Global anomalous transport of ICRH- and NBI-heated fast ions
George J. Wilkie, Istvan Pusztai, Ian G. Abel, William Dorland, Tünde Fülöp
GGlobal anomalous transport of ICRH- andNBI-heated fast ions
G. J. Wilkie , I. Pusztai , I. Abel , W. Dorland , T. F¨ul¨op Department of Physics, Chalmers University of Technology, Gothenburg, Sweden Princeton Center for Theoretical Science, Princeton, NJ, United States Department of Physics, University of Maryland, College Park, MD, United StatesE-mail: [email protected]
Submitted 25 Aug 2016
Abstract.
By taking advantage of the trace approximation, one can gain an enormouscomputational advantage when solving for the global turbulent transport of impurities.In particular, this makes feasible the study of non-Maxwellian transport coupled inradius and energy, allowing collisions and transport to be accounted for on similar timescales, as occurs for fast ions. In this work, we study the fully-nonlinear ITG-driventrace turbulent transport of locally heated and injected fast ions. Previous resultsindicated the existence of MeV-range minorities heated by cyclotron resonance, andan associated density pinch effect. Here, we build upon this result using the t3core code to solve for the distribution of these minorities, consistently including the effectsof collisions, gyrokinetic turbulence, and heating. Using the same tool to study thetransport of injected fast ions, we contrast the qualitative features of their transportwith that of the heated minorities. Our results indicate that heated minorities aremore strongly affected by microturbulence than injected fast ions. The physicalinterpretation of this difference provides a possible explanation for the observed synergywhen NBI heating is combined with ICRH. Furthermore, we move beyond the traceapproximation to develop a model which allows one to easily account for the reductionof anomalous transport due to the presence of fast ions in electrostatic turbulence.
Fast ions are an important component of a fusion device, being responsible fora portion of heating required to bring tokamaks and stellarators up to fusion-relevanttemperatures. Various phenomena can cause radial transport and hence a redistributionof this energy, including: neoclassical transport, transport from nonaxisymmetricmagnetic “ripples”, stochastic magnetic field regions, Alfv´en waves driven unstable bythe fast ions themselves, and microturbulence. This latter effect is what we focus onin this work, taking advantage of recently developed tools to study the coupled radius-energy phase space transport of trace non-Maxwellian species.With the ion cyclotron resonance heating (ICRH) technique, electromagnetic wavesare launched into the plasma at a frequency resonant with that of ion cyclotron motionat some locations. Under certain circumstances, a very small population of minorityions can very efficiently absorb the power and be heated up to MeV-range energies [1]. a r X i v : . [ phy s i c s . p l a s m - ph ] J a n nomalous transport of fast ions t3core transport simulations, contrastingthe behavior of heated and injected fast ions. Finally, in Sec. 3, we improve upon thetrace approximation by developing a simple model to adjust the bulk plasma anomaloustransport when fast ions are present in significant quantities.
1. Description of the problem
The method we use to solve the multiscale transport problem is based on the traceapproximation, in which the fast ions do not have an effect on the turbulence. Ina later section, we provide corrections to this approximation. For now, assuming thetrace approximation holds, the pitch-angle averaged low-collisionality transport equationreads [4, 5]: ∂F ∂t + 1 V (cid:48) ∂∂r V (cid:48) Γ r + 1 v ∂∂v v Γ v = S, (1)where S is the source; V ( r ) is the volume enclosed by the flux surface labelled by itshalf-width r ; Γ r is the turbulent radial flux of fast ions as a function of speed v ; andΓ v includes all forms of transport in energy: turbulent energy exchange, direct heating,collisional diffusion and slowing-down. Upon pitch-angle averaging to obtain Eq. (1),we eliminate the pitch-angle scattering from both collisions and turbulence. However,in order to obtain a closed equation for the pitch-angle averaged distribution function F , we neglect the pitch angle dependence of the turbulent diffusivities (which multiplyderivatives of F ). According to the scalings of Ref. [6], this assumption is incorrect.However, the numerical results of Ref. [7] seem to indicate a very weak dependence ofthe diffusivity on pitch angle, at least at high energy and for non-extreme pitch angles( e.g. ξ (cid:46) . nomalous transport of fast ions r = − D rr ∂F ∂r − D rv ∂F ∂v (2)Γ v = − D vr ∂F ∂r − vν s F − (cid:18) v ν (cid:107) + D vv + P n f m (cid:19) ∂F ∂v , (3)where ν s and ν (cid:107) are the slowing-down and parallel velocity diffusion collision frequencies[9] summed over all bulk species, P ( r ) is the heating power per unit volume, m is thefast ion mass, and n f is the local fast ion density. Whether the source S or injectedpower P is used to inject energy into the system is the difference between “injected”and “heated” fast ions discussed above, and this is a distinction we will continue