The constraint of plasma power balance on runaway avoidance Christopher J. McDevitt Nuclear Engineering Program University of Florida Gainesville FL 32611

2025-05-02 0 0 787.74KB 6 页 10玖币

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The constraint of plasma power balance on runaway avoidance

Christopher J. McDevitt

Nuclear Engineering Program, University of Florida, Gainesville, FL 32611

Xian-Zhu Tang, Christopher J. Fontes, Prashant Sharma

Los Alamos National Laboratory, Los Alamos, NM 87545

Hyun-Kyung Chung

Korea Institute of Fusion Energy, 169-148 Gwahak-ro, Yuseong-gu, Daejeon 34133, Korea

(Dated: December 9, 2022)

In a post-thermal-quench plasma, mitigated or unmitigated, the plasma power balance is mostly between

collisional or Ohmic heating and plasma radiative cooling. In a plasma of atomic mixture {nα}with αlabeling

the atomic species, the power balance sets the plasma temperature, ion charge state distribution {ni

α}with i

the charge number, and through the electron temperature Teand ion charge state distribution {ni

α},the parallel

electric ﬁeld Ek.Since the threshold electric ﬁeld for runaway avalanche growth Eav is also set by the atomic

mixture, ion charge state distribution and its derived quantity, the electron density ne,the plasma power balance

between Ohmic heating and radiative cooling imposes a stringent constraint on the plasma regime for avoiding

and minimizing runaways when a fusion-grade tokamak plasma is rapidly terminated.

The fast termination of a fusion-grade plasma in a tokamak

reactor is prone to Ohmic-to-runaway current conversion [1],

which is made extraordinarily efﬁcient by the avalanche

mechanism [2–4] due to the knock-on collisions between pri-

mary runaways and background free and bound electrons [5–

7]. Such fast shutdowns could be intentional, for safety upon

the detection of an inadvertent sub-system fault, for example,

or unplanned, as the result of a tokamak disruption. Disrup-

tions can have a variety of causes [8] including such a mun-

dane event as a falling tungsten ﬂake into the plasma. For the

relativistic energies characteristic of runaway electrons (RE),

their local deposition on the ﬁrst wall can induce severe sur-

face and sub-surface damage of plasma facing components.

A straightforward and perhaps ideal approach to mitigate RE

damage is to minimize the runaway population by avoiding

the runaway avalanche altogether. This is the so-called run-

away electron avoidance problem in a tokamak plasma.

The most troublesome feature of a fast shutdown, as in a

tokamak disruption, is the ease for a fusion-grade plasma to

rid its thermal energy in comparison with the plasma current.

The so-called thermal quench (loss of plasma thermal energy)

is often one to two orders of magnitude (if not more) shorter

than the current quench (decay of plasma current) [1]. In

a post-thermal-quench plasma, mitigated or not, the plasma

power balance is mostly between collisional or Ohmic heat-

ing and plasma radiation. This is usually the case because

the post-thermal-quench plasma temperature is clamped by

high-Z impurity radiation to be a very low value, likely in the

range of a few electron volts. Radial transport at such low

thermal energies is relatively slow, even in the presence of a

stochastic magnetic ﬁeld [9, 10]. The source of high-Z impu-

rities could be divertor/wall materials that are introduced into

the plasma through intense plasma-wall interaction during the

thermal quench when the bulk of the plasma thermal energy

is dumped on the plasma-facing components. In a mitigated

thermal quench, high-Z impurities, such as neon or argon, are

deliberately injected into the plasma via pellets or gas jets.

In the standard scenario where the thermal quench is fast

and the post-thermal-quench plasma is cold and rich in high-

Z impurities, an Ohmic-to-runaway current conversion is in-

evitable when a ﬁnite RE seed and large amount of plasma

current is present. This results in the formation of a runaway

plateau shortly after the thermal quench. An interesting dis-

covery, from experiments on both DIII-D [11] and JET [12],

is that the high-Z impurities can be purged by a massive deu-

terium injection in the runaway plateau phase. The resulting,

mostly deuterium plasma can expel the REs via a large-scale

MHD event leading to a globally stochastic magnetic ﬁeld.

Since this RE mitigation scheme does not rely on the strict

avoidance of REs, it offers the possibility of simultaneously

satisfying competing requirements such as thermal quench

and RE mitigation. The details of the underlying MHD in-

stabilities vary in DIII-D and JET experiments [13], but the

expectation that open ﬁeld lines lead to rapid runaway loss via

parallel streaming is robustly met in both devices. The added

beneﬁt is the experimental observation that the runaways are

broadly disbursed onto the ﬁrst wall so no appreciable local-

ized heating is detected. The so-called MHD ﬂush of the run-

aways after an impurity purge leaves the possibility that the

mostly deuterium plasma could reheat to sustain an Ohmic

current without crossing the avalanche threshold. This is the

topic of the current paper.

