A computational model for transport in partially ergodized magnetic fields
A 3-dimensional plasma uid transport problem for fusion edge plasmas is considered. Conventional numerical methods from uid dynamics or gas dynamics are not applicable, if the coordinate line along one of the main transport directions exhibits ergodic behaviour at least in some regions of the comput...
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| Date: | 2000 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | A computational model for transport in partially ergodized magnetic fields / D. Reiter, A. Runov, S. Kasilov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 33-38. — Бібліогр.: 11 назв. — англ. |
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| author | Reiter, D. Runov, A. Kasilov, S. |
| author_facet | Reiter, D. Runov, A. Kasilov, S. |
| citation_txt | A computational model for transport in partially ergodized magnetic fields / D. Reiter, A. Runov, S. Kasilov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 33-38. — Бібліогр.: 11 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | A 3-dimensional plasma uid transport problem for fusion edge plasmas is considered. Conventional numerical methods from uid dynamics or gas dynamics are not applicable, if the coordinate line along one of the main transport directions exhibits ergodic behaviour at least in some regions of the computational domain. In stellarator plasma edges, or in tokamaks with eld line perturbations as foreseen for TEXTOR under dynamic ergodic divertor (DED) operation, such complications can arise. We propose and discuss a novel "Multi-Coordinate System Approach" within the framework of a Monte Carlo procedure. A 3-dimensional plasma uid code is developed, benchmarked and applied to a model with geometrical and magnetic eld parameters chosen to fall in the range of parameters expected for TEXTOR with DED [1]. The resulting patterns in the computed plasma temperature eld near the perturbation coils are in accordance with experimental observations of the radial modulation of the temperature eld in TORE SUPRA, though under slightly di erent experimental conditions there.
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A COMPUTATIONAL MODEL FOR TRANSPORT IN PARTIALLY
ERGODIZED MAGNETIC FIELDS
D.Reiter1;2, A.Runov1;2, S.Kasilov3
1 Institute for Plasmaphysics, Forschungszentrum J�ulich GmbH
EURATOM Association, Trilateral Euregio Cluster, Germany
2 Institute for Laser and Plasmaphysics, Heinrich Heine University D�usseldorf, Germany
3 Institute of Plasma Physics, Kharkov, Ukraine
ABSTRACT
A 3-dimensional plasma
uid transport problem
for fusion edge plasmas is considered. Conventional
numerical methods from
uid dynamics or gas dy-
namics are not applicable, if the coordinate line along
one of the main transport directions exhibits ergodic
behaviour at least in some regions of the computa-
tional domain. In stellarator plasma edges, or in toka-
maks with �eld line perturbations as foreseen for TEX-
TOR under dynamic ergodic divertor (DED) opera-
tion, such complications can arise. We propose and
discuss a novel "Multi-Coordinate System Approach"
within the framework of a Monte Carlo procedure. A
3-dimensional plasma
uid code is developed, bench-
marked and applied to a model with geometrical and
magnetic �eld parameters chosen to fall in the range of
parameters expected for TEXTOR with DED [1]. The
resulting patterns in the computed plasma tempera-
ture �eld near the perturbation coils are in accordance
with experimental observations of the radial modula-
tion of the temperature �eld in TORE SUPRA, though
under slightly di�erent experimental conditions there.
I. INTRODUCTION
3 dimensional
uid simulations have become a
more or less standard tool in computational
uid me-
chanics. In magnetised plasmas, however, even 2-
dimensional approximations still appear to be at the
borderline of what is possible numerically. This due
to several plasma physical peculiarities, such as the
strong nonlinear character of transport coe�cients
themself, the high an-isotropicity, and the large dif-
ferences of relevant timescales involved in one prob-
lem (electron energy transport along B, compared to
particle transport across B, for example), leading to
very sti� equations. Finally the sources and sinks
(due to atomic ("chemical") processes) are, typically,
both nonlocal (kinetic) and nonlinear. Still, extract-
ing physical information from current magnetic fusion
experiments requires computational assistence, due to
the large number of individual processes competing
with each other (detailed bookkeeping). See [2] for a
recent review.
