Plasma and Fusion Research: Regular Articles

Volume 11, 2403023 (2016)

Fokker-Planck Simulation Study of Hot-Tail Eﬀect on Runaway
Electron Generation in ITER Disruptions∗)
Hideo NUGA, Akinobu MATSUYAMA1) , Masatoshi YAGI1) and Atsushi FUKUYAMA1)

Kyoto University, Kyoto-Daigaku-Katsura, Nisikyo-Ward, Kyoto 615-8540, Japan
Japan Atomic Energy Agency, 2-166 Omotedate-Obuchi, Rokkasho 039-3212, Japan(Received 30 November 2015 / Accepted 1 February 2016)

A Fokker-Planck code TASK/FP, which calculates the evolution of the relativistic momentum distributionfunction of electrons and the induced toroidal electric field, has been applied to the study of runaway electron(RE) generation in ITER disruptions. The hot-tail eﬀect on the RE generation is investigated. Hot tail is formed
during the thermal quench ahead of the Dreicer generation, which have an impact on the subsequent current andthe electric field evolution. The hot-tail aﬀects the RE current density profile even in the secondary RE dominantcase. In spite of the small ratio of the primary electrons, they are multiplied significantly by the avalanche eﬀect.
Consequently, the hot-tail eﬀect plays an important role in determining the total RE current density profile after
the current quench.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: runaway electron, disruption, ITER, hot-tail eﬀect, thermal quench
DOI: 10.1585/pfr.11.2403023

1. Introduction
The disruption is one of the most serious phenomena in tokamaks, which induces the huge electromagnetic
forces to the device and generates the high-energetic runaway electrons (REs) causing the damage of plasma facing
component [1]. In ITER, the RE current is considered toreach a few mega-amperes. If the thermal quench is sufficiently fast, the primary RE generation rate is enhanced
through the so-called “hot-tail eﬀect” [2, 3]. In order topredict the disruption dynamics in ITER, kinetic FokkerPlanck calculations are required by taking into account ofRE generation processes including the hot-tail eﬀect.
The aim of this paper is to compare the RE generation process, especially RE current density profile, including and excluding the hot-tail eﬀect in ITER. For thispurpose, the Fokker-Planck code, named TASK/FP [4], isapplied to simulations of the evolution of the relativistic
electron momentum distribution function, the RE current,
and the induced electric field in a self-consistent manner.
The code treats the electron distribution function in twodimensional momentum space for sub-relativistic region(less than 0.5 MeV). In the present calculation, although
the treatment of relativistic REs is simplified, the thermaland sub-relativistic electron distribution function is analyzed with a good resolution, being useful for detailed calculation of the primary RE generation rate.
As discussed above, the RE generation in a highcurrent tokamak like ITER is expected to be dominatedby the secondary generation. Nevertheless, our simulation
author’s e-mail: nuga@p-grp.nucleng.kyoto-u.ac.jp∗) This article is based on the presentation at the 25th International Toki
Conference (ITC25).

shows that the hot-tail formation plays an important role indetermining the RE current profile after the current quench.
Since even a small increment of the seed current density isamplified by the avalanche eﬀect, the secondary RE current density profile is modified by the hot-tail eﬀect.
In the rest of this paper, the following contents are discussed. Models used in our code are explained in Sec. 2.
The numerical results of RE generation with an ITER-likeparameters are shown in Sec. 3. Conclusion and Discussion of this paper is provided in Sec. 4.

2. ModelsIn this paper, we evaluate the RE generation rate byusing the kinetic Fokker-Planck code TASK/FP, which is
two dimensional in momentum space and one dimensionalin the minor radius. The equation can be written in theconservative form:∂f= −∇ · SC,E∂t qe E= −∇ · −D C · ∇ f + FC +f ,me↔

(1)

where ∇ is the derivative operator in the momentum space(p, θ), and the subscripts C denotes the collision term. pand θ denote the momentum
and the pitch angle. In Eq. (1),↔
the collision coeﬃcients D C and FC are determined withthe weak relativistic isotropic background collision model[5, 6]. It is assumed that the momentum distribution functions of the background plasma are Maxwellian for bothelectrons and ions.
In TASK/FP, the induced electric field is described by2403023-1

c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles
the following equations:
1 ∂ ∂E∂r= μ0 j,r ∂r ∂r∂t

Volume 11, 2403023 (2016)
Table 1 Plasma parameters employed in this paper [2, 12].

