Runaway electron drift orbits in magnetostatic perturbed fields

Transcription

1 Runaway electron drift orbits in magnetostatic perturbed fields G. Papp 1,2, M. Drevlak 3, T. Fülöp 1, P. Helander 3 1 Department of Applied Physics, Nuclear Engineering, Chalmers University of Technology and Euratom-VR Association, SE Göteborg, Sweden 2 Department of Nuclear Techniques, Budapest University of Technology and Economics, Association EURATOM, H-1111 Budapest, Hungary 3 Max-Planck-Institut für Plasmaphysik, Teilinstitut Greifswald, Germany papp@chalmers.se Abstract. Disruptions in large tokamaks can lead to the generation of a relativistic runaway electron beam that may cause serious damage to the first wall. To mitigate the disruption and suppress the runaway beam the application of resonant magnetic perturbations has been suggested. In this work we investigate the effect of resonant magnetic perturbations on the confinement of runaway electrons by simulating their drift orbits in magnetostatic perturbed fields and calculating the orbit losses for various initial energies and magnetic perturbation magnitudes. In the simulations we use a TEXTOR-like configuration and solve the relativistic, gyro-averaged drift equations for the runaway electrons including synchrotron radiation and collisions. The results indicate that runaway electrons are well confined in the core of the device, but the onset time of runaway losses closer to the edge is dependent on the magnetic perturbation level and thereby can affect the maximum runaway current. However, the runaway current damping rate is not sensitive to the magnetic perturbation level, in agreement with experimental observations. PACS numbers: Av, Dq, Fs, Ny, Fa Submitted to: Nuclear Fusion

2 Runaway electron drift orbits in magnetostatic perturbed fields 2 1. Introduction A serious problem facing ITER and similar devices is the occurrence of plasma-terminating disruptions. During a disruption a very quick cooling of the plasma takes place, so that the conductivity drops and, as a result, the toroidal electric field rises dramatically. This electric field can detach a fraction of the electrons (in velocity space) from the bulk of the plasma and accelerate them to very high energies (tens of MeV). The detached electrons are normally referred to as runaway electrons [1]. The beam of runaway electrons usually drifts to the wall of the tokamak, where it can cause serious damage. Unmitigated disruptions represent a severe risk for ITER [2] and should be avoided by reliable control of the plasma discharge. An extrapolation of the effects of a disruption from existing devices to ITER is affected by uncertainties, which should be minimized to guarantee the integrity of the machine. A good understanding and modelling capability of runaway electrons should aid in evaluating different methods aimed at mitigating their effects. The two most discussed disruption mitigation methods are massive gas injection (MGI) and killer pellet injection (KPI). In the first case (MGI), to avoid runaway generation a large amount of gas is needed (more than atoms in ITER [3]), which can have serious implications for the vacuum systems. In the second case (KPI), a sophisticated tailoring of the pellet size, velocity and composition is needed to obtain a short-enough current quench time and avoid runaway generation simultaneously [4, 5]. In addition to these techniques, resonant magnetic perturbations (RMP) are used for runaway electron mitigation. In this work we concentrate on the use of RMP to increase the runaway electron losses. This has been shown to work well in various experiments: in JT-60 it was shown that runaways were absent for a sufficiently high perturbation field [6]; in TEXTOR the runaway losses were enhanced by RMP and it was shown that the runaway avalanche can be suppressed [7, 8, 9, 10]; and also in Tore Supra applying an ergodic divertor led to enhanced losses [11]. However, recent results from JET indicate that RMP have not been successful in suppressing the runaway beam [12]. The reason for the difference in the experimental success of suppressing runaways in various devices is not yet properly understood. Previous theoretical work has shown that if the radial diffusion of runaway electrons is sufficiently strong, avalanches can be prevented and the magnetic perturbation level necessary for this has been estimated to δb/b = 10 3 for typical tokamak parameters

3 Runaway electron drift orbits in magnetostatic perturbed fields 3 [13]. However, the radial diffusion coefficient is fairly uncertain, and depends on many parameters. In the simplest picture, the electrons follow stochastic magnetic field lines and diffuse radially out of the plasma with the Rechester-Rosenbluth diffusion coefficient [14]. In reality, this represents an upper bound on the transport and runaway confinement is usually better since finite Larmor radius and magnetic drift effects reduce the diffusion coefficient. Due to the complexity of the effect of magnetic drift on diffusion, a reliable picture of how the runaway electrons are transported out of the plasma can only be obtained via three-dimensional numerical modelling of the runaway electron drift orbits. In this paper we present results of simulations of the runaway electron drift orbits in TEXTOR-like perturbed magnetostatic fields and evaluate the effect of RMP coils on runaway loss enhancement. This has been done by solving the equations of the runaway electron drift orbits including the effect of synchrotron radiation and plasma collisions. The main reason we use a TEXTOR-like configuration is that it has been shown experimentally on TEXTOR that runaway losses can be enhanced by the application of RMP. At sufficiently high perturbation levels a reduction of the runaway current was observed. It is useful to select a case where there are experimental observations on runaway losses that allow us to benchmark our calculations. Previous work [9, 10] analysed the dynamics of runaway electrons in TEXTOR plasmas with perturbed magnetic fields, by using a mapping method based on Hamiltonian guiding center equations [15, 16]. The runaway electrons were used as test particles to reveal the ergodized magnetic field structure, and it was noted that the radial runaway profile was modified due to preferential losses in the ergodic zone. In our work we focus on the runaway electron losses and their dependence on energy and perturbation amplitude. Furthermore, we examine the role of collisions, synchrotron radiation losses and an accelerating electric field in the runaway electron dynamics. The paper is organized as follows: In Sec. 2 the relativistic drift equations are described, together with the momentum loss caused by synchrotron radiation. In Sec. 3 the structure of the magnetic field and the perturbation are described for a specific TEXTOR-like equilibrium used in the simulations. In Sec. 4 the effect of RMP on the particles radial and velocity distribution is studied. In Sec. 5 the loss fractions of the runaway electrons are discussed and the effect of RMP and the toroidal electric field on the transport is described and compared with experimental observations. Finally, the results are summarized in Sec. 6.

