Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity
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| Zitieren: | Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2002. — № 5. — С. 69-71. — Бібліогр.: 16 назв. — англ. |
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Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. 2015-03-08T21:03:18Z 2015-03-08T21:03:18Z 2002 Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2002. — № 5. — С. 69-71. — Бібліогр.: 16 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/77885 This work was supported by the Russian Federal Program on Support of Leading Scientific Schools, Grant No. 00-15-96526, the Research Support Foundation of the State of Sao Paulo (FAPESP), University of Sao Paulo, and Excellence Research Programs (PRONEX) RMOG 50/70 Grant from the Ministry of Science and Technology, Brazil en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity Article published earlier |
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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. Basic plasma physics |
| title_short |
Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
| title_fullStr |
Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
| title_sort |
polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity |
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Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. |
| author_facet |
Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. |
| topic |
Basic plasma physics |
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Basic plasma physics |
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2002 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
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1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/77885 |
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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2002. — № 5. — С. 69-71. — Бібліогр.: 16 назв. — англ. |
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2025-11-25T14:06:05Z |
| last_indexed |
2025-11-25T14:06:05Z |
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| fulltext |
POLARIZATION CURRENT THRESHOLD MODEL OF
NEOCLASSICAL TEARING MODES IN THE PRESENCE OF
ANOMALOUS PERPENDICULAR VISCOSITY
M.S.Shirokov,1,2 A.B.Mikhailovskii,1 S.V.Konovalov,1 V.S.Tsypin3
1 Institute of Nuclear Fusion, RRC “Kurchatov Institute”, Moscow, Russia
2 Moscow Institute of Physics and Engineering, Moscow, Russia
3 Institute of Physics, University of São Paulo, S/N, 05508-900 SP, Brasil
1. The primary theory of neoclassical tearing modes
(NTMs) [1-3] predicted that, as a result of the boot-
strap drive, magnetic islands can be generated in
tokamak discharges with favorable current profile,
i.e. for ∆′ < 0, where ∆′ is the standard parameter
of tearing mode theory [4]. Then, the island width
W should be smaller than a maximal one Wmax
proportional to the parameter beta (the ratio of
plasma pressure to the magnetic field pressure), i.e.
W ≤Wmax, (1)
where
Wmax =
βCb
(−∆′)
, (2)
Cb is a constant.
However, the theory [1-3] did not explain, which
β should be substituted into this expression for
Wmax. Then one could suggest that generation of
NTMs is possible for arbitrarily low β. Meanwhile,
expreimental data on TFTR [5] have shown that
these modes are generated only if β exceeds some
critical (threshold) value βcrit,
β ≥ βcrit. (3)
Then the question arose: how one should modify
the theory [1-3] in order to explain the critical beta
for NTM onset.
As a result, two threshold models of NTMs, al-
lowing one to predict βcrit, have been formulated:
the polarization current threshold model [6] and
the transport threshold model [7] (such a termi-
nology has been introduced in [8]). The present
work is addressed to further development of the
first of these models, while the companion work [9]
summarizes recent developments of the transport
threshold model. The goal of the present work is
to incorporate anomalous perpendicular viscosity
into the polarization current threshold model.
2. The polarization current threshold model of
NTM looks as [6]
dW
dt
∼ ∆′
4
+ ∆bs + ∆p, (4)
where ∆bs and ∆p are responsible for the bootstrap
drive and polarization current effect, respectively.
Starting expression for ∆p is [10]
∆p =
1
cψ̃
(
2Ls
B0
)1/2 ∫ −∞
−ψ̃+ε
dψ
∮
J‖ cos ξdξ(
ψ̃ cos ξ − ψ
)1/2
.
(5)
Here J‖ is the oscillatory part of the parallel current
density, ψ is the magnetic flux function introduced
by
ψ = ψ̃ cos ξ − x2B0/2Ls, (6)
ψ̃ is a constant related to W by
ψ̃ = 16W 2B0/Ls, (7)
ξ is the island cyclic variable, Ls is the shear length,
B0 is the equilibrium magnetic field, c is the speed
of light, ε is a positive infinitesimal.
