Recent progress in models for electron cyclotron current drive

The recent progress in electron cyclotron current drive calculations with the adjoint technique is reviewed. The main attention is focused on such points as parallel momentum conservation in the like-particle collisions and the relativistic effects which are especially important for high-temperature...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2012
Hauptverfasser:	Marushchenko, N.B., Beidler, C.D., Maassberg, H.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Нагрев плазмы и поддержание тока
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/109096
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Recent progress in models for electron cyclotron current drive / N.B. Marushchenko, C.D. Beidler, H. Maassberg // Вопросы атомной науки и техники. — 2012. — № 6. — С. 38-42. — Бібліогр.: 23 назв. — рос.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-109096
record_format	dspace
spelling	irk-123456789-1090962016-11-21T03:02:32Z Recent progress in models for electron cyclotron current drive Marushchenko, N.B. Beidler, C.D. Maassberg, H. Нагрев плазмы и поддержание тока The recent progress in electron cyclotron current drive calculations with the adjoint technique is reviewed. The main attention is focused on such points as parallel momentum conservation in the like-particle collisions and the relativistic effects which are especially important for high-temperature plasmas. For moderate-temperature plasmas, also the finite collisionality effects become to be important. The effectiveness and accuracy of the developed numerical models are demonstrated by ray-tracing calculations. Сделан обзор недавних достижений в методах вычисления электронно-циклотронного тока увлечения. Основное внимание направлено на учёт сохранения продольного импульса в операторе столкновений, а также учёт релятивистских эффектов, существенных в высокотемпературной плазме. В плазме относительно небольшой температуры также эффекты конечной столкновительности становятся существенными. Эффективность и точность развитых численных моделей продемонстрированы результатами, полученными с помощью метода лучевых траекторий. Зроблено огляд недавніх досягнень в методах обчислення електронно-циклотронного струму захоплення. Головна увага спрямована на облік збереження продольного імпульсу в операторі зіткнень, а також на облік релятивістських ефектів, значних у високотемпературній плазмі. У плазмі відносно невеликої температури ефекти кінцевої зіткненості також стають істотними. Ефективність та точність развинутих моделей продемонстровано результатами, які були отримані за допомогою методу променевих траєкторій. 2012 Article Recent progress in models for electron cyclotron current drive / N.B. Marushchenko, C.D. Beidler, H. Maassberg // Вопросы атомной науки и техники. — 2012. — № 6. — С. 38-42. — Бібліогр.: 23 назв. — рос. 1562-6016 PACS: 52.55.-s, 52.55.Fa, 52.55.Hc, 52.55.Wq, 52.65.Ff http://dspace.nbuv.gov.ua/handle/123456789/109096 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	Нагрев плазмы и поддержание тока Нагрев плазмы и поддержание тока
spellingShingle	Нагрев плазмы и поддержание тока Нагрев плазмы и поддержание тока Marushchenko, N.B. Beidler, C.D. Maassberg, H. Recent progress in models for electron cyclotron current drive Вопросы атомной науки и техники
description	The recent progress in electron cyclotron current drive calculations with the adjoint technique is reviewed. The main attention is focused on such points as parallel momentum conservation in the like-particle collisions and the relativistic effects which are especially important for high-temperature plasmas. For moderate-temperature plasmas, also the finite collisionality effects become to be important. The effectiveness and accuracy of the developed numerical models are demonstrated by ray-tracing calculations.
format	Article
author	Marushchenko, N.B. Beidler, C.D. Maassberg, H.
author_facet	Marushchenko, N.B. Beidler, C.D. Maassberg, H.
author_sort	Marushchenko, N.B.
