MAGNETIC FIELD TOPOLOGY AND HEAT FLUX

Transcription

1 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 24

2 Referent: Prof. Dr. R. Wolf Korreferent: Prof. Dr. H. Soltwisch Dekan: Prof. Dr. R.-J. Dettmar Tag der mündlichen Prüfung: 23. Juni 24

3 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.

4 Contents Contents 1 1 Introduction Nuclear fusion Magnetic confinement The TEXTOR tokamak Power and particle exhaust concepts Limiter Poloidal divertor Ergodic Divertor Outline of this work Magnetic field lines in a tokamak Formulation of the field line equations Hamiltonian systems Integrable systems A Poincaré section Perturbed systems Mapping scheme Formulation of the Hamilton function Mapping method to integrate the field line equations Field lines in the ergodized edge DED setup

5 2 CONTENTS The DED coil design Divertor Target Plates The basics of the ergodization The program Atlas for the TEXTOR-DED Spectrum of the perturbation Spectrum of the perturbation in the toroidal model Description of the visualization methods Poincaré plot A method to characterize the laminar zone Structure of the laminar zone Magnetic footprints Private flux zone Topological properties of the flux tubes The variation of the ergodization Variation of the poloidal beta Variation of the plasma current Width of the ergodic and laminar zones Thermographic measurements Operational range of the TEXTOR-DED experiments Thermographic system Introduction into thermography The thermographic setup Calibration Temperature calibration Non-uniformity calibration Measured temperature distribution Heat flux analysis Modelled structures and the measurements Distortions in the measured patterns Heat fluxes in the plasma edge

6 CONTENTS Variation of the ergodization Variation of the plasma position Variation of the plasma current Variation of the poloidal beta Summary 15 Acknowledgements 19 A Magnetic equilibrium field 111 B The THEODOR code 113 Bibliography 115

7 4 CONTENTS

8 Chapter 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, ecological and political criteria. Nuclear fusion is one of the best candidates with respect 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 1D + 3 1T 4 3He + 1 n + E k (1.1) The E k = 17.6 MeV in the equation above is the amount of a kinetic energy carried by the α-particle (2 %) and the neutron (8 %). The high energy release is equivalent to the mass defect between reaction partners and products. One kilogram of D-T fuel would release about 1 8 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 (.15 gr of deuterium in 1 l 5

9 6 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 heat the deuterium and tritium to a temperature of: T 1 8 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 proportional to the strength of the magnetic field, so that in a strong field charged particles move along magnetic field lines. In tokamaks a magnetic field consists of three components: the toroidal field B ϕ produced by external, toroidal set of coils, the poloidal field B θ - resulting from the plasma current I p (in tokamaks B ϕ B θ ) and for equilibrium reasons a vertical field B v. These three components create a helical magnetic field configuration forming 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, JT6) it has an elongated D-shape. Because the tokamak plasma is bent around the major axis, the value of the poloidal and toroidal 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 on the figure 1.2). The value of the Shafranov

10 1.2. MAGNETIC CONFINEMENT 7 z z y x R r R I P Figure 1.1: The plasma in a tokamak is confined by the toroidal B ϕ, poloidal B θ and vertical B v (not shown in the figure) magnetic fields. The resulting structure consists of 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 pol. Poloidal beta is defined as: β pol = p ds/ ds B 2 a/2µ (1.2) where the integrals are surface integrals (p denotes the plasma pressure) over the poloidal cross-section and B a = µ I 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 = ϕ/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:

11 8 CHAPTER 1. INTRODUCTION Z Figure 1.2: The sketch illustrating the Shafranov shift. The centers of the flux surfaces (presented in the poloidal section) are shifted towards the LFS. denotes the Shafranov shift of the innermost (drawn) flux surface. q = rb ϕ R B θ. (1.3) Many modes (waves) in a plasma are aligned relative to the magnetic field lines. The knot of the modes are characteristic numbers; generally one denotes the toroidal mode number by n and the poloidal one by m. For this set of modes the q-value is often written as q = m. In the standard case for TEXTOR the q-profile is monotonic n with q(r = ).7 on the magnetic axis and q(r = a) 3.5 at the edge (see figure 1.3). If the surfaces is characterized by the rational value of q, a chosen magnetic field line makes a closed loop on its surface; on surfaces with the irrational safety factor field lines never close the loop. Because field lines with low rational numbers (e.g. q=1, 3/2, 2 etc) close quickly in themselves, they cover only a very small fraction of the flux surface. Therefore they are easily subject to perturbations,

12 1.2. MAGNETIC CONFINEMENT safety factor q major radius R [cm] Figure 1.3: The typical case of q-profile for TEXTOR, where the values are higher at the edge then in the center of the plasma (reproduced from [22]). hence low rational flux surfaces are easily destroyed. The analysis of the destruction by resonant perturbation fields will be a major part of this work. q = 3 q = Figure 1.4: Two examples of the surfaces with different safety factors. Left figure: q = 3 - field line after three toroidal points comes back to the starting position; right figure q = π is irrational - field line never comes back to the starting points

13 1 CHAPTER 1. INTRODUCTION The TEXTOR tokamak TEXTOR tokamak (Torus Experiment for Technology Oriented Research) in Jülich is operated in the frame of the Trilateral Euregio Cluster (TEC) by three plasma physics institutes from Belgium, Netherlands and Germany. The main scientific aim of the tokamak is to study plasma-wall interactions; however, research is also carried out in other fields like MHD or magnetic confinement. TEXTOR is a medium size fusion machine with a circular poloidal cross-section. The major radius is 1.75 m and the minor radius is.47 m. The plasma current induced by a transformer (equipped with an iron yoke) can be varied from 2 ka to 8 ka. The toroidal magnetic field (typically 1.6 T 3. T) is produced by 16 equidistant coils. For additional heating, TEXTOR is equipped with two neutral beam injectors, an ion and an electron cyclotron resonance heating system. All the systems together produce about 9 MW of heating power. The tokamak in Jülich (before the DED was installed) used to be a limiter machine with the pumped ALT-II limiter, which is still installed on the LFS 45 degrees below the equatorial midplane. Now it can be operated as a limiter or divertor machine. major radius minor plasma radius toroidal magnetic field plasma current long pulse capability additional heating R = 1.75 m a.47 m 1.6 < B ϕ < 3. T I p.8 MA t 1 s NBI, ICRH, ECRH Table 1.1: General features of the TEXTOR tokamak 1.3 Power and particle exhaust concepts In section 1.2 it was discussed that the core of the burning fusion plasma is insulated from the tokamak wall by the onion-like structure of magnetic flux surfaces.

