On single-mode equilibrium and self-organization of reversal field pinch
The MGD self-stabilization of kink instability of paramagnetic z-pinch with strong current and some features of the RFP-like quasi-single-mode self-organization as a result of plasma azimuth and z-convection generated by MGD oscillations are studied in single mode approximation on basis of a quasi-l...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2011
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| Cite this: | On single-mode equilibrium and self-organization of reversal field pinch / A.A. Gurin // Вопросы атомной науки и техники. — 2011. — № 1. — С. 29-31. — Бібліогр.: 2 назв. — англ. |
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irk-123456789-906132016-01-04T15:47:23Z On single-mode equilibrium and self-organization of reversal field pinch Gurin, A.A. Магнитное удержание The MGD self-stabilization of kink instability of paramagnetic z-pinch with strong current and some features of the RFP-like quasi-single-mode self-organization as a result of plasma azimuth and z-convection generated by MGD oscillations are studied in single mode approximation on basis of a quasi-linear model. Розглянуто стабілізацію нестійкості кінків парамагнітного пінча з сильним струмом та деякі особливості самоорганізації пінчів з оберненим полем під впливом полоїдального й тороїдального обертання, генерованого коливаннями в одномодовому наближенні на основі квазілінійної моделі. Рассмотрены стабилизация неустойчивости кинков парамагнитного z-пинча с сильным током и некоторые особенности самоорганизации пинчей с обращенным полем под влиянием полоидального и тороидального вращения, генерированного колебаниями в одномодовом приближении на основе квазилинейной модели. 2011 Article On single-mode equilibrium and self-organization of reversal field pinch / A.A. Gurin // Вопросы атомной науки и техники. — 2011. — № 1. — С. 29-31. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 52.35.Py, 52.55.Lf http://dspace.nbuv.gov.ua/handle/123456789/90613 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Магнитное удержание Магнитное удержание Gurin, A.A. On single-mode equilibrium and self-organization of reversal field pinch Вопросы атомной науки и техники |
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The MGD self-stabilization of kink instability of paramagnetic z-pinch with strong current and some features of the RFP-like quasi-single-mode self-organization as a result of plasma azimuth and z-convection generated by MGD oscillations are studied in single mode approximation on basis of a quasi-linear model. |
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Gurin, A.A. |
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On single-mode equilibrium and self-organization of reversal field pinch |
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On single-mode equilibrium and self-organization of reversal field pinch |
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On single-mode equilibrium and self-organization of reversal field pinch |
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On single-mode equilibrium and self-organization of reversal field pinch |
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On single-mode equilibrium and self-organization of reversal field pinch |
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on single-mode equilibrium and self-organization of reversal field pinch |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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On single-mode equilibrium and self-organization of reversal field pinch / A.A. Gurin // Вопросы атомной науки и техники. — 2011. — № 1. — С. 29-31. — Бібліогр.: 2 назв. — англ. |
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Вопросы атомной науки и техники |
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AT gurinaa onsinglemodeequilibriumandselforganizationofreversalfieldpinch |
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2025-07-06T18:48:22Z |
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2025-07-06T18:48:22Z |
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1836924486034653184 |
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ON SINGLE-MODE EQUILIBRIUM AND SELF-ORGANIZATION
OF REVERSAL FIELD PINCH
A.A. Gurin
Institute for Nuclear Research of NASU, Kiev, Ukraine
E-mail: gurin@kinr.kiev.ua
The MGD self-stabilization of kink instability of paramagnetic z-pinch with strong current and some features of the
RFP-like quasi-single-mode self-organization as a result of plasma azimuth and z-convection generated by MGD
oscillations are studied in single mode approximation on basis of a quasi-linear model.
PACS: 52.35.Py, 52.55.Lf
1. INTRODUCTION
At present, it is established that toroidal discharges
qualified as a reversal field pinches (RFP) with improved
confinement are described by the “quasi-single mode”
(QSM) nearly laminar kink oscillations with low-
amplitudes [1]. This gives a basis to think that the single
dominant mode in the MGD spectrum is inherent in the
RFP nature and may be used to found the dynamic quasi-
linear model of RFP discharges. In this paper, the
magnetic configuration of cylindrical z-pinch is
considered in the terms of general magneto-static
equilibrium rotB=(αB+βB×er)/B2 (B=(0, Bθ(r),B
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2011. № 1. 29
Series: Plasma Physics (17), p. 29-31.
