Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma
Eigen-frequencies and shapes of the non-local magnetic kink-modes m=1, n>1 in a field reversal pinch are obtained in cylindrical force-free approximation by shooting method. The mode profile and the resistive layer containing the resonant surface of the internal kinks are calculated numerically...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
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irk-123456789-881692015-11-09T03:02:21Z Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma Gurin, A.A. Магнитное удержание Eigen-frequencies and shapes of the non-local magnetic kink-modes m=1, n>1 in a field reversal pinch are obtained in cylindrical force-free approximation by shooting method. The mode profile and the resistive layer containing the resonant surface of the internal kinks are calculated numerically. Instability is revealed, both for external and internal modes, which is not effected by resistive layer in plasmas with high conductivity, it is determined by the current gradient due to the high paramagnetic parameter in the pinch core. Сформульована крайова задача та чисельним методом стрільби визначені власні частоти й профілі гвинтових кінк-мод m=1, n>1 в пінчі з оберненим полем. В циліндричному наближенні розраховані профілі нелокальних мод, включно з резистивними шарами в разі внутрішніх мод. Доведена нестійкість мод в плазмі з великою провідністю, яка визначається градієнтом струму й не пов’язана безпосередньо з наявністю резистивного шару внутрішніх мод у плазмі з великою провідністю. Сформулирована краевая задача и численным методом стрельбы определены собственные частоты и профили винтовых кинк-мод m=1, n>1 в пинче с обращенным полем. В цилиндрическом приближении рассчитаны профили нелокальных мод, включая резистивные слои в случае внутренних мод. Показана неустойчивость мод в плазме с высокой проводимостью, которая определяется градиентом тока и не связана непосредственно с наличием резистивных слоев внутренних мод в плазме с высокой проводимостью. 2009 Article Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma / A.A. Gurin // Вопросы атомной науки и техники. — 2009. — № 1. — С. 25-27. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 52.35.Py, 52.55.Lf http://dspace.nbuv.gov.ua/handle/123456789/88169 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Магнитное удержание Магнитное удержание |
| spellingShingle |
Магнитное удержание Магнитное удержание Gurin, A.A. Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma Вопросы атомной науки и техники |
| description |
Eigen-frequencies and shapes of the non-local magnetic kink-modes m=1, n>1 in a field reversal pinch are obtained
in cylindrical force-free approximation by shooting method. The mode profile and the resistive layer containing the
resonant surface of the internal kinks are calculated numerically. Instability is revealed, both for external and internal
modes, which is not effected by resistive layer in plasmas with high conductivity, it is determined by the current
gradient due to the high paramagnetic parameter in the pinch core. |
| format |
Article |
| author |
Gurin, A.A. |
| author_facet |
Gurin, A.A. |
| author_sort |
Gurin, A.A. |
| title |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| title_short |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| title_full |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| title_fullStr |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| title_full_unstemmed |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| title_sort |
kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2009 |
| topic_facet |
Магнитное удержание |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/88169 |
| citation_txt |
Kink macroinstabilities and resistive layer structure of internal kinks in cylindrical plasma / A.A. Gurin // Вопросы атомной науки и техники. — 2009. — № 1. — С. 25-27. — Бібліогр.: 2 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| first_indexed |
2025-07-06T15:52:13Z |
| last_indexed |
2025-07-06T15:52:13Z |
| _version_ |
1836913402787659776 |
| fulltext |
KINK MACROINSTABILITIES AND RESISTIVE LAYER STRUCTURE
OF INTERNAL KINKS IN CYLINDRICAL PLASMA
A.A. Gurin
Institute for Nuclear Research of NASU, Kiev, Ukraine, E-mail: gurin@kinr.kiev.ua
Eigen-frequencies and shapes of the non-local magnetic kink-modes m=1, n>1 in a field reversal pinch are obtained
in cylindrical force-free approximation by shooting method. The mode profile and the resistive layer containing the
resonant surface of the internal kinks are calculated numerically. Instability is revealed, both for external and internal
modes, which is not effected by resistive layer in plasmas with high conductivity, it is determined by the current
gradient due to the high paramagnetic parameter in the pinch core.
