Physics of the dynamic ergodic divertor
The Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and are fed individually outside the vessel. A perturbation field is c...
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| Zitieren: | Physics of the dynamic ergodic divertor / M.W. Jakubowski, K.H. Finken, S.S. Abdullaev, M. Kobayashi, M. Lehnen, R. Wolf // Вопросы атомной науки и техники. — 2002. — № 4. — С. 42-44. — Бібліогр.: 6 назв. — англ. |
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Jakubowski, M.W. Finken, K.H. Abdullaev, S.S. Kobayashi, M. Lehnen, M. Wolf, R. 2015-04-14T05:25:32Z 2015-04-14T05:25:32Z 2002 Physics of the dynamic ergodic divertor / M.W. Jakubowski, K.H. Finken, S.S. Abdullaev, M. Kobayashi, M. Lehnen, R. Wolf // Вопросы атомной науки и техники. — 2002. — № 4. — С. 42-44. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.55Fa https://nasplib.isofts.kiev.ua/handle/123456789/80260 The Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and are fed individually outside the vessel. A perturbation field is created by the electrical currents in the perturbation coils with Fourier components resonant to the magnetic surfaces. The stochastic boundary layer is generated in the outermost region of the plasma, which due to long and short connection lengths can be divided into ergodic and laminar regions. Field line tracing and mapping techniques were used to analyse properties of the TEXTOR-DED plasma boundary. The DED will operate with several frequencies (DC or AC up to 10 kHz). In the “dynamic” operation the convective heat flux is deposited to a large plasma-facing surface and forces are transferred to the plasma edge, what can introduce a differential rotation of the plasma. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement Physics of the dynamic ergodic divertor Article published earlier |
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Physics of the dynamic ergodic divertor |
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Physics of the dynamic ergodic divertor Jakubowski, M.W. Finken, K.H. Abdullaev, S.S. Kobayashi, M. Lehnen, M. Wolf, R. Magnetic confinement |
| title_short |
Physics of the dynamic ergodic divertor |
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Physics of the dynamic ergodic divertor |
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Physics of the dynamic ergodic divertor |
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Physics of the dynamic ergodic divertor |
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physics of the dynamic ergodic divertor |
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Jakubowski, M.W. Finken, K.H. Abdullaev, S.S. Kobayashi, M. Lehnen, M. Wolf, R. |
| author_facet |
Jakubowski, M.W. Finken, K.H. Abdullaev, S.S. Kobayashi, M. Lehnen, M. Wolf, R. |
| topic |
Magnetic confinement |
| topic_facet |
Magnetic confinement |
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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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The Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and are fed individually outside the vessel. A perturbation field is created by the electrical currents in the perturbation coils with Fourier components resonant to the magnetic surfaces. The stochastic boundary layer is generated in the outermost region of the plasma, which due to long and short connection lengths can be divided into ergodic and laminar regions. Field line tracing and mapping techniques were used to analyse properties of the TEXTOR-DED plasma boundary. The DED will operate with several frequencies (DC or AC up to 10 kHz). In the “dynamic” operation the convective heat flux is deposited to a large plasma-facing surface and forces are transferred to the plasma edge, what can introduce a differential rotation of the plasma.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/80260 |
| citation_txt |
Physics of the dynamic ergodic divertor / M.W. Jakubowski, K.H. Finken, S.S. Abdullaev, M. Kobayashi, M. Lehnen, R. Wolf // Вопросы атомной науки и техники. — 2002. — № 4. — С. 42-44. — Бібліогр.: 6 назв. — англ. |
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| first_indexed |
2025-11-26T23:30:22Z |
| last_indexed |
2025-11-26T23:30:22Z |
| _version_ |
1850781246815207424 |
| fulltext |
PHYSICS OF THE DYNAMIC ERGODIC DIVERTOR
M.W. Jakubowski†, K.H. Finken, S.S. Abdullaev, M. Kobayashi, M. Lehnen, R. Wolf
Institut für Plasmaphysik, Forschungszentrum Jülich GmbH,
Association EUROATOM–KFA, Trilateral Euregio Cluster, D-52425 Jülich Germany
† Also: Institute of Physics, University of Opole, 45 –052 Opole, Poland
The Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound
coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and
are fed individually outside the vessel. A perturbation field is created by the electrical currents in the perturbation coils
with Fourier components resonant to the magnetic surfaces. The stochastic boundary layer is generated in the outermost
region of the plasma, which due to long and short connection lengths can be divided into ergodic and laminar regions.