14 1.3. POWER AND PARTICLE EXHAUST CONCEPTS 11 However, also the plasma edge has to fulfil specific tasks: the power and the particles coming from the plasma core has to be removed from the plasma bulk at an adequate rate [33]. The first wall of a fusion device has to withstand and exhaust the α-particle heating power. Also the helium-ash, which is a product of the fusion reaction (see eq. 1.1), must be removed from the plasma. Moreover, the global confinement is affected by the plasma-wall interaction, therefore appropriate controlling of the plasma edge is an important field of research [6] Limiter There exist several different concepts of the exhaust of power and particles. One of the simplest ideas, is to define the plasma outer edge by a limiter (see figure 1.5a). The hot core is kept away from the chamber walls by inserting an annulus of solid material. The magnetic flux surfaces, which touches the limiter is called last closed flux surface (LCFS). The LCFS divides tokamak plasma into the confinement region, where the magnetic flux surfaces are closed and a scrape-off layer (SOL). The main advantage of using the limiter is its simplicity and direct impact on the plasma. But it has also disadvantages - a contact with an extremely hot environment can produce too high power loads. The limiter surface is facing the plasma directly; therefore recycling particles or eroded impurities stream easily into the main plasma. In order to avoid high heat loads at the limiter it can be combined with low Z impurity seeding inside the LCFS (e.g. neon). By this a thin radiating plasma mantle is created [47], where part of the power coming from the core is removed by the radiation of neon. The ionized particles, which leave the core, follow the magnetic field lines and hit the limiter. Some limiters take advantage from this strong flow of particles: Since the power decay length is shorter than the decay length of particle float, one can build a limiter such that nearly all the power is removed at a front surface. The more remote parts of the limiter can be quipped with so called scoops. A sufficient fraction (about 1 2%) of the incoming particles are guided into these scoops, neutralized and pumped away via a pumping duct. These limiters are called

15 12 CHAPTER 1. INTRODUCTION a) b) scrape-off layer plasma core first wall plasma core last closed flux surface (LCFS) limiter pump divertor pump I D Figure 1.5: Two concepts of controlling the plasma edge: a) limiter and b) poloidal divertor. pump limiters. TEXTOR is equipped with such a pump limiter ALT-II (Advanced Limiter Test). ALT-II is equipped with a turbo pump. This allowed for an intense programme on the removal of the helium (ash) from a tokamak Poloidal divertor The concept of a poloidal divertor (see figure 1.5b) uses the external current I D, parallel to the plasma current. Due to the additional field the magnetic field lines are deflected into a separate chamber. The superposition of the intrinsic poloidal field and the external divertor field creates a singular field line structure, the so called X - point, where the total poloidal field component vanishes. The X-points divides the plasma into two regions: the main confinement volume and the scrape-off layer. The plasma sink action is generated by intersection of the scrape-off layer with the target

16 1.3. POWER AND PARTICLE EXHAUST CONCEPTS 13 plates near I D. Energy and the particles flows crossing the separatrix flow towards the target plates along the magnetic field lines. The divertor configuration rather naturally allows to compress neutralized helium ash and impurities in the divertor region, from where they can easily escape into the pumping duct. According to the present planing, a future fusion reactor will be equipped with a poloidal divertor. With respect to power transfer, limiter and divertor face similar problems. The expected heat load of a fusion reactor is at the limit for all known materials. The main advantage of using the poloidal divertor is the accessibility to the H-mode [18]. The name H-mode denotes a significant improvement of the plasma confinement. This regime of operation is most easily attained during auxiliary heating of diverted tokamak plasmas when the injected power is sufficiently high. In an H-mode plasma an edge transport barrier is formed, which leads to a steep edge pressure gradients and a high pressure at the top of the barrier Ergodic Divertor In a fusion reactor, the heat load density to the plasma facing components is considered as a challenging problem [19]. The high heat load results from the narrow width of the characteristic decay length of the power, which is for all tokamaks of order of 1 cm. The wetted area of the divertor can be enhanced somewhat by inclination of the target tiles and the expansion of the magnetic flux near the target plates. It would be favorable to deconfine the edge of the plasma i.e. to degrade the magnetic confinement in the outer volume of the tokamak. This idea led to the concept of an ergodic divertor, where an additional magnetic perturbation field is created by external coils. For a given set of coils and a given magnetic equilibrium, the radial magnetic field exhibits a poloidal and toroidal spectrum. This spectrum governs the radial domain where these modes are resonant. Typically, these resonances are located at the plasma boundary. For small amplitude of the perturbation field the island chains are created on the resonant surfaces. With increasing perturbation these islands grow and at some point overlap. The volume, where this happens, is called ergodic. In the ergodic region the magnetic

17 14 CHAPTER 1. INTRODUCTION Figure 1.6: The sketch of an ergodized tokamak edge. The proper radial scale is not preserved. field lines, which in equilibrium case are restricted to a given flux surface, experience small deflections in front of the DED coils; the field lines remain over long distances in the ergodic volume before they finally reach the wall. This creates a volume in the plasma edge where the field lines have stochastic properties. Each of the field lines fills the volume of the ergodic layer. It can significantly change the plasma properties in the edge. An averaged radial heat (Q) and particle (Γ) transport is enhanced in the ergodic region, in particular electron heat transport, because electrons predominantly follow the magnetic field lines [52]. The enhanced transport leads to a flattening of the electron temperature. Moreover in the outermost boundary, where the near field effects are significant (i.e. the field lines are strongly deflected towards the divertor target plates), magnetic field lines have short wall-to-wall connection lengths 1. Here the parallel and perpendicular transport is clearly separated. The particles following the field lines hit the wall, before the field 1 By the connection length we understand the distance along a field line between two intersections with the tokamak walls.

