David Zarzoso

D Del-Castillo-Negrete

R Dumont

X Garbet

Y Sarazin

R Heinonen

TOWARDS THE PREDICTION AND QUANTIFICATION OF ENERGETIC PARTICLE TRANSPORT AND LOSSES IN FUSION PLASMAS: the case of alpha particle transport and losses in a JET size tokamak

and research institutions in France or

INTRODUCTION

Energetic particles (EP) are ubiquitous in both laboratory and astrophysical plasmas and exhibit velocities much larger than the thermal velocity of the bulk plasma. EP, such as the alpha particles, must be sufficiently well confined in order to transfer their energy to the bulk plasma through Coulomb collisions or to ensure the current drive efficiency [1,2]. Nevertheless, due to the curvature of the magnetic field lines, the trajectories of EP depart from the magnetic flux surfaces. This departure is called magnetic drift and is more pronounced when the energy of particles is high. The effect of the transport and losses of EP in the presence of single helicity electrostatic perturbations was already analysed in detailed in [3,4]. In the present work we focus on the impact of single helicity magnetic perturbations on the transport and losses of fusion-born alpha particles, with 3.5 MeV, for which taking into account the magnetic drift is of prime importance owing to their high energy. The magnetic drift introduces an intrinsic periodicity, which can couple to the periodicity of electro-magnetic perturbations, resulting therefore in higher order harmonics. Such phenomenon for EP generated by external heating in tokamaks was raised by Mynick [5] and later confirmed by Carolipio in DIII-D tokamak [6] and invoked by Garcia-Muñoz to explain the losses of NBI-generated EP in the presence of tearing modes in AUG tokamak [7]. Further analysis regarding the interaction of EP and tearing modes in phase space was presented in [8]. The most recent results obtained on the topic have been reported in [9] for the analysis of losses of passing energetic ions in the EAST tokamak. From a fundamental point of view, a tearing mode is a special class of instability occurring in magnetized laboratory and space plasmas in the presence of non-ideal effects and can tear apart the magnetic field lines, resulting in their reconnection. This process leads to a modification of the magnetic topology, characterized by the formation of magnetic islands. In magnetically confined plasmas, tearing modes enhance particle and energy IAEA-CN-1105 transport, degrading this way the overall confinement. In the present paper, we explore the mechanism highlighted in the seminal work of Mynick [5] and extend it to the transport and losses of alpha particles in JET-like tokamaks in view of the next JET experimental campaign. For this purpose, we will use a recently developed and upgraded Guiding Centre Tracking (GCT) code [3,4] to integrate the trajectories of tracers in the presence of an externally prescribed magnetic perturbation.

GUIDING-CENTRE TRACKING (GCT) CODE

GCT solves the time evolution of the guiding-centre coordinates (𝒙, 𝑢 ∥ , 𝜇) in a prescribed 3D electro-magnetic field (𝜙, 𝐴 ∥ ) for particles with mass 𝑚 and charge 𝑒𝑍. This is done by integrating the equations of motion in covariant formulation (in the absence of collisions in the present paper)

𝑑𝑥 𝑖 𝑑𝑡 = 𝑢 ∥ 𝑏 𝑖 + 𝑣 𝐷 𝑖 + 𝑣 𝐻 𝑖 + 𝑣 𝛿𝐴 𝑖 𝑚 𝑑𝑢 ∥ 𝑑𝑡 = -(𝑏 𝑖 + 𝑚𝑢 ∥ 𝑒𝑍𝐵 ∥ * 𝜏 𝑖 ) 𝜕 𝑖 [𝜇𝐵 𝑒𝑞 + ℎ] 𝑑𝜇 𝑑𝑡 = 0
Where (𝑥 1 , 𝑥 Representing, respectively, the magnetic drift velocity, the Hamiltonian drift velocity and the magnetic drift velocity modified by the parallel flow due to the magnetic potential. Note that the Hamiltonian drift velocity is the sum of the 𝐸 × 𝐵 drift velocity and a perpendicular velocity due to the magnetic field line bending. Finally, the co-variant component of a vector 𝒀 is given by 𝑌 𝑖 = 𝒀 • 𝛁𝑥 𝑖 . In the previous expressions, 𝐽 0 represents the gyro-average operator, which has been set to the identity in the present work for the sake of simplicity, although it can be activated in GCT.

