Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges
Using the 2D test particle description, that includes acceleration in the toroidal electric field and collisions with the plasma particles, the generation of suprathermal electrons is analyzed under conditions of working gas puffing close to the Doublet III-D (DIII-D, General Atomics, USA) quiesce...
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irk-123456789-1488442019-02-19T01:26:26Z Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges Pankratov, I.M. Bochko, V.Y. Магнитное удержание Using the 2D test particle description, that includes acceleration in the toroidal electric field and collisions with the plasma particles, the generation of suprathermal electrons is analyzed under conditions of working gas puffing close to the Doublet III-D (DIII-D, General Atomics, USA) quiescent runaway shot #152895 parameters. As the result of close collisions, the formation of trapped suprathermal electron population in a nonuniform tokamak magnetic field has been shown. Проведено аналіз генерації надтеплових електронів у токамаці DIII-D для параметрів, близьких до квазістаціонарного розряду з електронами-втікачами #152895, за умов напуску робочого газу. Були використані рівняння руху пробної частинки на двомірній фазовій площині з урахуванням прискорення тороїдальним електричним полем та зіткнень із частинками плазми. Показано, що в результаті близьких кулонівських зіткнень утворюється популяція надтеплових електронів, захоплених неоднорідним магнітним полем токамака. Проведен анализ генерации надтепловых электронов в токамаке DIII-D при параметрах, близких к квазистационарному разряду с убегающими электронами #152895, в условиях напуска рабочего газа. Использованы уравнения движения пробной частицы на двумерной фазовой плоскости с учетом ускорения ее тороидальным электрическим полем и столкновений с частицами плазмы. Показано, что в результате близких кулоновских столкновений образуется популяция надтепловых электронов, захваченных неоднородным магнитным полем токамака. 2018 Article Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges / I.M. Pankratov, V.Y. Bochko // Вопросы атомной науки и техники. — 2018. — № 6. — С. 8-11. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.55.Fa; 52.38.Ph; 52.65.Cс http://dspace.nbuv.gov.ua/handle/123456789/148844 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Магнитное удержание Магнитное удержание |
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Магнитное удержание Магнитное удержание Pankratov, I.M. Bochko, V.Y. Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges Вопросы атомной науки и техники |
| description |
Using the 2D test particle description, that includes acceleration in the toroidal electric field and collisions with
the plasma particles, the generation of suprathermal electrons is analyzed under conditions of working gas puffing
close to the Doublet III-D (DIII-D, General Atomics, USA) quiescent runaway shot #152895 parameters. As the
result of close collisions, the formation of trapped suprathermal electron population in a nonuniform tokamak
magnetic field has been shown. |
| format |
Article |
| author |
Pankratov, I.M. Bochko, V.Y. |
| author_facet |
Pankratov, I.M. Bochko, V.Y. |
| author_sort |
Pankratov, I.M. |
| title |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges |
| title_short |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges |
| title_full |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges |
| title_fullStr |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges |
| title_full_unstemmed |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges |
| title_sort |
momentum-space analysis of suprathermal electrons generation under conditions of gas puffing during runaway tokamak discharges |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2018 |
| topic_facet |
Магнитное удержание |
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http://dspace.nbuv.gov.ua/handle/123456789/148844 |
| citation_txt |
Momentum-space analysis of suprathermal electrons Generation under conditions of gas puffing during Runaway tokamak discharges / I.M. Pankratov, V.Y. Bochko // Вопросы атомной науки и техники. — 2018. — № 6. — С. 8-11. — Бібліогр.: 11 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT pankratovim momentumspaceanalysisofsuprathermalelectronsgenerationunderconditionsofgaspuffingduringrunawaytokamakdischarges AT bochkovy momentumspaceanalysisofsuprathermalelectronsgenerationunderconditionsofgaspuffingduringrunawaytokamakdischarges |
| first_indexed |
2025-07-12T20:26:02Z |
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2025-07-12T20:26:02Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №6(118)
8 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2018, № 6. Series: Plasma Physics (118), p. 8-11.