The situation is even more complicated in exper-
iments, in which, for physical reasons, the 2 dimen-
sional structure of the magnetic �eld is destroyed.
Intrinsic or even externally produced magnetic
perturbations can provide a stochastic �eld topology
in parts of fusion edge plasmas, both in tokamaks and
in stellarators. In these edge plasma regions, ergodic
zones, islands, and laminar zones can coexist and, cer-
tainly, mutually in
uence each other. Transport is in-
herently 3 dimensional here.
A further complication arises, since the so called
"laminar zones" are not necessarily those found from
�eld line tracing. Here we use the term "laminar zone"
for that part of the
ow �eld, in which the connection
length from the source to the sink of any extensive
quantity (say, for the plasma energy) along the mag-
netic �eld is short (only a few turns around the torus
or less). This generalised terminology accounts for the
physical properties and the in
uence of laminar zones
upon the
ow dynamics, distinct from the just geo-
metrical characterisation in the more commonly used
de�nition based upon �eld line length only.
In TEXTOR such stochastic regions will be pro-
duced deliberatly, by the the "Dynamic Ergodic Di-
vertor" (DED), in the near future. In stellarators they
will arise naturally, at increasing plasma-�.
We expect that TEXTOR-DED and stellarators
will be "similar" in this sense and di�erent from the
intrinsically 2 dimensional (at least in an idealisation)
axisymmetric divertor or limiter concepts.
Due to a 3D distribution of recycling sources
and sinks there may be a complicated pattern of re-
gions with short connection length to either a material
boundary (sink), or to a sink region caused by neutral
plasma interaction. In other words a dual character
of the scrape o� layer may be manifested, in which a
pattern of regions of long and short connection length
along the B-�eld to whatever sink (surface or volume
distributed recycling sink) is produced.
The simplest possible model for the plasma
ow
under such conditions, therefore, must at least have
the following ingredients:
� The complete �eld line pattern, including ergodic,
island and "laminar" zones.
� A model for transport in these �elds, with homo-
geneous boundary conditions only at the separa-
trix between perturbed (SOL) and unperturbed
region, consistently linking the various regions in-
side the SOL.
� A model for the 3D recycling process. Distinct
from regular (2D) boundary plasmas, as e.g., in
ideal axisymmetric divertors or limiters, here we
have to deal with a breaking of the symmetry be-
tween the magnetic topology (hence: the paral-
lel plasma
ow) on the one side and the vacuum
chamber (hence: recycling) on the other side.
It is known since long, even from intrinsically 2D sit-
uations, that this "misalignment" between the plasma
ow and the neutral particle recycling
ow can, po-
tentially, lead to complicated local
ow patterns with
ow reversal, perhaps even under globally low recy-
cling conditions as expected for TEXTOR [3]. The
terse physical cause here is an overloading of particu-
lar �eld lines (
ux bundles) with recycling of plasma
e�ux from other, neighboring �eld lines.
We wish to study the following simple model, care-
fully accounting for the above mentioned complica-
tions:
Field lines (
ux bundles) are fed from the unper-
turbed core plasma and, mutually, from each other
by anomalous cross-�eld di�usion normal to the �eld
lines. With this competing is the fast plasma trans-
port parallel to the magnetic �eld, which we assume
to be as described by classical plasma transport theory
(Braginskii).
Charged particles recombine upon impact on any
material surface (limiter, vacuum vessel,...) and neu-
tral atoms and molecules are released there, typically
at an almost equal rate.
These neutrals are reionised, and, most impor-
tantly, not necessarily within the same
uxtube as the
former ion.
