Runaway electron losses caused by resonant magnetic
perturbations in ITER
G. Papp1,2 , M. Drevlak3 , T. Fülöp1 , P. Helander3 and G. I.
Pokol2
1
Department of Applied Physics, Nuclear Engineering, Chalmers University of
Technology and Euratom-VR Association, SE-41296 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
E-mail: [email protected]
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 suppress the
runaway beam the application of resonant magnetic perturbations (RMP) 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 transport and orbit losses for various initial energies
and different magnetic perturbation levels. In the simulations we model the ITER RMP
configuration and solve the relativistic, gyro-averaged drift equations for the runaway
electrons including a time-dependent electric field, radiation losses and collisions. The
results indicate that runaway electrons are rapidly lost from regions where the normalised
perturbation amplitude δB/B is larger than 0.1% in a properly chosen perturbation
geometry. This applies to the region outside the radius corresponding to the normalised
toroidal flux ψ = 0.5.
PACS numbers: 28.52.Av, 52.20.Dq, 52.20.Fs, 52.27.Ny, 52.55.Fa
Submitted to: PPCF
Runaway electron losses caused by resonant magnetic perturbations in ITER
2
1. Introduction
The toroidal electric field generated due to the sudden cooling of the plasma in a disruption
can give rise to an intense runaway electron beam. If these fast electrons hit the plasma
facing components they can cause serious damage. Large tokamaks such as ITER could
be more susceptible to the formation of runaway electron beams than present tokamaks.
The reason for this is that the avalanche effect due to close Coulomb collisions with the
background electrons becomes more significant for larger tokamaks with high currents. It
can be shown that the ease with which runaways are generated increases exponentially
with plasma current [1]. Due to avalanche multiplication, runaway currents of several
mega-amperes may be generated in an ITER disruption. The uncontrolled interaction
of such a high energy electron beam with plasma facing components is unacceptable for
next-step devices and therefore the issue of how to avoid or mitigate the beam generation
is of prime importance for ITER.
Magnetic perturbations can be a possible way for runaway suppression [2].
Experiments of runaway suppression with externally generated perturbation fields have
been conducted in several machines. In many cases it has been shown that magnetic
perturbations could effectively suppress the runaway beam, e.g. in JT-60U [3] and
TEXTOR [4, 5, 6, 7]. But there are also examples where magnetic perturbations did
not affect the runaway beam, e.g. in JET [8].
Previous simulations of the runaway electron drift orbits in a TEXTOR-like
configuration with resonant magnetic perturbations (RMP) have shown that the loss of
high-energy runaways is dominated by the shrinkage of the confinement region, and is
independent of the perturbation current [9]. The losses are mostly due to the wide orbits
of the runaways that intersect the wall and therefore it is most effective for electrons
close to the edge. The results indicated that the runaways in the core of the device
are typically well confined; however, the onset time of runaway losses closer to the edge
is dependent on the magnetic perturbation level and can affect the maximum runaway
current. In larger tokamaks one may fear that magnetic perturbations would be less
effective, simply because the runaways that are generated close to the centre will not come
close to intersecting the edge stochastic region. However, in paper [9] it has been proposed
that in connection with MHD perturbations caused by e.g. gas injection, core runaways
could be transported towards the edge [10, 11]. Therefore edge magnetic perturbations
together with gas injection could be an effective way of suppressing the runaway beam
even in large tokamaks, such as ITER.
An extrapolation of the effect of magnetic perturbations on runaway beam
suppression from the existing experimental data to ITER is difficult, partly because the
experimental results are not entirely understood, but also because runaway dynamics is
expected to be different in ITER. The aim of the present work is to contribute to the
understanding of runaway dynamics in perturbed magnetic field in an ITER-like scenario.
