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Magnetic field topology and heat flux patterns under the influence of the dynamic ergodic divertor of the TEXTOR tokamak [Elektronische Ressource] / vorgelegt von Marcin Jakubowski

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121 pages
MAGNETIC FIELD TOPOLOGY AND HEAT FLUX PATTERNS UNDER THE INFLUENCE OF THE DYNAMIC ERGODIC DIVERTOR OF THE TEXTOR TOKAMAK DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften an der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum vorgelegt von Marcin Jakubowski aus Opole, Polen Jülich 2004 Referent: Prof. Dr. R. Wolf Korreferent: Prof. Dr. H. Soltwisch Dekan: Prof. Dr. R.-J. Dettmar Tag der mündlichen Prüfung: 23. Juni 2004 Abstract This thesis concerns the structures of the magnetic field induced by the Dynamic Ergodic Divertor (DED), which was recently installed at the TEXTOR tokamak in. Sixteen perturbation coils ergodize the field lines in the plasma edge and destroy the resonant surfaces. This creates an open chaotic system in the plasma edge. The structures of the magnetic field in the ergodic and the laminar region are systematically investigated using a set of codes, called “Atlas”. In Atlas, the field lines are traced using a mapping technique, which is based on the Hamiltonian formalism. This method is a fast and accurate algorithm to study the stochastic magnetic field lines. Typically, the ergodic region is a mixture of stochastic domains and island chains. The field lines in the stochastic domain have very long connection length and each field line fills the ergodic volume.
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MAGNETIC FIELD TOPOLOGY AND HEAT FLUX
PATTERNS UNDER THE INFLUENCE OF THE DYNAMIC
ERGODIC DIVERTOR OF THE TEXTOR TOKAMAK





DISSERTATION
zur
Erlangung des Grades eines
Doktors der Naturwissenschaften
an der
Fakultät für Physik und Astronomie
der Ruhr-Universität Bochum