MOMENTUM-SPACE ANALYSIS OF SUPRATHERMAL ELECTRONS
GENERATION UNDER CONDITIONS OF GAS PUFFING DURING
RUNAWAY TOKAMAK DISCHARGES
I.M. Pankratov1, 2, V.Y. Bochko1
1V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
2National Science Center “Kharkov Institute of Physics and Technology”,
Institute of Plasma Physics, Kharkiv, Ukraine
Using the 2D test particle description, that includes acceleration in the toroidal electric field and collisions with
the plasma particles, the generation of suprathermal electrons is analyzed under conditions of working gas puffing
close to the Doublet III-D (DIII-D, General Atomics, USA) quiescent runaway shot #152895 parameters. As the
result of close collisions, the formation of trapped suprathermal electron population in a nonuniform tokamak
magnetic field has been shown.
PACS: 52.55.Fa; 52.38.Ph; 52.65.Cс
INTRODUCTION
The energy of disruption generated runaway
electrons can reach as high as tens of mega-electron-volt
energy and they can cause a serious damage of plasma-
facing-component surfaces in large tokamaks like ITER
[1]. The precise measurement of runaway electron
parameters during disruptions is not so easy to carry out.
At the same time, the quiescent runaway electron (RE)
generation during the flat-top of DIII-D low density
Ohmic discharges allows accurate measurement of all
key important parameters to runaway electron excitation
[2, 3]. Precise measurements of RE distribution
functions and dissipation rates in the spatial, temporal
and energy domains were carried out, a new effective
diagnostic called the “Gamma Ray Imager” was
applied. Quantitative discrepancies between
experimental measurements and modeling were found
for all RE energies, but the most qualitative discrepancy
was found at low energy.
Our analysis of electron trajectories in the 2D
runaway region (p∥, p⊥) shows that the suprathermal
electron population with p∥ < p⊥ occurs (p∥ and p⊥ are
longitudinal and transversal components of momentum
with respect to the confining magnetic field,
respectively). In this case, the suprathermal electrons,
which are trapped in a non-uniform magnetic field, may
appear in tokamak [4]. A possibility of formation of
such suprathermal electrons during recent DIII-D
experiments [2, 3] is investigated in the paper.
1. RUNAWAY DIII-D EXSPERIMENTS
UNDER QUIESCENT CONDITIONS
In DIII-D, the behavior of REs were investigated
during flat-top stage of Ohmic discharges with the
parameters: toroidal magnetic field was Bt = 1.4 T,
plasma current was Ip = 0.8 MA and loop voltage was
Vloop = 0.6 V [2, 3]. Low density access led to the
generation of a primary RE population which was built
up over several seconds. Near the end of this discharge,
strong puff of working gas was used, which cause RE
parameter variations. During this puffing the value of
plasma density increased approximately from the value
of ne ≈ 0.5 1019 m-3 to the value of ne ≈ 1.5 10 19 m-3 and
the ion effective charge Zeff (t) dropped from the value of
2 to 1.25.
The primary generation mechanism was the
dominated mechanism during these experiments.
Specific behavior of the ECE signal was observed.
2. 2D-MOMENTUM-SPACE ANALYSIS
OF SUPRATHERMAL ELECTRON
GENERATION
Here we model this situation. To study qualitatively
the behavior of electron trajectories in runaway region
during duration of the gas puff (τ ≈ 0.5 s), the plasma
parameter evolution in time is given by the next
equations:
( ) ( ) ( ) ( )( )/ 0 1 0 / ,e e e en t n n n tτ τ= + − (1)
( ) ( ) ( ) ( )( )/ 0 1 0 / ,eff eff eff effZ t Z Z Z tτ τ= + − (2)
where ne(0) = 0.5 1019 m-3, ne(1) = 1.25 1019 m-3,
Zeff(0) = 2, Zeff(1) = 1.25. The zero of normalized time,
0, corresponds to gas puff start.
We use 2D equations (like [5]) of test electrons in
normalized form:
( ) ( )( )
( )3/22 2
0
1 2 ,e eff
cr
dp eE p
n t Z t
dt p p p
τ
⊥
= − +
+
� � �
�
(3)
( ) ( )( )
22
2 22 2
0
2 2 1 ,e
eff
cr
eE pn tdp
Z t
dt p p pp p
τ⊥
⊥⊥
= + − ++
� �
��
(4)
where ,p ⊥� → , 0/ crp p⊥� , the electron density
( )en t → ( ) ( )/ 0e en t n , t→ /t τ , E
� is the toroidal
electric field, e, me are the charge and rest mass of
ISSN 1562-6016. ВАНТ. 2018. №6(118) 9
electron, respectively, L is the Coulomb logarithm
(E|| = 50 mV/m, L = 15) and
( )2 3 2
0 00 / 4cr e ep e m n L Eπε=
�
. (5)
Here we analyze the suprathermal region that is why the
acceleration due to the toroidal electric field and the
effect of the collisions with the plasma particles are
taken into account in Eqs. (3) and (4) only.
