Advanced fluid modelling of transport in tokamaks
The need for and conditions for fluid modelling of tokamak transport is discussed. In particular emphasis is put on fluid closure, off diagonal transport fluxes and non- Markovian effects. Some recent results of a fluid model using a reactive closure are also discussed including theory and simulatio...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2002 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/80311 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Advanced fluid modelling of transport in tokamaks / J. Weiland // Вопросы атомной науки и техники. — 2002. — № 4. — С. 89-91. — Бібліогр.: 14 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860241661306601472 |
|---|---|
| author | Weiland, J. |
| author_facet | Weiland, J. |
| citation_txt | Advanced fluid modelling of transport in tokamaks / J. Weiland // Вопросы атомной науки и техники. — 2002. — № 4. — С. 89-91. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The need for and conditions for fluid modelling of tokamak transport is discussed. In particular emphasis is put on fluid closure, off diagonal transport fluxes and non- Markovian effects. Some recent results of a fluid model using a reactive closure are also discussed including theory and simulations of zonal flows and temperature and particle pinches.
|
| first_indexed | 2025-12-07T18:30:32Z |
| format | Article |
| fulltext |
ADVANCED FLUID MODELLING OF TRANSPORT IN
TOKAMAKS
Jan Weiland
Department of Electromagnetics, Chalmers University of Technology
and
EURATOM-VR Association, S-41296 Gëteborg, Sweden
The need for and conditions for fluid modelling of tokamak transport is discussed. In particular emphasis
is put on fluid closure, off diagonal transport fluxes and non- Markovian effects. Some recent results of a
fluid model using a reactive closure are also discussed including theory and simulations of zonal flows and
temperature and particle pinches.
Introduction
Although research on transport in magnetic confine-
ment devices has been a high priority research area
since the beginning of fusion research, the agreement
between theory and experiment has been poor un-
til the development of advanced fluid models started
in the end of the 1980s [1]. The reason for this has
mainly been the fact that toroidal effects play a very
important role in toroidal devices and that including
these fully, in principle, requires a kinetic description
[2–4]. Although it is relatively easy to apply a lin-
ear gyrokinetic description, this problem is quite dif-
ficult since a nonlinear kinetic description is needed
for transport calculations. Nonlinear kinetic codes
can not yet be run on the confinement timescale with
realistic computation times. Thus some kind of ap-
proximation of kinetic theory is needed. The most
successful methods so far has been the development
of advanced fluid models.
Advanced fluid models
As indicated above, we need to deal with the ki-
netic resonance in some approximate way. Simple
fluid models in general are applicable either in col-
lision dominated cases or in the adiabatic (fast pro-
cess) or isothermal (slow process) limits in collision-
less plasmas. Thus a fluid model that can be ap-
plied for intermediately fast processes in collisionless
plasmas can be defined as an advanced fluid model.
Due to the dramatic improvement in agreement with
experiments found when the flat density regime was
included, we can also tie the definition of advanced
fluid models to the ability to include this regime as
was done in Ref [4].
Energy equation and closure
The highest moment we will consider here is the heat
flow. We then write the energy equation for the col-
lisionless case (no thermal force)
2
3
n
(
∂
∂t
+ v · ∇
)
T + P∇ · v = −∇ · q. (1)
Here the choice of q defines the closure. In our
reactive model we use the Braghinskii diamagnetic
heat flux:
q = q∗ =
5
2
P
mΩ
(
e‖ × ∇T
)
. (2)
Gyrofluid models here add an imaginary part rep-
resenting the contribution to Landaudamping and
Magnetic drift resonances from all higher moments.
This imaginary part is fitted to linear kinetic theory.
We will shortly give a motivation for our choice of q in
a collisionless state. After replacing ∇ ·v by the con-
tinuity equation we find that convective diamagnetic
effects cancel. We then arrive at the temperature
perturbation:
δT
T
=
ω
ω − 5/3ωD
[
2
3
δn
n
− ω∗e
ω
(
2
3
− η
)
eφ
Te
]
. (3)
Here fluid resonance due to nonadiabatic part is kept!
δn/n−(ω∗e/ω)(eφ/Te) is a difference due to compres-
sion and (ω∗e/ω)η(eφ/Te) is a convective part.