(2)

j = σ E + ec(nrp + nrs ),(3)
dnrp∂f=∇ · SC,E dp = −dp,
(4)
dt
∂t

dnrs
E/EC − 1πϕ= nr×dt
τr ln Λ3(Zeﬀ + 1)⎛⎞−1/2⎜⎜⎜⎟⎟⎟4π(Zeﬀ + 1)2E⎜⎝1 − C +⎟⎠, (5)222E 3ϕ(Zeﬀ +1)(E /EC +4/ϕ −1)
where σ is the Spitzer conductivity σsp = 1.96 ne q2e τe /me(∼ 35.7 T e3/2 /Zeﬀ [MA/V · m, T e in keV]) with the neo3/2
4classical correction [7], τe = 3(2π)3/2 02 m1/2e T e /ne qe ln Λ,2 2 3
41/2
τr = 4πε0 m c /ne e ln Λ, ϕ = 1 − 1.46ε + 1.72ε, andε is the inverse aspect ratio. nrp and nrs are the primaryand secondary RE densities, and the critical electric field
EC = ne q3e ln Λ/4π 02 me c2 (∼ 9.17 × 10−22 ne [V/m, ne inm−3 ]), respectively. Equation (2) expresses the electricfield diﬀusion and this equation is closed by the Ohm’s law(Eq. (3)), where we assume that the current density consists of two components: the bulk current, σ E, and theRE current, ecnr . The boundary condition for the electricfield is obtained by the expression of E in vacuum region:E(r) ∝ ln(r/b) and E(r = b) = 0, where b is the location ofthe wall. The total RE density nr is evaluated from the sumof the primary (nrp ) and secondary (nrs ) REs; they are evaluated from Eqs. (4) and (5), respectively. In Eq. (4), therate of the primary RE generation, which is the seed generation for the subsequent secondary generation, is given
by calculating the flux through the boundary of momentum calculation domain: 0 < p < pmax , where (pmax )2 /2me
∼ 0.5 MeV. This means that the electrons whose momentum is greater than pmax are regarded as REs. Because ourresearch focuses on not the RE energy distribution but thenumber of REs generated in a disruption, the evolution off in the p > pmax region is not calculated. By the sametoken, we assume that REs travel with the velocity of lightin order to simplify the treatment of relativistic REs. The
rate of the secondary RE generation dnrs /dt [8], Eq. (5),which is caused by the avalanche mechanism, is given bya function of the RE density nr and the electric field normalized by the critical electric field E/EC rather than themomentum distribution. This is because the secondary REgeneration rate is insensitive to the primary RE energy [9].
To calculate
the conductivity, σ , and the collision↔
coeﬃcients, D C and FC , requires the background plasma
temperature, therefore, the evolution of the plasma temperature being given as a function of time. Here, the decay ofthe plasma temperature is modeled with the time constantof the thermal quench τq :T (t, ρ) = (T 0 (ρ) − T f (ρ)) exp −t/τq + T f (ρ), (6)T 0 (ρ) = (T 0 (0) − T 0 (1))(1 − ρ2 )2 + T 0 (1),(7)T f (ρ) = T f (0)(1 − 0.9ρ2 ),(8)Radii
R = 6.2 m, a = 2.0 m, b = 2.4 m
Initial current
15 MA
Current densityj0 (1 − ρ1.74 )3.23
Initial temperature T 0 (0) = 22.7 keV, T 0 (1) = 2.27 keV
Post-quench temp. T f (0) = 10 eV
Density profilen(ρ) = (n(0) − n(1))(1 − ρ2 )0.67 + n(1)n(0) = 1.06 × 1020 m−3n(1) = 1.06 × 1019 m−3
Eﬀective charge Zeﬀ = 1Ion species
Deuteronwhere ρ denotes the normalized minor radius, whereas T 0and T f are the initial and post-quench temperature, respectively. In Fokker-Planck simulations, electrons and ionsare assumed to have the same temperature.
For simulations, we choose ITER-like parameters tabulated in Table 1. The simulations were performed with
the number of the grids (Np , Nθ , Nr ) = (1024, 64, 32) andit was decomposed into 1024 cores by Message Passing
Interface.
In the followings, in order to examine the eﬀect of
the hot-tail, comparisons have been made for the results
including and excluding the hot-tail eﬀect. The results excluding the hot-tail eﬀect are obtained by the simulation
which uses the theoretical RE generation rate model for
Dreicer generation rate [10] instead of Eq. (4). The expression of the theoretical model is given by:
1 dnrp= CR (α, Zeﬀ )Ê −h(α,Zeﬀ )ne dt⎛⎞⎜⎜⎜ λ(α)⎟⎟2× exp ⎜⎝−−Γ(α, Zeﬀ )⎟⎟⎠ ,4ÊÊ(9)where, Ê = E/ED , α = Êmc2 /T ,
1h(α, Zeﬀ ) =α(Zeﬀ + 1) − Zeﬀ + 716(α − 1)α(1 + Zeﬀ )(α − 2) ,+2α−1
1 λ(α) = 8α α − − α(α − 1) ,2
2(1 + Zeﬀ )α2 π− arcsin 1 −Γ(α, Zeﬀ ) =,8(α − 1) 2
αand CR (α, Zeﬀ ) is an unknown constant of order unity inε. The unknown constant CR is chosen to be 0.35 for thefollowing calculation. This value is conventionally used in
RE simulations [11].

3. Results
The hot-tail eﬀect becomes eﬀective when the thermal
quench time is shorter than the electron-electron slowing
down time for a few times of the initial thermal velocity

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Volume 11, 2403023 (2016)

Fig. 1 RE currents versus thermal quench time are shown. Solidand dotted curves denote the RE currents including andexcluding the hot-tail eﬀect. Red, green, and blue curves
are the net RE, the primary RE, and the secondary RE
current, respectively.

Fig. 3 Current density profiles for τq = 4.0 ms, 2.0 ms, and 1.15
ms are shown. Solid and dotted curves denote the RE current density including and excluding the hot-tail eﬀect.
Green and blue curves are the primary and secondary RE
current density at the time when the induce electric field
vanishes after the current quench.
Fig. 2 Evolutions of total, RE, and ohmic current for τq =2.0 ms. Solids and dotted curves denote the results including and excluding the hot-tail eﬀect.

[4]. The e-e slowing down time is given by:τees (v) = 0.5

4π 02 m2e v3.
ne q4 ln Λ(10)Figure 1 shows the relation between the thermal quench
time to generated RE currents after the steady state isreached with RE current plateau in the simulations. The
horizontal axis denotes the thermal quench time and thevertical axis denotes the amount of the RE current, where
the total, primary, and secondary currents are plotted.
Dashed vertical lines show the e-e slowing down time
calculated with the test-particle velocity of 2vth and 3vth .
From Fig. 1, it is found that when τq becomes shorter thanτs (3vth ), the net RE current becomes larger than those calculated excluding the hot-tail eﬀect. The primary RE current is seen to be comparable with the secondary RE current when τq is shorter than 2 ms. Because the RE current
is generated to compensate the poloidal magnetic flux during the resistive decay of the ohmic current, the secondary
RE current generation becomes less pronounced for thecase where the primary RE current reaches the magnitudelarge enough to maintain the electric field below EC . Figure 2 shows the evolutions of total, RE, and ohmic currentfor τq = 2.0 ms in Fig. 1. It is found that the generation ofRE current and the decay of the ohmic current are triggered
earlier owing to the hot-tail eﬀect. This is because hot-tailelectrons can become REs easily even with the weak electric field due to the low collisionality.

Fig. 4 The primary RE current density profiles for 2.0 ms in thelogarithmic. Solid and dotted curves denote the RE current density including and excluding the hot-tail eﬀect,respectively.

Figure 3 shows the RE current density profiles for (a)τq = 4.0 ms, (b) τq = 2.0 ms, (c) τq = 1.15 ms, respectively.
From Fig. 3 (a), when τq is longer than τs (3vth ) 3.88 ms,
the primary and the secondary RE current density profiles
are in good agreement to those excluding the hot-tail effect. On the contrary, Fig. 3 (b) shows that the hot-tail effect changes not only the amount of RE current but also
the RE current density profile, being broader in the caseincluding the hot-tail generation. The enlargement of primary RE current density profiles are displayed in Fig. 4.
We see that the hot-tail eﬀect reduces the primary RE current density around magnetic axis and enhances it in outerregion (ρ > 0.4).
Such changes of the RE current density profiles canbe explained as follows. In Figs. 5 (b), (c), and (d) a small
peak of the primary generation rate at t ∼ 10 ms corresponds to the hot-tail generated REs. Because of the highvelocity of the hot-tail electrons, they can be REs evenwith the weak electric field. Conversely, if the hot-tail ef-2403023-3

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Volume 11, 2403023 (2016)

Fig. 6 Evolutions of (a) the induced electric field, (b) the RE
generation rate on axis, and (c) that on ρ = 0.52 for τq =1.15 ms. In Fig. (c), the RE generation rates excluding thehot-tail eﬀect are not displayed since they are negligibly
small in this scale.
Fig. 5 Evolutions of (a) the induced electric field, (b) the RE
generation rate on axis, (c) that on ρ = 0.33, and (d) thaton ρ = 0.52 for τq = 2.0 ms. In Fig. (d), the RE generation
rates excluding the hot-tail eﬀect are not displayed since
they are negligibly small in this scale.

fect is excluded, the first peak of the primary RE generation rate is yielded at t ∼ 15 ms [Figs. 5 (b) and (c)]. The
hot-tail REs are yielded at the earlier time than the Dreicer generation and they also trigger the secondary generation. Such an earlier onset on the secondary generation
maintains the induced electric field weaker (in Fig. 5 (a))
to be consistent with the conservation of the poloidal flux.
Hence the subsequent Dreicer generation at t ∼ 15 ms decreases since the Dreicer generation rate is sensitive to theelectric field strength. As seen in Fig. 5 (b), because thedecrement of the second peak (t ∼ 15 ms) is greater than
the increment of the first peak (t ∼ 10 ms), the primary REs
slightly decrease on the axis. From a similar comparison,
we see why the primary REs increase in the outer region(ρ > 0.4). Finally, small changes of the primary RE profile
in Fig. 4 is amplified by the avalanche eﬀect, and the secondary RE current density profile is modified as illustrated
in Fig. 3 (b).
Figure 3 (c) illustrates the calculated RE current density profile for τq = 1.15 ms, where the hot-tail eﬀect dominates RE generation on-axis. In this case, the primary
RE current density is evaluated to be much larger than in
the case excluding the hot-tail eﬀect for the whole radialzone. In the inner radial zone (ρ < 0.2), the resistive decay of the poloidal magnetic flux is compensated solely by
the primary RE generation rate owing to the high primary
generation rate as shown in Fig. 6 (b). This reduces theelectric field around the axis to nearly zero (in Fig. 6 (a)),

therefore the secondary generation being suppressed. On
the contrary, in the outer region (ρ ∼ 0.5), the electricfield is maintained enough to generate secondary REs (inFigs. 6 (a) and (c)). Consequently, the secondary RE current density profile becomes hollow.

4. Conclusion and Discussion
We have applied the kinetic Fokker-Planck simulation code TASK/FP to describe the evolution of electron
distribution function during the RE generation in ITERlike plasmas including the hot-tail eﬀects. The hot-tailbecomes important when the thermal quench time τq isshorter than the typical slowing time of electrons with afew times of the initial thermal velocity τees (2 − 3vth ). We
have confirmed that, if the thermal quench is suﬃciently
faster than the condition, the primary RE current carriedby the hot-tail electrons becomes larger than the secondary
RE current generated by avalanche eﬀect. We have foundthat the hot-tail eﬀect also strongly aﬀects the radial profile
of RE current density.
Since the generation of hot-tail electrons, inductive
toroidal electric field, and secondary electrons are strongly
coupled with each other, careful analyses are required forquantitative prediction of RE generation. For the mitigation of RE generation, massive gas injection (MGI) has
been proposed and tested in tokamak plasmas, since MGI
is expected to increase the plasma density and Zeﬀ and tosuppress the generation of REs by high collisionality. If
MGI also shorten the thermal quench time, however, thehot-tail eﬀect may enhance the primary RE generation andnullify the mitigation of REs. The analysis MGI is left for
future study including the modeling of thermal quench.

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