This will be achieved by a three-dimensional numerical modelling of the runaway electron
Runaway electron losses caused by resonant magnetic perturbations in ITER
3
drift orbits. We will present simulations of the runaway electron drift orbits in ITER-like
perturbed magnetic fields and evaluate the effect of magnetic perturbations on runaway
losses. In the simulations we use a time-dependent electric field obtained for an ITER-like
disruption scenario calculated with a model of the coupled dynamics of the evolution of
the radial profile of the current density (including the runaways) and the resistive diffusion
of the electric field [12]. We also include the effect of collisions along with synchrotron
and Bremsstrahlung radiation losses.
Previous studies on the ITER ELM perturbation system [13] aimed on ELM
suppression during normal operation, where the deep penetration of the perturbation
is undesirable. For the runaway mitigation, that is applied during a disruption, the aim is
different. Deeper penetration, in principle, can lead to more efficient runaway suppression.
In this article we study various perturbation current configurations based on the state of
the art design of the ITER RMP coil system.
The results indicate that runaway electrons can gain more than 100 MeV in an ITERlike disruption. They are rapidly lost from regions where the normalised perturbation
amplitude δB/B is larger than 10−3 . This applies to the region outside the radius
corresponding to the normalised toroidal flux ψ = 0.5. The losses are partly due to
the confinement volume shrinkage for large energies and partly due to the fact that the
stochasticity of the field lines leads to rapid radial transport. The numerically found
threshold δB/B ' 10−3 confirms earlier analytical [2] and numerical [14] estimates for
the perturbation level needed for runaway losses. The losses are sensitive to the chosen
perturbation configuration.
The paper is organised as follows: In Sec. 2 we give a description of the applied
numerical model. In Sec. 3 the ITER-like scenario used in the simulations is described,
including an estimate of the electric field. Also the structure of the magnetic field and the
perturbation are described. In Sec. 4 the effect of RMP on the particles’ orbits is studied
and the loss fractions of the runaway electrons are discussed. Furthermore, the effect of
RMP and the toroidal electric field on the transport is described. Finally, the results are
summarised in Sec. 5.
2. Numerical model
We solve the relativistic, gyro-averaged equations of motion for the runaway electrons
including the effect of synchrotron and Bremsstrahlung radiation with the ANTS (plasmA
simulatioN with drifT and collisionS) code [9]. 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. The collisions are modelled
with a collision operator that is valid for both non-relativistic and relativistic energies.
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
Runaway electron losses caused by resonant magnetic perturbations in ITER
4
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.
In the case of electrons with some hundred MeV energy the radiation losses cannot
be neglected. Therefore synchrotron radiation and Bremsstrahlung losses are included in
the simulations. Synchrotron radiation is included in the form derived in the Appendix of
our previous work [9]. The term arising from the guiding centre motion associated with
the major radius R = 6.2 m is neglected, because even at very large energies the Larmor
motion dominates the radiation for the ITER parameters, as shown in figure 1.
Ratio of toroidal / larmor component of synch. rad.
0
10
−1
Toroidal / Larmor
10
−2
10
−3
10
−4
10
−5
10
ITER
−6
10
0
50
100
150
200
Energy (MeV)
250
300
Figure 1. Comparison of the guiding centre term and the Larmor term in the
synchrotron radiation for the ITER parameters. The guiding centre term is at least
an order of magnitude smaller even at very high (> 200 MeV) energies.
Bremsstrahlung radiation is included in the form of a decelerating force, dp/dt =
−FB , dpk /dt = −FB pk /p [15], where
µ
¶
4
1
2 2
FB =
me c re ne (Zeff + 1)γ log 2γ −
,
(1)
137
3
me is the electron rest mass, c the speed of light, re the classical electron radius, ne the
plasma electron density, Zeff the effective charge and γ is the relativistic gamma factor
for the particle. It follows from the above that dp⊥ /dt = −FB p⊥ /p.