The same model is used for ions, trapped electrons
and impurity ions. We will now outline the derivation
of the closure (2) in the collisionless case.
Collisionless closure
Continuing the fluid hierarchy up to the heat flow
equation we have the highest moment:
〈vivjvkvl〉 = 〈vivk〉〈vjvl〉 + . . . + G,
where G = 〈vivjvkvl〉irr is the irreducible, genuine
fourth moment.
If the particle velocities are stochastic we can make
the Random phase approximation
〈vivk〉 = |vi|2ð(i − k), G = 0.
This is exactly fulfilled for a Maxwell distribution.
In a recent paper [5] it was shown by a Maximum
entropy principle that G = 0 defines a closure in
an isolated unmagnetized solid state plasma.This fo-
cuses interest on the external sources.
If we include only moments with external sources
(n and T ) we also arrive at the condition G = 0 un-
der the condition that no internal sources are created
for G. An example of when an external source of
a lower moment can create a source for a higher mo-
ment is Ohmic heating. However, this is a completely
random process which fulfills the Random phase ap-
proximation. In order to create a source for G we
need a source that generates correlations. An exam-
ple of this is instabilities driven by the external heat-
ing such as beam driven modes. Here the beam fills in
velocity space continuously so that flattening or par-
ticle trapping are balanced by the source in velocity
space. For modes driven by ideal particle and heat
sources we do not expect sources in velocity space to
play a role. Furthermore, since we include the low-
est order velocity resonances, nonlinear relaxation of
the resonance due to higher order moments will not
mean complete flattening of velocity space. If we fur-
ther assume isotropic temperature perturbations we
again arrive at the closure (2). The argument for do-
ing this is that there are nonlinear equilibration terms
for the parallel and perpendicular temperatures per-
turbations in the energy equations. Effects of a non-
linear closure were recently studied by generalizing
the method of Mattor and Parker [6] to include dif-
fusion [7].
Linear properties of ITG modes
Combining Eq (2) with the usual perpendicular fluid
drifts and parallel ion motion leads to a three pole
ion density response. Then using Boltzmann elec-
trons we arrive at an eigenvalue problem that can
Fig 1
be solved analytically in the strong ballooning limit.
The stability limits are shown in Fig 1.
The upper curve in Fig 1 shows the local stabil-
ity boundary, the tangent line is the nonlocal stabil-
ity boundary and the horizontal line is the stability
boundary for the adiabatic model. The ion conduc-
tivity can be written:
χi =
1
ηi
(
ηi − 2
3
− 10
9τ
εn
)
γ3/k2
x
(ωr − 5ωDi/3)2 + η2
. (4)
The three parts in paretesis represent in order diag-
onal, off diagonal and convective parts. We note that
the convective part gives a strong inward flux compo-
nent for hot ions The real frequency in the denomina-
tor (non-Markovian effect) increases stiffness and in-
troduces a selectivity regarding transport from modes
with different real eigenfrequency [8,9]. It comes from
the denominator of (3). At nonlocal marginal stabil-
ity the first factor in (4) and the denominator of (3)
both vanish.
Fig 2 shows typical radial profiles of χ for experi-
ments and simple fluid models. An example of this
is shown in Ref 10. The behaviour of advanced fluid
models was obtained from Eq (4) for the parameters
in Ref 10.
Fig 2
Pinch fluxes
In connection with radial E × B convection we can
write the temperature perturbation as:
δT = −ξ • ∇T + αξ • ∇n + βξ • ∇B, (5)
where ξ is the E×B displacement and the coefficients
α and β are compressibility coefficients. The fist term
on the right hand side is the convective temperature
perturbation, the second is due to expansion after
convection in the density gradient and the last term
is due to expansion after convection in the magnetic
field gradient. The last two terms will reduce the
temperature perturbation. Linearly they will create
a threshold in Ln/Lt or Lb/Lt. for instability. The
nonlinear fluxes are obtained by multiplying (5) with
the radial E × B velocity. The last two parts will
then give off diagonal fluxes that tend to counteract
the flux due to the first, diagonal term. If the off diag-
onal fluxes dominate we would get a pinch. However
then there is no thermal instability. If we, however,
have two different species that can give thermal in-
stability where one is stable and the other is unstable,
the unstable feedback loop will drive the stable one
due to E × B convection and we get a pinch from
the stable loop [11]. An example of an electron heat
pinch simulation with our transport model was pre-
sented in Ref 12. This was from off axis ECH heat-
ing in RTP where the ECH source was modulated
in time. The electron heat pinch was driven by the
ITG mode and both average temperature and the first
three harmonics of the modulation frequency were in
good agreement with the experiment.