Monte Carlo procedures are available (e.g. the
EIRENE code, [4]), which can describe this neutral gas
transport with su�cient accuracity, once the plasma
pro�les are given. Here we describe the conceptually
new second module of such a code-package, which com-
putes the plasma
ow �elds (e.g., temperature �elds),
for given recycling sources (e.g., ionisation pro�les).
The computational task is to solve di�usion-
advection equations for plasma transport in complex
3 dimensional magnetic �eld con�gurations includ-
ing ergodic layers, i.e., with potentially chaotic (non-
integrable)
ow �elds.
Existing 3D
uid codes for the stellarator pe-
riphery (EMC3, [5], BoRiS, [6]) are based upon one
"global" magnetic coordinate system. Magnetic sur-
faces must exist everywhere, and the coordinate sys-
tem must be perfectly aligned to them, also every-
where. Hence these procedures restricted to integrable
ow �elds, and are not suitable in the presence of er-
godic layers.
Instead, our Multi-Coordinate System approach
( see below) allows modelling of plasma transport in
general magnetic �eld structures and still accounts for
the detailed material boundary geometry of a device.
The main concepts followed here are:
1. A Monte-Carlo approach for di�usion-advection
equations (Lagrangian discretization) is em-
ployed. Every "test particle" (
uid parcel) per-
forms jumps (random steps) along and across the
magnetic �eld in it's own moving coordinate sys-
tem. Such sets of locally adequate coordinate sys-
tems (each one restricted to a sub-domain small
compared to the Kolmogorov-length, and sepa-
rated by cut-surfaces from the neighboring sub-
domains) can readily be established, in the same
way as Clebsch coordinates are constructed for
open magnetic con�nement con�gurations [7].
2. The link between two neighboring local coordi-
nate systems (handing
uid parcels over from one
to the next coordinate system) is done by a special
"interpolated cell mapping", a well established
procedure from nonlinear dynamics.
3. A second (global and more simple) Eulerian co-
ordinate system is used to evaluate plasma pa-
rameters from the trajectories of the
uid parcels.
This second mesh is chosen in order to represent
the grid boundaries: e.g., vacuum vessel, limiters,
divertor targets, etc., and, hence, the recycling
process in a convenient way.
The rough scheme of the method is as follows. We
start with an initial guess for the 3D pro�le of some
hydrodynamical quantity A, preferably a speci�c ex-
tensive quantity (energy density, momentum density,
mass density, etc..).
We then wish to �nd the result of the evolution of
A after a time interval �t. According to the Monte-
Carlo technique, we scatter a number of "
uid parcels"
("test particles", by abuse of language) each of which
represents some amount of A. The spatial distribu-
tion of the parcels must reproduce the initial pro�le
A(~r; t0). Then each parcel performs a jump (in it's
own, or local coordinate system) according to a law of
motion derived from the balance equation for A and
to the value of �t chosen for the discretisation in time.
Hence, we solve a determinsitic problem numeri-
cally by a Monte Carlo simulation of a concomitant
probabilistic problem.
The new spatial distribution of the particles rep-
resents the new pro�le A(~r; t0 +�t). This procedure
(excluding the initial discretization) can be repeated
either until a steady state is found, or the dynamical
behaviour of the system itself is studied.
The mathematical task here is to derive the ex-
pressions for the components of a suitable random
walk process, especially accounting for the particular
combination of mapping and �eld line tracing, which
we shall discuss below in more detail (see the next
section).
According to the point 2, we need a preparation
step to obtain all geometric (i.e., magnetic) informa-
tion (interpolation coe�cients for the mapping proce-
dure, the components of the metric tensor etc.).