In GCT Where (𝑚, 𝑛, 𝜔) are the poloidal and toroidal mode numbers and the frequency, respectively. The radial eigenfunction 𝐴 ∥ (𝑟) is calculated numerically using a shooting module, as done in [10], in a radial grid which can be as fine as required. Therefore, the integration can be done using interpolated values obtained from standard interpolation methods such as Lagrange polynomials. The toroidal and poloidal numbers and the frequency are fixed. The angular and temporal dependence can therefore be analytically evaluated. The integration of the trajectories is done using a 4 th order explicit Runge-Kutta method in toroidal coordinates. This method can raise the problem that the minor radius for particles close to the magnetic axis can take intermediate or final negative values, which is obviously mathematically forbidden. Therefore, for those particles the trajectories are integrated in cylindrical coordinates.

SIMULATION SET-UP

We will use as reference a Deuterium plasma with 𝑚 𝑟𝑒𝑓 = 𝑚 𝐷 , 𝑍 𝑟𝑒𝑓 = 𝑒. We also take as reference values for the temperature and magnetic field 𝑇 𝑟𝑒𝑓 = 5 𝑘𝑒𝑉, 𝐵 𝑟𝑒𝑓 = 3.5 𝑇. In addition, we select values typical of the JET tokamak, i.e. the minor and major radius are 𝑎 = 1 𝑚, 𝑅 0 = 2.9 𝑚, respectively. We focus our analysis on the transport and losses of fusion-born alpha particles, with energy 𝐸 𝛼 = 700𝑇 𝑟𝑒𝑓 , in the presence of a single helicity tearing mode with (𝑚, 𝑛) = (2, -1). In this paper, we follow the example of previous works [5,6,7] and focus our analysis on static magnetic islands. Therefore, for this specific case, we set 𝜔 = 0. For the sake of simplicity, we use analytical equilibrium magnetic field characterised by circular and concentric flux surfaces, although the results can be obtained also in the presence of triangularity, elongation and Shafranov shift. The present analysis is done using an equilibrium exhibiting a reversed safety factor above 1 in the core and close to 7 at the edge. This In the presence of this equilibrium, the total magnetic field 𝑩 = 𝑩 𝑒𝑞 + 𝛿𝑩 exhibits an island around the resonant surface. This island is also present in the Poincaré map of the trajectories of particles, since for deeply passing thermal particles or deeply passing particles with very low energy, the trajectories follow almost exactly the magnetic field lines. This is represented in FIG. 2 for an amplitude 𝐴 ∥ = 0.05.