4 Runaway electron drift orbits in magnetostatic perturbed fields 4 2. Model equations We solve the relativistic, gyro-averaged equations of motion for the runaway electrons including the effect of synchrotron radiation with the ANTS (plasma simulation with drift and collisions) code [17]. This code calculates the drift motion of particles in 3D fields and takes into account collisions with background (Maxwellian) particle distributions, using a full-f Monte Carlo approach. For the purposes of this paper it has been extended to include synchrotron radiation losses and a new collision operator that is valid for both low and high velocities. In the following we will briefly describe the equations we solve and their implementation in the ANTS code Relativistic drift equations The relativistic Lagrangian L = mc 2 1 v 2 /c 2 eφ + ea v is regarded as a function of r, ṙ and t, with the particle position r = R + ρ(ˆx cos θ + ŷ sin θ), R is the guidingcenter position, ρ is the gyroradius, (ˆx, ŷ, ˆb) is an orthogonal set of vectors aligned with the magnetic field B = Bˆb and Ṙ (ˆb Ṙ)ˆb, so that v = (ˆb Ṙ)ˆb + ρ θ( ˆx sin θ + ŷ cos θ). The gyro-average of the Lagrangian becomes L(R, ρ, θ, Ṙ, ρ, θ, t) = mc 2 1 [(ˆb Ṙ)2 + ρ 2 θ 2 ]/c 2 eφ+e(a Ṙ+Bρ2 θ/2)(1) and A and φ are evaluated at the guiding-center position R. The Euler-Lagrange equation ( ) d L = L dt Ṙ R (2) gives mˆb d(γv ) dt = e(e + Ṙ B) µ B γmv2 κ, (3) where µ is the magnetic moment, κ = ˆb ˆb = ˆb ( ˆb). The parallel component of this equation is d(γmv ) dt = ee µ B (4) and the perpendicular component (take the vector product with ˆb) gives E Ṙ = v ˆb ˆb+ B + ebˆb B+ µ ( ) γmv2 eb ˆb κ γm = v ˆb+ eb ˆb v 2 2 ln B + v2 κ (5)

5 Runaway electron drift orbits in magnetostatic perturbed fields 5 These equations are used in the ANTS code to follow the particle orbits. The momentum and energy loss due to synchrotron radiation is also included and the details are given in Appendix A The ANTS code The ANTS code is highly modular and flexible, written in C/C++, using MPI for paralellisation. It is able to use the entire range of coil types available from the ONSET coil optimisation package [18] in order to describe the external magnetic field. ONSET and ANTS (and also EXTENDER [19]) share the same range of classes for the representation of the mesh fields, which enhances compatibility. The magnetic field is defined on a mesh in the entire domain of computation, and the integration of the particle orbits is carried out in Cartesian coordinates. This approach provides the greatest flexibility and facilitates a faithful treatment of magnetic fields with islands and ergodic zones, since the existence of magnetic surfaces is not required. The code is able to use fully 3D fields. Our TEXTOR-like, circular magnetic configuration is stellarator symmetric (θ = θ and ϕ = ϕ) [20], and this symmetry reduces the memory demand of the calculations to some hundred MBs. The flexibility of ANTS allows one to run it with different sets of differential equations and Monte Carlo operators describing the drift motion and collisions of the test particles with the background. For the purposes of this paper the ANTS code has been extended to include synchrotron radiation, and a new collision operator valid for both low and high energies has been implemented (see Appendix B for details). The initialisation of the pseudo-random number generator can be reproduced for every single particle in every run, which enables the precise reproduction of every simulation regardless of the stochastic nature of the collisions. This feature offers several benefits. The comparison of the newly implemented radiation term and collision operator with the former description can be carried out with the highest possible precision. This also means that to achieve the necessary statistics in comparisons, a smaller amount of test particles are needed. 3. Magnetic field structure The runaway electron drift orbits have been calculated for typical TEXTOR-like discharges, chosen to be similar to the ones where the runaways were shown to be suppressed by resonant magnetic perturbations created by dynamic ergodic divertor

6 Runaway electron drift orbits in magnetostatic perturbed fields 6 (DED) coils. This divertor consists of 16 helical coils at the high field side of the device. Electrical currents flowing in these coils generate a perturbation field, and a sufficiently large perturbation result in an ergodization of the magnetic field lines. The different operational scenarios are indexed as 12/4, 6/2, 3/1 modes, depending on the current configuration chosen. For instance, in the m/n = 6/2 DC configuration four neighbouring coils are switched in parallel and the neighbouring bundles are fed with currents of opposing sign. In case of deep field penetration the perturbation field may excite tearing modes, however these do not seem to affect the runaway beam suppression [7]. In our calculations we use a TEXTOR-like post-disruption equilibrium, with major radius R 0 = 1.8 m, minor radius a = 0.46 m, toroidal magnetic field B t = 2.25 T (at R = 1.75 m), plasma current I p = 320 ka. In TEXTOR, the runaways were generated deliberately by injection of Argon atoms and therefore we assume that the postdisruption density has increased to 10 times its pre-disruption value. The density, pressure, temperature and q-profile we used are n e = n 0 (1 0.9ρ 2 ) 2, with n 0 = m 3, T e = T 0 (1 1.1ρ 2 ) 2 with T 0 = 10 ev and q = q 0 (1 βρ 2 ) α, with the parameters q 0 = 0.97, q a = 4, α = 0.8, β 1 (q 0 /q a ) 1/α, and ρ is the normalized flux. The unperturbed magnetic equilibrium has been calculated by VMEC [21] for the parameters above and the TEXTOR coils Magnetic field perturbations The magnetic field perturbations are modelled to be similar to the ones produced by DED-coils on TEXTOR in the 6/2 DC operation mode [22, 23]. The 6/2 DC mode has a 180 toroidal rotation symmetry, the generated islands have toroidal mode number of n = 2. Mode numbers of n = 1 and n = 4 can be reached with configurations possessing a symmetry of 360 and 90, respectively. For simplicity the terminals are ignored, so the coils in our simulations are stellarator-symmetric, which saves memory. Figure 1a shows a sketch of the DED coil configuration used in our simulations. These coils create magnetic perturbations at the plasma periphery on the high field side of the torus that decay radially toward the inside of the plasma as shown on figure 1b. In this work it is assumed that the field generated by the DED coils is significantly smaller than that from the toroidal field coils and the plasma current. Therefore, and because we neglect plasma shielding (see below), we approximate the perturbed magnetic field by simply superimposing the field from the DED coils on the field of the