In a plasma of atomic mix {nα}with αlabeling the atomic

species, the power balance between Ohmic heating and radia-

tive cooling sets the plasma temperature, ion charge state dis-

tribution {ni

α}with ithe charge number, and through the elec-

tron temperature Te, the ion charge state distribution {ni

α},

and the parallel electric ﬁeld Ek.Since the threshold elec-

tric ﬁeld for runaway avalanche growth Eav is also set by

the atomic mixture, charge state distribution and its derived

quantity, the electron density ne,the plasma power balance

between Ohmic heating and radiative cooling imposes a strin-

arXiv:2210.10925v2 [physics.plasm-ph] 8 Dec 2022

Figure 1. Transition between Ohmic and RE roots. The red curve

indicates the parallel electric ﬁeld on the Ohmic root, whereas the

green curve indicates the parallel electric ﬁeld on the RE root. The

temperature at which the curves intersect deﬁnes Tav . The deuterium

density was taken to be nD= 1021 m−3, the neon density nN e =

1019 m−3, and the current density was taken to be j= 2 MA/m2.

gent constraint on the plasma regime for avoiding and mini-

mizing runaways when a fusion-grade tokamak plasma is to

be terminated either intentionally or unintentionally. Robust

RE avoidance can be achieved if Ohmic heating is able to off-

set the radiative and transport losses, and reheat the plasma so

the parallel electric ﬁeld Ek=ηjkdrops below the runaway

avalanche threshold Eav.If this could be maintained over the

remainder of the current quench, effective runaway “avoid-

ance” would have been achieved. The key question is the crit-

ical deuterium density and the fractional neon impurity den-

sity below which such a scenario can be triggered. A second

question is whether the reheated plasma can be placed in the

regime that the Ohmic current quench falls within the known

design constraint for the current quench duration, which in the

case of ITER has an upper bound of 150 milliseconds (ms), for

limiting the halo current, and a lower bound of 50 ms in order

to avoid excessive eddy currents. [1, 14, 15]

This Letter lays out the basic physics considerations under-

lying the answers to both questions explained above, which

are of practical importance to a tokamak reactor like ITER.

From the plasma power balance between Ohmic heating and

radiative cooling, we ﬁnd that the operational space for plasma

reheating and runaway avoidance is highly constrained in

terms of the plasma density and the remnant impurity con-

tent. This can be illustrated by considering the quasi-steady

state parallel electric ﬁeld as a function of the electron tem-

perature, an example of which is plotted in Fig. 1. First con-

sidering the case in which a negligible number of runaway

electrons are present, the parallel electric ﬁeld will be given

by Ek=ηjk, with ηthe plasma resistivity and jkthe plasma

parallel current density [16]. Noting that the plasma resistivity

scales as η∝1/T 3/2

e, the electric ﬁeld will decrease rapidly

as Teis increased for a given plasma current density jk. Once

the magnitude of the electric ﬁeld has dropped below Eav ,

runaway electron ampliﬁcation by the avalanche mechanism

will no longer be possible. The electron temperature at which

this occurs will be referred to as Tav. For temperatures below

Tav, two distinct roots of the system are present. This can be

(a) (b)

Figure 2. Ohmic heating ηj2(dashed lines) with current car-

ried by background electrons, collisional heating Eav ·j(dashed-

dotted lines) with current carried by runaway electrons and Eav

the avalanche threshold ﬁeld, and radiative cooling rate Prad (solid

lines) are shown as a function of Teand for three deuterium den-

sities: nD= 1020 m−3(black), nD= 1021 m−3(red), and

nD= 1022 m−3(blue). There are 4 cases shown: (a) j=

2MA/m2, nN eon /nD= 5%; (b) j= 2 MA/m2, nN eon /nD= 1%;

motivated by considering an Ohm’s law, modiﬁed to account

for the presence of runaway electrons, of the form:

Ek=ηjk−jRE .

For jRE jk, the electric ﬁeld can again be approximated

by Ek≈ηjk, which yields the red curve shown in Fig. 1.

For Te< Tav this root can, however, be recognized to be

unstable. In particular, since Ek> Eav when Te< Tav ,

any seed RE population present in the plasma will be ampli-

ﬁed by the avalanche mechanism. As a larger fraction of the

plasma current is carried by REs, this will cause Ekto drop

until Ek≈Eav [4, 17]. This second root, which we will re-

fer to as the RE root, is stable for Te< Tav, and leads to the

formation of a current plateau. Thus, a sufﬁcient condition to

avoid RE formation is to maintain Te&Tav. The primary

challenge is to identify a solution whereby Te&Tav while

simultaneously adhering to the ITER requirement of a current

quench timescale in the range of 50-150 ms. [14]

The challenge of simultaneously satisfying these two con-

straints is made evident in the power balance curves illustrated

in Fig. 2. Here, the Ohmic heating rate is plotted along with

the radiative cooling rate Prad as a function of the electron

temperature Te. The bulk plasma heating can be estimated by

multiplying the parallel electric ﬁeld sketched in Fig. 1 by the

plasma current density. For the Ohmic root, this leads to the

familiar expression Pη≡ηj2

k. For the RE root, this leads

to the net energy transferred to the plasma being given by

PRE =Eavjk. While a small fraction of this energy will be

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TheconstraintofplasmapowerbalanceonrunawayavoidanceChristopherJ.McDevittNuclearEngineeringProgram,UniversityofFlorida,Gainesville,FL32611Xian-ZhuTang,ChristopherJ.Fontes,PrashantSharmaLosAlamosNationalLaboratory,LosAlamos,NM87545Hyun-KyungChungKoreaInstituteofFusionEnergy,169-148Gwahak-ro,Yuseong-gu...

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The constraint of plasma power balance on runaway avoidance Christopher J. McDevitt Nuclear Engineering Program University of Florida Gainesville FL 32611

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