title	Recent progress in models for electron cyclotron current drive
title_short	Recent progress in models for electron cyclotron current drive
title_full	Recent progress in models for electron cyclotron current drive
title_fullStr	Recent progress in models for electron cyclotron current drive
title_full_unstemmed	Recent progress in models for electron cyclotron current drive
title_sort	recent progress in models for electron cyclotron current drive
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2012
topic_facet	Нагрев плазмы и поддержание тока
url	http://dspace.nbuv.gov.ua/handle/123456789/109096
citation_txt	Recent progress in models for electron cyclotron current drive / N.B. Marushchenko, C.D. Beidler, H. Maassberg // Вопросы атомной науки и техники. — 2012. — № 6. — С. 38-42. — Бібліогр.: 23 назв. — рос.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT marushchenkonb recentprogressinmodelsforelectroncyclotroncurrentdrive AT beidlercd recentprogressinmodelsforelectroncyclotroncurrentdrive AT maassbergh recentprogressinmodelsforelectroncyclotroncurrentdrive
first_indexed	2025-07-07T22:33:52Z
last_indexed	2025-07-07T22:33:52Z
_version_	1837029273089605632
fulltext	38 ISSN 1562-6016. ВАНТ. 2012. №6(82) RECENT PROGRESS IN MODELS FOR ELECTRON CYCLOTRON CURRENT DRIVE N.B. Marushchenko, C.D. Beidler, H. Maassberg Max Planck Institute for Plasma Physics, EURATOM Association, Wendelsteinstr. 1, D-17491 Greifswald, Germany The recent progress in electron cyclotron current drive calculations with the adjoint technique is reviewed. The main attention is focused on such points as parallel momentum conservation in the like-particle collisions and the relativistic effects which are especially important for high-temperature plasmas. For moderate-temperature plasmas, also the finite collisionality effects become to be important. The effectiveness and accuracy of the developed numerical models are demonstrated by ray-tracing calculations. PACS: 52.55.-s, 52.55.Fa, 52.55.Hc, 52.55.Wq, 52.65.Ff INTRODUCTION The adjoint technique proposed in Refs. [1, 2] is an advanced and convenient method for calculation of the current drive (CD) in plasmas. Moreover, only this technique is directly applicable for stellarators while the bounce-averaged Fokker–Planck model explicitly assumes axisymmetric configurations. Formally, the applicability of the adjoint technique is limited by the natural condition that the plasma response to the rf source in electron cyclotron resonance heating (ECRH) remains linear, i.e., when the density of the rf power is sufficiently low in comparison with the rate of collisional thermalization, but practically, the standard ECRH/ECCD easily satisfy this condition. The central idea of the adjoint technique is exploiting a self-adjoint properties of the linearized collision operator to express the current through the adjoint Green's function, which is proportional to the linear plasma response in presence of parallel electric field that is formally identical to a solution of the Spitzer-Härm problem [3, 4]. This technique was subsequently applied to determine the current generated by NBI [1] and RF sources [2], for the electron cyclotron current drive (ECCD) in toroidal plasmas [5], and ECCD generated by asymmetric reflecting wall (passive ECCD) in toroidal plasmas [6]. At present, the adjoint technique is commonly used for calculations of ECCD in different ray- and beam-tracing codes [7-9]. The key point of the adjoint technique is the choice of the approach. Formally, in toroidal plasmas, the adjoint 4D drift-kinetic equation (3D in tokamaks) must be solved while taking into account such a factors as the geometry, the small but finite collisionality, conservation of parallel momentum, relativity, etc. Due to toroidicity, the problem can be reduced to an easily solvable level only for two opposite limits, the highly collisional (not interesting for toroidal plasmas) and low collisionality (“collisionless”) limit, where trapped particles play essential role. In the last limit, the trapped particles do not contribute to the current