7 Runaway electron drift orbits in magnetostatic perturbed fields 7 (a) 10 4 X (db/b ) 2 flux DED: 1 ka (b) Radial position (normalized flux) Figure 1. (a) Sketch of the DED coil configuration implemented in the simulation. Black coils are fed with +I DED, grey coils are fed with I DED. The torus marks the plasma. (b) Radial dependence of the flux surface average (δb/b) 2 for I DED = 1 ka. unperturbed VMEC solution. The radial variation of the flux-surface averaged magnetic perturbation level (δb/b) 2 for I DED = 1 ka is shown on figure 1b. For other perturbation amplitudes the values can be calculated in a straightforward way, since (δb/b)2 I DED. The magnetic perturbation level that is predicted to be necessary for runaway suppression is δb/b = 10 3 [13]. To achieve such a perturbation level at the flux-surface ρ = 0.7 requires I DED = 6.7 ka, while having δb/b = 10 3 in the center requires I DED = 10.6 ka. 6.7 ka is close to the technically achievable upper limit of 7.5 ka on TEXTOR in this mode. The magnetic flux surfaces in the perturbed equilibrium are shown in figure 2 for various DED currents. Clearly, the edge region becomes more ergodic with higher DED-currents. Particles outside the last intact magnetic surface are expected to leave the plasma rapidly, and this is indeed what happens, as will be shown later. Also, as the magnetic perturbation grows magnetic islands appear, the locations of which are correlated with rational values of the safety factor. In the simulations we neglect the effect of shielding of magnetic field perturbations by plasma response currents. This is motivated by the fact that the shielding is expected to be small in cold post-disruption plasmas. Estimates of the plasma shielding of RMP in non-disruptive DIII-D plasmas have shown that the plasma response currents significantly reduce the effect of the perturbation field on the magnetic configuration in the core plasma [24]. However comprehensive calculation of the shielding in disruptive plasmas is still lacking. Analysis of the shielding effect is beyond the scope of the present paper, but it is clear that it would reduce the runaway losses and therefore our results will have to be interpreted as an upper limit on actual losses.

8 Runaway electron drift orbits in magnetostatic perturbed fields DED: 0 ka Equilibrium LCFS 0.4 DED: 6 ka Equilibrium LCFS Z [m] Z [m] (a) -0.4 (b) R [m] R [m] Figure 2. Magnetic flux surfaces in a perturbed TEXTOR-like equilibrium for various DED-currents. The equilibrium last closed flux-surface are shown by red ellipses. (a) I DED = 0 ka. (b) I DED = 6 ka. The regions with magnetic islands are highlighted and the corresponding mode numbers are given. 4. Effect of RMP on particle radial and velocity distributions Experimental observations in several tokamaks show that magnetic perturbations can cause an enhanced radial loss of runaway electrons [6, 7, 8]. In TEXTOR, the application of DED perturbation fields resulted in a significant decrease of the runaway population [7]. Interestingly, the current decay rate showed no dependence on the type of perturbation and its magnitude. In most cases a threshold I DED /n = 1.4 ka could be identified beyond which there was full suppression of high energy runaways (E > 25 MeV). However, in a few cases runaways were not suppressed even when the DED-current was higher than the above mentioned threshold. In order to understand these experimental facts we analyse the particle positions, energies and pitches, together with their losses in the TEXTOR-like perturbed magnetostatic fields presented in the previous section, for various DED-currents and runaway electron energies. To be able to gain detailed information about the effect of RMP, we first focus on simulations without an accelerating electric field. The effect of the electric field will be explored in section 5.2. We begin by showing Poincaré plots of the particle orbits: one particle was launched at each radial position and followed for t = 30 µs simulation time. Various energies and

9 Runaway electron drift orbits in magnetostatic perturbed fields 9 DED currents are used to illustrate the structure of the perturbation and the shrinkage of the magnetic confinement region with increasing electron energy. Figure 3. Poincaré plots of the particle orbits for different runaway energies and DED currents. Left to right: 1, 10 and 30 MeV; top: I DED = 0 ka, bottom: I DED = 6 ka. Figure 3 shows the Poincaré plots of the particles with energies 1 MeV, 10 MeV and 30 MeV in cases where the DED currents are I DED = 0 ka and I DED = 6 ka. Note, that the particle orbits of runaway electrons with kinetic energy 1 MeV (figure 3a and d) are very similar to the Poincaré plots of the field lines (figure 2a and b), since low energy particles approximately follow field lines. We have already noted in figure 2 that for higher DED-currents the edge zone becomes ergodic. This can be observed on figure 3d as well. Note that the effect of the DED decreases with increasing particle energy: at 10 and 30 MeV (figure 3e and figure 3f) the edge stochastization is less visible. The shrinkage of the confinement zone plays an important role at high particle energies regardless of the DED. The effective boundary of the confinement volume for low energy (E 1 MeV) particles coincides with the equilibrium LCFS marked with the red line as shown on figure 3a. That the confinement volume shrinks with increasing particle energy. The orbits (in the unperturbed field) of the particles are circles that are displaced horizontally with respect to the flux surfaces, with a displacement that is proportional to the energy (for relativistic particles). This is the reason why the confinement region