PACS: 52.35.Py, 52.55.Lf
1. INTRODUCTION
At present, within the framework of quasi-single
helicity (QSH) modern conception for the reversed-field
pinch (RFP) laboratory plasma [1], the task is set to
explain the nature of the frequency spectrum and to
describe the space structure of dominant modes in QSH
states. These modes turn out to be internal ones and
answer to the condition of F = kBz - mBθ/r = 0 on a
resonant surface rs, 0<rs<a (a is the plasma column
radius). Therefore, the tearing instability of resonant
modes is considered commonly as a main basis for
description of mode stability. However, the tearing theory
does not determine the real frequencies of oscillations
observed in usual kHz MHD spectra giving only
increments of aperiodic instability which are low in the
MHD scale. Moreover, the theory provides solutions
localized close to the resonant surface, whereas the modes
of the QSH spectrum don’t demonstrate any real link to
resonant surfaces. Instead, they show the result of non-
local mechanism of mode excitation. In the theory of z-
pinches the question remains: what total profiles of real
modes actually occur with the presence of resonant
surfaces inside pinches?
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2009. № 1. 25
Series: Plasma Physics (15), p. 25-27.
In this report, the radial profiles and frequencies of
MHD modes, both external and internal ones, are
presented on basis of general MHD theory taking into
calculations Hall effect and the high plasma conductivity.
Eigen-value boundary problem is described and a few
shooting method results are presented.
2. THEORETICAL MODEL
We study the stability of a FRP configuration taking
into account the Hall-effect under arbitrary parameter
П = 4πе2а2N0/Mc2 (the “linear ion” introduced by
Braginskii as provided by ratios ωh:ωA:ωBi = 1:П1/2:П),
where a, M, N are pinch radius, ion mass and plasma
density correspondently, ωh is “helicon frequency”,
ωh = cB0/4πеа2N0, ωA and ωBi are commonly used Alfven
and ion cyclotron frequencies. In cylindrical coordinates
(r,θ,z), when ωh, a, N0(0), BB0(0) are assumed as an units for
ω, r, N(r), B=B0B (r)+ δB(r)e-iωt+iζ (ζ= kz-mθ is the helical
phase), oscillations are governed by equations:
δB = rot(ξ×BB0 + iωξ/П – iηrotB/ω),
ω2N0 ξ = П{(δB×rotBB0) + (B0B ×rotδB)}. (1)
Here ξ is a plasma displacement, η = (c2/4πσωha2) =
νe/ωBeN, where νe and ωBe are electron collisional and
cyclotron frequencies, σ is a plasma conductivity.
Fluctuations δN do not arrive in the set of equations (1)
because the plasma convection is not taken into account
and the “resistivity” η is assumed invariable. Also Eq. (1)
disregard the plasma pressure but take into account effects
of sharpened gradients of magnetic pressure in RFP.
In reality, in the PRP plasma core, the equilibrium is
very close to the force-free one, rotBB0×B0B = 0. We use
only cylindrical force-free configuration, rotBB0 = λB0B ,
where λ(r) is compatible uniquely with the real radial
distribution of the safety factor q(r)=arBz/RBθ [1] under
any choice of the aspect ratio R/a (R is major toroidal
radius). In our model of magnetic configuration λ(0) = 4
thus the considerable paramagnetic pinch effect is taken
into consideration, which is close to reality [1]. The radial
distributions of BBz and Bθ B used in our calculations are
plotted in Fig. 1 jointly with λ(r)/4, q(r) and F(r).
0.0 0.2 0.4 0.6 0.8 1.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
r/a
λ/4
Bz
B
θ
q
F
Fig.1. .Magnetic configuration Bz(r), Bθ(r) and shapes of
λ(r)/4, q(r), F(r) for internal mode m=1, n=10 (П=40)
To consider the situation with discrete toroidal modes
one can set k=n/R (n is major toroidal number) and write
the factor F through the safety factor: F= (kRBBθ/r)(q-m/n).