drive, but produce a non-negligible drag on the passing particles. This model, which is accepted as most relevant to ECCD calculations in toroidal plasmas [5-9] usually tends to underestimate the current drive efficiency as it neglects all effects due to (barely) trapped electrons. Historically, for calculation of ECCD the linearized collision operator was simplified by high-speed-limit (hsl) approach [10-12], which does not conserve parallel momentum in like-particle collisions. This approach is rather marginal even for moderate temperature plasmas and surely not sufficient for high temperature plasmas. The collisionless Spitzer problem with parallel momentum conservation (pmc) was considered in [5] and with relativistic effects taken into account for different physical mechanism of CD generation in [6, 13, 14]. There are some specific scenarios when existence of the barely trapped electrons can also be important for generation ECCD. The effect of small but finite collisionality in ECCD was considered in Refs. [15, 16] (see also the references therein). In the present paper, recent progress in ECCD calculations is reviewed. A comparative analysis of the different approaches and their applicabilities is presented. Considering the ITER scenarios, the role of parallel momentum conservation in like-particle collisions in high-temperature plasmas is illustrated. Also the role of finite collisionality effects is discussed. 1. ADJOINT TECHNIQUE In the ray-tracing codes, the toroidal current driven on the elementary arc-length of the ray-trajectory can be cal-culated. The key value is the current drive efficiency, η = 〈j\|\|〉/〈pabs〉, where 〈j\|\|〉 = -〈∫d3u(ev\|\|δfe)〉 is the density of current and 〈pabs〉 = me0c2 〈∫d3u(γ-1)QRF〉 is the density of the absorbed RF power, which are the functions only of the flux-surface label (here, ...〈 〉 denotes averaging over the magnetic surface). The current driven by the RF source can be formally calculated (in linear approach) by solving the drift kinetic equation (DKE), which describes the line-ar response of electrons to the RF source, ( ) ( ),lin e e RF eMv f C f f∇ δ − δ = −∇ ⋅u Γ (1) where ∇\|\| = ∂/∂l is the derivative along the field-line, δfe is the distortion of the electron distribution function from the Maxwellian, feM, and Clin the linearized collision operator, ΓRF the quasi-linear diffusion flux in u-space, ∇u = ∂/∂u, and u = vγ the momentum per unit rest mass. Exploiting self-adjoint properties of Clin, it's possible to express the current drive through the convolution of the RF source with the adjoint Green's function χ, which is solution of the adjoint kinetic equation, ISSN 1562-6016. ВАНТ. 2012. №6(82) 39 0( ) ( ) ,lin eM eM e e eM th v v f C f bf v ∇ χ + χ = −ν (2) where b=B/Bmax. Find that χ(-u\|\|) is formally identical (apart from normalization) to the generalized Spitzer function in toroidal geometry. The final expression for ECCD efficiency is 3 2 3 0 0 .RFth e e RF d uev b m b d u ∇ χ⋅∫〈 〉 η = − ⋅ ν 〈 〉 ⋅∫ u Γ v Γ (3) Being quite convenient for numerical representation, this form is the usual basis for ECCD calculations in the ray-tracing codes. Most important for a precise calculation of ECCD is the model chosen for the operator Clin. Historically, the high-speed-limit (hsl) approach [10-12] was commonly accepted as the standard tool for calculation of ECCD. This approach is based on the assumption that only the supra-thermal electrons with v >> vth are involved in the cyclotron interaction. Unfortunately, in high- temperature plasmas, the energy range of electrons which make the main contribution to ECCD is not so far from the bulk, and the hsl approach fails even for highly oblique launch. 