TAE induced alpha particle and energy transport in ITER
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1 TAE induced alpha particle and energy transport in ITER K. Schoepf 1, E. Reiter 1,2, T. Gassner 1 1 Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria; fusion@oeaw 2 Institute for Ion Physics and Applied Physics, University of Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria klaus.schoepf@uibk.ac.at Abstract. Mechanisms relevant to energetic-ion transport in tokamaks are numerically modelled for a qualitative as well as quantitative evaluation of their effects. For that the Fokker-Planck code FIDIT is used to describe the convective-diffusive transport of fast ions, while the perturbative PIC code HAGIS is employed to simulate the interaction of energetic particles and TAEs. Properly switched upon checking stability/instability criteria, the iterative running sequence of these codes enables the study of combined transport effects, i.e. the convectivediffusive loss of energetic ions that are redistributed by waves. Taking the standard H-mode ITER scenario with a constant DT fusion source we considered the presence of 15 TAE modes and evaluated synergetic transport effects caused by the co-action of wave-particle interplay and classical particle transport. 1. Introduction Essential for the realization of thermonuclear self-heating in fusion reactor plasmas is a comprehensive understanding of the confinement of energetic charged fusion products. For that it is important to evaluate the several processes which transport fast ions out of the hot plasma before they can collisionally transfer their excess energy to the background components. In this study mechanisms relevant to energetic-ion transport in tokamaks are investigated and numerically modelled for a qualitative as well as quantitative evaluation of their effects. Of particular interest herein is the synergy of classical and wave-induced transport [1-4], which mainly determines the evolution of the energetic particle distribution. For the corresponding modelling a coupled operation of the fast-ion Fokker-Planck transport code FIDIT [5] (3D constant-of-motion (COM) space) and the wave-particle simulation code HAGIS [6] is applied. The iterative running sequence of these codes is regulated by instability/stability switches, which allows for studies of combined transport effects, e.g. the convective-diffusive loss of energetic ions that are redistributed by waves. Moreover, the iterative HAGIS/FIDIT coupling renders possible a longer-time simulation of the transport behavior of fast ions in plasmas with MHD mode activity [7]. 2. Methods and models We investigated the evolution of an ensemble of 15 toroidicity-induced Alfvén Eigenmodes (TAEs) in the presence of fusion alphas and studied the emerging particle and energy transport effects. A plasma configuration based on the standard H-mode ITER scenario [8] was assumed for our calculations. The profiles of background electron density
2 and temperature, background ion temperature and the safety-factor are displayed in figure 1. The ion background in our simulation consists mainly of deuterons (45.7%) and tritons (45.7%), with impurities of Beryllium (2.3%), Argon (0.1%) and Helium (4.9% 4 He and 1.1% 3 He). The densities of the various background ions are assumed to have the same profile Figure 1. Left: Radial profiles of electron density and electron and ion temperature. Right: q-profile as a function of flux surface radius r. shapes as the electron density, but scaled to meet charge neutrality throughout the plasma according to their respective fractional population. External heating methods like ICRH or neutral beam injection were not considered for simplification. Supposing Maxwellian distributions of the fuel ions, a suitable fusion alpha source term was derived and introduced in the Fokker-Planck code FIDIT which yields a stationary distribution of energetic alphas at some time after switching on the constant alpha source. 2.1 Evolution of TAEs For the simulations performed with HAGIS the equilibrium reconstruction was calculated with HELENA. The radial eigenfunctions of 15 TAEs supported by the equilibrium were computed with CASTOR. As illustrated in figure 2, TAE modes with toroidal mode numbers ranging from n=4 to n=15 where found to occur, and two distinct eigenmodes one global and one core localized were identified for each toroidal mode number in the range n= Taking the equilibrium and plasma parameters of the standard H-mode ITER scenario as well as a constant d-t fusion source, the build-up of the fast-alpha distribution was modelled by the time-dependent FIDIT code [5]. A TAE instability criterion, based on an analytical expression of mode driving and damping rates, was implanted in FIDIT and indicated mode growth twice before an almost stationary fast-alpha distribution function
3 emerged in FIDIT about 1s after the d-t fusion source had become active (a numerical evaluation of the fast-ion driven mode growth, e.g. with CASTOR-K [9], was deemed here Figure 2. The ensemble of considered waves computed with CASTOR: Normalized electrostatic perturbation potential as a function of the radial coordinate s / for the 15 toroidal Alfvénic modes included in the HAGIS simulation. The various poloidal harmonics are plotted in different colours, beginning in blue for lower m up to higher modes marked in red. pol edge pol as computationally too intensive because of the required employment at each time step in FIDIT). In both cases the momentary alpha distribution function was transferred to the HAGIS code [6] for modelling the evolution of TAE amplitudes. Whereas the effect of the first instability was negligible due to a minor fast ion pressure, the second has already led to a significant redistribution of the alpha population. After amplitude saturation of the strongest TAEs the redistributed alpha ensemble, as simulated by HAGIS, was taken as input to FIDIT for modelling the alpha evolution up to the stationary distribution after about 1s upon
