TAE induced alpha particle and energy transport in ITER
K. Schoepf1, E. Reiter1,2, T. Gassner1
1Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria;
fusion@oeaw 2Institute for Ion Physics and Applied Physics, University of Innsbruck, Technikerstr. 25, 6020
Innsbruck, Austria
E-mail: [email protected]
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 convective-
diffusive 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
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% 4He and 1.1%
3He). 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=12-14.
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
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 /edge
pol pol 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.
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
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 0/B B is displayed in figure 3. The highest
amplitudes were found for the n = 11 Alfvénic mode with a saturation amplitude
40/ 6 10B B
and the global mode with n = 13 saturating at 3
0/ 3 10B B . Our
Figure 3. Evolution of the relative amplitudes 0/B B 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
with values of up to / 1.5drive damp , while in the present simulation this ratio is
/ 1.7 2drive damp for the strongest growing modes.
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,
= ion Landau damping, trapped electron collisional damping, radiative damping ,
damp
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 .
s / 0.2 0.6edge
pol pol . Inspecting the image on the RHS it is seen that the wave-
particle 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.
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.
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
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.
Total fast alpha population
Tota
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Tota
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Total fusion alpha energy content
Minimum alpha population
with newly born
high-energy alphas
FO losses of redistributed alphas Collisional loss of marginally confined α’s + TFR induced losses
Minimum alpha energy
content due to lost and
decelerating alphas
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 2014-2018 under grant
agreement No 633053. 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”.
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