tomake throughout this work. The form of the heating term in Eq. (3) is the same asthe equation used by Ref. [10] for the isotropic part of the ICRH-heated distribution.Equations (1), (2), and (3) form a closed two-dimensional partial differential equationand is of similar form to that used in Ref. [5], except our treatment is not a quasilinearmodel, but is rigorous for trace transport in full nonlinear turbulence. The tool weemploy to solve Eq. (1) is t3core , which was originally developed for alpha particletransport [11] and employs a finite-volume method and obtains the diffusion coefficientsby post-processing output from gs2 [12] nonlinear gyrokinetic turbulence simulations.This is done by comparing the fluxes of two different trace species with the same chargeand mass, but different radial gradients and/or temperatures. For each energy andradius, one can determine the unknowns D rr and D rv by algebraically solving Eq. (2)for each of the two species given the two different equilibrium distributions and thecorresponding calculated fluxes.The local properties of our nominal case at r = 0 . a , which is the same ITER-like scenario from Ref. [2], are as follows. The shape of the flux surface is such thatthe ellipticity is κ = 1 .
409 and triangularity is δ = 0 .
075 (with radial derivatives aκ (cid:48) = 0 . aδ (cid:48) = 0 . R = 3 . a , and R (cid:48) ( r ) = − . s = 0 . q = 1 .
27. The plasma is a mixture of 70% protons and 30% He by charge density,there is no local gradient in particle density for either electrons or ions. Electrons are adynamic kinetic species, and our simulations include only electrostatic fluctuations.This case is based off of an ITER hybrid scenario (case 20020100 from the ITPApublic database [13]), but with an increased ion temperature gradient scale length aT (cid:48) i /T i = − .
509 so that the ion temperature gradient (ITG) mode is marginallyunstable. The electron density is 9 . × / m , temperatures are T i = T e = 25 keV,and ρ ∗ = ρ i /a = 0 . P max = 140 kW / m is absorbed by the He minority, and is radially distributed asa Gaussian with a width of ∆ r = 0 . a : P ( r ) = P max exp (cid:2) − ( r − . a ) / r (cid:3) .This is an approximation of the simulated absorption profile in Ref. [2]. When we nomalous transport of fast ions Figure 1.
The diffusion coefficients of Eqs. (2) and (3) as functions of energy, forwhich the Larmor radius ρ is used a proxy. Determined from nonlinear gs2 simulationsfor the three types of trace fast ions considered in this work: alpha particles (blacksolid), NBI knock-on hydrogen (green dashed), and ICRH-heated He minorities (reddotted). Units for D rr are m / s, D rv and D vr are given in m / s , and D vv in m / s . consider the NBI-like injected case, these are protons injected at 1 MeV with an identicalradial power distribution as the ICRH ions: S ( r, v ) ∝ exp (cid:2) − m f ( v − v inj ) / T i (cid:3) ,where the constant of proportionality for each radius is numerically calculated so that (cid:82) S ( r, v ) ( m f v /
2) d v = P ( r ).The domain of our transport simulations is the annulus 0 . ≤ r/a ≤ . r = 0 . a . This is done in order to isolate theeffects of transport; although a transport simulation whose flux tubes are radially-varying is more reflective of reality, and certainly possible in our framework, it is anunecessary complication. Figure 1 shows the energy dependence of the four diffusioncoefficients as determined from these gs2 simulations. There, we compare several typesof fast ions which might be present: ICRH-heated He, NBI knock-on H, and fusionproduced alpha particles ( He). The horizontal axis is expressed as the ratio of theLarmor radius (at a given energy) to the thermal hydrogen Larmor radius ρ i . Thisis done to highlight the consistent physics between the species: the diffusion peaks atapproximately the same Larmor radius in all cases, and has similar scaling at highenergy. Although the coefficients differ by factors of order unity, qualitatively they aresimilar functions of energy, when properly scaled to account for different isotopes. Thesecoefficients form the basis of the transport simulations that follow. Note that despite D vr and D vv appearing large when written in SI units, they contribute negligibly tothe transport simulations because these terms are small compared to the correspondingradial transport terms and collisions. Nevertheless, these terms are retained in oursimulations for completeness. For a disussion of Onsager symmetry between D rv and D vr at high energy, see the appendix.From our gyrokinetic simulations of this case, we obtain a turbulent heat fluxof 0 . / m , which translates to about 100 MW of core heating power, which is nomalous transport of fast ions r = 0 . a , the population of minorities is held fixed as a Maxwellian at the local iontemperature and a density of n f = 10 / m , approximately 0.1% of the electron density n e = 9 . × / m . At the inner boundary r = 0 . a , the net flux of minorities intothe domain is zero at every energy. In velocity space, the flux through v = 0 vanishes,and the distribution is set to vanish at a suitably high speed v max . The temperature isheld fixed at T e = T i = 25keV throughout the domain.