Q
i
i-1
i+
1
TEXTOR-DED, poloidal section
ALT
DED
Fig.1: P oloidal cross section of TEXTOR model, showing:
ALT-II toroidal limiter, DED target and Poincar�e cut's
We will c hoose, below, cuts � = const: in a quasi-
toroidal coordinate system as the basis for the cell
mapping procedure. This choice is somewhat arbitray,
and it is made here because of the particular symmetry
of the TEXTOR limiters and perturbation coils. On
each such (Poincar�e-) section a regular 2D (r; �)-mesh
is de�ned. We solve (numerically) the equations of the
magnetic �eld line through the knots of such a mesh
from a given section to the next and to the previous
one, and interpolate the resulting transformation func-
tions (i.e., the coordinates of �nal points as functions
of the coordinates of the starting points) by means
of bicubic splines (hence the name: "interpolated cell
mapping").
It is useful to de�ne a "limiter shadow" or "dead
zone" on each section.
Moving from these "shadowed" zones along the
�elds results in the termination of the trajectory on a
material surface, before the next section is reached.
II. BASIC EQUATIONS
The main goal of the present work is to develop
a computational tool, which permits to solve a set of
prototypical balance equations of the type:
@��
@t
= r~�� + Q� (1)
Here � is a speci�c quantity (per unit mass), � is the
mass density, and ~�� is the
ux of �.
Q� comprises all terms (sources, sinks) that do
not �t into the divergence of the
ux ~��.
We specialise the
ux ~�� to di�usive-advective
form:
~�� = �~r(��) + ~V ��: (2)
� is the di�usivity (or: conductivity) coe�cient, and
~r(��) is the thermodynamic force driving the dissi-
pative part of the
ux.
Again, all remaining terms can be put into Q�.
This is the general di�erential equation governing
the evolution of the extensive quantity (per unit vol-
ume) ��.
Because of the geometrical and physical complica-
tions mentioned above, we will employ Monte Carlo
methods to �nd approximations to solutions of sets of
such conservation equations.
We do this by specifying a Markovian randomwalk
process Xrw, which approximates (in the limit of small
steps) a di�usion process Xd. We can then utilise rela-
tions between this di�usion process Xd and a Fokker-
Planck equation and see, that our Monte Carlo proce-
dure solves, indeed, our set of conservation equations.
For this purpose, we �rst have to rewrite the pro-
totypical equations in Fokker-Planck form.
The
uid equations, in this form and in some gen-
eral curvilinear coordinates xi read:
@�
@t
=
1
p
g
@
@xi
p
g
�
�ij
@�
@xj
� V i
Drift
�
�
+ Q�; (3)
where g is the metric determinant.
Here � is the di�usivity related to the conductivity
� by � = (�=m)� = n�. n denotes the number density.
The velocity components arising in the drift term are
V i
Drift
=
5
3
V i
�
+ �ij
@
@xj
logn; (4)
where V i
�
are
uid velocity components. The source
term Q� comprises all remaining terms from the bal-
ance equation.
In the particular case of the energy balance equa-
tion for electrons and ions in a single ion species
plasma we readily identify
�� = ue;i = 3nTe;i=2
u is the internal energy density. n and Te;i are
plasma density and electron and ion temperatures, re-
spectively. The di�usion tensor in (3) (omitting the
indices "e" and "i" from now on) is taken to be of the
form
�ij = �?g
ij +
�
�k � �?
�
hihj ; (5)
where gij = (rxi)(rxj) and hi = hrxi are con-
travariant components of the metric tensor and of
the unit vector along the magnetic �eld h = B=B.
�k;? = �k;?=(n), �k and �? are classical parallel
and anomalous perpendicular heat conductivity coef-
�cients, respectively.
III. MULTI COORDINATE SYSTEM APPROACH
(MCSA)
For the numerical solution of eqs. (3) a set of many
(typically M � 20) local magnetic coordinate systems
Sm; m = 1::M is used in combination with a map-
ping procedure derived from �eld line tracing [8]. This
permits to avoid arti�cial cross-�eld transport other-
wise caused by a numerical mixing of parallel and per-
pendicular
uxes. Every local coordinate system is
chosen such that for two coordinates (e.g., labelled 1
and 2) the condition
h � rxi
m
= 0; i = 1; 2: (6)
holds. I.e. two coordinates are chosen normal to B for
each coordinate system.