18 1.4. OUTLINE OF THIS WORK 15 lines can indicate chaotic behavior. This region is called laminar zone [37] and it is restricted to a small radial extent near the ergodizing coils, while the amplitudes of the magnetic perturbation show a strong radial decay with increasing distance from the coils (see chapter 2). The laminar zone is established by magnetic field lines which intersect wall elements after only a few toroidal turns (open ergodic system). Therefore the parallel transport (Q, Γ ) is dominant there. The characteristic diffusion length is of order of 1 cm for 3 m long flux tubes. In the well developed laminar zone diameter of the flux tubes exceeds 1 cm. Thus, the heat and particle transport in the plasma edge are dominated by the laminar zone which shows a well defined structure compared with that of the ergodic layer. The ergodicity of the magnetic field was investigated in many experiments, e.g JFT-2M [3], JIPP T-IIU [31], TEXT [28], W7-AS [56, 57] or TORE SUPRA [35, 12, 17]. Since June 23 the newly installed Dynamic Ergodic Divertor (DED) in the TEXTOR tokamak went into operation [32]. Sixteen perturbation coils ergodize the magnetic field in the edge of the TEXTOR tokamak. It was modelled (e.g. [3, 23, 34]), that the structure of the magnetic field in the edge of the TEX- TOR tokamak contains both: the ergodic and the laminar zone. Several numerical and analytical attempts have been performed to model the heat and particle transport and the plasma properties, in particular the Monte Carlo codes (E3D [13] and EM3D [26] codes), as well as a finite element method [9, 11]. The first experimental results confirm the modelling [25, 27]. 1.4 Outline of this work The aim of this work is to improve the understanding of the complicated threedimensional structure of the magnetic field in the edge of TEXTOR-DED experiment by means of modelling and to validate these models by measurements of the heat flux density deposition pattern on the divertor walls by means of the infrared thermography. In the ergodic divertor the transport in the edge is defined by the laminar zone. Therefore, it is necessary to have a comprehensive overview of the

19 16 CHAPTER 1. INTRODUCTION magnetic field structures generated by the DED in TEXTOR and to understand the general properties of the laminar zone topology. The numerical modelling, which was done up to now (e.g. [9, 26]), is time consuming and therefore can be performed only for the limited number of cases. An analytical approximation using a mapping technique, which gives fast and reliable method to study the stochastic magnetic field lines has been developed by S.S. Abdullaev [4]. In this work we use this method to perform a comprehensive study of the topology of the ergodic and laminar region. The modelling of the magnetic field structures is performed with the Atlas code, which was developed as part of this work. The Atlas, which is based on this new mapping technique, allows to visualize the three dimensional structure of the TEXTOR-DED magnetic field and to identify individual magnetic flux tubes in the laminar zone. According to the modelling, the heat deposition pattern on the divertor target plates is strongly connected to the structure of the laminar zone and in particular magnetic footprints. Measured heat fluxes on the DED target plates are compared with the field lines structure modelled with the Atlas. First, we will briefly introduce the principles of Hamiltonian dynamics (in particular KAM theory) and the mapping technique in chapter 2. In chapter 3 we will discuss the properties of the perturbation magnetic field using a simple cylindrical model and introduce the techniques used to visualize the ergodic and laminar regions. After that we perform a comprehensive study of the complex edge magnetic field topology in the TEXTOR-DED plasma edge. The temperature distribution measured on the divertor target plates using a fast infrared system is analyzed in chapter 4. From the measured temperature distribution pattern the power influxes are calculated using the 2D THEODOR code [44]. The measurements and evaluated heat fluxes are compared to the predictions of the Atlas.

20 Chapter 2 Relation of the magnetic field lines in a tokamak with the Hamiltonian dynamics 2.1 Formulation of the field line equations in the Hamiltonian formalism The magnetic field line is a curve, to which a vector of the magnetic field is always tangential. The general expression describing a field line is given by the equation: d x ds = B( x) B( x), (2.1) where x = (r(s), θ(s), ϕ(s)) represents the coordinates of the field line in toroidal geometry (see figure 1.1); s is a running coordinate along the field line and it simply labels the points along the field line. In a tokamak, in the idealized axisymmetric case, field lines form helical curves lying on nested flux surfaces. In reality, the topology of the field lines is affected by many factors, such as MHD instabilities. It is important to know the structure of the magnetic field, because the motion of the charged particles is strongly determined by the magnetic fields (via Lorentz force and drift motions). Coordinates systems exist [6, 4], in which the magnetic field line 17

21 18 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK equations are equivalent to the equations of dynamical systems in phase space. It is very favorable to present the field line equations in the Hamiltonian form, because the formalism of the Hamiltonian systems is well developed. In order to write the field line equations in Hamiltonian form a new coordinate system with Clebsch coordinates (ψ,ϑ,ϕ) is introduced : ϑ is an intrinsic poloidal angle, 2πψ is the toroidal magnetic flux enclosed by the flux surface ψ and ϕ is a toroidal angle. The geometrical coordinates are related to the intrinsic coordinates via the Fourier series: r(ϑ,ψ) = R [rm(ψ) c cos(mϑ) + rm(ψ) s sin(mϑ)] (2.2) θ(ϑ,ψ) = ϑ + m= ϑ s m(ψ) sin(mϑ), (2.3) m= where r c m, r s m and ϑ s m are the Fourier coefficients. In the case of an unperturbed magnetic field, the intrinsic angle is defined by the toroidal angle and the q-profile: ϑ = ϕ q + ϑ. A typical relation between the intrinsic angle and the geometrical one is shown in figure 2.1 for three different values of β pol. The change in the shape of the curves is caused mainly by the change of the field lines pitch angle. The spatial coordinates of the field lines are unique functions of the new coordinates x = x(ψ, ϑ, ϕ). Using intrinsic coordinates one can write a divergence-free magnetic field in a canonical form: B = ψ ϑ + φ χ(ψ,ϑ,ϕ)[6], where χ is the poloidal flux. The magnetic field line equations become: dϑ dϕ = χ ψ, dψ dϕ = χ ϑ (2.4) where we recognize the Hamiltonian formulation (a non-autonomous system with one and a half degree of freedom) with χ H,ϕ t. If the Hamiltonian depends only on ψ, the field lines are straight in the plane ϕ, ϑ, contrary to the appearance in the ϑ, ψ plane. In case of the axisymmetric equilibrium field in the tokamak, the Hamiltonian is a function of the normalized toroidal flux ψ via the safety factor q(ψ). χ = ψ dψ q(ψ) (2.5)