vorgelegt von

Marcin Jakubowski
aus Opole, Polen






Jülich 2004






























Referent: Prof. Dr. R. Wolf

Korreferent: Prof. Dr. H. Soltwisch

Dekan: Prof. Dr. R.-J. Dettmar

Tag der mündlichen Prüfung: 23. Juni 2004

Abstract

This thesis concerns the structures of the magnetic field induced by the Dynamic Ergodic
Divertor (DED), which was recently installed at the TEXTOR tokamak in. Sixteen perturbation
coils ergodize the field lines in the plasma edge and destroy the resonant surfaces. This creates an
open chaotic system in the plasma edge.
The structures of the magnetic field in the ergodic and the laminar region are
systematically investigated using a set of codes, called “Atlas”. In Atlas, the field lines are traced
using a mapping technique, which is based on the Hamiltonian formalism. This method is a fast
and accurate algorithm to study the stochastic magnetic field lines.
Typically, the ergodic region is a mixture of stochastic domains and island chains. The
field lines in the stochastic domain have very long connection length and each field line fills the
ergodic volume. After many turns around the torus they are deflected toward the divertor wall
and leave the ergodic zone via so-called fingers.
The field lines in the laminar zone are characterized by their very short connection lengths
(compared to the Kolmogorov length) between two intersections with the wall. The structure of
the flux tubes in the laminar zone is studied with Atlas. The flux tubes have the stagnation point
half way between the intersections. When hitting the wall, they form stripe-like strike zones in
front of the DED coils. At higher level of ergodization they split into pairs. In between these
pairs, a private flux zones are established. It is shown, that the topology of the laminar zone
resembles the structure of the scrape-off layer of the poloidal divertor. The detailed plasma
properties (e.g. local density, temperature, flux amplitude and direction) are, however,
complicated by the adjacent areas of flux tubes with different connection length.
The ergodic and laminar structures in the plasma boundary depend strongly on the global
plasma properties, in particular on the safety factor profile and the plasma pressure. These
quantities determine the location and the separation of the resonant surfaces. The ergodization is
systematically studied by varying the plasma current and poloidal beta. The width and structure
of the ergodic and laminar region is a nonlinear function of the global parameters. As general
tendency it has been found, that at the higher level of ergodization (i.e. at higher plasma current
and lower beta poloidal) the laminar zone is dominant in the perturbed volume, while at lower
level of ergodization the ergodic region dominates. Since the plasma pressure influences the pitch
of the magnetic flux tubes in front of the DED coils (at constant edge safety factor), the
resonance conditions of the flux surfaces vary as well. This leads to a systematic variation of the
ergodization level as function of plasma pressure.
A thermographic camera was set up and used to validate the predictions made with Atlas.
The system measures temperature patterns on the divertor target plates. The heat flux density
distribution is strongly non-homogenous, forming the expected stripe-like pattern. One observes
four helical strike zones, which are parallel to the divertor coils. The measured patterns are in
rather good agreement with the results from the modelling with the Atlas. The variation of the
structure of the heat flux density deposition pattern with the plasma current and poloidal beta is
measured. For increasing level of the ergodization the strike zone broadens and at some point
splits up as it was predicted with Atlas. The predicted splitting of the divertor strike zones was
clearly manifested. Imperfections of the alignment of the divertor target tiles were used to reveal
local flux directions; namely, each of the power flux stripe consists of two parts with different
direction of incoming heat and particle fluxes. Contents
Contents 1
1 Introduction 5
1.1 Nuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Magnetic confinement . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The TEXTOR tokamak . . . . . . . . . . . . . . . . . . . . . 10
1.3 Power and particle exhaust concepts . . . . . . . . . . . . . . . . . . 10
1.3.1 Limiter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Poloidal divertor . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Ergodic Divertor . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Outline of this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Magnetic field lines in a tokamak 17
2.1 Formulation of the field line equations . . . . . . . . . . . . . . . . . 17
2.2 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Integrable systems . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 A Poincar´e section . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Perturbed systems . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Mapping scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Formulation of the Hamilton function . . . . . . . . . . . . . . 23
2.3.2 Mapping method to integrate the field line equations . . . . . 25
3 Field lines in the ergodized edge 29
3.1 DED setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
12 CONTENTS
3.1.1 The DED coil design . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Divertor Target Plates . . . . . . . . . . . . . . . . . . . . . . 32
3.2 The basics of the ergodization . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The program “Atlas” for the TEXTOR-DED. . . . . . . . . . . . . . 37
3.3.1 Spectrum of the perturbation . . . . . . . . . . . . . . . . . . 38
3.3.2 Spectrum of the perturbation in the toroidal model . . . . . . 43
3.4 Description of the visualization methods . . . . . . . . . . . . . . . . 47
3.4.1 Poincar´e plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 A method to characterize the laminar zone . . . . . . . . . . . 51
3.5 Structure of the laminar zone . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Magnetic footprints . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.2 Private flux zone . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Topological properties of the flux tubes . . . . . . . . . . . . . . . . . 61