The third coordinates x3
m
are chosen such that
their coordinate surfaces x3
m
= constant are nowhere
tangent to B, e.g.: h � rx3
m
> 0. As a result, the par-
allel
ux has only one nonzero component. This can
be seen, e.g., from the form of the general di�usion
tensor,
Dij = D?g
ij +
�
Dk �D?
�
(hrx3)2�i3�
j
3
(7)
where �i3 is the Kronecker symbol.
Particular sets of local coordinate systems used be-
low are constructed in quasitoroidal coordinates r; �; '.
Here r is the small radius, � the poloidal and ' the
toroidal angle, respectively. The third variable x3
m
is
chosen to coincide with the poloidal angle, x3
m
= x3 =
�. The cuts ("Poincar�e sections") are introduced at
the surfaces
� = const: = �m � (m�1)��; m = 1; : : : ;M (8)
where �� = 2�=M (see �g.1).
Consider an arbitrary point P = (r; �; '). The co-
ordinates x1
m
(P ); x2
m
(P ) in the system Sm are de�ned
as the small radius and the toroidal angle of the pro-
jection along the magnetic �eld line to the Poincar�e
section m. Hence x1
m
; x2
m
are linked with the quasi-
toroidal coordinates by the characteristics of eq. (6),
x1
m
= �(r; �; '; �m); x2
m
= �(r; �; '; �m); (9)
which satisfy the magnetic �eld line equations,
@�
@�0
=
hr(�; �0; �)
h�(�; �0; �)
; (10)
@�
@�0
=
h'(�; �0; �)
h�(�; �0; �)
;
and initial conditions
�(r; �; '; �) = r; �(r; �; '; �) = ': (11)
Here hr(r; �; '), h�(r; �; '), h'(r; �; ') are contravari-
ant components of the vector h in quasitoroidal coor-
dinates.
IV. MONTE-CARLO PROCEDURE
The mathematical justi�cation for the Monte
Carlo procedure used here is well known from standard
textbooks. Essentially, we simulate random walks
from a discontinuous Markov-process, which approxi-
mates a (continuous) di�usion process. Averaging over
the trajectories we obtain an approximate solution to
this Fokker-Planck equation, hence also to the heat
balance equation. The internal energy contained in
the system is distributed between an ensemble of Np
"test particles". Thus, the internal energy density av-
eraged over a given cell is estimated as the sum of
the weights of particles in this cell devided by the cell
volume.
In order to derive the elementary time step we
rewrite (3) in conservative Fokker-Planck form for the
pseudoscalar density N of test particles N = u
p
g
=w.
w is the weight of test particles and
is the plasma
volume,
@N
@t
=
@
@xi
�
@
@xj
DijN � V i
c
N
�
+QN ; (12)
where
V i
c
= V i +
1
p
g
@
@xj
p
gDij : (13)
The source terms QN = Qu
p
g
=w are accounted for
by weight adjustment, and here we describe the dy-
namics of test particles only. The random process gov-
erning the test particle motion is
xi(t+�t) = xi(t) + �xi; (14)
where �xi are small random steps and �t � �min.
Here �min is the shortest relaxation time correspond-
ing to parallel relaxation: ��1
min
= Dk=L
2
k, Lk is the
parallel spatial scale of plasma and magnetic �eld pa-
rameters. The Fokker-Planck equation for the density
of test particles N subjected to random process (14)
coincides with (12) if
h�xi�xji = 2Di;j(x(t))�t; h�xii = V i
c
(x(t))�t;
(15)
The error in the representation of the distribution
function of test particles subjected to the above ran-
dom process is quadratic in �t after one time step.
