Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration
Electromagnetic drift instabilities in the plasma of a field reversed configuration are considered with no assumption of adiabatic response of ions and/or electrons in the range of perpendicular wave number values from k┴ < 1/ρTi up to k┴ ~ 1/ρTe (ρTi and ρTe are ion and electron thermal gyroradi...
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| Zitieren: | Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 75-77. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1107832017-01-07T03:03:35Z Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration Khvesyuk, V.I. Chirkov, A.Yu. Basic plasma physics Electromagnetic drift instabilities in the plasma of a field reversed configuration are considered with no assumption of adiabatic response of ions and/or electrons in the range of perpendicular wave number values from k┴ < 1/ρTi up to k┴ ~ 1/ρTe (ρTi and ρTe are ion and electron thermal gyroradiuses). Stabilization by finite plasma length is studied. Stabilising effect of low temperature gradients on electron mode is discussed. Электромагнитные дрейфовые неустойчивости впервые рассматриваются для плазмы обращенной магнитной конфигурации без использования приближения адиабатического отклика ионов и/или электронов в диапазоне значений перпендикулярного волнового числа от k┴ < 1/ρTi до k┴ ~ 1/ρTe (ρTi и ρTe – ионный и электронный тепловые гирорадиусы). Исследуется стабилизация конечной длинной плазмы. Обсуждается стабилизирующий эффект низких градиентов температуры на электронную моду. Електромагнітні дрейфові нестійкості вперше розглядаються для плазми зверненої магнітної конфігурації без використання наближення адіабатичного відгуку іонів і/або електронів у діапазоні значень перпендикулярного хвильового числа від k┴ < 1/ρTi до k┴ ~ 1/ρTe (ρTi і ρTe – іонний і електронний теплові гірорадіуси). Досліджується стабілізація кінцевою довжиною плазми. Обговорюється стабілізуючий ефект низьких градієнтів температури на електронну моду. 2008 Article Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 75-77. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 52.35.Qz; 52.55.Lf; 52.25.Fi http://dspace.nbuv.gov.ua/handle/123456789/110783 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics |
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Basic plasma physics Basic plasma physics Khvesyuk, V.I. Chirkov, A.Yu. Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration Вопросы атомной науки и техники |
| description |
Electromagnetic drift instabilities in the plasma of a field reversed configuration are considered with no assumption of adiabatic response of ions and/or electrons in the range of perpendicular wave number values from k┴ < 1/ρTi up to k┴ ~ 1/ρTe (ρTi and ρTe are ion and electron thermal gyroradiuses). Stabilization by finite plasma length is studied. Stabilising effect of low temperature gradients on electron mode is discussed. |
| format |
Article |
| author |
Khvesyuk, V.I. Chirkov, A.Yu. |
| author_facet |
Khvesyuk, V.I. Chirkov, A.Yu. |
| author_sort |
Khvesyuk, V.I. |
| title |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| title_short |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| title_full |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| title_fullStr |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| title_full_unstemmed |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| title_sort |
electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2008 |
| topic_facet |
Basic plasma physics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110783 |
| citation_txt |
Electron temperature gradient drift instability in the finite beta edge plasma of a field reversed magnetic configuration / V.I. Khvesyuk, A.Yu. Chirkov // Вопросы атомной науки и техники. — 2008. — № 6. — С. 75-77. — Бібліогр.: 14 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT khvesyukvi electrontemperaturegradientdriftinstabilityinthefinitebetaedgeplasmaofafieldreversedmagneticconfiguration AT chirkovayu electrontemperaturegradientdriftinstabilityinthefinitebetaedgeplasmaofafieldreversedmagneticconfiguration |
| first_indexed |
2025-07-08T01:07:10Z |
| last_indexed |
2025-07-08T01:07:10Z |
| _version_ |
1837038914103148544 |
| fulltext |
ELECTRON TEMPERATURE GRADIENT DRIFT INSTABILITY IN THE
FINITE BETA EDGE PLASMA OF A FIELD REVERSED MAGNETIC
CONFIGURATION
V.I. Khvesyuk, A.Yu. Chirkov
Bauman Moscow State Technical University, Moscow, Russia, e-mail: khves@power.bmstu.ru
Electromagnetic drift instabilities in the plasma of a field reversed configuration are considered with no assumption
of adiabatic response of ions and/or electrons in the range of perpendicular wave number values from k⊥ < 1/ρTi up to k⊥
~ 1/ρTe (ρTi and ρTe are ion and electron thermal gyroradiuses). Stabilization by finite plasma length is studied.