Fig. 1. The primary and secondary runaway regions (1)
are presented (normalized variables are used); Sr(0,1)
and Sa(0,1) (red) are separatrixes [5, 6] for plasma
parameters at t=0 or t=1.
The curve
0(2 / )e crp m c p p⊥ =
�
is shown by brown (2),
the locus of the knocked-on electrons lies below this
curve. Straight lines / 1 / 2p p ε⊥ =
�
(q = 1, 3/2, 2)
are marked by green (3), q is the safety factor. Typical
test electron trajectories (flowing around “virtual”
saddle point ( ||, ,,S Sp p⊥ )) are shown by blue, dots
correspond to starting points at t=0.5, directions of
electron motion are shown by arrows. The evolution of
the “virtual” saddle point location in time is shown by
dark blue (4)
For constant values of parameters ne and Zeff at t = 0
and 1 the separatrixes Sr(0,1) and Sa(0,1) separate
trajectories of test electron by usual way [5, 6]. Only
electrons with coordinates initially situated above Sr
may run away. For 0 < t < 1 these parameters ne and Zeff
are not constants, the dynamic situation takes place.
Coordinates of the saddle point (“virtual” saddle point)
change in time:
( )2
, ( ) ( ( ) 1) / ( ) 2S e eff effp t n t Z t Z t⊥ = + + , (6)
( )2
, ( ) / ( ) 2S e effp t n t Z t= +
�
. (7)
For trajectories near “virtual” saddle point the
inequality p∥ < p⊥ holds and the motion of electrons
here is not so fast in 2D plane, the time is the order of
(0.06…0.1) τ).
Recall, in accordance with the conservation laws of
energy and momentum, the knocked-on electrons of
secondary generation are arranged on elongated ellipses,
the major axes of which are equal to the momentum of
the incident mega-electron-volt electrons. Secondary
runaway region in the phase space (p∥, p⊥) is filled by
these ellipses. This region is bounded from the top by
the curve (see, e.g. [4])
0(2 / )e crp m c p p⊥ =
�
. (8)
3. BANANA ORBITS OF TRAPPED
SUPRATHERMALS
Confining magnetic field in tokamak is non-uniform
and is described by Eq. (9):
( ) 0,
1 cos
B
B r θ
ε θ
=
−
, (9)
where ε = r/R, r is the radius of magnetic surface, R is
the major radius, θ is the poloidal angle, the value θ = π
corresponds to low field side (lfs).
Straight lines
/ 1 / 2p p ε⊥ =
�
(10)
for the values of safety factor q=1, 3/2 and 2 are shown
in Fig. 1 (the data from Fig. 2h of Ref. [2] are used).
The entire range of locus of the knocked-on electrons in
2D plane (p∥, p⊥) lies above straight lines of Eq. (10).
These electrons may be trapped in a non-uniform
tokamak magnetic field. It is the necessary criterion, but
not sufficient condition. At t > 0.5 the crossing of saddle
point curve with the q = 3/2 straight line is visible.
As it is clear from trajectories analysis for primary
test electrons in Fig. 1, that for these electrons the
probability of such trapping in a non-uniform magnetic
field is not so high.
It is necessary to distinguish situation on the outer
(lfs) and inner sides of the tokamak discharge. The
suprathermal electrons are trapped in the lfs region.
Narrow banana orbits of these trapped suprathermal
electrons are shown in Fig. 2. More strong losses of
these trapped electrons may occur from the plasma
region where these electrons are located (outer part of
discharge). It is possible even formation of supertrapped
electrons (on the ripples of a longitudinal magnetic
field) which escape from the plasma owing to toroidal
drift.
Bounce period of trapped suprathermal electrons is
equal to Tb = 0.47μs (q = 1.5) and Tb = 0.59μs (q = 2)
for Fig. 2,a. For Fig. 2,b Tb = 0.34 μs (q = 1.5) and
Tb = 0.42 μs (q = 2). Note, the strong inequality
ωbs >> νeffcoll holds, where ωbs is the oscillation
frequency of the bounce motion of trapped suprathermal
electrons in a non-uniform magnetic field and νeffcoll is
the effective collision frequency (regime of banana
trajectories). Ratio ωbs / νeffcoll can reach about five
orders of magnitude. The pitch angle was taken into
account in estimation of the value of νeffcoll.