Bz(r)) is
the mean magnetic field). The single helical mode m=1,
n>>1 is taken into consideration which belongs to the
Alfven spectrum of unstable kinks of force-free
paramagnetic configuration with parameters α>1, β<<1
(1/α is the radius of paramagnetic pinching) to be close to
conditions observed experimentally for high current
pinches with low safety factor q =arBz/RBθ <<1 (a and R
are minor and major toroidal radiuses, the condition r=1
corresponds to the plasma column radius).
The frequency and increment of the kink as well as
all radial amplitude distributions are determined by
solution of the Hain-Lust linear diffusive pinch boundary
problem in formulation [2] taking additionally into
account arbitrary plasma convection. The mean
components (0, Vθ(r),Vz(r)), “poloidal” and “toroidal”
rotational velocities, to be generated by oscillations, are
determined as an the eigen-mode quadratic forms
averaged on oscillations. Similarly, the parameter α(r) is
determined by averaging of the B-projection of the Ohm’s
law for high conducting plasma. Substitution of the Vθ(r)
and Vz(r) into eigen-problem algorithm reveals the
stabilization of the kink instability. At the same time the
squared amplitude contribution of velocity and magnetic
field oscillations into α (magnetic dynamo “α-effect”)
results in the appearance of a weak negative B Bz in the edge
plasma region. The dominant mode may be found from
the condition of marginal stability under maximal
amplitude. It is characteristic that not high amplitudes
need for this: the velocity perturbations are measured in
the “milli-Alfven” scale whereas magnetic perturbations
are turn out about percents in comparison with BzB at the
pinch axis.
We discuss some aspects of presented quasi-linear
model of self-stabilization of a paramagnetic pinch to
conclude that it may describe the initial fast phase of the
RFP-equilibrium, whereas slow dissipative processes
have to be taken into account to find an appropriate
anomalous radial transport on the saturated amplitudes.
2. BASIC EQUATIONS
Basic equations are given by standard one-fluid MGD
description and, at first, its linear version for ideal plasma
implemented in the boundary problem for the plasma
radial displacement ξ (r) in a helical wave:
2 2
2 2 2 2 2 2
2
( ) 0, (1)
,
( ) ,
2 4 2 ,
( ) / ,
/ ,
.
A
A
z
A z
z z
z z
z z
d D dr r Q
dr D dr
D N F
D N r k r m B
d B dQ D B k B W r GW
dr r dr
W k B B V B N D
F k B mB r
G k rB mB
θ
θ θ
θ θ
θ
θ
ξ ξ
ω
ω
ω
+ =
′= −
′= − +
= − − −
′= +
= = −
= +
kB
Eq. (1) presents the well-known Hain-Lust-Goedbloed
equation in the form close to given in [2]. In contrast to
[2] Eq. (1) takes also into account the plasma rotation
introduced into ω΄ and, that is more essential, into W but
neglects all β-effects. Eq. (1) implies the equilibrium
equation that in a general case takes form β=dP/dr-Vθ2/r
where plasma pressure in ideal plasma obeys the equation
P=β0N-5/3 with β0~0.01. Because of α>1, and if only Vθ do
not run up to supersonic values, to follow effects of
plasma rotations on kink spectra, β may be put equal 0.
(Although some results for the case β≠0 without rotation
will be shown below). The Alfven scale is used to reduce
the description to the non-dimensional form: the
commonly used Alfven velocity and frequency,
VA=B(0)/(4πMN(0))½ (cm/s) and ωA=VA/a (s-1), are
assumed as an units for velocities and frequencies under
calculations. At that the physical values B(0) and N(0) are
chosen as units for magnetic field and plasma density at
any r. The mean density profile N(r) is assumed in the
form 1-0.9r2 in definitions to Eq. (1).
The displacement ξ defines the plasma radial velocity
disturbance δVr through Doppler shifted frequency:
( ) , ( ).zi t ik z im
rV i r e rω θδ ω ξ ω ω− + −′ ′= − = − kV
Boundary conditions are ξ(1)=0 and ξ(0)≠0 which are
admissible under additional condition m2=1 as it is true
for kinks. Some algebra gives all amplitude distributions
δN(r), δB(r), δV(r) in terms of functions y1(r) = ξ(r) and
y2(r) = (d/dr)(rξ(r)) thereby the non-zero value ξ(0) sets a
zoom for the overall picture of oscillations.