3.7 The variation of the ergodization . . . . . . . . . . . . . . . . . . . . 63
3.7.1 Variation of the poloidal beta . . . . . . . . . . . . . . . . . . 64
3.7.2 Variation of the plasma current . . . . . . . . . . . . . . . . . 66
3.8 Width of the ergodic and laminar zones . . . . . . . . . . . . . . . . . 67
4 Thermographic measurements 77
4.1 Operational range of the TEXTOR-DED experiments . . . . . . . . . 77
4.2 Thermographic system . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Introduction into thermography . . . . . . . . . . . . . . . . . 80
4.2.2 The thermographic setup . . . . . . . . . . . . . . . . . . . . . 81
4.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Temperature calibration . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 Non-uniformity calibration . . . . . . . . . . . . . . . . . . . . 85
4.4 Measured temperature distribution . . . . . . . . . . . . . . . . . . . 88
4.5 Heat flux analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Modelled structures and the measurements . . . . . . . . . . . . . . . 92
4.7 Distortions in the measured patterns . . . . . . . . . . . . . . . . . . 96
4.8 Heat fluxes in the plasma edge . . . . . . . . . . . . . . . . . . . . . . 97CONTENTS 3
4.9 Variation of the ergodization . . . . . . . . . . . . . . . . . . . . . . . 100
4.9.1 Variation of the plasma position . . . . . . . . . . . . . . . . . 100
4.9.2 Variation of the plasma current . . . . . . . . . . . . . . . . . 101
4.9.3 Variation of the poloidal beta . . . . . . . . . . . . . . . . . . 103
5 Summary 105
Acknowledgements 109
A Magnetic equilibrium field 111
B The THEODOR code 113
Bibliography 1154 CONTENTSChapter 1
Introduction
1.1 Nuclear fusion
Since many years the problem of an increasing energy need is well recognized [8].
The acceptable power source should satisfy - sometimes conflicting - economical,
ecologicalandpoliticalcriteria. Nuclearfusionisoneofthebestcandidateswithre-
spect to the international community demands. It is relatively clean as compared to
others, see e.g. [49]. The reaction of a helium nucleus synthesis from the deuterium
and tritium is assumed as the most promising for the thermonuclear reactor.
2 3 4 1D+ T→ He+ n+E (1.1)k1 1 3 0
TheE = 17.6 MeV in the equation above is the amount of a kinetic energy carriedk
by the α-particle (20 %) and the neutron (80 %). The high energy release is equiv-
alent to the mass defect between reaction partners and products. One kilogram of
8D-T fuel would release about 10 kWh of energy, which is sufficient for one day of
operation of an 1 GW power plant.
Although the fusion reaction reagents are not radioactive (except of the tritium)
the bombardment of the containment structure by energetic neutrons will induce
radioactive isotopes. However, one can select wall materials with a short decay time
such that the waste processing is no problem after a century decay time. Moreover,
the resources are either widely available like deuterium (0.15 gr of deuterium in 1 l
56 CHAPTER 1. INTRODUCTION
of water) or easily produced such as tritium from lithium.
Of course, at ”low” temperature the repulsive forces between two positively
charged particles make this fusion reaction extremely improbable, thus one needs to
8heat the deuterium and tritium to a temperature of: T & 10 K. The plasma can
provide conditions necessary for sustaining the D–T fusion process long enough to
get the positive power balance, that is, the fusion power produced by the plasma
exceeds the external heating power power injected into the plasma to maintain it at
thermonuclear temperatures.
1.2 Magnetic confinement
The fusion process can be achieved in a plasma environment at temperatures of
about one hundred millions K. Such temperatures can only be reached if the plasma
is very effectively insulated. The main stream of fusion research uses magnetic
confinement and here the tokamak line is the most developed one. The basis for
magnetic confinement of a plasma is the fact that charged particles spiral about the
magnetic field lines. The radius of spiral, so called gyro-radius, is inversely propor-
tionaltothestrengthofthemagneticfield,sothatinastrongfieldchargedparticles
move along magnetic field lines.
In tokamaks a magnetic field consists of three components: the toroidal field
B – produced by external, toroidal set of coils, the poloidal fieldB - resulting fromϕ θ
the plasma currentI (in tokamaksB B ) and for equilibrium reasons a verticalp ϕ θ
fieldB . These three components create a helical magnetic field configuration form-v
ing closed magnetic flux surfaces (see figure 1.1). For the TEXTOR tokamak the
shape of the flux surfaces is almost circular. In more confinement oriented devices
(ASDEX-Upgrade, JET, DIII-D, JT60) it has an elongated D-shape. Because the
tokamakplasmaisbentaroundthemajoraxis,thevalueofthepoloidalandtoroidal
field components is higher at the inner side of a torus (high field side – HFS) and
lower at the outer side of a torus (low field side – LFS). The centers of inner flux
surfaces are shifted more towards the LFS than the outer ones; this shift is called
Shafranov shift [48] (marked as Δ on the figure 1.2). The value of the Shafranov1.2. MAGNETIC CONFINEMENT 7
zz
y x
r
D
q
R R0
IP
Figure 1.1: The plasma in a tokamak is confined by the toroidal B , poloidal B andϕ θ
vertical B (not shown in the figure) magnetic fields. The resulting structure consists ofv
set of nested magnetic flux surfaces on which field lines have helical shape. In the upper
left corner of the figure a definition of the coordinates system is shown.
shifts depends on the plasma pressure, which can be expressed via β . Poloidalpol
beta is defined as: R R
p·dS/ dS
β = (1.2)pol 2B /2μ0a
where the integrals are surface integrals (p denotes the plasma pressure) over the
poloidal cross-section and
μ I0
B =a
l
where I is the plasma current and l is the length of the poloidal perimeter of the
plasma [61].
The pitch of a field line in the poloidal direction relative to the toroidal one is
named rotational transform and labelled ι. The safety factor q is defined as the
inverse of ι/2π; thus, q = Δϕ/2π. The safety factor is a number of toroidal turns,
which a magnetic field line makes during one toroidal turn. For a large aspect-ratio
tokamak with circular cross-section it can be approximated by: