Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces
Coefficients of neoclassical transport in tokamaks and stellarators with non-axisymmetric equipotential surfaces mis-aligned with magnetic flux surfaces are derived for the 1/ν regime. In the general case, they are given by integral expressions including field line integrals similar to those definin...
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irk-123456789-1959002023-12-08T12:59:11Z Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces Markl, M. Heyn, M.F. Kasilov, S.V. Kernbichler, W. Albert, C.G. Magnetic confinement Coefficients of neoclassical transport in tokamaks and stellarators with non-axisymmetric equipotential surfaces mis-aligned with magnetic flux surfaces are derived for the 1/ν regime. In the general case, they are given by integral expressions including field line integrals similar to those defining the effective helical ripple [V.V. Nemov et al. // Phys. Plasmas. 1999, v. 6, p. 4622]. For small mis-alignments in a tokamak with a simplified geometry, they are reduced to simple analytical expressions. Виведено коефіцієнти неокласичного переносу в токамаках та стелараторах з неспівпадаючими еквіпотенціальними та магнітними поверхнями для режиму 1/ν. У загальному випадку вони даються інтегральним виразом з інтегралами вздовж силових ліній, які подібні до тих, що визначають ефективну гвинтову модуляцію в роботі [V.V. Nemov et al. // Physics of Plasmas. 1999, v. 6, p. 4622]. Для малих неспівпадінь у токамаку зі спрощеною геометрією вони зводяться до простих аналітичних виразів. 2022 Article Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces / M. Markl, M.F. Heyn, S.V. Kasilov, W. Kernbichler, C.G. Albert // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 9-12. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.25.Dg, 52.25.Fi DOI: https://doi.org/10.46813/2022-142-009 http://dspace.nbuv.gov.ua/handle/123456789/195900 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement Markl, M. Heyn, M.F. Kasilov, S.V. Kernbichler, W. Albert, C.G. Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces Problems of Atomic Science and Technology |
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Coefficients of neoclassical transport in tokamaks and stellarators with non-axisymmetric equipotential surfaces mis-aligned with magnetic flux surfaces are derived for the 1/ν regime. In the general case, they are given by integral expressions including field line integrals similar to those defining the effective helical ripple [V.V. Nemov et al. // Phys. Plasmas. 1999, v. 6, p. 4622]. For small mis-alignments in a tokamak with a simplified geometry, they are reduced to simple analytical expressions. |
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Markl, M. Heyn, M.F. Kasilov, S.V. Kernbichler, W. Albert, C.G. |
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Markl, M. Heyn, M.F. Kasilov, S.V. Kernbichler, W. Albert, C.G. |
| author_sort |
Markl, M. |
| title |
Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement |
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Non-axisymmetric neoclassical transport from mis-alignment of equipotential and magnetic surfaces / M. Markl, M.F. Heyn, S.V. Kasilov, W. Kernbichler, C.G. Albert // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 9-12. — Бібліогр.: 5 назв. — англ. |
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Problems of Atomic Science and Technology |
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142).
Series: Plasma Physics (28), p. 9-12. 9
https://doi.org/10.46813/2022-142-009
NON-AXISYMMETRIC NEOCLASSICAL TRANSPORT FROM MIS-
ALIGNMENT OF EQUIPOTENTIAL AND MAGNETIC SURFACES
M. Markl
1
, M.F. Heyn
1
, S.V. Kasilov
1,2,3
, W. Kernbichler
1
, C.G. Albert
1
1
Fusion@OEAW, Institute of Theoretical and Computational Physics,
Graz University of Technology, Graz, Austria;
2
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics
and Technology”, Kharkiv, Ukraine;
3
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: markl@tugraz.at
Coefficients of neoclassical transport in tokamaks and stellarators with non-axisymmetric equipotential surfaces
mis-aligned with magnetic flux surfaces are derived for the regime. In the general case, they are given by
integral expressions including field line integrals similar to those defining the effective helical ripple [V.V. Nemov
et al. // Phys. Plasmas. 1999, v. 6, p. 4622]. For small mis-alignments in a tokamak with a simplified geometry, they
are reduced to simple analytical expressions.
PACS: 52.25.Dg, 52.25.Fi
INTRODUCTION
In the standard neoclassical theory, the equilibrium
electrostatic potential is assumed to be constant within
magnetic flux surfaces, i.e. its equipotential surfaces are
aligned with magnetic surfaces. An exception is for
strong plasma rotation in tokamaks where the
centrifugal force induces an axisymmetric miss-
alignment – a dependence of the potential on the
poloidal angle [1]. In stellarators, a small non-
axisymmetric potential miss-alignment always exists
due to finite ion orbit widths, which may affect
neoclassical transport of high-Z impurities (see, e.g. [2,
3]) but produces only a correction for the transport of
bulk plasma components. In tokamaks with external
resonant magnetic perturbations (RMPs), non-
axisymmetric miss-alignment of equipotential surfaces
and magnetic surfaces causes in resonant layers around
rational magnetic surfaces a significant radial transport
of electrons which may exceed the anomalous transport
[4]. In Ref. [4], this transport has been obtained in a
quasilinear limit for the straight cylinder tokamak model
ignoring the toroid city and, thus, also ignoring the
effects associated with particle trapping.
In the present work, we derive neoclassical transport
coefficients for general toroidal magnetic and
electrostatic field geometry staying, however, limited to
a specific transport regime ‒ the regime ‒ valid for
low plasma collisionality and low banana precession
frequency, which should not exceed the collisional
detrapping frequency. Normally, this is fulfilled for the
electron component being the main driver for particle
transport in tokamaks because the ion particle flux
adjusts itself to the electron flux by viscous momentum
transport so that the overall flux is ambipolar.
Combined with the studies of mechanisms leading to the
mis-alignment of equipotential surfaces, the result will
be used for investigations of the increased particle
transport caused by RMPs known as "density pump-
out".
The derivation here is based on the results of
Ref. [5], where the effective helical ripple modulation
has been derived. Those results are generalized here for
the case where equipotential surfaces are not necessarily
aligned with magnetic surfaces. This causes some re-
definitions but generally leaves the main steps of
Ref. [5] intact.
1. BASIC EQUATIONS
We start with the stationary drift-kinetic equation
̂ (1)
in guiding center variables ( ), where is
the guiding center position,
( ) is the
perpendicular adiabatic invariant containing the species-
specific mass , the perpendicular particle velocity
and the species-specific cyclotron frequency .
Further, is the gyrophase and
is
the total energy containing the electric charge . Here,
̂ is the linearized Coulomb collision operator, and the
guiding center velocity is given by
(2)
(
)
(3)
where the curl is computed assuming that
( )
(
( ( ) ( )) (4)
10 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
i.e. to be a function of the coordinates and constant
invariants of motion and . The Jacobian of the
phase space coordinates is
|
( )
( )
|
| |
(5)
Further, the flux surface averaged normal particle flux
density is defined as
∫ ∫ (6)
where is the distribution function and is the flux
surface area. For any distribution function, which,
within the flux surface, depends only on integrals of
motion but not on the coordinates (in particular, for a
local Boltzmann distribution function), it can be shown
that there is no flux surface averaged radial particle or
total energy flux. This can be checked directly for the
surface ( ) as follows
∫
∑ ∫
∫
( )
∑ ∫
[ ]
∫
[( ) ]
∫
(
| |
) (7)
where the surface integral of the curl is zero for both,
passing particles, which occupy the whole flux surface
area, and for trapped particles existing in the regions
. In these regions, the integral is reduced via the
Stokes theorem to the region boundaries where the sub-
integrand is zero due to . Similarly, one can
check that the total energy flux, which contains an extra
factor in the sub-integrand, is also zero.
Particle flux density (6) can be written via the
species-specific fluid velocity as
∫
⟨ ⟩
⟨| | ⟩
(8)
where the neoclassical flux surface average is defined in
flux variables ( ) as
⟨ ⟩ (∫
∫ √
)
∫
∫ √
(9)
Using Newcomb’s theorem, this expression can also be
written in terms of field line integrals in field aligned
variables ( ) as
⟨ ⟩
(∫
)
∫
(10)
where these variables are defined via the safety factor
as
(11)
such that labels the field lines. These variables have
the same √ , and as the usual ones, while
. Since , the product √ is constant
on flux surfaces so that √ can be replaced by in
(9), thus arriving at (10).
2. LMFP PARTICLE FLUX DENSITY
Within the local neoclassical ansatz, the drift
kinetic equation is solved by a series expansion over the
radial drift velocity up to linear order using the
ansatz .The equilibrium distribution
function is given by a local Boltzmann distribution
( )
̅ ( )
( ( ))
(
̅( )
( )
) (12)
where the parameter ̅is chosen so that the
neoclassically averaged density ̅ is the same as the
density computed from (12). By the above arguments,
the fluxes are solely produced by the perturbation
( ). In the long mean free path
(LMFP) regime, which is of interest here, the fluxes are
produced by the leading order of in the collision
frequency (its bounce averaged part) which is constant
along the field lines, ( ). Obviously,
in the passing particle domain, the leading order is
independent of as well and contributes to the fluxes
only in the trapped particle domain. Expressing the
particle flux density (7) via field line integrals (10),
where the field aligned coordinates are constructed
from Boozer coordinates with a re-defined flux surface
label such that ⟨| |⟩ , and performing similar
transformations to those in Ref.[5] this contribution is in
the general form
∫
[ ]
∫
[( ) ]
[( ) ]
(∫ √
)
∑
(13)
where
∫
| |
(14)
Here, the integration is along the field line
, and the index enumerates the segments
of this line where
, with
being the segment
ends or turning points. Equation (13) does not assume
that magnetic field and electrostatic potential
modulations within the flux surface are small. It allows
various classes of trapped particles including particles
blocked by non-axisymmetric perturbations of both, the
magnetic field and the electrostatic potential, which is
most easily achieved even for small perturbations in a
tokamak with perturbed toroidal symmetry near the
extrema of the main axisymmetric magnetic field. If we
ignore in those tokamaks the contributions of blocked
particles and consider only the bounce averaged radial
drift of usual bananas dominant at small perturbation
amplitudes, eq. (13) simplifies to
∫
[ ]
∫
[( ) ]
[( ) ]
(∫ √
)
∫
(15)
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 11
where denotes the main class of banana-trapped
particles and corresponds to the unperturbed
magnetic field. The heat flux density is obtained
similarly by adding the average kinetic energy ̅
in the sub-integrand. Note that the above expressions
are valid for the general case of the LMFP regime, not
necessarily the regime.
3. SOLUTION OF THE BOUNCE
AVERAGED EQUATION
In the regime, the perturbation satisfies the
linear bounce averaged equation. Using the Lorentz
model for the collision integral,
̂
(16)
where is the deflection frequency, this equation is
√
∫
| |
(17)
It corresponds to a particular class of trapped particles
with a number of classes tending to infinity with a field
line length . Neighbouring classes bounded by the
same boundary layer , where simultaneously
, and which have a minimum along the field line
(for the aligned equipotential surfaces this corresponds
to local maxima of B) are coupled together via boundary
conditions (detailed discussion can be found in Ref.
[5]). As shown in [5], all those boundary conditions are
satisfied by the following first integral of the kinetic
equation over ,
√
(18)
Here, the derivative of the local Boltzmann distribution
(12) is given as
(
̅
) (19)
where the thermodynamic forces are
̅
̅
̅
(20)
Thus, particle flux (13) for the general case takes the
form
√
∫
[ ]
(
̅
)∫
[( ) ]
[( ) ]
(∫ √
)
∑
(
)
(21)
This expression generalises the formula for the
particle flux density of Ref. [5] for the case of non-
aligned equipotential surfaces and reduces to that
formula in case of alignment. In the latter case, the
dependence of the quantities and on the kinetic
energy can be factorized, which allows to factorize the
integral over energy. Thus, all geometrical information
is contained in the integral over the normalized
perpendicular invariant
which can be expressed
in terms of the factor
which formally defines
effective helical ripple modulation .