3. ITER scenario
The simulations have been carried out for the ITER scenario #2 (15 MA inductive
burn) [16]. Inductive scenarios are expected to produce the largest and most energetic
populations of runaway electrons. We investigate “pure” disruptions without impurity
injection disruption mitigation. We use a post-disruption equilibrium with the parameters
shown in table 1 and figure 2, based on simulations with the ASTRA code [16, 17]. The
core plasma temperature was set to the estimated 10 eV post-disruption value [18].
Runaway electron losses caused by resonant magnetic perturbations in ITER
5
Table 1. The plasma parameters used in the simulations.
11
Notation
Value
Major radius
Minor radius
Magnetic field on axis
Effective charge
Normalised flux
Normalised radius
Plasma current
Density on axis
Pre-disruption temperature on axis
Post-disruption temperature on axis
Approx. temperature profile
q on axis
q on edge
q profile
R0
a
B
Zeff
ψ
r/a
Ip
n0
T0pre
T0post
Te (ψ)
q0
qa
q(ψ)
6.195 m
2m
5.26 T
1.6
Ψ(r)/Ψ(a)
√
r/a ' ψ
15 MA
1020 m−3
24.7 keV
10 eV
T0 (1 − 0.98ψ)
0.994
3.8
¡
¢−α
q0 1 − [1 − (q0 /qa )1/α ] · ψ 2
, α = 2.09
10
10
9
8
7
6
5
ITER SCEN #2 15MA IND. 10 eV (ASTRA)
ITER SCEN #2 15MA IND. (ASTRA)
(a)
0
Electron density
Ion density
0.2
0.4
0.6
0.8
1
Radial position (normalized flux)
Temperature [eV]
Density [1019/m3]
12
Parameter name
8
6
4
2
0
(b)
0
Electron temperature
Ion temperature
0.2
0.4
0.6
0.8
1
Radial position (normalized flux)
Figure 2. (a) Density and (b) temperature profiles for the electrons and ions as used in
the simulation.
3.1. Energy gain due to the electric field
The maximum energy E that a runaway electron can gain in a tokamak disruption is
limited by E ≤ ecδψ/R, where δψ is the change in the poloidal flux caused by the decay
of the plasma current. For large aspect ratio and circular cross section we have dψ/dr =
Ra
rB/q(r), so on the axis (where the drop in ψ is the largest) δψ ≤ B 0 r/q(r)dr . Ba2 /3
for a typical ITER inductive burn scenario q-profile. Thus the absolute upper limit of the
reachable energy is E ≤ eca2 B/3R ' 340 MeV for the simulation parameters (B0 = 5.3
T, a = 2 m, R = 6.2 m). This is significantly larger than the 20 MeV estimate on smaller
devices [9].
The self-consistent computation of the accelerating electric field during runaway
generation is non-trivial [19, 20]. For this work we used a time-dependent electric field
(shown in figure 3) which is taken from simulations of the evolution of the radial profile
Runaway electron losses caused by resonant magnetic perturbations in ITER
6
of the current density and the diffusion of the electric field [12]. In figure 3b the energy
Rt
gain of the particles up to t = 45 ms is shown, calculated as E(t) = −e 0 v(τ ) · E(τ )dτ '
Rt
−ec 0 Eφ (τ )dτ . From this figure it is clear that particles can reach energies in excess of
100 MeV in an ITER disruption. Due to the non-monotonic initial current profile [16],
the most energetic runaways are expected around mid-radius. Note that ' 100 MeV
is the maximum energy reachable by electrons. Once the avalanche starts, the electron
distribution will be dominated by lower (in the order of 10 MeV) energy particles.
Figure 3. (a) Temporal and radial evolution of the toroidal electric field Eφ . (b) Energy
gain of the particles due to the electric field as a function of time and radius.
3.2. Magnetic field structure and perturbations
The unperturbed magnetic equilibrium has been calculated by VMEC [21] for the
parameters above and the ITER scenario #2 PET separatrix [16]. The equilibrium used
in the simulations is presented in figure 4a along with the cross-section view of the ITER
RMP coils on the low field side of the plasma.