Field line tracing and mapping techniques were used to analyse properties of the TEXTOR-DED plasma boundary. The
DED will operate with several frequencies (DC or AC up to 10 kHz). In the “dynamic” operation the convective heat
flux is deposited to a large plasma-facing surface and forces are transferred to the plasma edge, what can introduce a
differential rotation of the plasma.
PACS: 52.55Fa
INTRODUCTION
The Dynamic Ergodic Divertor was developed for
spreading a convective heat flux over a larger plasma-
facing surface1. An ergodicity of the magnetic field is
created by the resonant components of the perturbation
magnetic field originating from the currents flowing in
the DED coils. For the low perturbation current
amplitude, magnetic islands centred on the resonant
magnetic surfaces are created. With increasing current,
the islands start to overlap destroying magnetic flux
surfaces. This creates a specific topology of the magnetic
field, where the field lines are not restricted to the flux
surfaces, but “diffuse” over the whole ergodic region. The
term “ergodic” means that the field line will pass
infinitesimally close to any pre-selected point of the
available space. The magnetic field in the edge is an open
system; field lines intersect the wall. A distance between
two intersections is called connection length. If the
connection length is large as compared to the
Kolmogorov length the corresponding field lines form a
proper ergodic zone; if it is smaller or of the order of the
Kolmogorov length, the field lines establish the laminar
zone. These terms will be explained in the following
sections
The magnetic perturbation field can be rotated with
different frequencies (up to 10 kHz). Applied the
magnetic field should:
a)Prevent the divertor plates from the overheating and
excessive erosion;
b)Decrease significantly an amount of impurities
entering the plasma core;
c) Create a plasma rotation (in the high frequency
operation), what may improve a plasma confinement
and delay disruptions.
In the following sections the DED arrangement is
described and a short introduction to the Hamiltonian
formalism for the calculations of the magnetic field
structure is provided. A discussion of this structure and of
the dynamic aspects is given.
THE EXPERIMENTAL SET-UP
The DED (as shown on a figure 1) consists of four
quadruples of coils. Each coil goes helically once around
the torus following field lines on a 3≈q surface. Every
coil is supplied individually from the outside of the vessel
by the 4-phase, either DC or AC (50 Hz, 1 – 10 kHz) with
the current amplitude up to 15 kA. The main perturbation
modes (m = 10,..,14 poloidal mode numbers, n = 4
toroidal mode number) are centred at the respective
magnetic flux surface with q = 12/4. The 12/4 mode was
chosen as a standard operational mode due to technical
constrains (such as current density, skin effect or heat
capacity) and physical requirements. One of the most
important physical requirement is not to disturb a q = 2
surface (this could lead to interactions with intrinsic MHD
modes and affects the plasma stability). The perturbation
field for the 12/4 mode scales like
r
coil
B
effm
r
r
δ ∝
ж ц
з ч
и ш
, where
meff ≈ 20. It decays very rapidly with distance from the
coils. The 4-phase current can be operated with several
frequencies : DC, 50 Hz and at seven frequencies in the
band 1 kHz — 10 kHz). These frequencies correspond to
the rotating perturbation field, where the phase velocity of
42 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 42-44
Fig. 1. Sketch of the DED coils arrangement.
the propagating field projected onto a poloidal plane
equals 12 m/s, 240 m/s and 2400 m/s respectively. The
flexibility of the power supply system allows to connect
coils in several ways creating m = 6, n = 2 and m =3, n =
1 modes. There is also a possibility to mix the 12/4 mode
with the 6/2 mode.
