Spontaneous generation of beta-limiting MHD modes in tokamaks
It is shown that under conditions typical of fusion reactors of tokamak type, a spontaneous generation of magnetohydrodynamic (MHD) modes is possible, which may limit plasma pressure in ideally stable plasma. Electromagnetic field of such modes consists of two different scale lengths: a large-scale...
Gespeichert in:
| Datum: | 2003 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/110342 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Spontaneous generation of beta-limiting MHD modes in tokamaks / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2003. — № 1. — С. 46-48. — Бібліогр.: 20 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-110342 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1103422017-01-16T12:35:02Z Spontaneous generation of beta-limiting MHD modes in tokamaks Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. Magnetic confinement It is shown that under conditions typical of fusion reactors of tokamak type, a spontaneous generation of magnetohydrodynamic (MHD) modes is possible, which may limit plasma pressure in ideally stable plasma. Electromagnetic field of such modes consists of two different scale lengths: a large-scale (MHD) component and a small-scale one, which is smaller than Larmor radius of thermal ions. It is suggested that these modes can be responsible for spontaneous generation of neoclassical tearing modes observed experimentally. The value of threshold beta for this mode onset was found to be similar to that in experiments showing NTMs. Показано, що в умовах термоядерних реакторів типу токамак можлива спонтанна генерація магнітогідродинамічних (МГД) збурень, що призводять до обмеження на тиск ідеально стійкої плазми. Електромагнітне поле таких збурень має не тільки великомасштабну МГД складову, а ще й дрібномасштабну, характерний розмір якої малий у зрівнянні з ларморівським радіусом іонів. Припущено, що вказані збурення можуть бути відповідальні за експериментально досліджену спонтанну генерацію неокласичних тиринг мод (НТМ). Величина порога по бета для збудження виявлених нестійкостей є близькою з характерними значеннями для експериментів з НТМ. Показано, что в условиях термоядерных реакторов типа токамак возможна спонтанная генерация магнитогидродинамических (МГД) возмущений, приводящая к ограничению на давление идеально устойчивой плазмы. Электромагнитное поле таких возмущений имеет не только крупномасштабную МГД составляющую, но и мелкомасштабную, характерный размер которой мал по сравнению с ларморовским радиусом ионов. Высказано предположение, что указанные возмущения могут быть ответственны за экспериментально наблюдаемую спонтанную генерацию неоклассических тиринг мод (НТМ). Величина порога по бета для возбуждения обнаруженных неустойчивостей оказывается близкой к характерным для экспериментов с НТМ значениям. 2003 Article Spontaneous generation of beta-limiting MHD modes in tokamaks / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2003. — № 1. — С. 46-48. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 52.55.Fa, 52.35.-g http://dspace.nbuv.gov.ua/handle/123456789/110342 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 |
| collection |
DSpace DC |
| language |
English |
| topic |
Magnetic confinement Magnetic confinement |
| spellingShingle |
Magnetic confinement Magnetic confinement Shirokov, M.S. Mikhailovskii, A.B. Konovalov, S.V. Tsypin, V.S. Spontaneous generation of beta-limiting MHD modes in tokamaks Вопросы атомной науки и техники |
| description |
It is shown that under conditions typical of fusion reactors of tokamak type, a spontaneous generation of magnetohydrodynamic (MHD) modes is possible, which may limit plasma pressure in ideally stable plasma. Electromagnetic field of such modes consists of two different scale lengths: a large-scale (MHD) component and a small-scale one, which is smaller than Larmor radius of thermal ions. It is suggested that these modes can be responsible for spontaneous generation of neoclassical tearing modes observed experimentally. The value of threshold beta for this mode onset was found to be similar to that in experiments showing NTMs. |
| format |
Article |
| author |
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. |
| author_sort |
Shirokov, M.S. |
| title |
Spontaneous generation of beta-limiting MHD modes in tokamaks |
| title_short |
Spontaneous generation of beta-limiting MHD modes in tokamaks |
| title_full |
Spontaneous generation of beta-limiting MHD modes in tokamaks |
| title_fullStr |
Spontaneous generation of beta-limiting MHD modes in tokamaks |
| title_full_unstemmed |
Spontaneous generation of beta-limiting MHD modes in tokamaks |
| title_sort |
spontaneous generation of beta-limiting mhd modes in tokamaks |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2003 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110342 |