22 2.2. HAMILTONIAN SYSTEMS ϑ/2π θ/2π Figure 2.1: Dependence of the intrinsic angle ϑ on the geometrical poloidal angle θ for different values of β pol : curve 1 β pol =., curve 2 β pol = 1., curve 3 β pol = 2. The pitch angle of the field lines in the ϕ ϑ plane is given by χ/ ψ = q(ψ) Hamiltonian systems The Hamiltonian theory can be used to describe the magnetic field topology of the tokamak with the ergodic divertor. In order to understand the dynamics of such a system, a brief review of Hamiltonian theory is presented. Only major points will be outlined, which are necessary to understand the mapping scheme used for the field lines (see chapter 3). We consider an autonomous Hamiltonian system ( H/ t = ) with N degrees of freedom. The state of the system is completely determined by the set of canonical variables (p(t),q(t)): p = p 1,...p N is the momenta and q = q 1,...q N is the position of system in 2N-dimensional phase space. The trajectory of system is determined

23 2 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK by the Hamilton equations: q = H(p,q) p, ṗ = H(p,q). (2.6) q The most basic structural property of Hamilton equations is that they are symplectic, the Hamiltonian flow conserves certain invariants in phase space, particularly 2-form [5] n dp i dq i = const (2.7) i=1 The equation of the magnetic field lines 2.4 corresponds to the non-autonomous Hamiltonian system with 1 + 1/2 degrees of freedom. For such a system the invariant 2.7 corresponds to the magnetic flux conservation Integrable systems The Hamiltonian system 2.6 is integrable, if there exist N constants of motion. For these systems one can introduce a set of new canonical variables in which equations have a simple form. A new set of canonical variables (ψ,ϑ) is called action-angle variables. If the variables are separable, then the action variable ψ can be expressed via the Poincaré integrals [5] ψ i = 1 p i dq i = const, (2.8) 2π c i The action variable is the integral invariant of system. The variable ϑ can be treated as an angular variable: ϑ i = ω i t + ϑ i, ω i = H ψ i (2.9) Real phase space coordinates (q,p) are periodic function of angle variable ϑ i with the period 2π. The motion is confined to a N-dimensional subspace of the 2Ndimensional phase space, as the momentum coordinates are constants of motion. The Hamiltonian is dependent only on action variable, H = H(ψ), and the trajectories of such a system lie on N-dimensional tori. An example for system with two degrees of freedom is shown in a figure 2.2.

24 2.2. HAMILTONIAN SYSTEMS 21 surface of section trajectory of system 1 Figure 2.2: Motion of a phase space point for an integrable system with two degrees of freedom. x A Poincaré section It is convenient to study the dynamics of the trajectories by means of a surface of a section (also called a Poincaré section). The surface of the section is created by choosing a proper subspace of the phase space which is reduced by one dimension. In the example shown in figure 2.2 it is a plane ψ 1, θ 1 (with θ 2 mod 2π = const). The trajectory intersects the surface of section periodically with a period t = 2πα(ψ 1 ). The quantity α is a ratio of the system s eigenfrequencies α = ω 1 ω 2, ω i = ϑ i. In the case of the field line equations α = q(ψ) 1. Depending on the value of α, the solutions in phase space (ψ 1, θ 1 ) are different. In the case, for which α( ψ 1 ) = n/m, where n,m are integers, the system after m iterations comes back to the initial position, which yields periodic solution of period m and the surface ψ 1 is called a resonant surface. If α( ψ 1 ) n/m, the solution becomes quasi-periodic and ψ 1 is a non-resonant surface. If the relation dα i di i > is valid, the system satisfies the twist condition and one can construct a twist map (dropping the subscript i): ψ n+1 = ψ n, ϑ n+1 = [ϑ n + 2πα(ψ n+1 )] mod 2π (2.1)

25 22 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK For the field lines equations 2.1 take the form: ψ n+1 = ψ n, [ ϑ n+1 = ϑ n 2π ] q(ψ n+1 ) mod 2π (2.11) If q( ψ n ) = m/n then ϑ n+m = ϑ n 2πn, i.e. m points appear in the plot on a circle representing the flux surface ψ (see figure 1.4). The twist map is area preserving, what translates to a flux conservation in the case of magnetic field lines Perturbed systems The class of integrable Hamiltonian systems is a small subset of Hamiltonian systems. The question arises what happens to the integrable Hamiltonian in the presence of a perturbation e.g. H(ψ,ϑ,ϕ) = H (ψ) + ɛh 1 (ψ,ϑ,ϕ), (2.12) where H represents integrable Hamiltonian and ɛh 1 is a small perturbation. The KAM theory 1 shows that for small perturbations non-resonant tori survive; they are only slightly deformed [5]. What happens to the resonant surfaces (e.g. with α = n/m)? The twist map 2.1 under perturbation is replaced by: ψ n+1 = ψ n + ɛg(ψ n,ϑ n ), ϑ n+1 = [ϑ n + 2πα(ψ n ) + ɛf(ψ,ϑ)] mod 2π, (2.13) where g and f are generating functions determined by the perturbing part of the Hamiltonian [46]. If α is growing with increasing ψ, there exist KAM tori outside the resonant surface α = n/m, which is mapped counterclockwise (α > n/m) after m steps. On other hand, inside the resonant surface there exist a KAM surface, which is mapped clockwise (α < n/m). According to the Poincaré-Birkhoff theorem [46], between these two surfaces there must be a torus, which does not change its angular coordinates, i.e. ϑ n + m = ϑ n. This curve, which is not a KAM surface, is depicted as a solid line in figure 2.3a. After the mth step this curve is mapped radially into 1 KAM refers to Komogorov, Arnold and Moser who proved the theorem