Stabilising effect of low temperature gradients on electron mode is discussed.
PACS: 52.35.Qz; 52.55.Lf; 52.25.Fi
Drift waves and drift instabilities driven by gradients
of plasma density, ion and electron temperatures are the
most significant phenomena closely connected with
turbulent transport of high temperature plasma in the
magnetic confinement devises [1]. From the classical
theory of drift instabilities [2], it is known that parallel
component of the wave vector k|| is much less then the
perpendicular wave number k⊥ (k|| << k⊥). The range of k||
in infinite plasma is determined only by the positive
growth rate solutions of the dispersion equation. In finite
length configuration instability satisfy the following
condition:
2π/k|| < L, (1)
where L is the length of plasma configuration along mag-
netic field force lines.
We analyze gradient-driven drift instabilities taking
into account of electromagnetic effects and condition (1)
for edge finite β plasma of a field reversed configuration
(FRC). Ion temperature gradient/electron temperature gra-
dient (ITG/ETG) instability takes into account non-adia-
batic responses for ions and electrons for any k⊥. The
analysis is carried out in the framework of the local elec-
tromagnetic kinetic approach. The local model of low fre-
quency (ω << ωci, ωci is the ion cyclotron frequency) drift
instabilities is based on the linearized Vlasov equation,
quasineutrality condition, and Ampere’s law for parallel
and perpendicular perturbations of the magnetic field [3–
8]. Basic equations are
j
j
j f
m
q
t 10 )(
∇⋅×+∇⋅+
∂
∂
vBvv =
j
j
j f
m
q
0)( ∇⋅×+−= BvE , (2)
03
1 =∑ ∫
j
jj dfq v , (3)
∑ ∫µ=×∇
j
jj dfq v310 vB , (4)
t∂
∂−ϕ− ∇= AE , (5)
AB ×∇= , (6)
)1)((),(0 xfxf jMjj ε−= vv , (7)
0
0
0
1
=
∂
∂
−=ε
x
j
j
j x
f
f . (8)
Here v is the velocity of the particle (variable of
integration); qj and mj are the charge and the mass of the
particle of kind j (j = i, e), respectively; µ0 is the magnetic
permittivity of the vacuum; f1j is perturbation of the
velocity distribution function; f0j is unperturbed velocity
distribution function; B0 is unperturbed static magnetic
field inside the plasma; E is the electric field of the wave;
B is the magnetic field of the wave; k is the wave vector;
ϕ is the scalar potential; A is vector potential; Coulomb
gauge 0=⋅∇ A is used; x is the coordinate along density
and temperature gradients; )(vMjf is the Maxwellian
velocity distribution function at x = 0.
Using standard integrating procedure one can obtain
the following system of equations, containing three
independent variables (ϕ, A⊥, A||)
0)( 3
0 =
Λ+
ϕ
∑ ∫
j
jjMj
jB
j dJhf
Tk
q
v , (9)
∑ ∫ ⊥⊥⊥ Λµ−=
j
jjjj dJhnqAk vv 3
10
2 )( , (10)
∑ ∫ Λµ−=⊥
j
jjjj dJhnqAk vv 3
||00||
2 )( . (11)
Here kB is the Boltzmann constant; Tj is the temperature;
)(0 jJ Λ and )(1 jJ Λ are the Bessel functions;
cjj k ω=Λ ⊥⊥ /v ; indexes ⊥ and || indicate respectively
perpendicular and parallel to static magnetic field
components of vectors; ⊥v is the perpendicular velocity;
||v is the parallel velocity; ωcj is the cyclotron frequency;
*
|| ||
j
j
Dj
h
k
w w
w w
+
= ґ
+ - v
[ ]
jB
Mjj
jj Tk
fq
JAJA )()()( 10|||| Λ−Λ−ϕ× ⊥⊥vv (12)
is non-adiabatic portion of the velocity distribution
function perturbation; DjDj Vk ⋅=ω is the magnetic drift
frequency; DjV is the magnetic drift velocity of the
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 6. 75
Series: Plasma Physics (14), p. 75-77.
particle;
−η+ω=ω
2
3
2
1
2
**
jB
j
jjj Tk
m v
;
njj
jB
j LBq
Tk
k
0
* ⊥=ω is the diamagnetic drift frequency;
Tjnjj LL /=η ; jjnj nnL ⊥∇−= / ; jjTj TTL ⊥∇−= / ; nj is
the unperturbed density; A⊥ = k⊥B||, B|| is the wave
magnetic field parallel to the static magnetic field.