2. ANALYTICAL LIMITS In Eq. (2), different time-scales exist: while the first term is characterized by the transit time, v\|\|∇\|\| ∝ τc-1, the collision operator is characterized by the collision time Clin ∝ τc-1. For ordering Eq. (2), we take into account that the ratio / * ( ) /t c e u R uτ τ ≡ ν = ν γ ι can vary significantly. When collisionality is high, * 1ν , i.e. t cτ τ , Eq. (2) reduces to the local problem. In this case, only the 1st Legendre harmonic of χ is necessary, i.e. 1χ = ξχ and 13 1 2 1 d−χ = χξ ξ∫ (here, ξ = v\|\|/v). Then, instead of Eq. (2), it becomes sufficient to solve the 1D integro-differential equation for 1χ , 1 1 0 1 1 1 1 ˆ ˆ( ) with ( ) ( ) / .lin lin lin e eM eM th uC C C f f v χ = −ν χ = χ γ (4) Here, 1 linC is the 1st Legendre harmonic of the linearized collision operator. This is the classical Spitzer problem for calculation of the plasma conductivity which has been thoroughly studied for both non- relativistic [3] and relativistic [4]. This solution being applied to Eq. (3) gives the upper limit for current drive efficiency, but is of no practical relevance for hot plasmas in toroidal devices. In the opposite (low collisionality or “collisionless”) limit, * 1ν , the impact of the trapped particles is important. In this case, the dimensionality of the problem can also be reduced to 2D since the spatial dependence appears only due to the coupling between the pitch, 1 bξ = σ −λ with 1σ = ± and the local magnetic field, ( )b l , through the (normalized) magnetic moment, λ . By averaging Eq. (2) over the magnetic surface, the Vlasov operator is annihilated and the problem is reduced to a 2D equation (5) This is the basic model for calculation of CD in the different ray- and beam-tracing codes. The form chosen for the collision operator is very important for the solution. Note also that in this approach the problem is antisymmetric (with respect to ξ), and, as a consequence, only the antisymmetric part of the quasilinear operator contributes in the current drive calculated by the convolution Eq. (3). Representing the solution of Eq. (5) as series of the eigenfunctions Φk(ξ), ( , ; ) ( ) ( )k kk odd u F u = χ λ σ = σ Φ ξ∑ , (6) one can obtain exact solution, where the coefficients Fk(u) must be calculated as solutions of the set of 1D integro-differential equations. Following this line, the numerical solver SYNCH was developed in Ref. [6]. It is possible also to simplify Eq. (5) without significant loss of accuracy. Since the pitch-scattering of electrons is the dominating process, all terms in the collision operator apart from the Lorentz term can be approximated by only the first Legendre harmonic [5], 1, 1 1 ˆ ˆ( ) ( ) ( ) [ ( ) ( ) ],lin lin e ee eeC u L C uχ ν χ + ξ χ + ν χ (7) where L(χ) is the Lorentz operator and 1, ( )χl eeC is the first Legendre harmonic of the linearized e/e collision operator. In this approximation, Eq. (5) can be solved analytically [6], 2 1 12 0 ( , ; ) ( ) ( ), ( ) (1 ) , 2 1 31 , 4 1 c c tr u H K u b dH h f b df f b b λ χ λ σ = σ λ 〈 〉 λ λ = −λ ∫ 〈 − λ 〉 λ λ = − = 〈 〉∫ 〈 − λ 〉 (8) where fc and ftr are the fractions of circulating and trapped particles, respectively, and h(x) is the Heaviside function. A function 12 3 0( )K u d= χ λ∫ , which is proportional to the Spitzer function, must be found as the solution of a 1D integro-differential relativistic equation, 1 0 ˆ ( ) ( ) ( ) .lin tr e e c th f uC K u K u f v − ν = −ν γ (9) In this approach, only the anti-symmetrical part of the quasilinear operator can contribute in the convolution Eq. (3) which gives the driven current. Nevertheless, when applied only to the collisionless limit, this approach is sufficiently accurate. Recently, a very fast and sufficiently accurate numerical model for calculating the ECCD efficiency was developed [14]. This model is based on the solution of the integro-differential equation, Eq.(9), where the Spitzer function, K(u), is calculated with parallel momentum conservation in the e/e collisions. In order to simplify and accelerate the numerical solution, relativistic effects are accounted through a power expansion in μ−1=Te /me0c2. 