10 Runaway electron drift orbits in magnetostatic perturbed fields 10 shrinks when the energy is large enough. Particles outside the confinement region follow drift orbits that intersect vessel components and are lost very rapidly. As we shall see, the losses therefore increase with increasing particle energy Particle positions We now turn to the examination of particle positions as functions of time. In figure 4 we have plotted the radial positions of 100 electrons with initial energy 1 MeV, launched at the flux-surface ρ = 0.7. In TEXTOR, the plasma current and the toroidal magnetic field are in opposite directions, and, of course, runaways always travel in the direction opposite to the plasma current. In TEXTOR this is therefore in the direction of the magnetic field, so that runaways in TEXTOR are co-passing (with respect to the magnetic field). Here we study both co- and counter-passing electrons, by simply reversing their launch direction to investigate possible differences and gain insights into the physics. This reversal is not likely to occur in practice. The particles are well confined and do not drift radially in the absence of collisions, as expected (this case is not shown in figure 4 but was checked to confirm the accuracy of our calculations). If collisions are switched on, a slightly asymmetric drift arise, as shown in figure 4. The time instant when this happens is when particles are converted (by collisions) from passing to trapped. This gives rise to an either outward or inward shift (with the size of the banana width) depending on the launch direction - outward for counter-passing electrons (figure 4a-c) and inward for co-passing ones (figure 4d-f). In addition to collisions, the runaways experience a reaction force from the emission of synchrotron radiation. Although this does not change the particle positions much, because the energy loss is very small (see Appendix A for details), it is nevertheless retained for purposes of completeness. radial positions [ρ] ρ t=0 = 0.7 I DED = 0.0 ka E t=0 = 1 MeV (CO) E = 0 V/m tor (a) time [ms] radial positions [ρ] ρ t=0 = 0.7 I DED = 3.0 ka E t=0 = 1 MeV (CO) E = 0 V/m tor (b) time [ms] radial positions [ρ] ρ t=0 = 0.7 I DED = 6.0 ka E t=0 = 1 MeV (CO) E = 0 V/m tor (c) time [ms]

11 Runaway electron drift orbits in magnetostatic perturbed fields 11 radial positions [ρ] ρ = 0.7 t=0 I DED = 0.0 ka E = 1 MeV (CU) t=0 E = 0 V/m tor (d) time [ms] radial positions [ρ] ρ = 0.7 t=0 I DED = 3.0 ka E = 1 MeV (CU) t=0 E = 0 V/m tor (e) time [ms] radial positions [ρ] ρ = 0.7 t=0 I DED = 6.0 ka E = 1 MeV (CU) t=0 E = 0 V/m tor (f) time [ms] Figure 4. Time dependence of the radial position of 100 electrons with initial energy 1 MeV launched at flux-surface ρ = 0.7 (a) I DED = 0 ka, (b) I DED = 3 ka, (c) I DED = 6 ka, all co-passing, with initial pitch v /v = 1 (TEXTOR case); (d)-(f) the same for counter-passing, with initial pitch v /v = 1. The simulations show that the magnetic perturbation of the DED scatters the particles radially. This happens even when collisions and radiation is switched off. In the cases shown on figure 4, with collisions switched on, there is a visible increase in the scatter of the particles due to the DED. Comparing figure 4b with figure 4c and figure 4e with figure 4f shows that the magnitude of the DED-current is not too significant in the width of the radial spreading of the particles. But as we will show later, the loss fraction will increase with the magnitude of the DED-currents, due to the stochastization of the plasma edge. Since the net radial displacement caused by the passing-trapped transition depends on the particle direction, the counter-passing particles reach the edge stochastic zone much faster than the co-passing particles. For this reason the co-passing (realistic) particles are less affected by the DED than the counter-passing ones, as will be shown later. These results illustrate that particle collisions play an important role in the transport processes. Including collisions in the simulation and using a collision operator that is valid for arbitrary energies is therefore a significant improvement over previous drift-only based simulations of the effect of RMP on runaway dynamics Thermalization The simulation of a test particle ends if it leaves the plasma or when it gets thermalized. Figure 5 illustrates how 1 MeV electrons launched at ρ = 0.7 are thermalized within 35 ms due to collisions. Figure 5a displays the time dependence of the energy and figure 5b shows how the anisotropic velocity distribution gets isotropic.

12 Runaway electron drift orbits in magnetostatic perturbed fields 12 energies [MeV] ρ t=0 = 0.7 I DED = 0.0 ka E t=0 = 1 MeV E = 0 V/m tor (a) time [ms] pitches [v / v] (b) time [ms] Figure 5. Time dependence of particle (a) energies and (b) pitch (v /v) for a 1 MeV particle distribution. Thermalisation occurs within 35 ms due to collisions. The background density plays a key role in the behaviour of the low energy (e.g. 1 MeV) runaway electrons. If we decrease the density by a factor of two, the thermalization time rises by a factor of two as well. If the particles are launched further in, for example at flux-surface ρ = 0.5, where the density is higher, most low energy particles are thermalized already after 20 ms. Since the time for thermalization grows with the energy, the higher energy particles will not be thermalized during the short timescale that is characteristic of a TEXTOR-disruption. Therefore the only loss mechanism for these particles consists in their orbits leaving the confinement region. The magnetic perturbation due to DED has practically no influence on the thermalization process. There is no notable difference between cases with and without DED if we investigate particle energy losses. The effect of the DED for co-passing runaways is smaller due to the inward shift at the passing-trapped transition than it would be for a counter-passing population. On the other hand, this transition kicks the particles to a higher density region, enhancing thermalization. In the simulations presented above, the toroidal electric field was set to be zero. To be able to analyse particle energy issues correctly, the accelerating field has to be taken account. Too fast thermalization for the low energy particles in the simulation can prevent possible orbit losses, leading to the underestimation of the DED induced drift orbit losses. The shrinkage of the confinement zone with increasing energy also plays an important role.