As in previous work [9], we neglect the effect of shielding of magnetic field
perturbations by plasma response currents. This approximation is expected to be valid
in cold post-disruption plasmas, and indeed, in smaller tokamaks such as TEXTOR it
was shown that RMP could effectively increase the radial transport of runaways, which
it would not have been able to do if it the perturbation had been shielded out from
the outer regions (ψ & 0.5). Clearly, if the effect of shielding were included, it would
reduce the perturbations and therefore also the runaway losses. Thus our results should
be interpreted as an upper limit on the actual losses.
The perturbed magnetic field is obtained by superimposing the field from the
perturbation coils on the field of the unperturbed VMEC solution. This approximation
is valid because the field generated by the perturbation coils is much smaller than that
from the toroidal field coils and the plasma current.
Runaway electron losses caused by resonant magnetic perturbations in ITER
7
4
3
Z [m]
2
1
0
-1
-2
-3
(a)
-4
4
5
6 7
R [m]
8
9
(b)
Figure 4. (a) Cross-section view of the VMEC equilibrium (green) and the PET
equilibrium separatrix (red). The RMP coils are marked with blue lines. (b) 3D Sketch
of the ELM perturbation coil configuration as implemented in the simulation.
(a)
Basic
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
ITER RMP: 60 kA
0.1%
+
n=3 (all)
n=9 mid-reversed
n=9 basic
0
+
0.2
0.4
0.6
0.8
Radial position (normalized flux)
1
(b)
+
Mid-reversed (”R”)
3
2
10 * sqrt[<(dB/B) >]
The runaway electron drift orbits have been calculated for the 15 MA inductive
scenario using the ELM perturbation coils as the source of RMP. The ELM perturbation
coil-set consists of 9 × 3 quasi-rectangular coils at the low field side of the device, shown
in figure 4b. Electrical currents flowing in these coils generate a perturbation field, and a
sufficiently large perturbation results in an ergodization of the magnetic field lines. These
coils create magnetic perturbations at the plasma periphery on the low field side of the
torus that decay radially toward the inside of the plasma.
+
+
p
Figure 5. (a) Radial dependence of the flux surface average h(δB/B)2 i for IRMP =
60 kA in n = 9 and n = 3 operational modes. (b) Sketches of the 1/9 part view of the
basic n = 9 and mid-reversed (“R”) n = 9 RMP configurations.
The radial variation of the flux-surface averaged magnetic perturbation level
Runaway electron losses caused by resonant magnetic perturbations in ITER
8
p
h(δB/B)2 i for Ipert = 60 kA is shown
on figure 5a. For other perturbation amplitudes
p
the values can be calculated from h(δB/B)2 i ∝ Ipert . The magnetic perturbation level
that is predicted to be necessary for runaway suppression is δB/B ' 10−3 [2, 14]. The
technically achievable upper limit of 60 kA can generate such a perturbation level up to
the flux-surface ψ = 0.45 − 0.5, while having δB/B = 10−3 in the centre would require
Ipert ' 180 kA.
Since the ITER RMP consists of 9 × 3 coils, the natural configurations have n = 9
or n = 3 lead toroidal modenumber. The two ways to operate the coils in the n = 9
case is to either have the same current direction in all the coils (basic) or to reverse the
current direction on the midplane (mid-reversed or “R”) as shown in figure 5b. In the basic
configuration the toroidal currents at the midplane more or less cancel out, that leads to
smaller perturbation level, as shown in figure 5a. The different n = 3 configurations that
will be shown later are constructed to have the same relative perturbation as the n = 9
“R” configuration (see figure 5a).
A Poincaré plot of the magnetic flux surfaces in the n = 9 perturbed and nonperturbed equilibrium are shown in figure 6. We did not apply any mapping technique.