| citation_txt |
Spontaneous generation of beta-limiting MHD modes in tokamaks / M.S. Shirokov, A.B. Mikhailovskii, S.V. Konovalov, V.S. Tsypin // Вопросы атомной науки и техники. — 2003. — № 1. — С. 46-48. — Бібліогр.: 20 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT shirokovms spontaneousgenerationofbetalimitingmhdmodesintokamaks AT mikhailovskiiab spontaneousgenerationofbetalimitingmhdmodesintokamaks AT konovalovsv spontaneousgenerationofbetalimitingmhdmodesintokamaks AT tsypinvs spontaneousgenerationofbetalimitingmhdmodesintokamaks |
| first_indexed |
2025-07-08T00:30:24Z |
| last_indexed |
2025-07-08T00:30:24Z |
| _version_ |
1837036601987825664 |
| fulltext |
SPONTANEOUS GENERATION OF BETA-LIMITING MHD
MODES IN TOKAMAKS
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. Neoclassical tearing modes (NTMs) are sug-
gested to be one of the main obstacles to achieve ac-
ceptable values of beta (ratio of plasma pressure to
magnetic field pressure) in fusion reactors of toka-
mak type [1]. The existing theory of NTMs [2] is
based on the notion that these modes are triggered
by other types of MHD (magnetohydrodynamic)
activity such as ELMs (Edge Localized Modes),
sawteeth and fishbones. Meanwhile, the observa-
tional data from ASDEX Upgrade [3] and TFTR [4]
show that NTMs can be generated spontaneously,
i.e. in the absence of any MHD activity. To explain
spontaneous generation of NTMs one should appeal
to a linear instability excited when the parameter
beta of ideally stable plasma exceeds a threshold
value. However, the existing linear theory [5] does
not predict such an instability.
The need to have a non-ideal instability excited
when beta exceeds a threshold value is felt also in
explanation of observational data on NTMs from a
series of tokamaks with hot-electron plasma (plasma
with electron temperature essentially larger than
the ion one) like T-10 [6], COMPASS-D [7] and
TCV [8]. The fact is that, after the revision of the
traditional NTM theory [9] confirmed by [10], it
became clear that the polarization current effect in
such a plasma is destabilizing. Therefore, the re-
vised theory does not predict a threshold beta for
NTM onset in these devices.
Certainly, in order to interpret these data from
the devices with hot-electron plasma one can turn
to the transport threshold models of NTMs (see
[11] and references therein). However, these models
deal with a rather obscure coefficients of anomalous
perpendicular transport. Therefore, though within
the scope of the transport models it is possible to
appeal to such an interpretation, it seems to be
doubtful that all totality of the data from different
devices can be satisfactorily explained.
Thus, a rather broad series of experiments using
the notions of NTM theory needs a β-limiting linear
instability.
In principle, preceding theoretical investigations
of linear modes in tokamaks include certain indica-
tions in favor of existence of beta-limiting instabil-
ities. Thus, it was shown in [12] that for q ' 1
an instability distorting the magnetic surfaces of
tokamak-type toroidal systems can be excited if,
qualitatively (see Eq. (3.29) of [12])
βp > β(0)
p ≡ s2L2
p/r
2
s . (1)
Here q is the safety factor, βp is the poloidal beta,
s is the shear, Lp is the characteristic scale of the
plasma pressure gradient, rs is the radial coordinate
of the rational magnetic surface where the mode is
localized. According to [12], for excitation of this
instability the presence of temperature gradient is
necessary.
The local approximation was used in [12], which
is insufficient to show the existence of the eigen-
modes. This defect of [12] has been corrected in
[13, 14]. The eigenmodes found in [13, 14] have
been called the beta-induced temperature gradient
(BTG) eigenmodes.
It was assumed in [12 - 14] that the BTG modes
are excited due to toroidal acoustic resonance, i.e.
for the condition ω∗ ' vTi/qR where ω∗ is the
characteristic diamagnetic drift frequency, vTi is
the ion thermal velocity, R is the torus major ra-
dius. Such a resonance is effective only for suffi-
ciently high poloidal and toroidal mode numbers,
m = nq = Lprs/ (qRρi) > 1, where ρi is the ion
Larmor radius. Experimental observation of BTG
modes on JET was reported in [15].
The analysis of [12 - 14] was based on the ap-
proximation that the characteristic radial scale of
the mode is larger than ρi or ρs, kxρi � 1, kxρs �
1, where ρs is the ion Larmor radius calculated for
the electron temperature, kx is the perpendicular
projection of the wave vector ( the variable x is
defined by x = r − rs). Nevertheless, in order to
reveal the eigenmodes the authors of [13, 14] have
been forced to allow for the formally small terms of
the order of (kxρi)
2 and (kxρs)
2.
At the same time, the MHD-like modes, includ-
ing those with kxρi ≥ 1, kxρs ≥ 1, are subject
of the theory of semicollisional modes [5, 16 - 18].