40 ISSN 1562-6016. ВАНТ. 2012. №6(82) 3. LOCAL ECCD EFFICIENCY IN TOKAMAKS For comparison of the considered approaches, the local dimensionless ECCD efficiency, 3 * 02 e e abs jne T p ζ = πε , (10) (here, we use the definition and notations from Eq. (10) in Ref. [12]), is calculated for X-mode, second harmonic with the different values of N\|\| = 0.34, 0.42, and 0.5. The calculations were performed for a circular tokamak with a magnetic field B = B0/(1+ε cosθ) and ε = r/R0 = 0.2 (here, θ is the poloidal angle). The plasma parameters, ne = 2×1019 m−3, Te = 5 keV, and Zeff = 1, are chosen in such a way that the main contribution in current drive is generated by electrons with u ~ 2vth, where parallel momentum conservation starts to be important. Fig. 1. Dimensionless ECCD efficiency for X2-mode for different launch angles as a function of normalized magnetic field, nY = nωce /ω. The data are calculated with different approaches for circular tokamak (see [8]) The results of calculations are shown in Fig.1, where the local ECCD efficiency, ζ, is plotted as a function of the normalized magnetic field, which actually defines the location of the resonance line in the phase space for the given N\|\|. The calculations were performed for poloidal angle θ = 0, i.e. for the minimum of B on the given magnetic surface. One can see that ζ calculated with the hsl model, significantly differs from the pmc values. Note also that ζ* calculated by exact [6] and approximate [14] pmc models are in satisfactory agreement. 4. COMPARISON OF THE MODELS The practical importance of performing accurate calculations of the current drive can be illustrated for quite typical ECCD scenarios in ITER. In Fig. 2, the results of ray-tracing calculations for the ITER reference scenario-2 are presented [8], where the angle- scan for the equatorial launcher is depicted. Three different codes were applied: TORAY-GA [17] with hsl model, TRAVIS [18] with both hsl and pmc models, and the Fokker–Planck code CQL3D [19]. For calculations by TRAVIS, both “exact” and “approximate” numerical solvers were applied, which are based on the fully relativistic splining and the weakly relativistic polynomial fit, respectively. One can see in Fig. 2 that the results obtained by both TORAY-GA and TRAVIS with the same hsl approach applied are in perfect agreement. On the other hand, a comparison with the results obtained with the pmc model shows that the hsl model significantly underestimates the ECCD efficiency (especially for small and moderate angles) and the convergence with pmc results is observed only for very high obliqueness. For the angles which are of the main interest, i.e. 20...40°, the discrepancy between the hsl and pmc models varies from 10 to 30 %. Fig. 2. ECCD efficiency as a function of the toroidal launch angle for ITER, equatorial launcher, obtained by different codes using different approaches (see [8]) From comparison of the results obtained by TRAVIS with the pmc model and by CQL3D, one finds that these results also coincide well. It can be pointed out that the accuracy of the pmc models [6, 14] for solving Eq. (5) is well confirmed by Fokker–Planck calculations. In a gedanken experiment [7] with a smaller launch angle and absorption close to the axis, i.e., a smaller fraction of trapped electrons, a strong discrepancy (current drive differs more than in two times) between the codes with a parallel momentum conserving collision term and the widely used hsl model was obtained. All collision terms in the simple hsl do not conserve parallel momentum leading to a smaller ECCD at moderate electron velocities in the wave absorption; see Ref. [8] for more details. 5. FINITE COLLISIONALITY EFFECTS The case, when the plasma parameters and magnetic equilibrium are such that the effects of finite collisionality can be important for generation of the current drive, is most complex. In this case, dimensionality of Eq. (2) cannot be reduced and Spitzer problem must be solved accounting the “mixing” of the variables in real and momentum spaces. The code NEO2 [20] solves this problem for arbitrary collisionality and the local generalized Spitzer function necessary for ECCD can be calculated [15], which, contrary to the collisionless model Eq. (5) where only the invariants of motion necessary, depends also from the local variables. ISSN 1562-6016. ВАНТ. 2012. №6(82) 41 Apart from solution obtained by NEO2, the Spitzer function can be obtained also by