26 2.3. MAPPING SCHEME 23 the dashed curve in figure 2.3a. As the twist map is area preserving, areas enclosed by both curves are equal and generally they intersect at an 2m number of points. Thus instead, of an infinite number of fixed points at the resonant surface only few of them remain even for a small perturbation. The non-resonant tori tend to move towards the elliptic points and form the islands; the points near the elliptic fixed points rotate around them [5]. Thus, if we examine the small region around one of the elliptic points of a periodic orbit, we will see that qualitatively it is similar to the picture of nested tori. The islands show similar behavior like the main tori, but on the finer scale. The hyperbolic points, contrary to the elliptic ones, are unstable. The trajectories of the system nearby the hyperbolic points become stochastic. In the case of magnetic islands in the tokamak the so called X-point of island is created at the hyperbolic point and the O-point around the elliptic point (see figure 2.3b). More detailed discussion about these points can be found in [38, 46, 5]. For growing perturbation amplitude of the islands grow until the neighboring island chains overlap. At that point all the KAM tori in between overlapping resonances are destroyed. The motion become even more complex then in the case of separated island chains. 2.3 Mapping scheme for field line equations in tokamaks Formulation of the Hamilton function As it was shown in section 2.1, the field line equations can be transformed into Hamilton equations (eq. 2.4) by introducing Clebsch coordinates (ψ,ϑ,ϕ). For any non-axisymmetric perturbation field in a tokamak, the Hamiltonian functions can be presented in the form: χ(ψ,ϑ,ϕ) = χ (ψ) + ɛχ 1 (ψ,ϑ,ϕ), (2.14)

27 24 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK mth iterate of solid curve Elliptic a) Hyperbolic = n/ m = 3/1 O-point b) X-point Figure 2.3: a) Illustration of the Poincaré-Birkhoff theorem. The intersection of the solid and dashed curves form hyperbolic and elliptic points. Adapted from [46]. b) The sketch of islands created on the destroyed tori.

28 2.3. MAPPING SCHEME 25 where ɛ is a small dimensionless parameter and χ 1 is a perturbed part of the Hamiltonian. The perturbation field can be written as a Fourier series: χ 1 (ψ,ϑ,ϕ) = mn h mn cos(mϑ nϕ). (2.15) The terms ɛh mn cos(mϑ nϕ) represent the resonant magnetic perturbations. The coefficients h mn are mainly determined by the toroidal component of the vector potential [29]: 2π 2π 1 h mn (ψ) = Re R( x(ψ, ϑ, ϕ)) (2πR ) 2 A ϕ ( x(ψ,ϑ,ϕ)) exp( mϑ + inϕ)dϑ dϕ (2.16) Usually, it is not easy to obtain an analytical expression for the Fourier components of the perturbation h mn. In the thesis the Fourier components of the perturbation are calculated numerically. The discussion of the method is given in section Mapping method to integrate the field line equations The behavior of the field lines in the tokamak in the presence of the perturbation can be presented in the form of Hamilton equations. Direct integration of the equations 2.4 using the integration schemes (e.g. the Runge-Kutta scheme) are not ideal in this application. They need large computational time and are not flux-preserving. The latter may lead to completely different long term behavior. To study the trajectories of the field lines a symplectic mapping method was developed [4, 1, 2, 29]. Correctly constructed map scheme is flux preserving and much faster compared to traditional integration methods. Usually maps are created using the Hamilton-Jacobi theory and the classical perturbation theory [46]. The mapping scheme for the magnetic field lines in the case of the tokamak magnetic field is constructed in the following way: first one needs to introduce poloidal sections at ϕ k = 2πk/s, k = 1, ±1, ±2,...,m, where s is a number of steps to proceed a full toroidal turn (e.g. s = 8). A relatively small number of steps, which is necessary to trace a field line for one toroidal turn, makes the mapping method much faster compared to the traditional integration techniques.

29 26 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK k, k, k k+1, k+1, k+1 Figure 2.4: The sketch presenting the integration of the magnetic field line with the mapping method. Here, it is presented a case with four-fold symmetry. For a given ϕ k, field line coordinates are (ψ k,ϑ k ), the map relates them to the coordinates at the next poloidal section by applying s times a set of equations: Ψ k = ψ k ɛ S k ϑ k, Θ k = ϑ k + ɛ S Ψ k, Ψ k+1 = Ψ k, Θ k+1 = Θ k + ϕ k+1 ϕ k q(ψ k ) ψ k+1 = Ψ k+1 + ɛ S k + 1 ϑ k+1, ϑ k+1 = Θ k+1 ɛ S k+1 Ψ k+1 (2.17) where S k = S(ϑ, Ψ,ϕ) ϕ=ϕk is a generating function and in the first order it is defined by S(ϑ, Ψ,ϕ) = (φ φ ) mn h mn (a(ξ mn ) sin φ mn + b(ξ mn )cosφ mn ) + O(µ), (2.18) where φ mn = mϑ nϕ, ξ mn = (m/q(ψ) n)(ϕ ϕ ),. The terms a and b are defined as follows: a(ξ) = 1 cos ξ, b(ξ) = sinξ ξ ξ. The free parameter ϕ lies in the interval ϕ k ϕ ϕ k+1. The term O(µ) represents corrections of order of µ and higher, µ = ɛ(φ φ ) δ 1, δ 1. Therefore, with sufficiently small step the map 2.17 can be applied for moderately large perturbation.

30 2.3. MAPPING SCHEME 27 If the ɛ =, then the map 2.17 becomes a standard twist map 2.11.