2. Comparing heated and injected ions: results and interpretation
Previous results indicated that microturbulence has the effect of causing a particle pinchof ICRH-heated minority ions against their own strong temperature gradient [2]. Thisconclusion was reached with a fixed fast ion distribution and radial profiles, while herewe let these evolve consistently with the turbulence around the peak heating location.For comparison, we will be comparing this to ions injected at high energy, reflectingthe behavior of neutral beam knock-on ions. Specifically, the velocity distribution ofICRH-type ions is more “globally” affected and moments are generally more sensitiveto the turbulence, whereas for injected fast ions, turbulence has a localized effect wheretransport is the strongest, causing an inversion (“bump-on-tail”) in some cases.Figure 2 shows a sample of our results, plotting the radial profile of ICRH ions asa function of radius, showing the effect of decreasing the diffusivities. Naturally, wedefine the temperature for a non-Maxwellian distribution as T f = (cid:82) ( mv / F d v /n f .For sufficiently weak transport (or sufficiently strong power injection), one obtains astrong peak in temperature that depends quite strongly on the turbulence. When theturbulence is sufficiently suppressed so as to allow a sharp peak in heated minoritytemperature, we observe the expected density pinch. In the case of reduced turbulence,where the diffusion coefficients are scaled down by factors of ten, the density of He isenhanced around the heating location due to this “thermodiffusion” (a flux of particlesagainst a temperature gradient). Eventually, particle diffusion balances this effect tocreate the steady-state profiles shown.We find that the NBI-like “injected” fast ions are less sensitive to the turbulencethan the ICRH ions: the peak density only changes by a factor of two when theturbulence is weakened by the same factor of ten, while the temperature changes byless than 15% (300 keV for the nominal case). We demonstrate in Fig. 3 that the peakpressure of ICRH-heated ions is more sensitive to the turbulence than NBI-type ions nomalous transport of fast ions Figure 2. (a) Gaussian heat deposition profile applied in the t3core transportsimulations, and the resulting (b) density, and (c) temperature profiles of the ICRH-heated He minority, the latter have the turbulent diffusivitives two scaled by thefactors indicated.
Figure 3.
Maximum pressure of fast ions (normalized to the electron pressure)as a function of the turbulence amplification factor. The “inconsistently heated”case is where a constant minority density is assumed throughout the domain whenimplementing the heating term. and, by extension, alpha particles. The latter are capable of reaching higher pressuresat a given amplitude of turbulence.The microturbulence also has an effect on the velocity space distribution (see Fig. 4),and there again we see ICRH-heated fast ions being more sensitive. Note in Fig. 4 that,even when the turbulence is weak, it affects the velocity distribution of heated ionsat all energies, while the effect on the distribution of injected ions is more localizedaround where D rr is dominant over collisions. This localization in velocity space is seeneven more clearly in the distribution of alpha particles [11], which extends to yet higher nomalous transport of fast ions Figure 4.