13 Runaway electron drift orbits in magnetostatic perturbed fields Effect of toroidal electric field A simple estimate of the toroidal electric field during the disruption is [13]: E L di 2πR dt µ 0 di 2π dt, (6) where we have approximated the plasma inductance by L µ 0 R. If we use the experimental value of di/dt 70 MA/s [7] we arrive to E = 14 V/m. This is equivalent to a force on an electron of N. The friction force due to collisions can be estimated from the energy loss. From figure 5a, the energy loss is 30 MeV/s for the 1 MeV electrons, which is equivalent to a N friction force. This is two orders of magnitude smaller than the accelerating force. Thus, for a proper treatment of the low energy particles the accelerating force has to be taken into account. We have to keep in mind that the self-consistent computation of the accelerating force during runaway generation is non-trivial [25, 26]. For this work we used the estimate E = 14 V/m electric field in the simulations. We only study co-passing electrons, since, as mentioned above, the induced electric field is always antiparallel to the plasma current and thus in the same direction as the magnetic field. There is a maximum energy E that a runaway electron can gain in a tokamak disruption, since de dt = ev E ec A t (7) for any charged particle accelerated by an inductive electric field. In a tokamak A φ = ψ φ, where 2πψ is the poloidal magnetic flux, so the energy is limited by E ecδψ/r, where δψ is the change in the poloidal flux caused by the decay of the plasma current. If the aspect ratio is large and the cross section circular we have dψ/dr = rb/q(r), so on the axis (where the drop in ψ is the largest) δψ B a 0 r Ba2 dr q(r) 4 (8) for a typical TEXTOR q-profile. Thus the maximum reachable energy is E eca2 B 4R 20 MeV (9) for the simulation parameters. If the initial 320 ka plasma current decays with di/dt 70 MA/s, the current decay time is 4.6 ms, and the calculated E = 14 V/m acting for 4.6

14 Runaway electron drift orbits in magnetostatic perturbed fields 14 ms can accelerate a particle to reach an additional energy up to 19.3 MeV. This shows the consistency of the energy limit estimation. Simulations with a constant accelerating electric field show unrealistic results after 5 ms simulation time. The next section focuses on the runaway losses. At first we consider the losses of electrons of a specific energy due to RMP, in which case the electric field is set to zero. In this case the simulation is done for electrons launched at ρ = 0.7 and followed for 100 ms. Then we will discuss runaway losses due to RMP for electrons launched at the center and including the effect of the electric field. 5. Runaway losses 5.1. Without accelerating electric field To determine the loss fraction of the runaways, 100 runaway electrons with initial energies of 1 MeV, 5 MeV, 10 MeV and 30 MeV were launched parallel to the magnetic field (both co- and counter-passing) at the magnetic flux surface ρ = 0.7 for various DED currents. The particles were followed for 100 ms and were considered to be lost when they left the last closed flux surface. These initial tests were performed without an accelerating electric field, which allows us to study the losses of electrons of a specific energy due to RMP. Table 1. Number of lost particles (out of 100 launched particles at ρ = 0.7) after 100 ms as function of DED current and runaway energy. Co-passing (TEXTOR) Counter-passing I DED \E 1 MeV 5 MeV 10 MeV 30 MeV 1 MeV 5 MeV 10 MeV 30 MeV 0 ka ka ka It is clear from the magnetic and particle Poincaré figures that the magnetic structure is increasingly stochastized with increasing DED current and that the confinement region shrinks with increasing particle energy. The number of lost particles (after a given time) thus increases with energy, as indeed observed numerically. As stated in the discussion of figure 3, high energy particles are less sensitive to the DED than the ones with low energy. This is confirmed in table 1: the DED results in only a few percent increase in the losses for 10 MeV and above.

15 Runaway electron drift orbits in magnetostatic perturbed fields 15 For low energies the co- and counter-passing particles have to be treated separately. As discussed above, the co-passing particles first experience a net inward drift due to collisions, and thermalization stops the population before geometrical losses could occur. For the counter-passing particles, the drift is outward, and in this case the effect of the DED is clearly visible. At 1 MeV after 100 ms we lose four times as many particles if I DED = 6 ka as if I DED = 3 ka. At 5 MeV this factor decreases by a factor two. Without the DED there are practically no losses. To better understand particle losses we plot the total fraction of lost electrons as a function of time. In figure 6a the time dependence of the loss fraction is shown for 1 MeV for various DED-currents, both co- (solid line) and counter-passing (dashed line). The five lines represent the three non-zero loss cases in table 1 plus two simulations with I DED = 20 ka. The direction plays an important role as was stated before. The main difference is the onset time of the particle losses, it decreases with the DED current and for very high perturbation magnitude (here 20 ka) the losses start immediately. After 35 ms the particles are thermalized and no more geometrical losses occur. In figure 6b the behaviour of 5 MeV particles is shown. Only the counter-passing particles are shown here, since in the co-passing case there were practically no losses. Note, that already at 5 MeV immediate particle losses dominate, which is the result of the shrinkage of the confinement region. There are some particle losses at 0 ka as well, but only after 90 ms. % of particles lost (a) 60 ρ = 0.7 E tor = 0 V/m E init = 1 MeV (CO/CU) ka CU 20 ka CO 20 6 ka CU 3 ka CU 6 ka CO time [ms] % of particles lost (b) ρ = 0.7 E tor = 0 V/m E init = 5 MeV (CU) 6 ka 3 ka 0 ka time [ms] Figure 6. Time dependence of particle losses. (a) Loss fractions for 1 MeV, I DED = 3 ka, I DED = 6 ka and I DED = 20 ka. Dashed line stands for counter-, solid line for co-passing particles. (b) Loss fractions for 5 MeV, I DED = 0 ka, I DED = 3 ka and I DED = 6 ka, only counter-passing. While the onset time depends on the DED current, the slope of the loss fraction (the