31 28 CHAPTER 2. MAGNETIC FIELD LINES IN A TOKAMAK

32 Chapter 3 Topological properties of the magnetic field in the ergodized edge of TEXTOR with the DED 3.1 Technical description of the Dynamic Ergodic Divertor In this section some technical information about the DED design is given. However, it is not an ambition of this work to give full description of the technology used to construct the DED coils in TEXTOR. More detailed information can be found in [14, 41]. From the technical point of view the role of the DED is to create alternatively: 1. static, 2. and rotating multi-polar helical magnetic perturbation field of different mode patterns. In particular (m/n) = 12/4, (m/n) = 6/2 and (m/n) = 3/1 base modes and specific combinations of these. The DED coils are realized by a set of 16 helical coils installed inside the vacuum vessel located on a high-field side of the torus (see figure 29

33 3 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE 3.1). One reason for installing the coils at the high-field side is the avoidance of interference with a large number of diagnostics and heating systems. The coils cover about 3 % of the inboard vessel surface in poloidal direction and full circumference in toroidal direction. Each coil makes one toroidal turn around the torus and is parallel to the field lines with q 3. The sketch of the divertor coils is shown in a figure 3.1. a) b) Figure 3.1: Sketch of the DED coils: a) in an ideally symmetric configuration; b) in a non-symmetric configuration. In the ideal configuration (fig. 3.1a) all sixteen coils including their input and output feeders are placed symmetrically around the torus. Due to technical reasons the coils are bundled into four quadruples (see figure 3.1b). This bundling creates an asymmetry. Because of the non-ideal feeding two additional coils are mounted to guarantee a stable vertical position. The compensation coils are positioned immediately above and below the DED coils. The system is designed to operate at the following frequencies: DC, few Hz, 5 Hz, 1 khz, 2 khz, 2.8 khz, 4 khz, 5 khz, 7 khz and 1 khz. The peak current in each of the coils is 15 ka for the operation below f=7 khz. The highest ergodization - in the static case - is reached with the distribution of the currents in the DED coils like the one presented in table 3.1. By choosing in the first (and last) coils opposite to their neighbors one avoids to use compensation coils. The penetration of the perturbation field in the 12/4 mode is very weak. ( ) meff 1 r The amplitude of the perturbation field scales like δb r r coil, where m eff 2[4]. The deviation of m eff from the mode number m is a toroidal effect. It results from the lower inclination of the magnetic field lines at the high field

34 3.1. DED SETUP 31 Coil: I c [ka] Table 3.1: Distribution of the current flowing in coils creating 12/4-mode for eight of the sixteen coils. side with respect to the average pitch angle. Because of the high multi-polarity the field decays within a few centimeters. According to equation above, much deeper penetration of the perturbation field can be obtained with the 3/1 or 6/2 modes. At I DED =15 ka a power produced by the current flowing in the coil reaches about 6 kw. In order to fulfill the various mechanical, electrical, high frequency and thermal requirements a novel type of a coil has been developed The DED coil design Each of the eighteen coils is about 1 m long. They consist of 294 insulated twisted copper wires, which are again combined into six bundles. The wires and bundles are twisted helically to minimize the influence of the skin effect and reduce eddy-current losses. This structure is supported from the inside by a central corrugated stainless steel tube acting as a pipe for water cooling. The wires are insulated by polyamide of thickness of.5 mm. The insulation is protected against the mechanical damages by glass-fiber bundles, glass-fiber taping and a glass-fiber tube. The whole structure is enclosed by a corrugated stainless steel tube. The maximum overall heat on the coils is of order of 3 MJ per pulse [14]. Cooling of the coils is realized by a combination of water and helium. The remaining space among the conductors and wires is filled with slowly streaming helium gas (p He =.2MPa;1 o C < T He < 25 o C), which ensures an efficient radial heat transfer from the conductor to the water-cooled central tube. The cooling water inside the central tube transports the energy along the coil to the outside cooling circuit. In the case of a vacuum leak of the outer steel tube, only helium enters the vacuum vessel of the tokamak. Operation without cooling with water is possible for a reduced repetition rate of the DED pulses.

35 32 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE Ø= 35 mm corrugated stainless steel tube for cooling water insulated twisted copper wires glass-fibre tube space for helium cooling Kapton layer glass-fibre bundle glass-fibre tape corrugated stainless steel tube Figure 3.2: Coil with twisted and insulated copper wires Divertor Target Plates In order to shield the coils from plasma contact, they are covered by the divertor target plates. The plates are mounted to the vessel by the poloidal support structure, which is fixed onto the vessel wall. Each tile is made of a graphite (IG-11U), 15 mm thick and have a toroidally shaped surface. For better protection of the coils from the overheating, a zirconium oxide layer of 5 mm thickness is attached to the rear side of the graphite plates. The scheme of the whole structure is shown in figure 3.3.

36 3.1. DED SETUP 33 Figure 3.3: Poloidal cross-section at the equatorial plane, showing the divertor tiles and the thermal insulation.

37 34 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE 3.2 The basics of the ergodization Let us assume a sheared magnetic field in the plasma edge with the shear value of s = r2 (dq/dr) q 2 R. The shear guaranties the existence of more than one resonant surface (i.e. with rational q-value). For a given set of and a magnetic equilibrium the radial magnetic perturbation field consists of a poloidal and toroidal spectrum (typically one toroidal and few poloidal modes), which is equivalent to the spatial Fourier transform of e.g. the radial magnetic field component. To assure proper performance of the ergodic divertor the applied perturbation field has to fulfill several requirements: the toroidal n an poloidal m spectrum of the perturbation field has to be in resonance with the flux surfaces at the plasma boundary (typically m/n 2); it must not perturb the q = 2 surface, because the system flips from stabilized MHD (2,1) mode to destabilized (2,1) mode [36, 2]. It can lead to disruption, therefore it should be avoided; the radial extent of the stochastic domain has to be wide enough in order to decouple the divertor region, which determines the plasma wall interactions, from the core (i.e. the width should be of order of ionization scale). In order to describe, the effect of the external perturbation field on the magnetic field topology let us consider growing perturbation field from zero: 1. If the spectrum of the perturbation is sufficiently broad and the amplitudes of the modes are high enough, initially few flux surfaces are destroyed by the resonant radial perturbation. According to the Poincaré Birkhoff theorem (see Sec ) magnetic field lines on the flux surfaces with q-value resonant to the perturbation spectrum (e.g. q = m i, where i = 1,...,k) create island n chains. The field lines belonging to the islands always stay in the volume defined by the island chain boundaries. The flux surfaces, which have safety factors non resonant to the perturbing magnetic field remain closed.