Drift frequency can be presented in the form
α−ω−=ω ⊥ 2
||
2
* 2
vv
R
jB
j
j
B
nj
Dj Tk
m
L
L
, (13)
where 00 / BBLB ⊥∇= , RLBR /=α , 1/R is the particle
orbit averaged curvature of the magnetic force line.
The scale of magnetic field gradient is connected with
plasma density gradient according the equation
∑
βη+
=
j nj
jj
B LL 2
)1(1
, (14)
where 2
00 /2 BTkn jBjj µ=β is local beta parameter
calculated by local static magnetic field in the plasma;
total local β is
0
0
1 β−
β
=β=β ∑
j
j , (15)
where ∑µ=β
j
jBj
V
Tkn
B2
0
0
0
2
; B0V is external (vacuum)
magnetic field; 000 1 β−= VBB .
For low β plasma (β → 0) in homogeneous magnetic
field effects of magnetic drift and vector potential are
negligible, i.e. electrostatic approximation is available. It
includes quasineutrality equation and the solution of the
Vlasov equation with no perturbed magnetic field. The
electrostatic approximation is available if ϕ< <ATjv ,
where jjBTj mTk /=v . From Ampere’s law one can
estimate ∑
=
µ
eij
jj dfqAk
,
3
10
2 ~ v
eB
Tee Tk
een ϕµ v0~ and
ϕ
µ
⊥ eB
TjTee
Tj
Tkk
ne
A 2
2
0~
vv
v . Electrostatic approximation is
available at
Tj
Te
Te
eBe
e k
B
Tkn
v
v2
2
0
0 )(22 ρ< <µ=β ⊥ . (16)
For ions Eq. (16) is satisfied for high β at ETG range
of k⊥ and for very low β at ITG range. For 1.0~>β e this
condition is satisfied for electrons and ions at 1>ρ⊥ Tek .
So, instability can be considered in framework elec-
trostatic limit with appropriate accuracy at the ETG range.
Fig. 1. Growth rates at moderate β0: ––––– –
electromagnetic solution, – – – – – electrostatic
approximation. k⊥ρTi = 1, Te/Ti = 1, ηi = ηe = 2
Fig. 2. Growth rates vs β0 (moderate β0): 1 – k⊥ρTi = 1, k||
Ln = 0.1; 2 – k⊥ρTi = 5, k||Ln = 0.07; 3 – k⊥ρTi = 10,
k||Ln = 0.025; 4 – k⊥ρTi = 15, k||Ln = 0.05.
Te/Ti = 1, ηi = ηe = 2
Fig. 3. Growth rates vs β0 (high β0): 1 – k⊥ρTi = 1, k||
Ln= 0.1; 2 – k⊥ρTi = 5, k||Ln = 0.07; 3 – k⊥ρTi = 43 (k⊥ρTe =
1), k||Ln = 0.1. Te/Ti = 1, ηi = ηe = 2
76
Fig. 4. Upper boundary of k||Ln (solid lines) and
approximate down boundary (dashed lines) of instability
in FRC vs k⊥ρTi: 1 – ηe = 2, ηi = 0.1, Te/Ti = 0.5; 2 – ηe =
1, ηI = 0.1, Te/Ti = 0.5; 3 – ηe = 2, ηi = 0.1, Te/Ti = 0.1
In Fig. 1, the comparison with electromagnetic and
electrostatic solutions are presented for moderate values
of β0 (β0 < 0.1). As a scale of the real frequency ωR and
growth rate γ we use )/(0 TiniB eBLTk ρ=ω . In Figs. 2
and 3, results of electromagnetic calculations for modes
with fixed k⊥ are shown. Values of k|| for presented modes
are close to maximum of the growth rate at fixed k⊥.
To calculate instability parameters we use typical
plasma conditions in FRC experiments [9-13]: ηi ≈ 0.1, η
e ≈ 1–2, Te/Ti ≈ 0.5 (for typical regimes), Te/Ti ≈ 0.1 (hot
ions and cold electrons). Examples of the results of calcu-
lated (k||Ln)b are presented in Fig. 4. For not very elongat-
ed FRCs 2πLn/L ~ 0.3, i.e. for finite length stabilized
modes (k||Ln)b < 0.3. The dashed line in Fig. 4 corresponds
to this approximate condition of stabilization.