16 Runaway electron drift orbits in magnetostatic perturbed fields 16 loss rate) as function of time is fairly insensitive to it at least after an intial transient dominated by prompt orbit losses. As we have seen before in figures 4, the magnitude of the DED current does not affect significantly the radial scatter of the particles and therefore the loss rate should not be significantly higher with higher DED-currents. Interestingly, this result correlates with the experimentally observed fact that the runaway current damping rate is independent of the DED-current. Without electric field, for the 1 MeV particles launched at the edge the simulated current damping rate is around MA/s, which is comparable to, but somewhat smaller than the experimentally observed damping rate of MA/s [7]. % of particles lost MeV CU 30 MeV CO 10 MeV CU 10 MeV CO ρ = 0.7 E tor = 0 V/m I DED = 0 ka time [ns] Figure 7. Time dependence of immediate particle losses for 10 and 30 MeV, both co- and counter-passing (marked with different linestyles). All the particle losses occur before 45 ns (immediate losses). The magnetic perturbation or the electric field plays practically no role. Figure 7 shows the time-dependence of the loss fractions for different initial energies (10 and 30 MeV) launched at the flux-surface ρ = 0.7. For such high energies, a large fraction of the particles leave the confinement region due to immediate drift orbit losses, within ns. However, for those particles that remain inside the confinement region the further losses are less than 2%. The DED does not influence these immediate losses, the curves are the same for every DED current. The electric field plays no role in the immediate losses and the curves are the same with and without applied electric field. High energy immediate losses depend on the particle energies. The four lines correspond to the two energies and two directions. The results clearly show that high energy particles launched at the edge are fairly insensitive to the magnetic perturbations. The losses are caused by the shrinkage of the confinement zone, which is larger for high particle energies.

17 Runaway electron drift orbits in magnetostatic perturbed fields Runaway losses with accelerating electric field Our simulations show that the electric field does not play a role in the immediate particle losses, but it does play a role afterwards. Since the acceleration exceeds the friction due to collisions, thermalization does not occur. It is very interesting to study the time dependence of the particle losses. % of particles lost (a) 40 0 ka 1 MeV 6 ka 1 MeV 0 ka 10 MeV 20 ρ = ka 10 MeV E tor = 14 V/m 0 ka 30 MeV E init = 1/10/30 MeV 6 ka 30 MeV time [ms] % of particles lost ρ = 0.7 E tor = 14 V/m E init = 1 MeV 20 0 ka 1 MeV (b) 6 ka 1 MeV time [ms] Figure 8. (a) Time dependence of particle losses with electric field, launched at ρ = and 6 ka case with 1, 10 and 30 MeV initial energies. DED influences the long-term behaviour of particle losses in the case of high energy particles. For low energy particles the DED influences the onset time of the losses, just as in the case without electric field. The gradient of the curves depends on the particle energy. (b) Zoomed version of (a) showing only 1 MeV particles. In figure 8 the loss fractions are shown for 0 and 6 ka DED current in the case of 1, 10 and 30 MeV initial particle energies for co-passing particles launched at ρ = 0.7. For low energy particles the DED, as in the case without electric field, influences the onset time of the losses. There is a 0.5 ms shift in the 1 MeV case. This shift may play a role in avalanche suppression. It is interesting to note that for the 1 MeV electrons, both without DED and with a 6 ka DED-current, all the particles get lost within 4 ms, and the curves hit 100% at roughly the same time. The time dependence of the losses of higher energy electrons (10 MeV and 30 MeV) is more interesting. As in the case for the 1 MeV electrons 85-90% of the particles get lost within 3.5 ms, but the remaining particles are confined for up to 25 ms (30 MeV), and interestingly the effect of DED can even increase the confinement time. This puzzling result is an artifact of the interplay of the monoenergetic distribution launched at ρ = 0.7 and the spreading of the particles. Regardless of DED, the particles launched at the same flux-surface have random banana orbit phase (dictated

18 Runaway electron drift orbits in magnetostatic perturbed fields 18 by birth place on flux surface) and hence get an instant spread dictated by the banana width. The banana width is larger for higher energy particles, and therefore higher energy particles moving inwards can penetrate much deeper in the plasma and thus kept confined for a longer amount of time before they finally get lost. The additional spreading due to DED may even increase the inward drift and consequently may then lead to an artificial confinement increase. This does not happen if the particles are launched in the plasma center, but in that case they do not reach the plasma boundary within the 5 ms lifetime of the accelerating electric field. The detailed parameter scans carried out lead to some fundamental conclusions. The higher the energy of the particle, the lower the effect of the resonant perturbation on it. The higher the initial energy, the shorter is the onset time for particle losses - this is due to the fact that the losses mainly occur through the shrinkage of the confinement zone. However, the time between the first and last particle loss decreases with smaller initial energies. The reason for this is that the high energy particles reach the plasma edge faster, but are less sensitive to the magnetic perturbations. The statistical significance of the simulations can be improved by increasing the number of particles. The standard deviation can be estimated with σ(n) N that is for 1000 particles 0 < σ < 32. We have verified the results presented in this paper by increasing the number of launched particles in a few test cases with up to a factor of 10. Our results show that there is no significant difference between the results with 100 and 1000 particles. Increasing the number of test particles does not influence the startor endpoints, nor the slope of the loss rate curves. The only effect is that the lines are somewhat smoother, but the increase in the required CPU time is too large to motivate this small improvement. 6. Conclusions In this paper we have studied the effect of perturbed magnetostatic fields on runaway electrons. In the simulations we have used a TEXTOR-like equilibrium and calculated the runaway drift orbits to study the losses for various runaway electron energies and magnetic perturbations. The calculations are done with the ANTS code extended for the purposes of this work to include synchrotron radiation losses and a collision operator valid for arbitrary particle energy (but neglecting Bremsstrahlung). The effect of the collision operator was illustrated in simulations without an accelerating electric field.