Under our choice of magnetic configuration the value k=2
determines the boundary between external kinks m=1:
k<2, F(r)<0 (0<r<1), n=9, 8, 7,…., or internal ones: k>2,
F(rs)=0 (0<rs<1), n=10, 11, 12,… The case a/R=0.2285,
k=2.285 chosen in Fig.1 corresponds to the internal mode
mailto:gurin@kinr.kiev.ua
with toroidal number n=kR=10. The resonant surface is
given by equality F(r)=0 at r= rs= 0.254.
The differential problem (1) may be reduced to the
more convenient form by means of non-divergent
representation of B most suitable for helical analysis:
B = ∇ χ×s + B
26
Bss,
s = rs2 ∇ ζ×er. (2)
Owing to the resistivity smallness, η<<1, the ideal
consideration can be assumed for external modes, as well
as for internal ones outsides resonant layers. By setting
η=0, one can reduce the set of equations (1) to the
ordinary two-order differential equation,
,1
22
2
2 δχδχδχ Q
sr
A
dr
dArs
dr
d
rs
=−⎟
⎠
⎞
⎜
⎝
⎛ (3)
where A and Q are an rational functions of ω with
coefficients depending on r through the magnetic field
components, plasma density N and their r-derivatives. Eq.
(1) introduce the two point value problem under
conditions δχ = r|m| at r→0 and δχ(1) = 0.
In a vicinity of the resonance r=rs the condition η > 0
must be taken into account. Eqs. (1) need more analysis
in order to be reduced to the standard form suitable for
numerical solution. The transformation is achieved by
introducing the additional variable:
.2
0
*
0
0 srs B
dr
di
B
BiB δ
ω
ηξωξδ −⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
Π
−=Ξ
rv
(4)
Here BB0*=B0B ×er. Then eqs. (1) can be transcribed as a
general ordinary differential problem for four-component
vector-function y=(δΨ, δΨ΄, δBBs, δΞ),
y′(x) = Ây, (5)
where x=r-rs. Â is a 4×4 matrix. The limit η=0 introduces
the singularity into the problem imposing the unphysical
requirement of wave reflection at the resonance surface:
δχ(rs) = 0 if η=0. In reality, η = 10-4–10-7, so this zero
constrain disappears but very small resistivity allows only
very low values of complex magnitudes δχ(rs) to be
possible inside resistive layer.
The smallness of η introduces high stiffness into the
set of equations (5) so to obtain solution inside the
resistive layer we engage in calculations the stiff program
from collection [2]. Being irreversible the stiff algorithm
need positive definition of the matrix Â. Unfortunately
the  loses its positive definition at x=0, and thus to
avoid numerical errors we use two-way shooting method
continuously sewing together the oppositely directed ideal
trajectories y(x) outside some vicinity of the resonance at
points x=–Δx and x=Δx with the stiff solutions inside the
interval (-Δx,Δx). The calculations show no noticeable
effects of the Δx on the shape δχ(r), they do show effect
of the η on the deepness of the profile dip inside resistive
layer. The presented results are obtained for η = 10-4.
3. RESULTS
Figs.2a-d illustrate radial distributions of absolute values
of the complex amplitude |δBBr(r)| = |δχ(r)|/r for different
kinks calculated under the parameter П=40 according to
the actual experimental data. They relate to the class of
non-local perturbations which cover entire plasma volume
bound by high conductive walls.
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
ab
s(
B
r(r
))
r/a
a
(m=1, n=7)
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
ab
s(
B r(r
))
r/a
m=1, n=10
b
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
ab
s(
B
r(r
))
r/a
m=1, n=11
c
0,0 0,2 0,4 0,6 0,8 1,0
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
ab
s(
B r(r)
)
r/a
d
m=1, n=12
Fig.2. The shape |δBBr(r)| of external kink: a) m=1,
n=7; b) m=1, n=10; c) m=1, n=11 ; d) m=1, n=12
(П=40)
All shapes are normalized to the value δBBr(0) =1. In
Fig. 2a the typical shape of external non-local mode
(m=1, n=7) is plotted which has no nulls or dips within
interval (0<r<1). In contrast to the simple case of external
modes, in the case of internal ones, Fig. 2b-d, displays
complicated non-monotonous behavior of the radial
distribution |δBrB (r)| of internal modes. Resonant dips
arrive against respective rs for every internal mode. The
The instability occurs under k<2 as far back as k=0,156 in
the case of external modes as it follows from Fig. 3.
dips, shown in Fig. 2b-d, locate at the rs: b ─ rs = 0.265;
c ─ rs = 0.364; d ─ rs = 0.394.