mailto:gurin@kinr.kiev.ua
In the network of standard one-liquid MGD descrip-
tion non-the linear balance equations may be obtained:
2
2
1div
/ 2 ,
div
/ 2 . (2)
r
r
ex
z
ex z r z
N V NV V
r
BNV rB B B
r r r r
N V
NV B B B B
r r z
θ θ
θ θ θ
θ
ν
θ
ν
+ =
∂ ∂ ∂
− + + −
∂ ∂ ∂
=
∂ ∂ ∂
− + + −
∂ ∂ ∂
V
kB
V
kB
30
Eqs. (2) are the mean quasi-state momentum θ- and z-
equilibrium where the fields N, V, B contain a kink
perturbation and angle bracket notes the averaging on its
oscillation period (that is the same, angle averaging on the
magnetic surface r=const). The momentum equilibriums
are maintained with the friction force caused by ion-
neutral collisions stopping the “spin-up” process caused
with the disbalance of magnetic and inertial forces. The
charge exchange collision frequency νex is assumed as
playing main role. Upon Alfven scaling, all dissipative
frequencies are very small, the νex can achieve maximal of
the order of 10-3 that we mean below.
Saving contributions of main fields N, ‹V›=(0,Vθ(r),
Vz(r)), ‹B›=(0,BBθ(r),BzB (r)) and field perturbations in Eqs.
(2), one can obtain the explicit expressions for rotational
velocities in the linear squared amplitude approximation:
2
2
1 2Re[ ]
1 1 2Re[ ],
1 2Re[ ]
1 1 2Re[ ].
r r
ex
z z
r z r z
ex
V N V
N
d r B B N V V
N r dr
V N V
N
d r B B N V V
N r dr
θ θ
θ θ
δ δ
δ δ δ δ
ν
δ δ
δ δ δ δ
ν
∗
∗ ∗
∗
∗ ∗
= − +
−
= − +
−
(3)
In the Eqs. (3), values Vθ, Vz are determined substantially
by terms containing νex in denominator. They ask very
small amplitudes to have scale <<1 because rotational
velocities of order 1 in the Alfven scaling are
experimentally unobserved. It becomes also clear from
the Ohm law for weakly non-ideal plasma averaged on
oscillations:
0 ( ) rot 0.z rE r δ δ η′−Φ + × + × − =e e V B V B B (4)
Pressure effects are omitted in Eq. (4), E0 is the “toroidal”
electric field driving the discharge (E0 is constant in the
straight cylinder case), Ф is an electrostatic potential,
η = (c2/4πσ)/(aVA), σ is a plasma conductivity. The
“resistance” η is very small parameter (up to 10-7
accordingly to estimations; in calculations η = 10-6 is
implied). Then B-projection of Eq. (4) gives
0
1 ( , ) ( , )z r r r rB V Bα α δ δ δ δ
η
= + × − ×B B e V B e , (5)
where α0=4πσaE0/cB(0) (all values are in absolute
physical units) is the parameter of paramagnetic force-
free model of diffusive pinch introduced as far back as
before tokamak epoch. For the force-free model,
rotB = αB/B2, which is true in the β = 0 approximation,
the safety factor q equals 2α0a/R at r=0 whereas
q(0)≈0.5a/R in RFP [1]. So the value α0=4 is adopted in
our calculations.
It is follows from Eq. (5) the amplitudes must be of
the order of η1/2 in order to the contribution of last terms
on right side in (5) might be comparable with α0. Thereby
the parameter ξ(0) must be specified with the gage 10-3 in
alpha-effect modeling. Otherwise the velocity amplitude
are being measured in “milli-Alfven” units.
On other hand, B×er-projection of Eq. (4) gives the
effect of oscillations on the mean radial velocity :
2 2
0/ / ( , ) ( , )r r rV B B B Vθη α δ δ δ δ= − + −V B B B / .B (6)
I. e. the mean radial velocity turns out of order of “micro-
Alfvens” in contrast with the mean Vθ, Vz which are about
“milli-Alfvens” so far as νex≈η1/2. It proves the derivation
of Eq. (3) in which a mean radial transport was neglected.
3. RESULTS
Fig. 1 illustrates effects plasma pressure on
increments, γ=Im(ω), of the right-hand kink (m=1) with
different toroidal numbers n=(R/a)kz in plasma without
any rotations. These results are obtained by solution of
Goedbloed equation [2] by shooting method and include
the case β0=0 when Eq.(1) is sufficient and the increment
turns out maximal. The spectrum envelops the region
about kz=2, thereby the particular toroidal number,
n = (R/a)kz= 8 (if the aspect ratio R/a=4) plays especial
role. It is “neutral” mode in which F(rs)=kB=0 at rs=0.