TRAJECTORIES OF ALPHA PARTICLES IN THE PRESENCE OF A TEARING MODE

When this magnetic island is formed, the trajectories of charged particles are modified with respect to the unperturbed trajectories, as we have just seen in the previous section. One can naively think that all particles follow the magnetic field lines (with a more or less pronounced drift due to the magnetic field inhomogeneity depending on the energy of the particles), leading to a flattening of the radial profiles inside the region where the magnetic island lies. However, as was invoked in [5], one single helicity magnetic island can lead to several islands in the trajectory space of particles. This can be explained as follows. The large energy of fusion-born alpha particles results in a large magnetic drift. We use in this section the angle-action description based on the existence of a set of variables (𝜶, 𝑱), where 𝑱 = (𝐽 1 , 𝐽 2 , 𝐽 3 ) are the invariants in the unperturbed motion and 𝜶 = (𝛼 1 , 𝛼 2 , 𝛼 3 ) are the three angles representing the three quasi-periodic directions of motion. In this description, the first action is proportional to the magnetic moment and the third action is exactly the toroidal canonical momentum. The first angle is the gyro-phase, the second angle is related to the poloidal angle as 𝜃 = 𝛼 2 + 𝜃 ̂(𝛼 2 ) and the third angle is related to the toroidal angle as 𝜑 = 𝛼 3 + 𝑞𝜃 ̂(𝛼 2 ), where 𝜃 ̂ is a periodic function of 𝛼 2 . The perturbed magnetic potential can be written as 𝑒 𝑖(𝑚𝜃+𝑛𝜑) = 𝑒 𝑖(𝑚𝛼 2 +𝑛𝛼 3 ) 𝑒 𝑖(𝑚+𝑛𝑞)𝜃 ̂(𝛼 2 ) . The complex exponential of a periodic function can be written via the Jacobi-Anger identities using the Bessel functions. This introduces naturally multiple harmonics of the form 𝑒 𝑖((𝑚+𝑝)𝛼 2 +𝑛𝛼 3 ) , with 𝑝 ∈ ℤ. The amplitude of these harmonics increases with the energy of the particles and with the amplitude of the magnetic perturbation. If these harmonics overlap, they can potentially lead to chaos and facilitate the transport of alpha particles from the inner region to the outer region of the tokamak, resulting in losses, as reported in [5,6,7].

We have confirmed these findings with GCT for the case of alpha particles, with mass and charge 𝑚 𝛼 = 2𝑚 𝑟𝑒𝑓 , 𝑍 𝛼 = 2𝑍 𝑟𝑒𝑓 , respectively. This is represented in FIG. 3 ⁄ and 𝜃 and on the right panel the Poincaré map in the poloidal cross-section. For thermal passing particles (FIG. 3) it is observed that the trajectories follow almost exactly the magnetic field lines. In the poloidal cross-section, only a single island is observed. However, in the left panel the higher harmonic islands can be observed, as they are amplified by the 𝑟 2 scale. When the energy of alpha particles is increased, two phenomena are observed. First, the trajectories and the islands are deformed, which is consistent with the increase of the magnetic drift. Second, regions where the islands overlap appear. These regions become wider when the energy and the pitch angle increase, as observed in FIG. 5, where a chaotic region appears around the main island and extends up to the edge. The existence of such chaotic regions can increase the transport of alpha particles from the core to the edge, resulting in losses. This is analysed in the next sections.

MODIFICATION OF THE INITIAL DENSITY PROFILE

In the presence of this magnetic island (𝐴 ∥ = 0.05), we now follow 50 million macro-particles with 𝐸 = 𝐸 𝛼 , pitch angles Λ uniformly distributed in the interval [0,1] and mass and charge 𝑚 𝛼 = 2𝑚 𝑟𝑒𝑓 , 𝑍 𝛼 = 2𝑍 𝑟𝑒𝑓 . These macroparticles are distributed uniformly in the whole poloidal cross-section, but initialized only at 𝜑 = 0. Each of these macro-particles represents a number of physical particles provided a given density profile of fusion-born alpha particles.

We can consider an initial profile of fusion-born alpha particles which is given in figure 15 of Ref. [10]. Such profile is peaked in the inner region of the tokamak a decreases down to 0 at the edge. For our case, we can model it using an analytic profile of the form

𝑛 𝛼 [× 10 16 𝑚 -3 ] = 2 (1 -tanh 𝑟 𝑎 ⁄ -0.4 0.17 )
We have run two simulations: without and with the tearing mode. The result is summarized in FIG. 6, where the radial dependence of different density profiles is plotted. The solid black line represents the initial profile of born alpha particles. In the absence of tearing mode, it is observed a significant modification of the profile, given by the dashed blue line. Such modification is due to the so-called prompt losses, a.k.a. first orbit losses. This simulation has been run during 2 • 10 5 reference cyclotron periods, or 1 ms. Beyond this time, the density profile is not modified any longer. In the presence of the tearing mode, the simulation has been performed up to 10 7 reference cyclotron periods, or equivalently 50 ms. During this time, a depletion of the density profile occurs in the region 0.3𝑎 ≤ 𝑟 ≤ 0.8𝑎, whereas in the inner region the modification is reduced to an oscillating behaviour.