To find J‖ one should solve the current continu-
ity equation
∇‖J‖ + ∇⊥ · j⊥ = 0, (8)
where j⊥ is the perpendicular current density, ∇‖
and ∇⊥ are the parallel and perpendicular gradi-
ents. For simplicity we neglect drift effects. Then,
for obtaining ∇⊥ ·j⊥ one can use the perpendicular
projection of the single-fluid motion equation
ρ0
d0V
dt
=
1
c
[j⊥ ×B]−∇p−∇ ·
(
π⊥ + π‖
)
, (9)
where
d0/dt = ∂/∂t+ V ·∇, (10)
V is the plasma velocity (its structure is explained
below), ρ0 is the equilibrium plasma mass density, p
is the plasma pressure, π⊥ and π‖ are the perpen-
dicular and parallel viscosity tensors, respectively,
B is the total magnetic field.
In the Braginskii [11] approximation, the per-
pendicular viscosity tensor gradient is given by
∇ · π⊥ = −ρ0µ⊥
(
∇2
⊥V⊥ + 4b∇2
⊥V‖
)
, (11)
where µ⊥ is the perpendicular viscosity coefficient,
b is the unit vector along the total magnetic field.
On the other hand, according to equation (19.6) of
[12], the parallel viscosity tensor gradient can be
expressed in terms of the parallel viscosity scalar
π‖ by
∇ · π‖ =
3
2
b (b ·∇)π‖ −
1
2
∇π‖
+
3
2
π‖ [b∇ · b + (b ·∇)b] . (12)
This scalar satisfies the relation (see, for details,
chapter 19 of [12] and, in particular, Eqs. (19.19),
(19.73), (19.78) of [12])
3
2
〈
π‖
∂ ln
√
gm
∂θ
〉
θ
= rsρ0χ̂θ
(
Vy +
ε
q
V‖
)
. (13)
Here gm is the metric tensor determinant, 〈. . .〉θ is
the averaging over the poloidal angle θ,
χ̂θ =
q2
ε1/2
(
d0
dt
+
νi
ε
)
. (14)
The y - direction is defined by the unit vector y =
b× x, x is the unit vector along the x - direction,
where x = r − rs, rs is the radial coordinate of the
rational magnetic surface, where the island chain is
localized, Vy is the y - projection of the cross-field
velocity V⊥ given by
V⊥ = c [b0 ×∇φ] /B0, (15)
φ is the electrostatic potential. Remaining defini-
tions in (13), (14) are: q is the safety factor, ε is the
inverse aspect ratio for r = rs, νi is the ion collision
frequency. The function V‖ is the oscillatory part
of the parallel plasma velocity.
As a result, according to [12], the value ∇⊥ · j⊥
is given by
∇⊥ · j⊥ = −cρ0
B0
∂
∂x
[
χ̂θ
(
Vy +
ε
q
V‖
)]
. (16)
To find V‖ we use the parallel projection of Eq.
(9). Then, allowing for (13), one can find (see, for
details, [12, 13])(
d0
dt
+
ε2
q2
χ̂θ − 4µ⊥
∂2
∂x2
)
V‖ +
ε
q
χ̂θVy = 0. (17)
3. We consider the problem of interest qualita-
tively, i.e. changing ∂2/∂x2 → −1/W 2, d0/dt →
−iω. It then follows from (17) that
ReV‖ = gqVy/ε, (18)
where
g =
ν2
i
[
1 + ε3/2
(
1 +W 2
µ/W
2
)
W 2
µ/W
2
]
+ ε1/2ω2
ν2
i
(
1 +W 2
µ/W
2
)2 + ω2/ε
(19)
and
Wµ '
(
µ⊥
ε1/2νi
)1/2
(20)
is the characteristic island width governed by the
perpendicular viscosity. The function
g = g (νi, ω, µ⊥) (21)
characterizes the collisionality dependence of the
polarization current effect. It is introduced by the
relation
∆p = g∆∞
p , (22)
where
∆∞
p = ∆p|νi→∞ . (23)
In the limit of vanishing perpendicular viscosity,
Wµ/W → 0, Eq. (19) reduces to [14]
g =
ν2
i + ε1/2ω2
ν2
i + ω2/ε
. (24)
Then the function g is given by [14]
g =
ε3/2, νi/ (εω) < C0,
εν2
i /ω
2, C0 < νi/ (εω) < ε−3/2,
1, νi/ (εω) > ε−3/2,
(25)
where
C0 ' ε3/4. (26)
In the limit Wµ/W →∞ one has from (19)
g = ε3/2, (27)
which is the same as the first line of the right-hand
side of (25). Thus, for sufficiently large perpendic-
ular viscosity one deals with the minimal value g
relevant to the limiting case of weak collisions.