The field lines are directly integrated in Cartesian coordinates similar to the particles for
the same reasons as described before. For better visibility, the Poincaré plots obtained
in the (R, Z) plane are converted to flux coordinates (θ, ψ). An approximation for these
coordinates was obtained by extracting the radial coordinate from the unperturbed VMEC
equilibrium and using θ = atan2(Z, R). The determination of the exact flux coordinates in
perturbed fields is complicated and is outside he scope of the present paper. As expected,
the edge region becomes distorted as a result of the RMP. As the magnetic perturbation
grows, magnetic islands appear, the locations of which are correlated with rational values
of the safety factor. At n = 9 these islands are very small and have high poloidal modenumber, therefore there is no significant overlapping or broad ergodic zone generation.
Figure 6. Poincaré plot of the magnetic field lines in the (a) unperturbed case and (b)
in the n = 9 “mid-reversed” configuration for Ipert = 60 kA. No significant ergodization
of the edge region.
Runaway electron losses caused by resonant magnetic perturbations in ITER
9
-
+
+
+
-
-
+
+
-
-
-
+
+
-
+
-
+
-
+
-
+
-
+
-
+
+
-
-
-
+
-
+
+
+
-
-
Figure 7. Sketches of the 120◦ view of the four different n = 3 configurations (A–D
from left to right).
Islands created by lower mode-number perturbations are considerably larger. The
four possible n = 3 configurations (shown in figure 7) have equal relative perturbation
that is almost the same as in the n = 9 configuration (figure 5) but can create much larger
islands and ergodic zones.
Figure 8. Poincaré plot of the magnetic field lines in four different n = 3 configurations
for Ipert = 60 kA.
Interestingly, the various n = 3 configurations give rise to quite different magnetic
Runaway electron losses caused by resonant magnetic perturbations in ITER
10
structure and that in turn will give rise to different loss rates. Poincaré plots of the four
different n = 3 configurations are shown in figure 8. Configurations “A” and “B” create
broad ergodic zones at the plasma periphery, while “C” and “D” generates a series of nonoverlapping islands that are expected to keep the particles confined for longer time than
in the other two cases.
4. Runaway losses
4.1. Confinement volume shrinkage
The drift topology for high energy particles can significantly differ from the magnetic
topology in both perturbed and unperturbed magnetic fields [22, 23]. For this reason we
begin by showing Poincaré plots of particle orbits. The plots were converted from (R, Z)
to (θ, ψ) coordinates the same way as the magnetic field lines, and the particles are shown
in the magnetic- and not the drift coordinate system. One particle was launched at each
radial position and was followed for t = 100 µs simulation time. We illustrate the structure
of the perturbation and the shrinkage of the confinement volume for various configurations
and particle energies. Even in the unperturbed case, the confinement volume shrinks and
the particle population is shifted towards the Low Field Side (LFS) with increasing energy,
see figure 9.
Figure 9. Poincaré plot of the particles at Ipert = 0 kA (a) 10 MeV (b) 100 MeV (c)
200 MeV.
Figure 10 shows the Poincaré plots of 10 MeV particles in the four n = 3
configurations with perturbation current Ipert = 60 kA. Clearly, the edge region becomes
ergodic and particles in this region are expected to be lost rapidly. In the case of “C” edge
islands confine the particles.
Runaway electron losses caused by resonant magnetic perturbations in ITER
11
Figure 10. Poincaré plot of 10 MeV particles at Ipert = 60 kA for the four n = 3
configurations.
The confinement volume shrinkage due to increasing particle energies and the RMP
is illustrated in figure 11 for two different particle energies. Here we switched off energy
modification terms (electric field, collisions, radiation) to study the behaviour of monoenergetic particle species and to explore the confinement shrinkage phenomenon as a
function of particle energy. The particles were followed only up to 1 ms, during which
they would not gain or lose too much energy in any way (see figure 3b).