Below, we are interested in the case of weak
potential perturbations in a tokamak, where one can
ignore additional trapped particle classes and where the
energy integral can be factorized as well. Expression
(15) for this case gives
√
(∫ √
)
∫
[ ]
(
̅
)
∫
[( ) ]
[( ) ]
∫
(
)
(22)
4. SMALL ELECTROSTATIC
PERTURBATIONS AND AXISYMMETRIC
MAGNETIC FIELD
In the case of small electrostatic perturbations and
an axisymmetric magnetic field, we can simplify
∫
( )
(22)
∫
( )
(23)
where is the particle velocity module and
(
)
( ). Presenting the potential
in the form
( ) ∑ ( )
(24)
and computing the integral over , equation (21)
takes the form
̅
√ √
(∫ √
)
∫
√ ( )
∫
(∫
( )
)
∑
|∫
( )
|
(25)
where
( ). Note that the normalized heat
flux is given by expression (25) with an
extra factor in the sub-integrand.
Thus, we obtain a matrix of Onsager-symmetric
transport coefficients determined via
thermodynamic forces as
12 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
̅ ( )
̅ ( ) (26)
If we approximate the collision as
,
where is the collision frequency for particles with
energy , then transport coefficients are related by
. (27)
The simplest expression for is obtained for near-
resonant potential perturbations, ( ) (
) with | | , in a large aspect ratio circular
tokamak with √ , and ( ( ))
where and are values at the magnetic axis and
. In this case, we arrive at
√
(
| |
)
CONCLUSIONS
We have derived an integral formula for non-
axisymmetric neoclassical transport fluxes in the
regime in general toroidal devices with non-aligned
equipotential and magnetic surfaces. The formula
considers all possible trapped particle classes allowing
for both, magnetic and electrostatic trapping. In the case
of aligned surfaces, it reduces to the result of Ref. [5]
defining the effective helical ripple. The formula can be
evaluated using the field line integration technique
similar to the one realized in the code NEO [5].
AKNOWLEDGEMENTS
This work has been carried out within the framework of
the EUROfusion Consortium, funded by the European
Union via the Euratom Research and Training
Programme (Grant Agreement № 101052200 –
EUROfusion). Views and opinions expressed are
however those of the author(s) only and do not
necessarily reflect those of the European Union or the
European Commission. Neither the European Union nor
the European Commission can be held responsible for
them.MM gratefully acknowledges support from NAWI
Graz and funding from the KKKÖ at ÖAW and the
Friedrich-Schiedel Stiftung für Energietechnik.
REFERENCES
1. P. Helander, D.J. Sigmar. Collisional Transport in
Magnetized Plasmas. Cambridge University Press,
2002, 310 p.
2. H.E. Mynick // Phys. Fluids. 1984, v. 27, p. 2086-
2092.
3. J.M. Garcia-Regana et al// Nuclear Fusion. 2017,
v. 57, p. 056004.
4. M.F. Heyn et al // NuclearFusion. 2014, v. 54,
p. 064005.
5. V.V. Nemov et al // Phys. Plasmas. 1999, v. 6,
p. 4622-46.
Article received 08.10.2022
АКСІАЛЬНО-НЕСИМЕТРИЧНИЙ НЕОКЛАСИЧНИЙ ПЕРЕНОС ВНАСЛІДОК
НЕСПІВПАДАННЯ ЕКВІПОТЕНЦІАЛЬНИХ ТА МАГНІТНИХ ПОВЕРХОНЬ
М. Маркль, М.Ф. Хайн, С.В. Касілов, В. Кернбіхлер, К.Г. Альберт
Виведено коефіцієнти неокласичного переносу в токамаках та стелараторах з неспівпадаючими
еквіпотенціальними та магнітними поверхнями для режиму . У загальному випадку вони даються
інтегральним виразом з інтегралами вздовж силових ліній, які подібні до тих, що визначають ефективну
гвинтову модуляцію в роботі [V.V. Nemov et al. // Physics of Plasmas. 1999, v. 6, p. 4622]. Для малих
неспівпадінь у токамаку зі спрощеною геометрією вони зводяться до простих аналітичних виразів.
|