19 Runaway electron drift orbits in magnetostatic perturbed fields 19 We found that runaway electrons in the core of the plasma are likely to be well confined. Particles in the plasma center do not get lost either during the lifetime of the toroidal electric field, or in the longer time interval without significant toroidal electric field that follows. For low-energy ( 1 MeV) particles closer to the boundary the onset time of the losses is dependent on the amplitude of the magnetic perturbation, and this should affect the maximal runaway current. The runaway current damping rate is however insensitive to the magnetic perturbation level, and its experimentally measured value is consistent with our simulations.. As expected, the energies of the particles are not affected by the magnetic perturbation. Synchrotron radiation emission does not contribute much to the losses, mainly due to the fact that the time-scale of these losses is much longer than the runaway loss time. Our results indicate that the loss of high-energy ( 10 MeV) runaways in the simulation is mostly due to the fact that their orbits are wide, which allows them to intersect the wall. The loss is dominated by the shrinkage of the confinement region, which is independent of the DED current. We have illustrated this in particle Poincaré plots, where the shrinkage is clearly visible for high energies. Note that this only stands for particles launched close to the plasma periphery. We expect the experimental results to be reproducible only by more complex simulations including the temporal evolution of the magnetic structure and the indirect effects of the RMP on runaway electron generation and loss processes. According to Izzo et al [27] the loss of the core runaways is enhanced by MHD perturbations onset by the disruption. These perturbations can expel a significant amount of runaways from the core to the edge in a small scale device like Alcator C-Mod or TEXTOR. The RMP, as described in this paper, can enhance the losses of the runaways once those are close to the edge. This could explain the observed enhancement of runaway losses due to the RMP on smaller tokamaks. For larger devices such as JET or ITER the MHD perturbations are not sufficient to expel the particles from the core. This could be the reason why RMP does not seem to be effective in JET: its minor radius is large enough that the runaway orbits never come close to intersecting the edge stochastic region. Adding RMPs in JET, or in other large tokamaks, might create a few islands or edge ergodic zone, but these would then be too thin to cause significant core orbit losses. Acknowledgments This work was funded by the European Communities under Association Contract between

20 Runaway electron drift orbits in magnetostatic perturbed fields 20 EURATOM, Vetenskapsrådet, HAS and Germany. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors would like to thank G. Pokol for fruitful discussions and M. Lehnen and O. Schmitz for providing the TEXTOR data. Appendix A. Momentum and energy change due to synchrotron radiation Due to the synchrotron radiation losses, the momentum and energy of the particles are not conserved. The Abraham-Lorentz force is [28] K = α { v + 3γ2 γ2 3γ2 v v v + (v v + c2 c2 c 2 ( ) } v v) 2 v, (A.1) where α = e 2 γ 2 /6πɛ 0 c 3. The acceleration of the particle gyrating in a magnetic field is approximately v ev B/(γm 0 ) so that v v = 0. Then, the Abraham-Lorentz force simplifies to ] ( ) K = α [ v + γ2 (v v)v = e4 B 2 v c2 6πɛ 0 m 2 0c 3 + γ2 v 2 v, c 2 where we used that v = e v B/(γm 0 ) = (eb/γm 0 ) 2 v. The average rate of change of perpendicular and parallel momentum due to synchrotron radiation can be calculated from K = m 0 cdp/dt: dp dt = e4 B 2 p 2 p 6πɛ 0 γ(m 0 c) 3 = p2 p γτ r (A.2) dp dt = e4 B 2 p (1 + p 2 )p 6πɛ 0 γ(m 0 c) 3 = (1 + p2 )p γτ r (A.3) where p = γv/c is the normalized momentum and we introduced the radiation time scale τ r = 6πɛ 0 (m 0 c) 3 /e 4 B 2. Equations (A.2-A.3) agree with the rate of change of momentum and pitch-angle given in Ref. [29] in the limit of small Larmor radius. Appendix A.1. Energy change due to synchrotron radiation The energy loss due to synchrotron radiation, as given in Ref. [29] is E = K v = m 0c 2 pṗ, (A.4) γ