HAMILTONIAN FORMALISM FOR FIELD
LINE TRACING
The Hamilton formalism is used to investigate the
magnetic field of tokamaks and stellarators. For the
destroyed magnetic flux surfaces, as it is the case for the
TEXTOR-DED, the perturbation theory can be used. The
field line equations can be written in the toroidal
coordinates system as:
1 1, Z RB BdZ dR
R d B R d Bϕ ϕϕ ϕ
= =
By replacing the time by the toroidal direction ϕ, one can
introduce the Hamiltonian-Jacobi equations. Supposing
that
0 0 0 0/ lnZB B R R A B R Rϕ = = −Ю
and introducing new variables:
( )00 0, / , ln(1 )/ zR R R p xz Z R x − = += =
one obtains:
, z
z
dz H dp H
d p d zϕ ϕ
∂ ∂
= = −
∂ ∂
where 2 2
0 0/H RA B Rϕ= , R0 is the major radius of the
tokamak and B0 is the magnetic field at R0. One can
introduce the perturbation field originating from the DED
coils by dividing the Hamiltonian into two parts:
0 1H H Hε= +
where H0 is an unperturbed part and
1 cos( )mn
m
H b m n tε ε θ ϕ
Ґ
= − Ґ
= − + Ωе
is a perturbed part written in the form of a Fourier series
(ϕ, θ are toroidal and poloidal angles respectively).2 If the
spatial Fourier spectrum has more than one frequency,
island chains are created. The islands are created at the
( ) /mnq r m n= flux surfaces by the perturbation field
which has a spatial Fourier component bmn at rmn. A
standard parameter to describe the degree of the
ergodization is the Chirikov parameter
1, ,
,
1, ,
2
1, ,
( )
2
8 ( )
m n m n
m n
m n m n
m n m n
sh t
r
r r
r b b mqR
m L B r
σ +
+
+
∆ + ∆
=
−
=
+ ж цЧз ч
и ш
It is the ratio between the islands width and the distance
between island chains (∆m,n is the island width at the
radius rm.n, Lsh – shear characteristic length) A region of
the plasma is ergodic if 1σ і , what simply means that
islands from different island chains are overlapping. The
Kolmogorov length is related to the Chirikov parameter.
4 / 3
02KL qRπ σ −= Ч
It simply characterises the distance from where on
initially neighbouring field lines starts to separate
exponentially.
STRUCTURE OF THE MAGNETIC FIELD IN
THE PLASMA BOUNDARY
For the considerations of chaotic systems so called
Poincaré plots are used, which allow a two dimensional
presentation of the magnetic field structure. It is created
by marking intersections of field lines with a chosen
poloidal plane of the torus. One example for the
superposition of the TEXTOR equilibrium field and the
perturbation field going from the DED is shown in Fig. 2.
The figure shows an outermost region of the poloidal
plane, which is bent to a rectangular shape. The low field
side is at the left and right side of the graph, high field
side in the centre of the picture. The DED coils and target
plates are presented as a rectangle at the top of the figure.
Presented an example is the case of a strong ergodization
of the edge magnetic field. The island chains are indicated
by the corresponding mode numbers. One can see typical
features of the magnetic field structure. The inner zone is
the confinement zone with non-perturbed magnetic flux
surfaces and some well separated island chains. The flux
43
Fig. 2. Poincaré plot for the magnetic field lines in the
edge at the full perturbation current
Fig. 3. The Chirikov parameter versus a plasma radius for
β
pol
= 0.0 and 1 – r
res
= 0.43 m, 2 – r
res
= 0.46 m
surface and island chains are bent due to the Shafranov
shift. The requirement of the Chirikov parameter less than
one for the m/n = 8/4 flux surface is fulfilled (see Fig. 3)
The ergodic region
For the second case in the Fig. 3 the region 42 cm ≤ r ≤
46 cm forms the ergodic layer, which shows a high level
of stochasticity with some remaining island chains. The
Chirikov parameter reaches a value above 4 at the very
edge.
The flux surfaces are destroyed and field lines “diffuse”
over the whole ergodic region. Connection lengths are of
order of 10000 meters. It enhances the radial transport
coefficient significantly. It is found that ergodization is
not restricted to a regime close to “design parameters” in
the operational space ( pol res1, r( 3) 42 cmqβ = = = ), but
that it covers large operational space. Key parameters to
modify the degree of ergodization are variations of the
perturbation field amplitude, of the plasma current, of βpol,
and of the plasma position3. The perturbation spectrum
depends on the pitch angle of the field lines, which can be
adjusted by the value of beta poloidal. By modifying βpol
and Iplasma one can shift the ergodic layer either outward or
inward in the range 0.9 1.0r
a
Ј Ј keeping the same level
of ergodization. The ergodic region is connected with the
walls only via very thin structures called fingers.