CONCLUSIONS
Our calculations have shown that under FRC
experiment conditions typical ITG instability (k⊥ <% 1/ρTi)
appears to be hardly restricted by finite size of FRC
devises, but in the range of ETG instability (k⊥ >~ 1/ρTe)
instability can exist. Parameters of such an instability are
seems to be close to ETG mode, but ion effects for FRC
experiment conditions are significant at k⊥ρTe < 1.
Maximum of growth rate is located at k⊥ρTe ~ 1.
Calculated values and real frequencies for the ETG
range agree well with data measured by Carlson on TRX-
2 device [14].
Special calculations are shown that to decrease
growth rate of ETG instability (and ETG driven turbulent
transport) one can decrease ηe. To sustain of low-ηe
configuration the heating of electrons in the plasma edge
can be used.
The work was supported by RFBR grant 08-08-
00459-a and President grant MK-2082.2008.8.
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4. Y.-K. Pu, S. Migliuolo // Phys. Fluids. 1985, v. 28,
p. 1722.
5. A.Y. Aydemir, H.L. Berk, V. Mirnov, O.P. Pogutse,
M.N. Rosenbluth // Phys. Fluids. 1987, v. 30, p.
3088.
6. K.T. Tsang, C.Z. Cheng // Phys. Fluids. 1991, v. B3,
p. 688.
7. M. Artun, W.M. Tang // Phys. Plasmas. 1994, v.1,
p. 2682.
8. F. JenkoW. Dorland, V. Kotschenreuter, B.N. Rogers
// Phys. Plasmas. 2000, v. 7, p. 1904.
9. N.A. Krall // Phys. Fluids. 1989, v. B1, p. 1811.
10. A.L. Hoffman et al. // Proc. 11th Int. Conf. Plasma
Physics and Controlled Nuclear Fusion Research/
IAEA, Vienna, 1987. v. 2, p. 541.
11. D.J. Rej et al. // Nucl. Fusion, 1990, v. 30, p. 1087;
A.L. Hoffman, J.T. Slough // Nucl. Fusion. 1993,
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12. L. Steinhauer // US-Japan Workshop on FRC. Niiga-
ta, 1996.
13. K. Kitano et al. // 25th EPS Conf. on Contr. Fusion
and Plasma Phys. Prague, 1998.
14. A.W. Carlson // Phys. Fluids. 1987, v. 30, p. 1497.
Article received 22.09.08.
ЭЛЕКТРОННАЯ ТЕМПЕРАТУРНО-ГРАДИЕНТНАЯ ДРЕЙФОВАЯ НЕУСТОЙЧИВОСТЬ В КРАЕВОЙ
ПЛАЗМЕ ОБРАЩЕННОЙ МАГНИТНОЙ КОНФИГУРАЦИИ С КОНЕЧНЫМ БЕТА
В.И. Хвесюк, А.Ю. Чирков
Электромагнитные дрейфовые неустойчивости впервые рассматриваются для плазмы обращенной магнит-
ной конфигурации без использования приближения адиабатического отклика ионов и/или электронов в диапа-
зоне значений перпендикулярного волнового числа от k⊥ < 1/ρTi до k⊥ ~ 1/ρTe (ρTi и ρTe – ионный и электронный
тепловые гирорадиусы). Исследуется стабилизация конечной длинной плазмы. Обсуждается стабилизирующий
эффект низких градиентов температуры на электронную моду.
ЕЛЕКТРОННА ТЕМПЕРАТУРНО-ГРАДІЄНТНА ДРЕЙФОВА НЕСТІЙКІСТЬ У КРАЙОВІЙ ПЛАЗМІ
ЗВЕРНЕНОЇ МАГНІТНОЇ КОНФІГУРАЦІЇ З КІНЦЕВИМ БЕТА
В.І. Хвесюк, О.Ю. Чирков
Електромагнітні дрейфові нестійкості вперше розглядаються для плазми зверненої магнітної конфігурації
без використання наближення адіабатичного відгуку іонів і/або електронів у діапазоні значень
перпендикулярного хвильового числа від k⊥ < 1/ρTi до k⊥ ~ 1/ρTe (ρTi і ρTe – іонний і електронний теплові
гірорадіуси). Досліджується стабілізація кінцевою довжиною плазми. Обговорюється стабілізуючий ефект
низьких градієнтів температури на електронну моду.
77
REFERENCES
|