4 switching on the fusion alpha source. This stationary alpha distribution has then been transferred to the HAGIS code for modelling the evolution of TAE amplitudes shown in figure 3. It is to be mentioned here that the distribution transfer requires proper coordinate transformation due to the different COM space variables in FIDIT and HAGIS [7]. All 15 modes shown in figure 2 were included in the HAGIS simulation of TAE interaction with energetic alphas. HAGIS was run long enough for reaching saturation of the modes having the highest amplitudes. With B denoting the amplitude of the perpendicular component of the perturbed magnetic field and B0 representing the magnetic field on axis, the time evolution of the relative mode amplitudes B / B 0 is displayed in figure 3. The highest amplitudes were found for the n = 11 Alfvénic mode with a saturation amplitude 0 4 B / B 6 10 and the global mode with n = 13 saturating at 0 3 B / B Our Figure 3. Evolution of the relative amplitudes B / B0 of 15 fusion alpha driven TAEs in ITER (standard H-mode scenario) as self-consistently simulated in HAGIS starting with a stationary alpha distribution delivered by FIDIT. simulation delivers results similar to calculations [10] with the code NOVA-K [11], where the initial alpha distribution was expressed analytically. There the largest ratio between mode growth rate and damping rate in ITER scenario 2 was predicted for modes with n = 10-12
5 with values of up to / 1.5, while in the present simulation this ratio is drive damp / for the strongest growing modes. drive damp Though the common HAGIS version provides reliable results for scenarios with mode frequencies locked to the plasma equilibrium, the suppression of collisionality restricts seriously its applicability for modeling nonlinear peculiarities of mode evolutions. In reality there appear chirping modes which exhibit a sequence of amplitude bursts, whereby the mode frequency sweeps during each burst. This is due to collisions which restore the unstable distribution function where it is otherwise flattened by the mode. Hence collisional interaction will result in additional free energy and consecutively in nonlinear mode evolution. 1D and 2D models of wave-particle interaction including drag and diffusion illustrate the formation of bursts as well as mode frequency sweeping, but do not yield realistic estimates of the evolving fast particle distribution function in full phase space [12,13]. We modified HAGIS to operate with 3-dim. B-fields and tried to extend HAGIS to include collisional effects [14], but did not succeed in implanting a collision-induced diffusion module consistent with the Hamiltonian structure of HAGIS. Nevertheless, since the HAGIS model yields a reasonable description of nonlinear evolution of marginally stable modes already in its collisionless form [15], we use this simpler version for demonstrating the synergy of TAE induced redistribution of fast ions and subsequent loss mechanisms. 3. Redistribution of the alpha distribution As previously mentioned, after ~ 75 ms upon starting the d-t fusion alpha source and simulating the build-up of the energetic alpha distribution in FIDIT, a linear growth of modes is indicated by a linear instability criterion and has been observed in a first HAGIS run at this time step [14]. However, due to the insufficient particle density at that time, the growth of the mode amplitudes until saturation is too small in order to significantly alter the fast-ion distribution. Upon amplitude saturation a new FIDIT sequence was launched, which developed the alpha distribution, which had been unsubstantially modified at 75 ms by interacting with TAEs, for further 175 ms assuming a constant d-t fusion source. At this point in time, 250 ms after activating the alpha source, the instability criterion in FIDIT indicated now a significant growth of TAEs. This criterion is based on the assessment of the growth rate of a single TAE, damp = ion Landau damping, trapped electron collisional damping, radiative damping,
6 where describes the interaction of fast alphas with a wave and may be positive or negative, depending on the shape of the distribution function. The analytical expressions for and the various damping rate terms damp can be found in [16,17]. A range of modes can be tested by the stability check module in FIDIT, and whenever one of them features a positive growth rate, the relevant output is produced for use in HAGIS while the FIDIT run is suspended. The higher particle pressure now triggered a stronger interaction of the fast alphas with the ensemble of TAEs, which effected a significant redistribution of the alpha particles. As evident from figure 4, the strongest radial redistribution occurs in the range Figure 4. Perturbation of the fast alpha distribution in ITER as induced by the 15 HAGIS modelled TAEs. Left: Radial dependence of the perturbation as a function of time; Right: Dependence of the perturbation on time and pitch angle cosine v // / v. edge pol pol s / Inspecting the image on the RHS it is seen that the waveparticle interaction is, as expected, strongest for trapped alphas. The impact on co-passing particles is noticeably weaker, and the distribution of counter-passing particles remains almost unaffected by the considered TAEs. Upon mode saturation the alpha distribution function is transferred again to FIDIT as an initial input to follow its evolution towards a stationary distribution (after about 1s) sustained by the constant d-t fusion source. This stationary alpha population forms the basis for studying the eventual TAE induced redistribution in HAGIS as well as the subsequent classical transport processes. A compact view of the redistribution effects of wave-particle interaction is provided by figure 5, where the spatial densities of fast alphas before and after the HAGIS simulation can be compared. The wave-induced transport is seen to lead to an outward shift of energetic alphas towards the low B-field side of the tokamak and to a significant depletion of the fast alpha density in the core plasma. Of interest is also the corresponding variation of the alpha energy density as illustrated in figure 6.