Modification of the distribution function of NBI-knock-ons (top) andICRH-heated ions (bottom) at the peak heating radius r = 0 . a . To compare withthe over-driven nominal case (solid black), the diffusion coefficients are scaled down byfactors of two (dashed blue) and ten (dotted cyan) in each. energies, where it is less affected by microturbulence.To understand the physical reason for this qualitative difference, in Fig. 5 examinethe paths through phase space that each type of fast ion takes. Recall that particles withmoderately suprathermal energies (where v ∼ v ti ) are most affected by the turbulence.Injected ions slow down via collisions before they reach these energies, while heated ionsmust “pass through” the turbulence-dominated part of velocity space before becomingpart of the distribution at high energy. If the turbulence is too strong, heated particlesare transported radially before reaching high energies.
3. Fast ion dilution model
The trace approximation is needed when using the diffusive form for the turbulent fluxesin Eqs. (2) and (3). However, even when fast ions are not strictly trace, they may stillnot participate directly in the drift wave dynamics. Instead, via quasineutrality of theequilibrium, they take the place of the thermal ions which are driving the turbulence. Ifthis near-trace behavior can be easily predicted, one could rescale the anomalous fluxeswithout running additional turbulence simulations. This section seeks to develop sucha model for electrostatic turbulence. nomalous transport of fast ions Figure 5.
Stream plots showing trajectories of fast ions for the injected (left) andheated (right) case. The slope at each point is the normalized ratio Γ v a/ Γ r v ti . Thecase shown here is where the turbulence intensity is scaled down by a factor of ten.For reference, the injection energy of NBI ions is at v inj ≈ . v ti . When solving for the perturbed electrostatic potential φ , quasineutrality reads [14]: eφ = − (cid:80) s Z s (cid:82) (cid:104) h s (cid:105) r d v (cid:80) s Z s (cid:82) ( ∂F s /∂E ) d v , (4)where h s is the non-adiabatic part of the perturbed distribution of species s , F s is its equilibrium distribution function, and (cid:104)(cid:105) r signifies the gyro-average at fixedposition in space. Suppose we have a simulation which contains only singly-chargedthermal ions and adiabatic electrons in quasineutrality so that n i = n e . In thiscase, eφ ∝ n i / ( n i /T i + n e /T e ) = T i / (1 + τ ), where τ ≡ T i /T e . Now, introduce apopulation of adiabatic fast ions with charge density Z f n f such that now n i + Z f n f = n e .Then, eφ ∝ n i / (cid:0) n i /T i + n e /T e + Z f n f /T f (cid:1) = T i / (1 + τ eff ) , which serves to define an“effective temperature ratio” τ eff , and T f is a suitable “effective temperature” of thefast ions, which may or may not be Maxwellian. In order for the computed φ to bethe same under these circumstances, we can account for the fast ions by adjusting thetemperature ratio of the original simulations accordingly: τ eff = (cid:18) τ + T i T f Z f n f n e (cid:19)(cid:30)(cid:18) − Z f n f n e (cid:19) , (5)where τ is the original (physical) temperature ratio. This second term in the numeratorcan be neglected for sufficiently hot ions provided that T f (cid:29) Z f T i .The case used in this section is at r = 0 . a of ITER scenario 10010100 from theITPA database [13], an ELMy H-mode case of 50/50 deuterium/tritium. The physicalparameters at this flux surface are: q = 1 .
54, ˆ s = 0 . κ = 1 . aκ (cid:48) = 0 . δ = 0 . aδ (cid:48) = 0 .
44, and a Shafranov shift derivative of R (cid:48) ( r ) = − . T i = 0 . T e , and nominally have a/L T i = 1 . a/L T e = 1 .
732 and a/L ne = a/L ni = 0. The fast ions are considered to be deuteriumat a temperature of T f = 100 T i and varying density, with no density or temperaturegradients. Due to the large number of simulations required these simulations were ofrelatively low resolution. The spatial grid is defined by N x = N y = 24, L x ≈ L y = 10 πρ i nomalous transport of fast ions Figure 6. (a) Turbulent deuterium heat flux, scanning in a/L
T i showing severalfast ion concentrations. Dashed lines are the fitted diffusivities. (b) The thermal iondiffusivity as a function of fast ion concentration (black diamonds), a fitted (blackdashed) line representing the presented model, and the dominant linear growth rate(red stars). Also shown are the results as determined from higher resolution nonlinearsimulations (cyan squares). perpendicular to the magnetic field, and N θ = 18 along the magnetic field. Velocityspace has 10 points in energy, and 26 pitch angles.Figure 6(a) shows the results of deuterium heat flux from nonlinear gs2 simulationsnear ITG marginality at several fast ion concentrations. It is evident that dilution haslittle effect on the critical gradient length scale: a/L T i ≈ . ± .