One of the main mathematical results of this theory
is a rather complicated general dispersion relation
(see, for details, Eq. (24.33) of [5] and Eq. (9)
of the present paper). This dispersion relation con-
tains a dimensionless parameter ν which in the case
T0i = T0e = T0 (T0i and T0e are the equilibrium ion
and electron temperatures, respectively) is given by
ν2 =
1
4
− (ω − ω∗e) (ω − ω∗i)
2k2
yρ
2
iω
2
A
. (2)
Here ω is the mode frequency, ωA = svA/ (qR) is
the Alfven frequency, vA is the Alfven velocity, ω∗e
and ω∗i are the electron and ion diamagnetic drift
frequencies, respectively, ky = m/rs is the poloidal
projection of the wave vector. Up to now, the anal-
ysis of the dispersion relation (24.33) of [5] was per-
formed only for the particular case
∣∣ν2 − 1/4
∣∣ � 1
In this case it describes the semicollisional internal
kink and tearing modes [17] and the semicollisional
ballooning modes [18]. All these modes do not be-
long to the class of the β-limiting modes.
The goal of the present paper is to discover a
new linear instability excited for a beta value larger
than a critical one and to show that this critical
beta is of the same order as that necessary for ex-
planation of the observational data from [3, 4, 6-8].
2. Turning to dispersion relation (24.33) of [5],
one can see that it is satisfied for
ν = 0. (3)
Then one has from (2) and (3)
(ω − ω∗e) (ω − ω∗i)− 2ω2
∗eβ
(0)
p /βp = 0. (4)
We call the modes described by (4) the “sub-Larmor”
modes.
It follows from (4) that in the case of suffi-
ciently low βp, βp � β
(0)
p , the mode frequencies
prove to be essentially larger than both the elec-
tron and ion diamagnetic drift frequencies. There-
fore, in this case the modes can not be excited by
the electron or ion diamagnetic drift effects. How-
ever, with increasing the βp the mode frequency (4)
decrease and prove to be of the order of ω∗e or ω∗i
for the condition (1). For such βp the dissipative
electron/ion drift effects can excite the sub-Larmor
modes.
3. Now we consider the case of vanishing ion
temperature. Similar to section 24.1 of [5], we start
from the current continuity equation written in the
Fourier space kx. We take
∇⊥ · j⊥ = − iω
4π
k2
xε⊥φ. (5)
Here φ is the electrostatic potential, ε⊥ = c2f/v2
A
is the perpendicular plasma permittivity, c is the
speed of light, f = f (ω) is the toroidal renormal-
ization of perpendicular inertia [5], j⊥ is the per-
turbed electric current density across the equilib-
rium magnetic field B0, ∇⊥ is the perpendicular
(with respect to B0) gradient. We use the parallel
Ohm’s law (cf. (22.1) of [5])
E‖ −
T0e
een0
(
∇‖ñ+
Bx
B0
n′0
)
=
j‖
σ
. (6)
Here E‖ = −∇‖φ + iωA‖/c is the perturbed par-
allel electric field, A‖ = 4πj‖/
(
ck2
x
)
is the parallel
projection of the perturbed vector potential, j‖ is
the perturbed parallel electric current density, σ
is the plasma electric conductivity, Bx = ikyA‖ is
the x-projection of the perturbed magnetic field,
ky is the poloidal projection of the wave vector, ee
is the electron electric charge, n0 is the equilibrium
plasma density, the prime is the radial derivative, ñ
is the perturbed plasma number density, ∇‖ is the
parallel gradient. The perturbed plasma number
density ñ is assumed to satisfy the electron conti-
nuity equation
−iωñ+ VExn
′
0 +∇‖j‖/ee = 0, (7)
where VEx = −ickyφ/B0 is the x-projection of the
perturbed cross-field velocity.
Far from the resonant point r = rs we use the
“constant ψ approximation”, where ψ ≡ −A‖. In
the Fourier space this means that for kx → 0
φ ∼ 1−∆′/kx, (8)
where ∆′ is the standard tearing mode theory match-
ing parameter.
As a result, following the approach explained in
chapter 24 of [5], we arrive at the dispersion relation
Γ2
(
− 1
4 + ν
2
)
Γ2 (−ν)Q (ν)
Γ2
(
− 1
4 −
ν
2
)
Γ2 (ν)Q (−ν)
=
(
4κ
β̂2
)ν
. (9)
Here Γ is the gamma function,
Q (ν) = 1 +
κ−1/2
8rs∆′
Γ2
(
− 1
4 −
ν
2
)
Γ2
(
− 5
2 −
ν
2
) (
ν2 − 1
4
)
, (10)
the value ν is given by (cf. (2))
ν2 =
1
4
− ω (ω − ω∗e)
k2
yρ
2
sω
2
A
, (11)
κ = fk2
yρ
2
s/ (1− ω∗e/ω), β̂ = (−iωγR)1/2
/ (kyρsωA),
γR = c2k2
y/ (4πσ) is the characteristic resistive de-
cay rate.