For chosen parameters boundary kz =2 separates the zones
of external modes, n<8, and internal ones, n>8, on the
axis kz. The cut line in Fig.1 shows the forbidden zone for
finding stable modes in terms of formulations [2] or (1)
because a singularity arrives within interval (0,1) if
0<rs<1.
Fig.1. Dependence of the kink increments on kz and
plasma pressure in motionless plasma
Fig. 2 illustrates the process of stabilization of kink
modes under the influence of plasma rotations which are
generated by all theirs own field oscillations inside
plasma in accordance with expression (3) for the radial
distributions Vθ(r), Vz(r) under condition β0=0. The
growth rates are plotted for some set of internal modes
including the neutral mode against the amplitude
parameter ξ(0) with gage η1/2. The most instable neutral
mode is saturated under most amplitude η1/2 and can be
claimed as a “dominant” mode in the β=0 model.
The complete stabilization of internal modes is not
achievable in the shooting algorithm because an approach
to the cut line breaks the iteration process. Note, the
modes under considerations are non-local ones when the
radial profile ξ(r) has no nodes inside plasma, within
interval (0<r<1). Why internal modes demonstrate so
abrupt relaxation with growing of small amplitude ξ(0) is
understandable if to look the Fig. 3 where full set of
amplitude functions, y1=ξ(r) and y2=(d/dr)(rξ(r)), are
shown for internal mode kz= 2.2 on an approach to
saturation. The stabilization process extremely sharpens
the amplitude profiles near r=rs for internal modes.
Fig. 4. Deformation of equilibrium by oscillations
CONCLUSIONS
We made sure that the non-local kink-modes, to be
unstable in the weakly non-ideal plasma of paramagnetic
pinches, can experience the relaxation of growth rates
due to the plasma rotation generated by own field kink
perturbations. Some mode may be dominant in the
spectrum of discrete toroidal numbers n. Theoretical
quasi-linear model determining the rotational velocities
and the algorithm taking into account the plasma rotation
are developed. The process of the mode saturation is
characterized by arising of a reversed z-field nearby
plasma surface.
Fig. 2. Stabilization of different modes in the case β0=0
However the presented model don’t able to describe a
static maintenance of amplitudes as well to find the value
of real frequency of resonant modes in the limit γ=0.
More dissipative processes must be involved into the
model as well as all β-effects. Some closing in the theory
of tearing modes ought to take place. Without such a
development the model cannot to explain also haw a
dominant mode can provide the positivity of mean radial
velocity in terms of the approach (6).
Fig. 3. Shapes of amplitude function y1, y2 under
parameters kz=2.2, γ=0.05
So, even if ξ(0) is very small the local values at rs become
sufficient to show considerable effect on solution of the
problem (1) in this case. So the results may be considered
as of physical interest so far the local rotational velocities
nearby resonant surfaces do not become supersonic with
diminishing of γ. Fig. 4 demonstrate the effect of rotation
on the paramagnetic radial distributions arriving under
parameters of Fig. 3. Some reversed z-field appears near
plasma edge and configuration becomes more close to the
RFP pattern than the initial paramagnetic model.
REFERENCES
1. P. Martin, et al.// Nucl. Fusion. 2003, v. 43, p. 1855.
2. J.P. Goedbloed // Phys. Fluids. 1972, v. 15, p. 1090.
Article received 07.10.10
ОБ ОДНОМОДОВОМ РАВНОВЕСИИ И САМООРГАНИЗАЦИИ ПИНЧА С ОБРАЩЕННЫМ ПОЛЕМ
А.А. Гурин
Рассмотрены стабилизация неустойчивости кинков парамагнитного z-пинча с сильным током и некоторые
особенности самоорганизации пинчей с обращенным полем под влиянием полоидального и тороидального
вращения, генерированного колебаниями в одномодовом приближении на основе квазилинейной модели.
ПРО ОДНОМОДОВУ РІВНОВАГУ ТА САМООРГАНІЗАЦІЮ ПІНЧА З ОБЕРНЕНИМ ПОЛЕМ
А.А. Гурин
Розглянуто стабілізацію нестійкості кінків парамагнітного пінча з сильним струмом та деякі особливості
самоорганізації пінчів з оберненим полем під впливом полоїдального й тороїдального обертання, генерованого
коливаннями в одномодовому наближенні на основі квазілінійної моделі.
31
CONCLUSIONS
REFERENCES
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