The runaway energy E ≥ 25 MeV was deduced in
Ref. [2] from the DIII-D experimental data analysis. It
10 ISSN 1562-6016. ВАНТ. 2018. №6(118)
means that in the DIII-D experiment [2, 3] the
secondary runaway generation process should take place
with avalanching time tav [6]
12 (2 ) / 9av e efft m cL Z eE≈ +
�
. (11)
For the DIII-D experiments [2] tav ≈ 1 s. The value
of tav ≈ 1 s is the same order of the value of duration of
gas puff (τ ≈ 0.5 s). However, because of the trapping of
the knock-on electrons, the avalanching process may be
suppressed in part.
Fig. 2. Narrow banana orbits of suprathermal electrons
(banana width is (0.3…0.6) cm).
The values of p⊥ and p∥ corresponds to point θ = π,
Est is the energy of suprathermals: a) p⊥/p∥ = 1.6/0.5
= 3.2, Est ≈ 50 keV,
b) p⊥/p∥ = 2.2/0.8 = 2.75, Est ≈ 100 keV.
Numbers 1, 2 corresponds to orbits near q = 1.5 and
q = 2, respectively. Plasma edge is shown by 3, the
direction of electron poloidal motion along the outer
banana part is shown by arrow
Recall, due to the radial viewing geometry of the
ECE radiometers on DIII-D, these diagnostics probe the
high pitch-angle RE population [2, 3]. This non-thermal
electron cyclotron emission (ECE) must be strongly
enhancement due to existence of the suprathermal
electron population with high values of the p⊥
momentum, p⊥> p∥. Note, for fixed maximum runaway
energy the amount of the knocked-on electrons
decreases with plasma density increasing [7].
Our comment to Fig. 14 in [2] and Fig. 3 in [3],
where ECE emission signal drop was observed after
exceeding of a pre-set trip level. In our opinion, the
more detail study of the influence of the trapped
suprathermal electrons on the plasma stability is needed
(see, e.g. Chapter 16 in Ref. [8]. Detail investigations of
such instabilities for suprathermal electrons are planned
in the future.
If trapped knock-on electrons are created far enough
from the magnetic axis (the DIII-D case), they may be
detrapped and run away [9] because of these trapped
electrons drift radially inwards due to the Ware pinch
effect [10]. Analysis of the time it takes for initially
trapped electrons to become runaways [9]
||
W
B
dt R
E
θ ε= ⋅∆ (12)
shows that for the DIII-D quiescent runaway
experiments [2,3] this time dtW is the order of 0.7s.
(E|| = 50 mV/m, Bθ ≈ 0.2 T, R = 1.67 m, Δε ≈ 0.1). Here
( ) ( ),r rε ε ε ′∆ = − r is the radial position where the
electron was trapped and r´ is the radial position where
the electron stay detrapped and run away. For
E|| = 5 V/m (disruption case) this time will be the order
of 7 ms. For electric field 40 V/m this time will be about
870 μs which gives up to 200 banana turns before
detrapping.
The inequality 3 5CH CHE E E< <
�
holds in the DIII-
D case [2], where 3 2 2
0/ 4CH e eE e n L m cπε= [11]. That it is
why the nonrelativistic Eqs. (3, 4) are used. It was
verified that presented results obtained from relativistic
equations practically coincided with nonrelativistic one.
CONCLUSIONS
The analysis of electron trajectories in the 2D
runaway region (p∥, p⊥) are carried out for parameters
close to the DIII-D experiments [2, 3]. The formation
population of trapped suprathermal electron with
p∥ < p⊥ is shown during gas puffing, when plasma
density ne and Zeff are changed in time. This
phenomenon is strong for knocked-on electrons. Such
population exists also before gas puff, but during gas
puff, the test electron trajectories are modified in
comparison with case of constant plasma parameters.
Main conclusions:
-The trapping of knocked-on suprathermal electrons
(banana orbits) in non-uniform magnetic field must be
taken into account. The avalanching (secondary
runaway generation) process may be suppressed in part.
-The effective collision frequency is much smaller in
comparison with the oscillation frequency of the bounce
motion of trapped suprathermal electrons.
