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: 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 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

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Total fusion alpha energy content

Minimum alpha population

with newly born

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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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