Particles were launched equidistantly in r/a (and not in normalised-flux), hence the
loss percentage stands for shrinkage measured in r/a, not as the exact shrinkage of the
√
confinement volume itself. Since r/a ' ψ, the shrinkage of the confinement volume
can be expressed as y2 = 1 − (1 − y1 )2 where y1 denotes the losses shown on the left hand
side vertical axis of the loss figures 11a-b. The confinement volume shrinkage is shown
on the right hand side axis (y2 ). The position of the last confined flux surface in the flux
coordinate (ψ) can be expressed as (1 − y1 )2 . Hence for a 25% particle loss on the figure
the Last Closed Flux Surface (LCFS) is ψ = 0.56, and the confinement volume shrinkage
is ' 44 %. As expected, at lower energies (10 MeV, figure 11a) the particles are mostly
confined in the unperturbed case and in the cases “C” and “D”. In the other perturbed
60
50
40
30
20
10
2
10
0
3
10
40
n=3 "A"
35
n=3 "B"
n=3 "C"
30
n=3 "D"
25 UNPERT.
20 Etor = 0 V/m
Einit = 100 MeV
15 IRMP
= 60 kA
10
5
0
-3
-2
-1
0
1
10 10 10
10 10
time [us]
12
60
50
40
30
20
(b) 10
2
10
% of conf. V. shrinkage
(a)
% of particles lost
40
n=3 "A"
35
n=3 "B"
n=3 "C"
30
n=3 "D"
25 UNPERT.
20 Etor = 0 V/m
Einit = 10 MeV
15 IRMP
= 60 kA
10
5
0
-3
-2
-1
0
1
10 10 10
10 10
time [us]
% of conf. V. shrinkage
% of particles lost
Runaway electron losses caused by resonant magnetic perturbations in ITER
0
3
10
Figure 11. Confinement volume shrinkage for the four different n = 3 perturbations at
(a) 10 MeV and (b) 100 MeV particles. The largest and fastest shrinkage is caused by
the “D” configuration.
cases they are less confined, depending on the chosen magnetic configuration, although the
configurations were constructed so that the normalised magnetic perturbation amplitudes
are the same. The most efficient of the four cases from figure 8 is case “B”, and therefore
in the following we will study the losses caused by this perturbation configuration.
Figure 11b shows the fraction of lost 100 MeV particles. This energy is close to the
maximum achievable during the disruption, which will not be reached by many particles.
As expected, even the unperturbed case will lead to particle losses, that is a clear indication
of the confinement volume shrinkage due to the increased particle energy. The new LCFS
for the 100 MeV particles will be around the original ψ ' 0.5 surface (∼50% confinement
volume shrinkage). The different configurations converge to the unperturbed case: with
increasing energy the effectiveness of the RMP drops, but the particles outside the new
LCFS associated with their energy are lost anyway due to their increasing energy. Since
the time axis is logarithmic, the “A” and “B” configurations are preferable to “C” and “D”,
because they cause losses orders of magnitude faster than the latter ones. Rapid losses
are beneficial because that makes the avalanche generation less effective. Furthermore,
the faster the particles are lost the less energy they can obtain from the electric field.
4.2. Runaway losses
In the unperturbed case, the particles that reach the edge of the confinement volume
will get lost. The confinement volume shrinks with increasing particle energy. Since the
particles can gain a significant amount of energy (up to ∼ 100MeV) during the disruption,
roughly the outer 40-50% of the confinement volume will be lost (see figure 11b) and the
new LCFS will be around the original ψ ' 0.5. For this reason there is no significant
difference in the losses if we launch particles with 10 keV, 100 keV or 1 MeV: the difference
in the starting energy is almost negligible compared to the energy gain. Losses start
roughly 1 ms sooner in the 10 MeV unperturbed case as compared to 1 MeV, but the
Runaway electron losses caused by resonant magnetic perturbations in ITER
13
loss dynamics is the same. Launching particles with even higher energy is unrealistic,
because only a negligible amount of such energy particles is expected before the disruption.