4. The expressions for the function g given by
Eq. (25) were found in the linear approximation
[14]. Let us show that they are qualitatively valid
also in the nonlinear regime.
One can find that Eq. (17) with µ⊥ = 0 and χ̂θ
of form (14) leads to (
ε1/2νi − ω
∂h
∂x
∂
∂ξ
)
V‖
=
ω
ky
q
ε1/2
(
νi − εω
∂h
∂x
∂
∂ξ
) (
∂h
∂x
−
〈
∂h
∂x
〉)
, (28)
where h = h (ψ) is the electrostatic potential profile
function [10]. In the limit of weak collisions, νi → 0,
it hence follows that
V‖ = ε1/2q
ω
ky
(
∂h
∂x
−
〈
∂h
∂x
〉)
. (29)
This corresponds to g = ε3/2, see the first line of
the equation (25).
In the opposite case of strong collisions, νi →
∞, instead of (29), one has from (28)
V‖ =
q
ε
ω
ky
(
∂h
∂x
−
〈
∂h
∂x
〉)
. (30)
This yields g = 1, see the third line of (25).
To analyze (28) for finite νi/ (εω) we represent
V‖ as the sum of the cosine and sine parts, i.e. the
even and odd parts (with respect to the variable ξ),
V‖ = Vc + Vs. (31)
Then we arrive at the following two equations
ω
∂h
∂x
∂Vc
∂ξ
= ε1/2νiVs + ε1/2q
ω2
ky
∂h
∂x
∂2h
∂ξ∂x
, (32)
ω
∂h
∂x
∂Vs
∂ξ
= ε1/2νi
[
Vc −
q
ε
ω
ky
(
∂h
∂x
−
〈
∂h
∂x
〉)]
.
(33)
The polarization current is defined by the function
Vc.
Note that for νi/ (εω) < ε−3/4 the contribu-
tion of Vs into (32) can be neglected. Then Vc
proves to be the same as for νi/ (εω) < 1, i.e. Vc
is given by the right-hand side of (29). On the
other hand, if the ratio νi/ (εω) lies in the interval
ε−3/4 < νi/ (εω) < ε−3/2, the equation system (32)
and (33) reduces to
ω
∂h
∂x
∂Vc
∂ξ
= ε1/2νiVs, (34)
ω
∂h
∂x
∂Vs
∂ξ
= −νi
ω
ky
q
ε1/2
(
∂h
∂x
−
〈
∂h
∂x
〉)
. (35)
It hence follows that in order of magnitude
Vc ' q
ν2
i
ωky
(
∂h
∂x
−
〈
∂h
∂x
〉)
. (36)
This corresponds to g = εν2
i /ω
2, cf. the second line
of (25).
The above nonlinear substantiation of of the col-
lisionality dependence (24) for µ⊥ → 0 allows one
to suggest that by means of more complicated non-
linear analysis for µ⊥ 6= 0, one can justify qualita-
tively behavior of the function g (νi, ω,W ) given by
(19).
5. According to [6], dependence of βcrit on the
function g, βcrit (g), characterizing the collisional-
ity dependence of NTMs, is given by
βcrit ∼ g1/2. (37)
Such a collisionality dependence was the subject
of experimental studies [15]. Following [6], it was
assumed in [15] that the function g has the step-like
form with a jump in a region of sufficiently large
νi/εω. Then the authors of [15] have concluded
that their experimental data corroborate the theory
[6]. However, according to [15], the form of the
function g suggested in [6] is unadequate, so that,
instead of g given by [6], one should use g of form
(25). Then one can find that experimental data of
[15] is in disagreement with the polarization current
threshold model.
According to (19), (27), such a disagreement is
redoubled in the presence of anomalous perpendicu-
lar viscosity. As a whole, this decreases attractive-
ness of the polarization current threshold model.
Then, in order to find βcrit one should appeal to
the transport threshold model [9] or to the theory
of β-limiting sub-Larmor modes [16].
Acknowledgments
This work was supported by the Russian Federal
Program on Support of Leading Scientific Schools,
Grant No. 00-15-96526, the Research Support Foun-
dation of the State of São Paulo (FAPESP), Uni-
versity of São Paulo, and Excellence Research Pro-
grams (PRONEX) RMOG 50/70 Grant from the
Ministry of Science and Technology, Brazil.
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