FIG. 6: Radial profiles of the initial density of alpha particles (solid black line) and the final density obtained in the presence (solid red line) and in the absence (dashed blue line) of a tearing mode.

IAEA-CN-1105

ANALYSIS OF THE TRANSPORT AND LOSSES OF ALPHA PARTICLES DEPENDING ON THEIR TRAJECTORIES

The net modification of the density profile that we have seen in the previous section is due to the losses of particles in the presence of the tearing mode. It is insightful to characterize the contribution of the different classes of particles to the losses. For that purpose, we determine the type of trajectory for each particle (co-passing, counterpassing or trapped) and calculate the probability that an alpha particle born at a position (𝑟, 𝜃) is lost. The result is illustrated in FIG. 7 for counter-passing (left panel), trapped (middle panel) and co-passing (right panel) particles.

FIG. 7: Probability of loss for an alpha particle born at a given position depending on the class of trajectory.

It is observed the formation of up-down symmetric structures for passing alpha particles, reminiscent of the poloidal structure of the tearing mode. The shift of the loss probability is also consistent with the magnetic shift of the passing trajectories. The inner region of the tokamak is characterized by a very low or even vanishing probability, whereas the outer region is characterized by a higher probability. For trapped alpha particles a band structure is formed, with higher loss probability for barely trapped particles. It is important to characterize the losses in terms of the exit time, i.e. the time an alpha particle takes to leave the system from its initial position. Of course, since for each initial position in real space one can have several values of exit time, the useful quantity is the mean exit time. This time is represented in ms in FIG. 8 The fact that the mean exit time and the loss probability are anti-correlated indicates that the transport and losses of alpha particles are characterized by rare events. Indeed, the probability that alpha particles leave the system at large times is low. Nonetheless, these events characterize the transport of alpha particles in the presence of a single-helicity magnetic mode, like a tearing mode. The PDF of the exit time is calculated and plotted in FIG. 9.

The left panel shows the analysis following the same classification of trajectories: counter-passing, co-passing and trapped. It is observed that in both cases the PDFs exhibit heavy tails and depart from a Gaussian. This indicates that the transport of alpha particles is not diffusive, as explained in [4]. In particular, it is observed an algebraic decay of the form 𝑃 𝑒𝑥𝑖𝑡 ~𝑡-𝜇 𝑒 , with 𝜇 𝑒 depending on the class of trajectory (𝜇 𝑒 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 > 𝜇 𝑒 𝑡𝑟𝑎𝑝𝑝𝑒𝑑 ). The right panel shows the same analysis, but based on the initial radial position of particles and the pitch angle. Similar algebraic decay is observed, the lower exponent is observed for particles born in the outer region (𝑟 𝑎 ⁄ ≥ 0.5) and for particles born with Λ ≥ 0.7 (independently of the radial region where they are born). FIG. 9: PDF of the exit time in log-log scale using different classifications of trajectories: counter-/co-passing/trapped (left panel) and initial radial position and pitch angle (right).

Finally, a scan on the amplitude of the tearing mode has been performed for four values of 𝐴 ∥ = [0.01, 0.025, 0.033, 0.05]. For each of these values, a simulation up to 50 ms has been run and the total number of lost particles has been calculated, compared to the number of prompt losses. The result is summarized in FIG. 10, where the fraction of lost particles is plotted as a function of the amplitude of the perturbation. In this calculation, we have taken into account the prompt losses. Therefore, the fraction that is plotted is calculated as