Therefore from now on we will present particles with 1 MeV initial energy. Increasing
the effective charge of the plasma an order of magnitude does not make any significant
difference in the loss dynamics. The same stands for the Bremsstrahlung and synchrotron
radiation losses, these two effects do not have significant influence on the loss dynamics.
If we launch the particles on a flux surface that is within the confinement zone but
will be outside it at large energies e.g. ψ = 0.7, we will observe particle losses even without
perturbation. This is caused solely by the high energy that the particles reach during he
disruption. The fraction of lost particles in the unperturbed case is shown in figure 12a.
The particle is lost if its energy reaches the critical energy that is required for its actual
position to be outside the LCFS associated with that energy. Launching particles within
the ψ ' 0.5 limit will cause no particle losses, as confirmed by direct particle simulations.
100
ψ = 0.7
Etor (t)
80 E
init = 1 MeV
IRMP = 0 kA
% of particles lost
% of particles lost
100
60
40
20
(a)
0
9.8
UNPERT 1 MeV
10 10.2 10.4 10.6 10.8 11 11.2
time [ms]
80
(b)
60
Etor (t)
Einit = 1 MeV
n=3 'B', 60kA
40
20
0
-3
10
-2
10
-1
ψ=0.7
ψ=0.6
ψ=0.5
Log(t) fit
0
1
10
10
time [ms]
10
2
10
Figure 12. Particle losses (a) without and (b) with perturbation.
In the perturbed case the ergodic zone arising at the edge will cause losses several
orders of magnitude faster than in the unperturbed case. As shown on figure 12b,
particles launched at ψ = 0.7 start to get lost already after 1 µs, and losses continue
with logarithmic temperature dependence until ∼ 0.1 ms (note the logarithmic time axis
on the figure). At around 0.1 ms already 95% of the particles are lost, but the remaining
5% takes up to 3 ms to get lost. Similar dynamics is observable if the particles are
launched at the flux-surface ψ = 0.6. The particle losses start an order of magnitude
later at 10 µs, and dynamics of the losses is the same: logarithmic losses up to ∼ 0.2 ms,
where around 95% of the particles are lost, followed by a longer period during what the
remaining particles are also lost within 10 ms. If we go one more ∆ψ = 0.1 step further
in, the losses start again 10 times later at 100 µs, and the logarithmic dependence will be
the same. In this case not all the particles will be lost, since the high energy LCFS is in
the vicinity of the ψ = 0.5 surface. Particles launched further in, e.g. at ψ . 0.5 will not
get lost even with strong RMP.
The swift onset followed by logarithmic losses is a clear result of the RMP, although
it seems to be less effective after a given amount of time. This is due to the fact that
Runaway electron losses caused by resonant magnetic perturbations in ITER
14
particles launched at a certain flux-surface get an instant spread dictated by the drift
orbit width. Also those particles that initially drift inwards will be transported outwards
eventually, but it takes longer for them to get lost.
The logarithmic loss dynamics show that most of the particles are lost during the
early phases of the losses, which seems to be favourable from the avalanche generation
point of view. Also, the particles lost due to RMP will have low energy (see figure 3b),
while the losses caused by the shrinkage of the confinement zone result in lost particles in
the 100 MeV energy range. Thus, RMP not only increases the amount of the particles lost
outside ψ=0.5, it might also significantly weaken the avalanche generation in that region
and results in lost particles at several orders of magnitude lower energies. Increased losses
at the edge can also help to prevent a runaway electron sheet formation close to the
boundary in the case of vertical displacement events (VDE) [24]. All of these results
are beneficial from the runaway electron suppression point of view. However, losing
fast electrons from the edge may lead to a larger inductive field in the centre of the
plasma, making the avalanche stronger there. Therefore, quantitative conclusions about
the magnitude of the total runaway current can only be drawn from simulations where
both the evolution of the electric field and losses due to RMP are included self-consistently.