Within this precision during one time step one can
model separately di�erent types of transport processes
using the independent sets of random numbers for each
process. The di�usion and convection can be modelled
separately as well,
�xi = �xi
D
+�xi
C
: (16)
The convection step is carried out as in any conven-
tional deterministic Lagrangian discretisation scheme.
The non-trivial part is the term with the second deriva-
tive. This di�usion step, in our local coordiantes
x1;2, is carried out by a stochastic perturbation of La-
grangian trajectories. It is given by
�xi
D
=
p
2�t�ik�k +
Di3
D33
�x3
D
; i; k = 1; 2: (17)
Here �ik is the square root matrix,
�ik�jl�kl = Dij � Di3Dj3
D33
; i; j; k; l = 1; 2 (18)
and �i is a set of random numbers, satisfying the con-
ditions < �i >= 0 and < �k�l >= �kl, �kl is Kronecker
symbol. The matrix on the right hand side of (18)
is non-negatively de�nite since the di�usion tensor is
non-negatively de�nite. Therefore the square root ma-
trix �ik is real.
Up to the precision of the map f�; �g the
random process introduced above induces no ar-
ti�cial cross �eld transport , i.e., our procedure
is perfectly aligned despite of the possibly er-
godic character of the magnetic �eld. The maps
f�(r; �m; '; �m+1); �(r; �m; '; �m+1)g,
Q
Bumper BumperALT
r, cm
θ-π π
Fig.2: Typical Poincar�e plot for a TEXTOR-
DED model magnetic �eld at ' = const
f�(r; �m; '; �m�1); �(r; �m; '; �m�1)g, m = 1; : : : ;M ,
are precomputed by numerical integration of eqs. (10)
for the mesh of r; ' values. They are then re-
constructed during the Monte Carlo computation by
means of bicubic splines which provide su�cient map-
ping accuracy.
We introduce time intervals �t small compared
with characteristic relaxation time of the relevant
plasma parameters. The transport coe�cients re-
quired are computed from the estimates of plasma pa-
rameters from the previous time step and are kept con-
stant during �t. This "explicit" procedure is repeated
until a stationary solution is reached.
V. SAMPLE APPLICATIONS
The new code E3D is employed here to study the
heat transfer in the edge plasma, expected during the
DED operation in TEXTOR.
We have to assume, so far, that the dominant heat
transport mechanism along the B-�eld is conduction,
i.e., that a certain temperature gradient parallel B is
maintained. Under typical (low recycling) limiter con-
ditions, parallel ion heat transport is convection lim-
ited by the electrostatic sheath in front of the limiter.
In the applications described below, addressing elec-
tron heat transfer, the convective component along B
is omitted.
The magnetic �eld enters into our procedure via
the �eld line equations, i.e., numerically, via �eld line
tracing. All formulae above are written for the most
general case, i.e., for the completely numerical treat-
ment of the equations of the �eld lines (10) (as, e.g.,
described in [1] using the DIVA- GOURDON code
combination there). However, for special purposes (as,
e.g., fast parameter studies, program debugging, code
validation and comparison to theoretical results etc.)
we have implemented a simple analytical model for
the magnetic �eld as well, which preserves all main
features of the real con�guration, ranging from almost
unperturbed magnetic surfaces through chains of is-
lands to the complete ergodicity, depending upon the
parameters of that model-�eld.
Results presented here have been obtained with
such a "model magnetic �eld", see �gure 2.
In addition to the typical TEXTOR data [1], we
have chosen the following input parameters (boundary
conditions) for the simulations shown here:
� radius of the perturbation coils rc = 53cm;
� number of Poincar�e sections M=19;
� heat
ux from core into SOL (r = 42cm) Qe;i =
0:45MW in electrons and ions each;
� cross �eld heat conduction coe�cient �? =
n�? with �? = 5m2=s;
� plasma density n = 7� 1012cm�3;
� perturbation �eld ~B varied from 0 to 0.1 T.