21 Runaway electron drift orbits in magnetostatic perturbed fields 21 where E = γm 0 c 2 = m 0 c p 2. If we substitute the momentum change ṗ = (p 2 /τ r) 1 + p 2 in Eq. (A.4), we find E = m 0c 2 p γ p τ r p = m 0 c 2 2 τ r γ p2 p2 + 1 = p2 E. γτ r Using that p = γv/c p = γv /c, the energy change becomes (A.5) E = p2 γτ r E = γv2 c 2 τ r E. (A.6) This energy change is equivalent to the synchrotron radiation power loss: Psynch(B, loss v, v ) E = γ(v, v )v 2 E(v c 2, v ). τ r (B) Equation (A.6) describes the energy loss due to synchrotron radiation. timescales suppose that d(γv 2 )/de = 0, so we can write ( ) E(t) = E t=0 exp γv2 t c 2 τ r (A.7) For short (A.8) which clearly illustrates that τ r is the time-scale of the radiation loss. These equations have been verified by ANTS-simulations. In our case τ r 1 s, and the energy loss of a 1 MeV particle in 1 ms due to the synchrotron radiation is 17 ev, which is negligible. Appendix B. Collision operator valid for arbitrary energies To be able to model the whole runaway process, a collision operator that is valid both for low velocities (near the thermal velocity) and for very high, strongly relativistic velocities. For this purpose a new collision operator has been constructed, so that it is correct both in the low and high velocity limits. (v v t ) is given in Ref. [30] as with The collision operator for superthermal electrons C(f) = 1 ( p 2 p p2 A(p) f ) p + F (p)f + 2B(p) L(f), (B.1) p 2 A(p) = Γv2 t cv, 3 B(p) = Γ 2cv ( 1 + v 2 t (B.2) ) v 4 1, (B.3) v 2 F (p) = Γv2 t T e v 2, (B.4)

22 Runaway electron drift orbits in magnetostatic perturbed fields 22 and Γ = n e e 4 ln Λ/4πɛ 2 0, with v normalized to c so γ 2 = 1/(1 v 2 ) and v 2 t = T e /mc 2. Equation (B.1) should be asymptotically matched to the collision operator valid for arbitrary but non-relativistic electrons given in [31]. This means that the operators should coincide in the region where they are both valid. To achieve this, we take the collision operator to have the form given by Eq. (B.1), with ( ) 2 ( ) A(p) = Γv2 t v v cv 2 G, (B.5) 3 v T e v T e B(p) = Γ [ ( ) ( ) ] v v φ G + v 2t v 2, (B.6) 2cv F (p) = Γv2 t T e v 2 2 ( v v T e v T e ) 2 ( v G v T e v T e ), (B.7) where vt 2 e = 2c2 vt 2. In the expressions above φ(x) 2 x π 0 e y2 dy is the error function and G(x) φ(x) x { φ (x) 2x 3, x 0 π (B.8) 2x 2 1, x. 2x 2 is the Chandrasekhar function. This form of the collision operator is in agreement with [32] for high energies and with [31] for low energies. It is useful to write C(f) in terms of normalized momentum q = γv, γ = 1 1 v 2 (B.9) and the relativistic collision time τ 1 = Γ/(m 2 ec 3 ). Denoting x e = q/ 2ɛ(1 + q 2 ) and v 2 t = ɛ we arrive at C(f) = [ ( ) ( ) ] 1 + q 2 q q q 2 Z τq 3 eff + Φ G + ɛ L(f) 2ɛ(1 + q2 ) 2ɛ(1 + q2 ) 1 + q 2 { ( ) [ + 1 ]} q 2 τq 2 q ɛ(1 + q 2 ) G q 1 + q (1 + q f )f + ɛ (B.10) 2ɛ(1 + q2 ) q q where the effect of ions, with an effective charge number Z eff has also been included. The ions are assumed to be massive and therefore contributing only to the pitch-angle scattering term. Appendix B.1. Monte-Carlo operator Monte-Carlo calculations model the behaviour of the plasma or one of its constituents by simulating the behaviour of a finite number of discrete particles. Therefore, the collision operator defined for action on a density distribution in continuous phase space needs to be

23 Runaway electron drift orbits in magnetostatic perturbed fields 23 converted into a Monte-Carlo operator acting on individual particles. For future brevity, we introduce the following notation: ( ) q 2 J (q) = ɛ(1 + q 2 ) G q, (B.11) 2ɛ(1 + q2 ) P(q) = ɛ 1 + q 2 3, (B.12) q [ ( ) ( ) ] I(q) = 1 + q τq 3 Z eff + Φ q 2ɛ(1 + q2 ) G q 2ɛ(1 + q2 ) q 2 + ɛ (B.13) 1 + q 2 Let q µ λ ν be the moments of the density distribution defined according to q µ λ ν = 2π q µ λ ν f(q, λ)q 2 dqdλ, n (B.14) so d dt qµ λ ν = 2π n q µ λ ν C(f)q 2 dqdλ. (B.15) where n is the number of particles and λ is the pitch (v /v). Integration by parts, using f(q, λ) = δ(q q 0 )δ(λ λ 0 ), yields the pitches and normalized momenta and their variances for the particles d dt ˆλ = I(ˆq)ˆλ, d dt ˆσ2 λ = I(q)(1 ˆλ 2 ) d dt ˆq = 1 { J (ˆq)(1 + ˆq 2 ) + } τ ˆq 2 ˆq [J (ˆq)P(ˆq)] d dt ˆσ2 q = 2 τ ˆq J (ˆq)P(ˆq) 2 (B.16) (B.17) (B.18) (B.19) References [1] HELANDER, P., ERIKSSON, L.-G., and ANDERSSON, F., Plasma Physics and Controlled Fusion 44 (2002) B247. [2] SHIMADA, M., CAMPBELL, D., ND M. FUJIWARA, V. M., et al., Nucl. Fusion 47 (2007) S1. [3] HENDER, T. C., WESLEY, J., BIALEK, J., et al., Nucl. Fusion 47 (2007) S128. [4] GáL, K., FEHéR, T., SMITH, H. M., FüLöP, T., and HELANDER, P., Plasma Phys. and Controlled Fusion 50 (2008) [5] SMITH, H. M., FEHéR, T., FüLöP, T., GáL, K., and VERWICHTE, E., Plasma Phys. and Controlled Fusion 51 (2009) [6] YOSHINO, R. and TOKUDA, S., Nuclear Fusion 40 (2000) [7] LEHNEN, M., BOZHENKOV, S. A., ABDULLAEV, S. S., and JAKUBOWSKI, M. W., Phys. Rev. Lett. 100 (2008)