momentum-correction technique [21]. To do it, the mono-energetic DKE Eq. (2) is solved (in 3D for stellarators and in 2D for axisymmetric tokamaks) by the drift-kinetic equation solver (DKES) [22]. In DKES, the distribution function is expanded in a Fourier series with respect to the poloidal and toroidal angles in Boozer coordinates and in Legendre polynomials with respect to the pitch, p = v\|\|/v. Fig. 3. Generalized Spitzer function χ vs pitch p = v\|\|/v for v/vth = 1, 2, calculated with different approaches for circular tokamak with ε = 0.13 at the poloidal angle θ = 90°. Plasma parameters: ne = 1020m-3, Te = 1.56 keV, Zeff = 1.5 In Fig. 3, the pitch dependencies of the mono- energetic Spitzer function are shown. Calculations were performed for the circular tokamak for the poloidal angle θ = 90° at the magnetic surface ε = r/R0 = 0.13. This point is interesting since the up-down asymmetry [23, 15] is well seen. Plasma parameters are chosen in such a way that the finite collisionality effects would be sufficiently pronounced. One can find also that these results are in good agreement with those obtained by NEO2 (see [15]). However, both NEO2 and DKES are too “expensive” and cannot be used as permanent tool ECCD calculations. Instead, the “off-set” approximation which significantly simplifies calculations of ECCD with finite collisionality was developed. Following Ref. [21], the “effective” circulating particle fraction can be expressed through the mono-energetic longitudinal conductivity, 33( )D ν , 33 2 3 ( ) ( ( ))( ( )) , 2 ν ν ν = 〈 〉 eff e c u D uf u u B (10) which is equivalent to the conductivity normalized to its Pfirsch-Schlütter value. In the highly collisional limit, ( ) 1eff cf ν →∞ = , and ( * 0)eff c cf fν → = is the circula- ting particle fraction for collisionless limit, considered before. The generalized Spitzer function given by Eq. (7) can be reformulated then as [16] ( , ; ) ( ; ( )) ( ).eff eff effu H u K uχ λ σ = σ λ ν (11) Here, the equation for ( )effK u coincides formally with Eq. (8) where cf and trf must be replaced by ( )eff cf u and ( ) 1 ( )eff eff tr cf u f u= − , respectively. Then * * * ( )2( ; ( )) ( ) (1 ) 3( ) ( ) eff eff c c eff eff c c f fH u H h f f δ ν λ ν = λ + −λ ν ν , (12) with eff eff c c cf f fδ = − and ( )H λ given by Eq. (8). This formulation being supported by pre-calculated the mono-energetic transport coefficient data-base has been in TRAVIS code implemented. In Fig. 4, the results of ray-tracing calculations for O2-mode in stellarator W7-X are shown. From the absorption rate is seen that contributions from both passing and trapped electrons are important. Contrary to this, in ECCD only the passing electrons contribute. If the finite collisionality effects are not negligible, the barely trapped electrons, due to their diffusion into the passing domain, create an enhancement of the distribution function for the passing particles. As consequence, driving the current is changed for those electrons which are in resonance close to the boundary passing/trapped particles that is seen in Fig. 4 (right). Fig. 4. The absorption rate, dPabs/ds (on the left) and current drive, dIcd/ds (on the right) along the tray trajectory for the O2-mode in W7-X; ne(0) = 1020 m-3, Te(0) = 2 keV, Zeff = 1.5. Left: the full line – total rate of absorption, dash-dotted – contribution from passing electrons, and dashed – from trapped ones. Right: current drive calculated in low collisionality limit (dash-dotted) and with the “off-set” model (full line) a b 42 ISSN 1562-6016. ВАНТ. 2012. №6(82) CONCLUSIONS In this paper, the different approaches necessary for calculations of the electron cyclotron current drive in plasmas with low and finite collisionalities and recent progress in numerical modelling has been reviewed. All the formulations based on the adjoint technique are oriented for usage in ray- and beam-tracing codes, which at the present time are the main tools for numerical studies of ECRH and ECCD physics. The main attention was focused on parallel momentum conservation in the like-particle collisions which is much