One can see that the dispersion relation (9) is
satisfied for the condition (3). It follows from (11)
that in this case the mode frequency is determined
by the dispersion relation similar to (4):
ω (ω − ω∗e) = k2
yρ
2
sω
2
A/4. (12)
Equation (12) yields
ω = ω± = −ω∗e
2
[
1±
(
1 +
2s2L2
n
βpr2s
)1/2
]
, (13)
One can see that, for the condition (1) the mode
propagating in the electron drift direction has fre-
quency of the order of electron diamagnetic drift
frequency.
4. It is possible that nonlinear development of
the “sub-Larmor” modes leads to NTMs. It is then
of interest to estimate the linear growth rate γ of
the modes considered and to elucidate the collision-
ality dependence of the beta threshold.
One can suggest that the growth rate γ of the
mode of type (13) is determined by
γ = γe + γi, (14)
where γe is the electron growth rate and γi is the
ion decay rate given by, respectively,
γe ' fe
(
νe
εω∗e
)
|ω∗e| , (15)
γi ' −
β
(0)
p
βp
νi. (16)
Here fe [νe/ (εω∗e)] is a small dimensionless param-
eter whose explicit form can be found turning to
[19], νe and νi are the electron and ion collision fre-
quencies, respectively. Then one can see that for
finite νi the condition (1) is insufficient for exci-
tation of the “sub-Larmor modes”. In this case,
instead of (1), one should use the estimate
βp > βcrit
p , (17)
where
βcrit
p =
{
β
(0)
p , νi < γe,
β
(0)
p νi/γe, νi > γe.
. (18)
To determine βcrit one was forced to appeal to
the polarization current threshold model [2] (see
also [20]) or to the transport threshold model. In
this context, Eq. (18) is an alternative to these
models in determining the βcrit.
The estimates, following from (18), prove to be
compatible with the experimental data from [3, 4,
6-8].
Acknowledgments
The authors would like to express their grati-
tude to Dr. S. E. Sharapov for discussions stimu-
lating this work.
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.
References
[1] ITER Physics Expert Group on Disruptions, Plasma
Control, and MHD, ITER Physics Basis Editors 1999
Nucl. Fusion 39 2251
[2] Wilson H R et al 1996 Plasma Phys. Control.
Fusion 38 A149
[3] Gude A et al 1999 Nucl. Fusion 39, 127
[4] Fredrickson E D 2002 Phys. Plasmas 9 548
[5] Mikhailovskii A B 1998 Instabilities in a Con-
fined Plasma (Bristol: Institute of Physics)
[6] Kislov D A et al 2001 Nucl. Fusion 41 1473
[7] Gates D A et al 1997 Nucl. Fusion 37 1593
[8] Reimerdes H et al 2002 Phys. Rev. Lett. 88
105005
[9] Waelbroeck F L and Fitzpatrick R 1997 Phys.
Rev. Lett. 78 1703
[10] Mikhailovskii A B, Pustovitov V D, Smolyakov
A I and Tsypin V S 2000 Phys. Plasmas 7 1214
[11] Shirokov M S, Mikhailovskii A B, Konovalov S
V, Tsypin V S, 2002 Development of Transport Thresh-
old Model of Neoclassical Tearing Modes, this confer-
ence
[12] Mikhailovskii A B 1973 Nucl. Fusion 13 259
[13] Mikhailovskii A B and Sharapov S E 1999 Plasma
Phys. Rep 25 803
[14] Mikhailovskii A B and Sharapov S E 1999 Plasma
Phys. Rep 25 838
[15] Sharapov S E et al 1998 Theory of Fusion Plas-
mas. Proc. Int. Varenna-Lausanne Workshop on Fu-
sion Plasmas (Varenna, 1998) eds Connor J W, Sindini
E and Vaclavic J, Bologna; Compositori, p. 215
[16] Drake J F and Lee Y C 1977 Phys. Fluids 20
1341
[17] Pegoraro F and Schep T J 1986 Plasma Phys.
Control. Fusion 28 647
[18] Mikhailovskii A B, Novakovskii S V and Churikov
A P 1988 Sov. J. Plasma Phys. 14 536
[19] Kuvshinov B N and Mikhailovskii A B 1998
Plasma Phys. Rep. 24 245
[20] Shirokov M S, Mikhailovskii A B, Konovalov
S V, Tsypin V S, 2002 Polarization Current Threshold
Model of Neoclassical Tearing Modes in the Presence of
Anomalous Perpendicular Viscosity, this conference
|