7 Figure 5. Density build-up of alphas with energies E 100 kev in ITER by a constant d-t fusion source after redistribution due to interaction with 15 TAEs. Following the redistribution the density is displayed at selected times, demonstrating the effect of collisional ripple-induced transport. Figure 6. Variation of fusion alpha energy density at various time steps before and after redistribution due to interaction with 15 TAEs. The energy density build-up is sustained by a constant d-t fusion source.
8 As visible in figures 5 and 6, the major loss of energetic alphas occurs in the first 10 ms, where marginally confined alphas escape from the plasma by collisional ripple transport. Since mainly the toroidally trapped ions interact strongly with the TAEs, the fast alphas redistributed by this wave-interaction to the low B-field side are apparently mostly trapped ions as can be also concluded by inspection of figure 5. Those trapped alphas at the outer plasma edge are in addition to collisional diffusion subjected to TF ripple induced transport. Immediately after the redistribution by TAEs the fast-alpha population is seen to strongly peak in the plasma core and then to radially spread out by Coulomb collisions. The tendency of profile flattening with time is hampered by the constant alpha source that builds up a distribution similar to that before the interaction with the waves. 4. Evolution of alpha population and total alpha energy Further insight into the synergy between wave-induced and classical transport of fast ions and the consequences for alpha heating is provided by a comparison of the differing temporal evolutions of the total number of alpha particles and their energy content in the confined ITER plasma, as depicted in figure 8 after redistribution by TAEs. While the total energy content of alphas with E 100 kev increases after ~45 ms, their total particle number was still decreasing until about 100 ms after the redistribution. It is therefore concluded that alphas with highest energies are removed first from the plasma due to ripple diffusion at the plasma edge. This transport happens slower for particles with lower energies. Since the fusion source is active all the time and new alphas with energies ~ 3.5 MeV are continuously born in the plasma, the alpha energy content increases earlier than the alpha particle number, as at times > 45 ms after redistribution mainly alphas with lower energies are lost from the plasma. 5. Concluding remarks The observed synergy of TAE-induced redistribution of fusion alphas towards the low B-field periphery and subsequent enhanced collisional transport has already been proposed and analytically quantified in refs. 1 and 2. Here the iterative employment of the FIDIT and the HAGIS codes, coupled via an analytical instability switch, proves to be an appropriate and most valuable tool for studying synergetic effects of wave-induced redistribution of energetic ions and their diffusive/convective transport in real tokamak geometries. The co-action of TAE driven and collisional ripple transport is seen to result in a detrimental loss mechanism: High-energetic are rapidly lost from the plasma core, practically
9 Total fast-alpha particle number FO losses of redistributed alphas Collisional loss of marginally confined α s + TFR induced losses Minimum alpha population with newly born high-energy alphas Total fast alpha population Total fast-alpha energy, MeV Minimum alpha energy content due to lost and decelerating alphas Total fusion alpha energy content Figure 7. Variation of fusion alpha population and energy density, supplied by a constant d-t fusion source, as a function of time after redistribution due to interaction with 15 TAEs. without heating noticeably the background plasma. Referring to a burning d-t plasma in ITER scenario 2 with a constant fusion source and the presence of 15 TAEs, the total particle and energy balance after wave-particle interaction delivers the following account: 3.6 % of fusion alphas are redistributed by the TAEs to orbits promptly lost, another 7.2% are subsequently lost by collisional ripple transport. Thus almost 11% of the fusion alphas in the stationary FIDIT distribution are removed from the plasma within 10ms after interaction with the waves, while in the same time period 14% of the alpha energy content prior to redistribution is lost. Finally we hint again at the deficiency of the presented dynamic evolution of fusion alpha distribution in the presence of TAEs, which is due to non-consideration of collisions in the HAGIS code applied. Collisions, even those effecting only a diminutive alteration of the alpha velocity, may remove the particle from the resonance domain, which will result in a break-down of the previously excited mode. On the other hand, a scattered ion can suddenly meet the resonance condition. Therefore the incorporation of a collisional δf- model in HAGIS is subject of our current research effort and is expected to produce a different pattern of nonlinear mode evolution, which may attest the collisionless HAGIS version to overestimate the TAE induced redistribution of fast ions.