38 3.2. THE BASICS OF THE ERGODIZATION If the amplitude of the spectrum grows then the islands width grows. At some point the last closed flux surface between two neighboring island chains is destroyed and islands start to overlap. The volume, where this happens is called stochastic or ergodic. The natural measure of the ergodicity is the Chirikov parameter, defined as: σ Chir = m+1,n + m,n r m+1,n r m,n, (3.1) where is the half width of islands and r is a minor radius of the island chain. The width of the islands is a function of the resonant component in the perturbation spectrum: m,n = 4 ɛh mn (ψ) dq 1 /dψ 1/2, (3.2) where ψ represents the resonant flux surface and h mn is an amplitude of the Fourier component in the perturbation spectrum defined in equation (2.16). A transition to stochasticity occurs, when σ Chir 1, i.e. the islands from neighboring chains start to overlap. The flux surfaces between the overlapping islands are destroyed and the ergodic layer is created. According to the ergodic hypothesis 1 field lines will fill all the available volume. The charged particles follow the field lines, hence transport will be different in the stochastic domain. One should expect an enhancement of the effective radial heat and particle diffusion coefficients, which in a quasi-linear approximation are proportional to the product D FL v th, where D FL is the field line diffusion coefficient and v th the thermal velocity. In the highly ergodized case (with σ Chir (m,n) > 1) D FL can be defined as: D FL = resonant(m,n) πqr h m,n B ϕ where B ϕ is the toroidal magnetic field at the axis [42]. 2, (3.3) 1 If the dynamical system is ergodic than a trajectory of any point in a phase space after a sufficiently long time will come infinitesimally close to any other point of the phase space. [46]

39 36 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE At the outermost boundary, where the near field effects are significant, magnetic field lines have short wall-to-wall connection lengths L c. In chaos theory a characteristic length for the separation of the neighboring orbits is the Kolmogorov length [36]: ( πσchir ) L K = πqr 2 (3.4) In general it is proven, that stochasticity prevails if L C (r,θ,ϕ) L K (r). If the L C (r,θ,ϕ) L K (r) the connection to wall is a dominant feature. laminar region perturbation coils ergodic region unperturbed plasma core divertor target plates tokamak vessel reference sections Figure 3.4: The sketch of the different regions created by the superposition of the tokamak equilibrium and the DED perturbation field presented in the poloidal crosssection In the latter case the transport of energy and particles is not of diffusive character like in the ergodic zone; it is of convective/conductive character. In contrast to the proper ergodic zone, the connection lengths of the magnetic field lines have smooth and continuous properties, however, with some sharp boundaries. This region has been termed as a laminar volume [16]. The structure of the different magnetic regions created by the DED is presented in figure 3.4. The goal of the DED in the static case is to redistribute the heat fluxes to larger areas. The particle and heat distribution pattern is strongly dependent on

40 3.3. THE PROGRAM ATLAS FOR THE TEXTOR-DED 37 the topology of the flux tubes in the laminar zone [11, 26, 13]. To avoid the heat peak loads the rotation of the perturbation field with low frequencies (e.g. 2-5 Hz) is foreseen. The heat load is smeared out while the heat deposition pattern follows the rotation of the perturbation field. 3.3 The program Atlas for the TEXTOR-DED Many local plasma parameters (e.g. plasma density or temperature) are correlated to the topology of the magnetic field. Therefore, for the upcoming analysis of the experimental data, it is crucial to have a good knowledge about the topology of the magnetic field lines in the ergodized edge of the tokamak. The program Atlas was developed [45] to visualize versatile kind of structures induced by the DED in the outermost volume of the TEXTOR plasma. The Atlas consists of both: several codes, and a set of pre-calculated data. The codes are based on the mapping technique described in the section To use this technique, one needs to formulate the proper generating function (eq. 2.18), thus to calculate the Fourier coefficients of the perturbation spectrum h mn. As it was already mentioned in section 2.3.1, it is difficult to formulate the analytical expression for h mn. In the thesis the Fourier coefficients of the perturbation field as well as the values of the safety factor are obtained numerically. They are calculated by integrating a set of field lines for one poloidal turn using a Runge-Kutta fifth order integration scheme [51]. Field lines start at ψ = ψ + w ψ, ϑ =, ϕ =, ψ = ψn ψ, where 1 ψ = ψ(r =.3m), ψ n = ψ(r = a) and w =,...,1. The intermediate values are calculated using the cubic spline method. The calculations are done in two steps. In the first step the spectrum of the perturbation, q-profile and the coefficients for transformation between the toroidal (r, θ, ϕ) and intrinsic (ψ,ϑ,ϕ) coordinates are calculated and stored. These values are later used for integrating the field lines in other applications, like visualization of ergodic or laminar regions. Such a scheme of calculations (see also figure 3.5) significantly safes computational time compared to a case, when all coefficients would be calculated every time they are necessary.

41 38 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE The magnetic field is obtained be superposition of the equilibrium field (see appendix A) with the perturbation field [4]. δb r (r,θ,ϕ) = m B m (r,θ) cos(mθ nϕ), (3.5) where the Fourier components of the perturbation magnetic field are defined as follows: ( ) m 1 r B m (r,θ) = B g m () r DED B = µ I DED n θ DED r DED. R R + r cos θ ( 1 ) r cos θ, 2m(R + r cos θ) Here, B is the amplitude of the perturbation, r DED radius of the DED coils, θ DED a poloidal extension of the DED coils, I DED amplitude of the DED currents, R - major plasma radius. Term g (k) m sin(m m (k) m = ( 1) m+m(k) g (k) defines shape of the perturbation spectrum: )θ DED (m m (k) )π, m (k) = (2k + 1)m, where m = 12, n = 4. More detailed discussion of the perturbation spectrum is given below Spectrum of the perturbation In order to discuss the properties of the perturbation spectrum created by the currents flowing in the DED coils we will use the simplest cylindrical model to explain the principles. Such a formulation allows us to discuss the properties of the spectrum without loosing generality, i.e. the conclusions are also valid for the perturbation spectrum derived (more accurate) for the toroidal case by [2]. In Atlas the latter one is used. The perturbation, which is mainly determined by the axial component of the vector potential and is presented as a Fourier series A z = m A m (r)e imθ, (3.6)