The numerical results illustrate the in
uence of
the perturbation �eld on the temperature pro�le and
the power load distribution on the TEXTOR limiters,
see �gure 3a and 3b for typical 3D temperature �elds,
plotted in one selected poloidal cross section. The par-
allel heat transport is fast enough for electrons and
ions to result in regular radial and poloidal patterns,
following the �eld structure. Related �ndings have al-
ready been reported from the ergodic divertor experi-
ments at TORE SUPRA [10]. The heat load pattern
on the TEXTOR limiters (�gure 4) is also found to
be strongly in
uenced by the e�ect of the perturba-
tion �eld. Even with maximal foreseen currents in the
perturbation coils with stationary (i.e. not yet dy-
namical) �elds, clear heat load patterns on the �rst
wall components prevail, however with some redistri-
bution between the toroidal ALT pump-limiter, the
inner bumper (i.e., "divertor target") and the vessel.
This is an indication of near �eld e�ects, i.e., the ex-
istance of so called "laminar zones"
The results seem to con�rm the qualitative picture
of the heat
ux patterns on the bumper limiter ("di-
vertor target"), [11], essentially as a consequence of the
�eld structure in the "laminar region" alone. Quan-
titatively, however, we �nd that the geometric origin
�eld lines carrying the main heat, which is deposited
on the "divertor target" is located in the ergodic parts
of the SOL, i.e., on �eldlines with a rather deep radial
penetration, distinct from the ad hoc pre-selection of
�eld lines of short geometrical length. Furthermore,
and as expected, we �nd a redistribution of global heat
load from the ALT limiter to the wall and bumper.
However, due to the localization of the heat
ux into
narrow stripes, the peak power load on the bumper is
increasing about 2 times faster with increasing pertur-
bation �eld. The proper balance between requirements
for pumping (strong localisation of plasma
uxes) and
heat removel (spreading of plasma e�ux over a large
area) seems to remain a critical issue even for fusion
reactor design, also with ergodic divertors.
We conclude from this �rst applications to a
strongly simpli�ed (and hence: still physically obvi-
ous) case that the procedure seems to work both cor-
rectly and economically (turn around time: 2 hours on
CRAY-T3E with the fully parallelized version).
T ,eVe
T ,eVi
r, cm
r, cm
ϑ
ϑ
Fig.3a,b: top: electron temperature �eld
bottom: ion temperature �eld
⊥
χ = 5 m /s2
B = 0
∼
B = max
∼
B = max/5
∼
Fig.4: heat load pattern on inner divertor targets
top: no perturbation �eld
middel: 20 % perturbation �eld
bottom: maximum perturbation �eld
VI. CONCLUSIONS
A fully parallelized 3D MC
uid code has been de-
veloped for the solving transport equations in tokamak
edge plasmas with arbitrary magnetic �eld topology
(including ergodic zones, islands and laminar zones).
The e�ciency of the code is su�cient for modelling
the DED operation in TEXTOR on CRAY T3E (turn
around time about 2 hours).
The computational model should prove general
enough to allow also the simulation of partially ergodic
stellarator edge plasma conditions without a principal
modi�cation of the existing code.
It has been shown that the power
ux to the
bumper limiter in TEXTOR is signi�cantly increasing
with the increase of the amplitude of the perturbation
magnetic �eld. Further studies with varied radial posi-
tion of the main (ALT-) limiter still need ot be carried
out to identify optimal operation windows.
At the same time, the power
ux to bumper limiter
becomes localized within a few helical stripes. This
produces an increase in peak power
ux value, approx-
imately two times faster than the total
ux to bumper
limiter. Dynamic divertor operation, however, even at
low frequencies (below 100 Hz) stongly reduces this
peak power loading again.
REFERENCES
1. K. H. Finken, et al. Special issue:\Dynamic Er-
godic Divertor". Fusion Engineering and Design,
37(3), 1997.