04, but it does reducethe stiffness. The value of χ i obtained for several fast ion concentrations is shown inFig. 6(b). There, we also show that χ i scales similarly to the linear growth rate, asmight be expected from critical balance arguments [15]. We choose a power law fit ofthese simulations to reflect the behavior of Eq. (5), which results in the following model: χ i ≈ χ i, (cid:18) − Z f n f n e (cid:19) . , (6)where χ i, is the thermal ion diffusivity without any fast ions. Note that the experimentalscaling of thermal diffusivity with respect to temperature ratio χ i ∝ τ − [16] isremarkably consistent with our results when applying Eq. (5). We expect this modelto be valid to the extent that the following assumptions hold: the turbulence isdominated by electrostatic dynamics; adiabatic electrons and fast ions remain a goodapproximation; the temperature ratio continues to obey the scaling of Ref. [16]; andone is in the limit of high “temperature” fast ions ( i.e. the fast ion term vanishes inthe denominator of Eq. (4)). These latter two assumptions can be relaxed by replacingthe power in Eq. (6) and/or by including the additional term in the numerator ofEq. (5). Furthermore, we note that previous work done on fast ion dilution [17, 4] isalso consistent with this model.By applying Eq. (6), a transport simulation of the bulk plasma can easily andconsistently respond to the stabilizing presence of fast ions, at least to a leading nomalous transport of fast ions Conclusion
In this work, we explored the global non-Maxwellian transport of isotropic heatedminority ions, allowing their radial and velocity distribution to arise naturally fromthe physics of collisions, gyrokinetic microturbulence, and direct heating. Our aim wasto highlight the qualitative differences between their response to microturbulence andthat of fast ions such as those generated by NBI or fusion reactions. Despite their verysimilar diffusion coefficients, these differences fundamentally limit how robustly theycan mimic the turbulent transport of alpha particles, and this limitation is primarilydue to how each type gains energy. However, their relatively strong sensitivity couldallow them to be a useful probe of the turbulence, and as experimental validation fortransport tools such as t3core .It has been experimentally observed that simultaneously applying NBI and IRCHheating simultaneously results in more heating than the sum of either of them alone[18]. Our results provide a possible explanation for this phenomenon. ICRH-heatedminorities are more affected by the turbulence because they have the opportunity totransport radially before being heated. Therefore, if one heats ions via ICRH that are already at high energy (via, for example, NBI), where turbulence has less effect, thenone avoids the strong-transport region of phase space altogether.Two important caveats to our results are the electrostatic and isotropicassumptions. Electromagnetic fluctuations, even in primarily electrostatic turbulence,could be important for trace transport at sufficiently high plasma beta. Also, the fastions considered here are generally not isotropic in velocity space. In order for our resultsto faithfully reflect the isotropic part of such a velocity distribution, we had to ignore thepitch-angle dependence of the turbulent diffusion coefficients. We believe that our coreresult (that ICRH ions are more sensitive to the microturbulence, having an impact onthe distribution at all energies) is robust to relaxing this assumption, but further studyis expected and welcome.The transport results presented here rely upon fast ions being trace, which is arobust assumption for electrostatic turbulence at small concentrations. Where fast ionsmight make up a more significant fraction of the plasma, we presented a model to accountfor their effect on the bulk plasma transport as a next approximation. Although the t3core simulations presented here do not employ this model, we believe it will be veryuseful for future transport simulations.The authors would like to thank Yevgen Kazakov for the original inspiration ofhigh-energy heated minority and Torbj¨orn Hellsten for helpful discussion. Simulationswere performed on the SNIC cluster Hebbe (project nr. SNIC2016-1-161) and theNERSC supercomputer Edison. This work was supported by the Framework grant nomalous transport of fast ions