-The ECE signal must be strongly enhanced due to
existence of the suprathermal electron population with a
high value of transversal momentum, when p⊥ > p∥.
-The plasma instability on trapped suprathermal
electrons may occur and take effect on such electrons
loss, on the ECE signal behavior and the RE distribution
function changes in the region of low energies.
-Additional losses of such electrons may take place
from outer part of discharge.
ISSN 1562-6016. ВАНТ. 2018. №6(118) 11
REFERENCES
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Fusion. 2007, v. 47, p. S128-S202.
2. C. Paz-Soldan, N.W. Eidietis, R. Granetz, et al.
Growth and decay of runaway electrons above the
critical electric field under quiescent conditions //
Physics of Plasmas. 2014, v. 21, p. 022514.
3. C. Paz-Soldan, C.M. Cooper, P. Aleynikov, et al.
Resolving runaway electron distributions in spase, time,
and energy // Physics of Plasmas. 2018, v. 25,
p. 056105.
4. N.T. Besedin, I.M. Pankratov. Stability of a runaway
electron beam // Nuclear Fusion. 1986, v. 26, p. 807-
812.
5. V. Fuchs, R.A. Cairns, C.N. Lashmore-Davies, et al.
Velocity-space structure of runaway electrons // Phys.
Fluids. 1986, v. 29, p. 2931-2936.
6. I.M. Pankratov, R. Jaspers, K.H. Finken, et al.
Secondary generation of runaway electrons and its
detection in tokamaks // Proc. 26th EPS Conf. on Contr.
Fusion and Plasma Physics (Maastricht, 1999).
European Physical Society, 1999, v. 23J, p. 597-600.
7. M.N. Rosenbluth, S.V. Putvinski. Theory for
avalanche of runaway electrons in tokamaks // Nuclear
Fusion. 1997, v. 37, p. 1355-1362.
8. A.B. Mikhailovskii. Instabilities of plasma in
magnetic traps. Moscow: “Atomizdat”, 1978 (in
Russian).
9. E. Nilsson, J. Decker, N.J. Fisch, et al. Trapped-
electron runaway effect // J. Plasma Physics. 2015,
v. 81, p. 475810403.
10. A.A. Ware. Pinch effect for trapped particles in a
tokamak // Physical Review Letters. 1970, v. 25, p. 15-
17.
11. J.W. Connor, R.J. Hastie. Relativistic limitation on
runaway electrons // Nuclear Fusion. 1975, v. 15,
p. 415-424.
Article received 18.09.2018
АНАЛИЗ В ИМПУЛЬСНОМ ПРОСТРАНСТВЕ ГЕНЕРАЦИИ НАДТЕПЛОВЫХ ЭЛЕКТРОНОВ
ПРИ НАПУСКЕ ГАЗА В РАЗРЯДЫ ТОКАМАКА С УБЕГАЮЩИМИ ЭЛЕКТРОНАМИ
И.М. Панкратов, В.Ю. Бочко
Проведен анализ генерации надтепловых электронов в токамаке DIII-D при параметрах, близких к
квазистационарному разряду с убегающими электронами #152895, в условиях напуска рабочего газа.
Использованы уравнения движения пробной частицы на двумерной фазовой плоскости с учетом ускорения
ее тороидальным электрическим полем и столкновений с частицами плазмы. Показано, что в результате
близких кулоновских столкновений образуется популяция надтепловых электронов, захваченных
неоднородным магнитным полем токамака.
АНАЛІЗ В ІМПУЛЬСНОМУ ПРОСТОРІ ГЕНЕРАЦІЇ НАДТЕПЛОВИХ ЕЛЕКТРОНІВ
ПРИ НАПУСКУ ГАЗУ В РОЗРЯДИ ТОКАМАКА З ЕЛЕКТРОНАМИ-ВТІКАЧАМИ
І.М. Панкратов, В.Ю. Бочко
Проведено аналіз генерації надтеплових електронів у токамаці DIII-D для параметрів, близьких до
квазістаціонарного розряду з електронами-втікачами #152895, за умов напуску робочого газу. Були
використані рівняння руху пробної частинки на двомірній фазовій площині з урахуванням прискорення
тороїдальним електричним полем та зіткнень із частинками плазми. Показано, що в результаті близьких
кулонівських зіткнень утворюється популяція надтеплових електронів, захоплених неоднорідним магнітним
полем токамака.
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