𝑓 𝑙𝑜𝑠𝑡 = 𝑁 𝑙𝑜𝑠𝑡 -𝑁 𝑝𝑟𝑜𝑚𝑝𝑡 𝑁 𝑡𝑜𝑡𝑎𝑙 -𝑁 𝑝𝑟𝑜𝑚𝑝𝑡
Where 𝑁 𝑙𝑜𝑠𝑡 is the total number of particles lost at 𝑡 = 50 ms, 𝑁 𝑝𝑟𝑜𝑚𝑝𝑡 is the number of particles lost in the absence of tearing mode (𝐴 ∥ = 0, i.e. prompt losses) and 𝑁 𝑡𝑜𝑡𝑎𝑙 is the total number of particles present in the system at 𝑡 = 0. The number of particles is calculated taking into account the density profile used in the previous sections in order to determine the number of physical particles that each macro-particle represents. Similar behaviour as that reported in figure 11 of Ref. [6] is observed. In the case of fusion-born alpha particles in JET, our GCT collisionless simulations predict losses that can represent up to 17% of the total population (having subtracted the prompt losses) after 50 ms. It is to be noted that this time remains below the expected slowingdown time (~200 -350 ms in the inner region 𝑟 𝑎 ⁄ ≤ 0.4 for advanced and hybrid scenarios) and also below the expected thermalization time (~300 -500 ms) [START_REF] Dumont R | IAEA Fusion Energy Conference[END_REF]. code to integrate the trajectories of guiding-centres in the presence of an imposed single helicity magnetic perturbation representing a tearing mode. Such perturbation leads to the formation of a magnetic island. It has been confirmed that the large magnetic drift of alpha particles leads to the formation of higher poloidal harmonics in phase space, which can eventually overlap and result in chaotic regions. These regions can increase the radial transport of fusion-born alpha particles and produce significant losses. An extrapolated density profile for alpha particles in a DT plasma of the JET tokamak has been used to quantify and characterize the transport and losses of alpha particles. It has been shown that the transport is not diffusive and that the probability of losses as well as the exit time depend strongly on the class of trajectory and the position at which the particle is born. It is to be noted that the work reported here needs to be extended to take into account the continuous generation of alpha particles for a realistic prediction of the modification of the radial profiles in the presence of a tearing mode. Also, including collisions and gyro-average effects will be done in the near future.

  safety factor profile can be representative of hybrid scenarios in the JET tokamak. The radial profile of the safety factor is given in the left panel of FIG. 1. The position of the resonant surface 𝑞 = -𝑚 𝑛 ⁄ is represented by the vertical dashed line. This safety factor profile provides a set of higher order resonances for 𝑞 = 3, 𝑞 = 4, 𝑞 = 5, 𝑞 = 6, which can potentially overlap with each other leading to chaos. This safety factor profile is also characterized by a positive Δ′, i.e. it is tearing unstable. The resulting magnetic potential for an (𝑚, 𝑛) = (2, -1) tearing mode is given in the right panel of the same figure.

FIG. 1 :

 1 FIG.1: Radial profiles of the safety factor (left) and the magnetic potential (right) obtained using the shooting method. The radial coordinate is normalized to the reference Larmor radius. The resonance surface is represented by a vertical dashed line.

IAEA-CN- 1105 FIG. 2 :

 11052 FIG. 2: Poincaré map of trajectories of deeply passing thermal particles in the presence of an (m,n)=(2,-1) tearing mode.

FIG. 4 :

 4 FIG. 3: Poincaré map of trajectories for (𝐸, Λ) = (𝑇 𝑟𝑒𝑓 , 0)

FIG. 8 :

 8 FIG. 8: Mean exit time for an alpha particle born at a given position depending on the class of trajectory.

FIG. 10 :

 10 FIG. 10: Fraction of lost fusion-born alpha particles after 50 ms as a function of the amplitude of the perturbation 7. CONCLUSIONS AND FUTURE DIRECTIONS

ACKNOWLEDGEMENTS

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 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. All the simulations were performed on the MARCONI supercomputer (CINECA) under project reference FUA34_REMOTE and on the SKL Irene partition of the TGCC Joliot-Curie HPC, under project reference ra5409. D.d.-C.-N. was sponsored by the Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy under Contract no. DE-AC05-00OR22725.