2. W. Arter Numerical simulation of magnetic fusion
plasmas Rep. Prog. Phys, 58, pp1-59, 1995.
3. D. Reiter J. Nucl. Mat., 196-198, p241, 1992
4. D. Reiter The EIRENE Code: User Manual In-
ternet: www.eirene.de, 2000
5. Y. Feng, et al. J. Nucl. Mat., 241-243, p930, 1997.
6. R. Schneider, M. Borchardt, J. Riemann,
A. Mutzke, S. Weber Contr. Plasma Physics,
40, 2000.
7. D'haeseleer, W.D., Hitchon, W.N.G., Callen,
J.D., Shohet, J.L. Flux Coordinates andMagnetic
Field Structure, Springer-Verlag, 1991.
8. S. V. Kasilov, V. E. Moiseenko, and M. F. Heyn.
Phys. Plasmas, 4, p2422, 1997.
9. A. Runov, D. Reiter, S. Kasilov et al. Phys. Plas-
mas, 2000 (accepted)
10. P. Ghendrih, et al. invited lecture at 14th PSI
Rosenheim, May 2000 J. Nucl. Mat., to appear,
2000.
11. K. H. Finken, T. Eich, and A. Kaleck. Nucl. Fus.,
38(4), p515{529, 1998.
|
| id | nasplib_isofts_kiev_ua-123456789-82363 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T03:47:27Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Reiter, D. Runov, A. Kasilov, S. 2015-05-29T05:54:04Z 2015-05-29T05:54:04Z 2000 A computational model for transport in partially ergodized magnetic fields / D. Reiter, A. Runov, S. Kasilov // Вопросы атомной науки и техники. — 2000. — № 3. — С. 33-38. — Бібліогр.: 11 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82363 533.9 A 3-dimensional plasma uid transport problem for fusion edge plasmas is considered. Conventional numerical methods from uid dynamics or gas dynamics are not applicable, if the coordinate line along one of the main transport directions exhibits ergodic behaviour at least in some regions of the computational domain. In stellarator plasma edges, or in tokamaks with eld line perturbations as foreseen for TEXTOR under dynamic ergodic divertor (DED) operation, such complications can arise. We propose and discuss a novel "Multi-Coordinate System Approach" within the framework of a Monte Carlo procedure. A 3-dimensional plasma uid code is developed, benchmarked and applied to a model with geometrical and magnetic eld parameters chosen to fall in the range of parameters expected for TEXTOR with DED [1]. The resulting patterns in the computed plasma temperature eld near the perturbation coils are in accordance with experimental observations of the radial modulation of the temperature eld in TORE SUPRA, though under slightly di erent experimental conditions there. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Мagnetic Confinement A computational model for transport in partially ergodized magnetic fields Article published earlier |
| spellingShingle | A computational model for transport in partially ergodized magnetic fields Reiter, D. Runov, A. Kasilov, S. Мagnetic Confinement |
| title | A computational model for transport in partially ergodized magnetic fields |
| title_full | A computational model for transport in partially ergodized magnetic fields |
| title_fullStr | A computational model for transport in partially ergodized magnetic fields |
| title_full_unstemmed | A computational model for transport in partially ergodized magnetic fields |
| title_short | A computational model for transport in partially ergodized magnetic fields |
| title_sort | computational model for transport in partially ergodized magnetic fields |
| topic | Мagnetic Confinement |
| topic_facet | Мagnetic Confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82363 |
| work_keys_str_mv | AT reiterd acomputationalmodelfortransportinpartiallyergodizedmagneticfields AT runova acomputationalmodelfortransportinpartiallyergodizedmagneticfields AT kasilovs acomputationalmodelfortransportinpartiallyergodizedmagneticfields AT reiterd computationalmodelfortransportinpartiallyergodizedmagneticfields AT runova computationalmodelfortransportinpartiallyergodizedmagneticfields AT kasilovs computationalmodelfortransportinpartiallyergodizedmagneticfields |