% of particles lost
100
80
Einit = 1 MeV
IRMP= 60kA
ψ = 0.7
Etor (t)
n=3 "A"
n=3 "B"
n=3 "C"
n=3 "D"
n=9 "R"
60
40
20
0
-3
10
-2
10
-1
0
10
10
time [ms]
10
1
Figure 13. Particle losses in the four different n = 3 and in the n = 9 mid-reversed
configuration, Ipert = 60 kA in each case. Initial launch surface is ψ = 0.7.
As was already shown in figures 6, 8, 10 and 11, the perturbation geometry plays
an important role in the behaviour of the magnetic structure and the losses, even though
the relative perturbation magnitude is the same (figure 5a). This was also confirmed
by direct particle simulations as shown in figure 13. As was postulated before, the “B”
configuration causes the strongest losses, followed by “A”. “C” is roughly an order of
magnitude less effective, while “D” and the stronger “mid-reversed” n = 9 configuration
causes just a slight increase in the particle losses as compared to the unperturbed case.
In the n = 9 case, the first losses occur after 10 ms, as a result of the inevitable
confinement volume shrinkage with increasing energy. This result shows that a careful
choice of the perturbation configuration (the current distribution in the 9×3 coils) is
Runaway electron losses caused by resonant magnetic perturbations in ITER
15
extremely important for the runaway suppression. Option “B” is the best out of the
tested configurations.
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 100 particles 0 < σ < 10. 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 start- or
endpoints, nor the slope of the loss rate curves.
5. Conclusions
In this paper we have studied the effect of magnetic perturbations on runaway electrons
in ITER. The calculations were done for the ITER inductive scenario #2 [16, 17] and
the simulations were performed with the ANTS code [9], taking into account synchrotron
and Bremsstrahlung radiation losses and collisions. In the simulations we used a timedependent electric field calculated with a code that solves the coupled dynamics of the
evolution of the radial profile of the current density (including the runaways) and the
diffusion of the electric field [12]. As expected, we found that runaways in the core
(ψ . 0.5) are well confined, since the normalised magnetic perturbation is not strong
enough there. However, runaways are rapidly lost if δB/B & 10−3 , which corresponds
to the region outside the normalised flux ψ = 0.5. The losses are caused partly by
the confinement volume shrinkage that the high-energy electrons experience as they are
accelerated by the electric field [9, 22], and partly by the increased radial transport in the
stochastic region. We performed simulations for several perturbation configurations and
concluded that runaway losses are quite sensitive to the perturbation configuration. We
identified one of the possible n = 3 perturbations to be the most efficient in this respect.
The results indicate that the presence of RMP not only increases the amount of lost
particles but may also counteract the avalanche generation at the edge, since it leads to
earlier losses of particles with lower energies. However, the runaway losses are sensitive
to the evolution of the electric field, thus the effect of the RMP on the whole runaway
electron population and dynamics can only be estimated with more complex simulations
that take into account the electric field dynamics self-consistently. This could be achieved
e.g. by the ARENA code [19], using the results presented in this paper as inputs, possibly
in a form of radial transport coefficients and/or time-dependent losses at the edge.
Runaway electron diffusion may also be affected by micro-scale magnetic turbulence,
which can be large in disruptive plasmas. However, Ref. [25] showed that their diffusivity
scales inversely with the energy E −1 for magnetic transport or even with E −2 in case
finite gyroradius effects become important. Therefore runaway electrons are usually well
confined in disruptive plasmas, as was observed in JET disruption experiments, unless
measures are taken to increase their radial transport.
Runaway electron losses caused by resonant magnetic perturbations in ITER
16
Acknowledgments
This work was funded by the European Communities under Association Contract between
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 S. Putvinski for providing the ITER RMP data and H. M. Smith for providing
the time and space-dependent electric field.
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