more precise for calculations of ECCD than the high-speed-limit (especially in hot plasmas). It was shown that an accurate kinetic solution of the Spitzer problem with parallel momentum conservation in like- particle collisions is of high importance in ECCD physics and may give a significant effect. A models for ECCD calculations with small yet finite collisionalities has also been described. In particular, there is considered the numerical model which adds to the collisionless solution of the drift- kinetic equation for the parallel conductivity a simple “off-set” contribution only in the passing particle domain. The basic information needed is the effective circulating particle fraction which is equivalent to the mono-energetic parallel conductivity coefficient normalised to the Pfirsch-Schlüter value; these values can be simply interpolated from databases of mono- energetic transport coefficients calculated, e.g., with the DKES code. Both for tokamaks and stellarators, this approach is very fast and can be directly implemented in ray-tracing codes (the test-version of the “off-set” model in already implemented in the code TRAVIS). REFERENCES 1. S.P. Hirshman // Phys. Fluids. 1980, v. 23, p. 1238. 2. T.M. Antonsen and K.R. Chu // Phys. Fluids. 1982, v. 25, p. 1295. 3. F.L. Hinton and R.D. Hazeltine // Rev. Mod. Phys. 1976, v. 48, p. 239. 4. B.J. Braams and C.F.F. Karney // Phys. Fluids B. 1989, v. 1, p. 1355. 5. M. Taguchi // Plasma Phys. Control. Fusion. 1989, v. 31, p. 241. 6. S.V. Kasilov and W. Kernbichler // Phys. Plasmas. 1996, v. 3, p. 4115. 7. R. Prater et al. // Nuclear Fusion. 2008, v. 48, p. 035006. 8. N.B. Marushchenko et al. // Phys. Plasmas. 2011, v. 18, p. 032501. 9. D. Farina et al. // Nuclear Fusion. 2012, v. 52, p. 033005. 10. N.J. Fisch // Rev. Mod. Phys. 1987, v. 59, p. 187. 11. R.H. Cohen // Phys. Fluids. 1987, v. 30, p. 2442. 12. Y.R. Lin-Liu et al. // Phys. Plasmas. 2003, v. 10, p. 4064. 13. C.F.F. Karney et al. // AIP Conf. Proc. 1989, v. 190, p. 430. 14. N.B. Marushchenko et al. // Fusion. Sci. Technol. 2009, v. 55, p. 180. 15. W. Kernbichler et al. // Contrib. Plasma Phys. 2010, v. 50, p. 761. 16. H. Maassberg et al. // Phys. Plasmas. 2012, v. 19, p. 02501. 17. K. Matsuda // IEEE Trans. Plasma Sci. 1989, v. 17, p. 6. 18. N.B. Marushchenko et al. // J. Plasma Fusion Res. 2007, v. 2, p. S1000. 19. R.W. Harvey and M.G. McCoy // Proc. of IAEA Techn. Comm. Meeting, 1992. IAEA, Vienna, 1993, p. 498. 20. W. Kernbichler et al. // J. Plasma Fusion Res. 2008, v. 3, p. S1061. 21. H. Maassberg et al. // Phys. Plasmas. 2009, v. 16, p. 072504. 22. W. I. van Rij and S. P. Hirshman // Phys. Fluids B. 1989, v. 1, p. 563. 23. P. Helander and P.J. Catto // Phys. Plasmas. 2001, v. 8, p. 1988. Article received 21.09.12 РАЗВИТИЕ МОДЕЛЕЙ ДЛЯ ВЫЧИСЛЕНИЯ ЭЛЕКТРОННО-ЦИКЛОТРОННОГО ТОКА УВЛЕЧЕНИЯ Н.Б. Марущенко, C.D. Beidler, H. Maassberg Сделан обзор недавних достижений в методах вычисления электронно-циклотронного тока увлечения. Основное внимание направлено на учёт сохранения продольного импульса в операторе столкновений, а также учёт релятивистских эффектов, существенных в высокотемпературной плазме. В плазме относительно небольшой температуры также эффекты конечной столкновительности становятся существенными. Эффективность и точность развитых численных моделей продемонстрированы результатами, полученными с помощью метода лучевых траекторий. РОЗВИТОК МОДЕЛЕЙ ДЛЯ ОБЧИСЛЕННЯ ЕЛЕКТРОННО-ЦИКЛОТРОННОГО СТРУМУ ЗАХОПЛЕННЯ Н.Б. Марущенко, C.D. Beidler, H. Maassberg Зроблено огляд недавніх досягнень в методах обчислення електронно-циклотронного струму захоплення. Головна увага спрямована на облік збереження продольного імпульсу в операторі зіткнень, а також на облік релятивістських ефектів, значних у високотемпературній плазмі. У плазмі відносно невеликої температури ефекти кінцевої зіткненості також стають істотними. Ефективність та точність развинутих моделей продемонстровано результатами, які були отримані за допомогою методу променевих траєкторій.

Recent progress in models for electron cyclotron current drive

Institution

Ähnliche Einträge