection of untrapped par- ticles should take place, in our notation, at the external part of the considered magnetic surface, i.e., at 0  : max , , 0 0 0 4 ( ) ( ,0) 1 1 1 3 u s u s s r r r r r r R R                       0 0 4 1 1 1 r r R R                   . (35) It should be noted that the features of the drift trajec- tories of negatively charged untrapped electrons are opposite with respect to ions, i.e., the field-aligned elec- trons, with s=+1, drift into the inner part of the magnetic surfaces and vice versa. Fig. 3. The trajectories of the trapped particles in an axisymmetric tokamak with circular magnetic surfaces In contrast to untrapped particles, the trajectories of the trapped particles have the ‘banana’-forms. Oscillat- ing between the stop-points, both the positively and negatively charged trapped particles change the sign of parallel velocity, 1s   , during one bounce period. As a result, the banana-orbit widths of the trapped ions and electrons are doubled due to their drift both inward from 32 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) the surface for s=+1 and outward for s=-1. The deflection of trapped particles from the magnet- ic surface can be determined by a simple expression after integration in Eq. (31): , 0 0 0 4 ( , ) 1 cos 1 1 3 t s s r r r r R R                       . (36) Thus, the maximal deflection of the trapped particles max , ,( ) ( ,0)t s t sr r r r (i.e., half of their maximum orbit- width in the equatorial plane of the torus, at 0  ) is estimated by max , 0 0 0 4 ( ) 1 1 1 3 t s s r r r r R R                     . (37) The banana trajectories of trapped particles in Fig. 3 are plotted at different levels of the nondimensional magnetic moment  for particles starting at stop-points on the magnetic surface, shown as a dashed line. The banana sizes and the values of the stop-points of trapped particles depend substantially on  , according to Eq. (29). As can be seen, strongly trapped particles (un- der large  ) have smaller sizes and orbit-widths. CONCLUSIONS The pressureless 2D toroidal current-carrying plas- ma model has been described to develop the kinetic theory of low-frequency oscillations in axisymmetric tokamaks with circular magnetic surfaces and large aspect ratios. The steady-state distribution function of plasma electrons and equilibrium magnetic field are self-consistent, satisfying Maxwell’s equations. If the toroidal magnetic field is changed to longitu- dinal cylindrical z-projection, 0 0zH H  , zh h  , and 0R  , our 2D toroidal model is transformed into a cylindrical magnetized current-carrying plasma model in the helical magnetic field. The drift-kinetic equations for the perturbed distri- bution functions of the trapped and untrapped (passing, circulating) particles are derived accounting for the magnetic drift effects in the first-order over the magnet- ization parameters, proportional to the Larmor radius gyration of ions and electrons moving along the equilib- rium magnetic field lines. The characteristical equations in the drift-kinetic equations allow us to estimate the finite orbit-widths of the ‘banana’-trajectories of both the trapped and untrapped particles. Analytical expressions are derived for the particle deflections from the magnetic surfaces. Since the toroidal drift deflections of untrapped and trapped particles are defined by the poloidal magnetic field, the corresponding orbit-widths are much larger than their Larmor radius in H0-field. REFERENCES 1. F. Porcelli, R. Stankiewicz, W. Kerner, H.L. Berk. Solution of the drift-kinetic equation for global plasma modes and finite particle orbit widths // Physics of Plasmas. 1994, v. 1(3), p. 470-480. 2. P.J. Catto, J. Lee, A.K. Ram. A quasilinear operator retaining magnetic drift effects in tokamak geometry // J. Plasma Physics. 2017, v. 83, p. 905830611-10. 3. B.B. Kadomtsev, O.P. Pogytse. Turbulent processes in toroidal systems. New York: Review of plasma physics / M.A. Leontovich, ed. “Consultants Bu- reau”. 1970, v. 5, p. 249-400. 4. R.G. Littlejohn. Variational principles of guiding centre motion // J. Plasma Physics. 1983, v. 29, p. 111-121. 5. A.B. Mikhailovskii. Instabilities of a confined plas- ma. Bristol: “IOP Publishing Ltd”. 1998, p. 462. 6. A. Smolyakov, X. Garber. Drift kinetic equation in the moving reference frame and reduced magneto- hydrodynamic equations // Physics of Plasmas. 2010, v. 17(4), p. 042105-9. 7. N.I. Grishanov, F.M. Nekrasov. Influence of to- roidicity on the longitudinal permittivity of a mag- netized plasma // Fizika plazmy. 1987, v. 13(1), p. 113-117 (in Russian). 8. N.I. Grishanov, F.M. Nekrasov. Dielectric constant of a toroidal plasma // Sov. J. Plasma Physics. 1990, v. 16(2), p. 129-134. 9. A.G. Elfimov, C.A. de Azevedo, A.S. de Assis, N.I. Grishanov, F.M. Nekrasov, I.F. Potapenko, V.S. Tsypin. Alfvén wave heating and current drive analysis in magnetized plasma structures // Brazilian Journal of Physics. 1995, v. 25(3), p. 224-240. 10. N.I. Grishanov, N.A. Azarenkov. Ion-cyclotron ab- sorption of fast waves in a cylindrical current-carrying plasma // Problems of Atomic Science and Technolo- gy. Series “Plasma Physics”. 2021, № 1, p. 36-40. 11. N.I. Grishanov, N.A. Azarenkov. On the fast waves in a cylindrical current-carrying plasma // Physics of Plasmas. 2021, v. 28(4), p. 042106-10. 12. N.I. Grishanov, C.A. de Azevedo, A.S. de Assis. Longitudinal permittivity of a tokamak plasma with elliptic and circular magnetic surfaces // Physics of Plasmas. 1998, v. 5(3), p. 705-715. 13. N.I. Grishanov, N.A. Azarenkov. About the cyclo- tron resonance conditions in magnetized current- carrying plasmas // Physics of Plasmas. 2019, v. 26(7), p. 122501-9. Article received 25.05.2023 ДРЕЙФОВО-КІНЕТИЧНІ РІВНЯННЯ У ЗАМАГНІЧЕНІЙ ПЛАЗМІ ЗІ СТРУМОМ М.І. Гришанов, М.О. Азарєнков Кінетичні моделі замагніченої плазми зі струмом розроблені для вивчення впливу ефектів магнітного дрейфу на взаємодію хвиля-частинка у токамаках та циліндричних плазмових системах із гвинтовим магні- тним полем. Отримано дрейфово-кінетичні рівняння для збурених функцій розподілу захоплених і проліт- них частинок у двовимірній осесиметричній тороїдальній плазмі з урахуванням їх баунс-коливань і кінцевої ширини орбіт їхніх бананових траєкторій. https://www.researchgate.net/journal/Physics-of-Plasmas-1089-7674

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