The modes considered in the frame of our model turn
out unstable. The complex eigen-frequencies, ω+iγ,
calculated by the shooting method are characterized by
the increments γ of order of 1 in the Hall scale, whereas
frequencies ω turn out of order 0.01. The eigen-
frequencies of basic internal modes are given in the Table.
CONCLUSIONS
We ensured that the non-local kink-modes are
unstable, i.e. be capable of self-excitation in the RFP,. in
the frame of the weakly non-ideal model (1). The
increments are found to be of order of ωh. They are
introduced by the magnetic gradient including second
radial derivatives of magnetic components B
Eigen-frequencies ω+iγ of internal modes (m,n)
B0(r), i.e. by
high current gradients in paramagnetic RFP plasmas,
independently from presence of resistive layers. This
conclusion can be analytically confirmed by integration of
(2) in the limit П→∞. However, in this case the real
frequencies ω vanish, and therefore calculated and
observed 10-100 kHz kink spectra can be entirely linked
with the Hall term in Eq. (1). Without doubt, plasma
convection (rotation and radial transport) must be
involved into consideration to explain the actually
observed stable spectra as well as a smaller importance of
the resonant peculiarities than when it is obtained in the
frame of the basic MHD description (1).
(m,n) (1,10) (1,11) (1,12)
ω+iγ 0.010+i1.69 0,0047+i1.71 0.0037+i1.53
1.0 1.2 1.4 1.6 1.8 2.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
ω
, γ
k
ω
ω
γ
γ
ω
γ
REFERENCES
1. P.Martin et al // Nucl. Fusion. 2003, v.43, p.1855-1862.
2. W.H. Press et al // Numerical recipes in FORTRAN77.
“Cambridge University Press”, 1997. Fig. 3. The dispersion of complex frequency ω+iγ for
external modes, 1<k<2, (П=40)
Article received 29.10.08
УСТОЙЧИВОСТЬ ВИНТОВЫХ КОЛЕБАНИЙ И СТРУКТУРА
РЕЗИСТИВНОГО СЛОЯ ВНУТРЕННИХ МОД В ЦИЛИНДРИЧЕСКОЙ ПЛАЗМЕ
А.А. Гурин
Сформулирована краевая задача и численным методом стрельбы определены собственные частоты и
профили винтовых кинк-мод m=1, n>1 в пинче с обращенным полем. В цилиндрическом приближении
рассчитаны профили нелокальных мод, включая резистивные слои в случае внутренних мод. Показана
неустойчивость мод в плазме с высокой проводимостью, которая определяется градиентом тока и не связана
непосредственно с наличием резистивных слоев внутренних мод в плазме с высокой проводимостью.
СТАБІЛЬНІСТЬ ГВИНТОВИХ КОЛИВАНЬ ТА СТРУКТУРА
РЕЗИСТИВНОГО ШАРУ ВНУТРІШНІХ КІНКІВ В ЦИЛІНДРИЧНІЙ ПЛАЗМІ
А.А. Гурин
Сформульована крайова задача та чисельним методом стрільби визначені власні частоти й профілі
гвинтових кінк-мод m=1, n>1 в пінчі з оберненим полем. В циліндричному наближенні розраховані профілі
нелокальних мод, включно з резистивними шарами в разі внутрішніх мод. Доведена нестійкість мод в плазмі з
великою провідністю, яка визначається градієнтом струму й не пов’язана безпосередньо з наявністю
резистивного шару внутрішніх мод у плазмі з великою провідністю.
27
CONCLUSIONS
REFERENCES
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
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/OtherNamespaces [
<<
/AsReaderSpreads false
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/OmitPlacedBitmaps false
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>>
<<
/AddBleedMarks false
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/PresetSelector /MediumResolution
>>
/FormElements false
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>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|