Zonal flows
Nonlinearly generated zonal flows were recently found
to give a nonlinear upshift in the critical temperature
gradient in nonlinear gyrokinetic simulations of ion
heat transport in a D- III-D H-mode [13]. We have
recently simulated this case with our fluid model [14].
We verified the nonlinear (Dimits) upshift within 10%
of the fluxtube nonlinear gyrokinetic simulations in
Ref 13. This is particularly interesting since the
zonal flows are excited resonantly just above the lin-
ear threshold. The resonance is due to the fluid res-
onance in the energy equation and is thus sensitive
to the fluid closure. We note that the IFS-PPPL-
98 model had a deviation of 50% in the nonlinear
upshift in the simulations in Ref 13. It is also impor-
tant that the general transport level in our model, as
compared to the nonlinear gyrokinetic results in Ref
13, was verified in Ref 14.
References
[1] B.A. Carreras, IEE Trans. Plasma Sci. 25, 1281
(1997).
[2] A. Jarmen, P. Andersson and J. Weiland, Nuclear
Fusion 27, 941 (1987).
[3] R.E. Waltz, R.R. Dominguez and G.W. Ham-
mett, Phys. Fluids B4, 3138 (1992).
[4] J. Weiland, Collective Modes in Inhomogeneous
Plasmas, IoP, Bristol 2000.
[5] A.M. Anile and O. Muscato Phys. Rev. B51,
17628 (1995).
[6] N. Mattor and S. Parker, Phys. Rev. Lett. 79,
3419 (1997).
[7] I. Holod, J. Weiland and A. Zagorodny, Phys.
Plasmas 9, 1217 (2002).
[8] J. Weiland and H. Nordman, Proc. Varenna Lau-
sanne workshop, Chexbres 1988, p 45.
[9] A. Zagorodny and J. Weiland, Phys. Plasmas 6,
2359 (1999).
[10] S.D. Scott et.al. Phys. Rev. Lett. 64, 531
(1990).
[11] J. Weiland, Physica Scripta T82, 84 (1999).
[12] P. Mantica et al., Investigation of Heat Pinch
Effects in Tokamak Plasmas, EU-US Transport
Task Force Workshop, Cordoba September 9-12
2002.
[13] A.M. Dimits et. al., Phys. Plasmas 7, 969
(2000).
[14] S. Dastgeer, Sangeeta Mahajan and J. Weiland,
Zonal flows and Transport in Ion Temperature
Gradient Turbulence, Accepted for publication
in Physics of Plasmas.
|
| id | nasplib_isofts_kiev_ua-123456789-80311 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:30:32Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Weiland, J. 2015-04-14T17:48:20Z 2015-04-14T17:48:20Z 2002 Advanced fluid modelling of transport in tokamaks / J. Weiland // Вопросы атомной науки и техники. — 2002. — № 4. — С. 89-91. — Бібліогр.: 14 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/80311 The need for and conditions for fluid modelling of tokamak transport is discussed. In particular emphasis is put on fluid closure, off diagonal transport fluxes and non- Markovian effects. Some recent results of a fluid model using a reactive closure are also discussed including theory and simulations of zonal flows and temperature and particle pinches. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Advanced fluid modelling of transport in tokamaks Article published earlier |
| spellingShingle | Advanced fluid modelling of transport in tokamaks Weiland, J. Basic plasma physics |
| title | Advanced fluid modelling of transport in tokamaks |
| title_full | Advanced fluid modelling of transport in tokamaks |
| title_fullStr | Advanced fluid modelling of transport in tokamaks |
| title_full_unstemmed | Advanced fluid modelling of transport in tokamaks |
| title_short | Advanced fluid modelling of transport in tokamaks |
| title_sort | advanced fluid modelling of transport in tokamaks |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80311 |
| work_keys_str_mv | AT weilandj advancedfluidmodellingoftransportintokamaks |