10 Acknowledgement This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme under grant agreement No The views and opinions expressed herein do not necessarily reflect those of the European Commission. Further the authors are grateful to the F. Schiedel- Stiftung für Energietechnik which facilitated this study by financially supporting the project TAE induced fast-ion transport in tokamak plasmas. References [1] Hladschik T. and Schoepf K., Effect of synergetic TF-ripple and TAE alpha diffusion on ITER ignition dynamics, Proc ICPP, Foz do Iguacu, Brazil, Vol. 1 p 29 (1994) [2] Hladschik T. and Schoepf K., TF ripple loss reduced alpha-heating, Proc. 11 th Topical Meeting on the Technology of Fusion Energy, New Orleans, USA, p 278 (1994) [3] Schoepf K., Synergetic transport effects by the co-action of MHD modes and collisional ripple transport in ITER, 3 rd Int. Workshop on Fast Ion Modelling and Diagnostics (FIMAD-3), Feb 2013, Innsbruck, Austria [4] Khan M., Schoepf K. and Goloborod ko V., Resonance and synergy effects on fast ion transport in tokamaks, Lambert Academic Publishing (2013) [5] Goloborod ko V. and Gassner T., Techn.Report (FIDIT code documentation), Institute for Theoretical Physics, University of Innsbruck (2011) [6] Pinches S. et al., The HAGIS self-consistent nonlinear wave-particle interaction model, Comput. Phys. Commun. 111 pp (1998) [7] Gassner T., Time evolution of fast ion distributions in MHD-active tokamka plasmas based on HAGIS/Fokker-Planck code coupling, PhD thesis, Faculty of Mathematics, Informatics and Physics, University of Innsbruck, Austria (2013) [8] Green B.J., ITER: burning plasma physics experiment, Plasma Phys. Contr. Fusion (2003) [9] Borba D. and Kerner W., CASTOR-K: Stability analysis of Alfvén eigenmodes in the presence of energetic ions in tokamaks, J. Comp. Phys. 153 (1) 101 (1999) [10] Gorelenkov N.N. et al., Study of thermonuclear Alfvén instabilities in next step burning plasma proposals, Nucl. Fusion (2003) [11] Fu G.Y., Cheng C.Z. and Wong K.L., Stability of the toroidicity-induced Alfven eigenmode in axisymmetric toroidal equilibria, Phys. Fluids B 5(11) 4040 (1993) [12] Berk H., Breizmann B. and Pekker M., Nonlinear dynamics of a driven mode near marginal stability, Phys. Rev. Letters (1996) [13] Lilley M., Breizmann B. and Sharapov S., Destabilizing effect of dynamical friction on fastparticle driven waves in a near- threshold nonlinear regime, Phys. Rev. Letters (2009) [14] Reiter E., Transport schneller Ionen in Tokamakplasmen mit MHD-Moden, Project Report to Friedrich-Schiedel-Stiftung, Institute for Theoretocal Physics, University of Innsbruck (Nov 2014) [15] Pinches S. et al., Spectroscopic determination of the internal amplitude of frequency sweeping TAE, Plasma Phys. Contr. Fusion (2004) [16] Fülüp T. et al., Finite orbit width stabilizing effect on toroidal Alfvén eigenmodes excited by passing and trapped energetic ions, Plasma Phys. Contr. Fusion (1996) [17] Connor J. et al., Non-ideal effects on toroidal Alfvén eigenmode stability, Proc. 21st EPS Conf. on Contr. Fusion and Plasma Physics, Montpellier, France ECA Vol 18 B Part II (1994).
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