Appendix A. Onsager symmetry for quasilinear phase space transport
The astute reader might notice a type of Onsager symmetry between D rv and D vr athigh energy in Fig. 1. A proof of Onsager symmetry for quasilinear radial transport ofa Maxwellian species has already been shown [19], but this is not the case for nonlinearturbulence [20]. In this appendix, we’ll show that this symmetry holds for quasilineartransport even in r - v phase space, as has been observed here and elsewhere [5]. This ismore than coincidence, and here we show that this is rigorous as long as the magneticdrift velocity is dominant over the nonlinear drift velocity in the gyrokinetic equation.The operator L is the left-hand side of the collisionless gyrokinetic equation, sothat when acting on the non-adiabatic perturbed distribution h : L [ h ] ≡ ∂h∂t + (cid:0) v (cid:107) b + v d + v φ (cid:1) · ∇ h, (A.1)where v d is the magnetic drift velocity including the ∇ B and curvature drifts, and v φ ≡ ( c/B ) b × ∇ (cid:104) φ (cid:105) R . The unit vector b points in the direction of the equilibriummagnetic field.What makes our transport calculations possible is that, for a trace species only, L is a linear operator, although it does in general depend on φ . Note that this is not equivalent to the quasilinear approximation used, for example, in Refs. [3] and [5], and ismore generally applicable. The quasilinear approximation is when one further assumesthat the v φ term in Eq. A.1 is negligible. This becomes more accurate at high energy,when v d (cid:29) v φ .Now, consider the following expressions for the diffusion coefficients: D rr = (cid:42)(cid:88) σ (cid:107) (cid:90) L − [ v φ · ∇ r ] ( v φ · ∇ r ) J v d λ (cid:43) t (A.2) D rv = (cid:42)(cid:88) σ (cid:107) (cid:90) L − (cid:20) ∂ (cid:104) φ (cid:105) R ∂t (cid:21) Zemv ( v φ · ∇ r ) J v d λ (cid:43) t (A.3) D vr = (cid:42)(cid:88) σ (cid:107) (cid:90) L − [ v φ · ∇ r ] Zemv (cid:18) ∂ (cid:104) φ (cid:105) R ∂t (cid:19) J v d λ (cid:43) t (A.4) D vv = (cid:42)(cid:88) σ (cid:107) (cid:90) L − (cid:20) ∂ (cid:104) φ (cid:105) R ∂t (cid:21) Z e m v (cid:18) ∂ (cid:104) φ (cid:105) R ∂t (cid:19) J v d λ (cid:43) t . (A.5)Here, J v is the velocity space Jacobian for coordinates v and λ = v ⊥ /v , σ (cid:107) is the signof the parallel velocity (which would otherwise ambiguous in these coordinates), and (cid:104)(cid:105) t represents a time-average and a spatial average over the flux-tube. Consider the nomalous transport of fast ions L does not depend on φ (thus has no explicit time dependence), then we can show that D rv = D vr by Fouriertransforming in time. For convenience, let D ≡ L − ∂/∂t . Recalling the definition oftime average to be an integral over time: (cid:90) L − [ v φ ( t ) · ∇ r ] ∂ (cid:104) φ (cid:105) R ∂t dt (A.6)= (cid:90) ∂ (cid:104) φ (cid:105) R ∂t (cid:90) ˜ L − [ v φ ( t ) · ∇ r ] ( ω ) e iωt dω dt (A.7)= (cid:90) ∂ (cid:104) φ (cid:105) R ∂t (cid:90) ˜ v φ ( ω ) · ∇ r − iω + D e iωt dω dt (A.8)= (cid:90) (cid:90) (cid:90) ∂ (cid:104) φ (cid:105) R ∂t v φ ( t (cid:48) ) · ∇ r − iω + D e iω ( t − t (cid:48) ) dt (cid:48) dω dt (A.9)= (cid:90) (cid:90) (cid:90) v φ ( t ) · ∇ r ∂ (cid:104) φ (cid:105) R /∂t (cid:48) − iω + D e iω ( t (cid:48) − t ) dt (cid:48) dω dt (A.10)= (cid:90) L − (cid:20) ∂ (cid:104) φ (cid:105) R ∂t (cid:48) (cid:21) v φ ( t (cid:48) ) · ∇ r dt (cid:48) , (A.11)which shows that Eqs. A.3 and A.4 are equivalent. The key approximation was that D does not depend on time, which is the case for the quasilinear approximation. Therefore,while not generally applicable in nonlinear turbulence, Onsager symmetry is expectedat high energy to the extent that v φ (cid:28) v d . References [1] Ye.O. Kazakov, D. Van Eester, R. Dumont, and J. Ongena. On resonant ICRF absorption inthree-ion component plasmas: a new promising tool for fast ion generation.
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