The laminar zone
The magnetic field of TEXTOR-DED is an “open”
system. It means that field lines can intersect tokamak
walls, and ions following the field lines will be
neutralised there. The consideration of the connection
lengths (relative to the Kolmogorov length) elucidates
specific structure of the plasma very edge. It is found that
despite of the strong ergodization, rather large continuous
areas exist, with relatively short connection lengths: 4
flux tubes with LC = 1 poloidal turn (because of four fold
toroidal symmetry of the system), 4 with LC = 2 poloidal
turns continuous areas. There exist areas with even higher
value of LC, but their dimensions are small as compared to
the Larmor ion orbits and have therefore no meaning for
the particle orbits. Due to short connection lengths field
lines in the laminar zone are not visualised on the
Poincaré plot (white region in the Fig. 2). A laminar plot
was developed to investigate properties of the laminar
zone. The idea is similar as for the Poincaré plot, but
additionally connection lengths are marked with different
colours or grey tones. One example is shown in Fig. 4,
each shade of grey denotes a zone with different number
of poloidal turns (1 poloidal turn ≈ 30 m). The plasma
current is 470 kA and βpol = 1. Very short connection
lengths in the very edge (< 2 poloidal turns) suggest to
treat that region in a similar way as the SOL of the
poloidal divertor. The laminar zone comparing to the
ergodic region has relatively simple structure and
transport is mostly convective, along the flux tubes, what
is reflected in the power deposition pattern onto divertor
tiles4.
Dynamic aspects
The dynamic option of the DED has been introduced in
order to distribute the heat flow to the divertor target plate
to a large area. During the static DED operation, the heat
flux is guided towards the divertor target plate; the
divertor strike points follow helically the DED coils. By
the rotation of the perturbation field, the heat load is
distributed over the whole target area.
The high frequency aspect of the DED-field has been
analysed in cylindrical approximation5. It has been shown
that the „low frequency“ (relative to Ωi) electromagnetic
wave of the DED effectively propagates in the area between
coils and resonance layer as the compressional Alfvén wave
(fast wave)6. At the resonance layer of the plasma, different
approximations are described either by a resistive annulus
or by tearing modes. The interaction of the external rotating
field with the current driven in the shielding layer results in
the transfer of angular momentum between the DED-coils
and the plasma6. The maximum poloidal torque applied to
the plasma amounts to about 50 Nm; this maximum occurs
at a frequency which seems to depend mainly on the width
of the current layer. In detail it depends on the assumed
plasma temperature, on the applied frequency and on the
assumed island or ergodization width. The toroidal
projection of the applied force has about the same value as
the one imposed by tangential NBI. Due to the combined
action of the DED and NBI torques, a differential rotation
may be induced. It may be speculated whether the
differential rotation suppresses convective cells and thus
improves the confinement.
44
Fig. 4. The topology of the field line connection lengths of
the laminar zone on the HFS of the plasma.
0
1
2
3
4
5
1 5 5 1 6 0 1 6 5 1 7 0 1 7 5 1 8 0 1 8 5 1 9 0 1 9 5 2 0 0 2 0 5
4 2
4 3
4 4
4 5
4 6
4 7
4 8
θ [ d e g . ]
r
[c
m
]
1Dynamic Ergodic Divertor (special issue), Fusion Engineering and Design 37 (1999) 335-448 (15 contributions)
2 S.S.Abdullaev et al., Phys. Plasmas, 6 (1999)
3 K.H.Finken et al., Nucl. Fusion, 39 (1999), 637
4 Th.Eich et al., Nucl. Fusion, 40 (2000), 1759
5 Finken, K.H., Nucl. Fusion, 30 (1999) 707
6 Faulconer, D.W., Koch, R., Fusion Engineering and Design, 37 (1997) 399
Introduction
The experimental set-up
Hamiltonian formalism for field line tracing
Structure of the magnetic field in the plasma boundary
The ergodic region
The laminar zone
Dynamic aspects
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