42 3.3. THE PROGRAM ATLAS FOR THE TEXTOR-DED 39 Spectrum Calculation of the perturbation spectrum ( Amn, hmn), q-profile, conversion coefficients c s s ( r, r, ) m m m output files input files Poincaré plot Topology of the ergoidc zone Laminar plot Topology of the laminar zone and magnetic footprints Trajectories Field line tracing, topological properties of flux tubes Figure 3.5: The scheme presenting the organization of the Atlas codes. The Fourier components of the perturbation spectrum (eq. 2.12), transformation coefficients (eq. 2.2) and values of safety factor are calculated and stored. These files are used in several applications, e.g to obtain a Poincaré plot. where θ is the poloidal angle. The amplitudes of the Fourier components are: A m = π π A m(r) sin mθ sin m θ f(θ)dθ, (3.7) where m = 12 is the central mode number and A m(r) is the radially dependent part of the poloidal mode. The function f gives the poloidal localization of the perturbation field and it is defined as follows: 1 if θ c θ θ c f = (3.8) if θ < θ c and θ > θ c

43 4 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE where θ c θ θ c is the poloidal extension of the perturbation currents on the surface of the cylinder. The solution of the integral 3.7 is A m = A m(r) sin(m m )θ c (m m )θ c. (3.9) The poloidal spectrum of the perturbation A m is localized near the central mode m = 12 and it consists of several modes: m 1,...,14. In linear approximation the interaction of the perturbation field with the resonance surfaces can be treated as the interaction of single modes with the corresponding flux surfaces; thus we can analyze the Fourier components of A z separately. To find the power of the radial dependence one needs to solve the equation: ( A z ) =, r r coil (3.1) for the amplitude A m of the single poloidal mode. The radial and azimuthal components of the vector A z are: A m r = ima m e imθ (3.11) A m θ = A m r (3.12) After including these components in equation 3.1 one obtains the linear differential equation for the radial part of the Fourier coefficients: d 2 A m dr r da m dr m2 r 2 A m =. (3.13) The second order differential equation has the solution: A z = A r m +B r m. In order to avoid singularity, A = for r > r coil and B = for r < r coil. The constants A and B are derived from the boundary conditions such that δb r is continuous at r = r coil and δb ϕ fulfils the jumping condition resulting from B = j. The radial decay of the poloidal modes is very rapid and what is important to notice the modes with higher m-number decay faster with the distance from the DED coils. The perturbation field is localized within the range of few centimeters on the highfield side of the tokamak. According to the equation 3.2 the width of the islands depends on the amplitude

44 3.3. THE PROGRAM ATLAS FOR THE TEXTOR-DED 41 1 m = 12 m = r/r c a) plasma current [ka] 1 m = 1 m = r/r c b) toroidal magnetic field [T] Figure 3.6: The relative position of two resonant surfaces (m indicates poloidal mode number, n in both cases equals 4)as a function of: a) the plasma current with fixed B ϕ = 2.25 T and b) the toroidal magnetic field with fixed I p = 44 ka

45 42 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE of the corresponding Fourier component of the perturbation field. Because of the very strong decay of the poloidal spectrum with the distance from the perturbation coils, the width of the islands centered at a given resonant flux surface strongly depends on the distance between the flux surface and the perturbation coils. This distance is given by the q-profile and the differential Shafranov shift. As it is discussed in chapter 1 the latter quantity shifts the centers of the flux surfaces towards the low field side, thus increasing the distance of the resonant surfaces from the origin of the perturbation field. The q-profile (see eq. 1.3) depends on the toroidal component of the magnetic field and the plasma current. For fixed toroidal magnetic field (B ϕ ) the radius of the resonance surface increases with the growing plasma current (I p ); this dependence is presented in figure 3.6a. The relative positions of the m = 12, 1 surfaces changes almost linear with the plasma current. Figure 3.6b shows the dependence of the radial position of the rational surface as the function of the toroidal magnetic field (B ϕ ) for fixed I p. As it can be expected from equation 1.3 the radius of the flux surface decreases with increasing B ϕ. Therefore, adjusting the q-profile, one can locate the resonant surface closer to the DED coils and thus increase the island size. As we can see in figure 3.6 for the standard toroidal magnetic field in TEXTOR B ϕ (r = ) = 2.2 T the q = 12/4 surface is close enough to the DED coils to be ergodized for relatively high plasma currents I p > 5 ka; however typically a lower plasma current in the TEXTOR tokamak is used. Hence, for the TEXTOR-DED operation, the B ϕ on the tokamak axis is lowered to 1.9 T in order to achieve bigger DED effect. The ergodic region is created when the island from neighboring island chains overlap. The width of each island is proportional to the different mode of the perturbation. In the typical case of the q-profile (see figure 1.3) the resonances with lower m number lie closer to the plasma center, thus the width of the island with lower m is smaller than the width of the one characterized by m+1. The distance between the overlapping island chains is defined by the magnetic shear. With the smaller shear the distance of the resonances is larger. If one fixes an outer resonant surface by an appropriate plasma shift to a given radius, then the separation of the other, inner resonances move away from the DED coils. Because of the strong radial

46 3.3. THE PROGRAM ATLAS FOR THE TEXTOR-DED 43 decay of the DED field these resonances react by smaller islands and therefore the ergodization becomes weaker. In toroidal geometry, the effect explains the weaker ergodization for plasmas with an increased value of the plasma pressure, indicated by β pol Spectrum of the perturbation in the toroidal model The first order toroidal corrections as a result of more accurate derivation of the vector potential of the perturbation field gives the following equation [2]: A ϕ (r,θ,ϕ) = m m 1 ra mn (r,θ) cos(mθ nϕ). (3.14) For large aspect ratio tokamaks, i.e. with R /a 1, the Fourier coefficients can be presented in the asymptotical form [24]: A mn (r,θ) B c r c g m 1 + r cos θ/r ( r r c ) m, (3.15) where B c = 2µ I c n/θ c r c =.23 T is a characteristic strength of the magnetic perturbation determined by the divertor current I c = 15 ka, the minor radius of the coils r c = cm and the angular poloidal extension of the coils θ c 2π/5. The ( 1 ) 1 factor + r cosθ/r corresponds to the first order toroidal correction. In the toroidal approximation the poloidal spectrum is localized near the central mode m = 2πn/θ c 2 instead of 12. The reason for the increased exponent m relative to the cylinder result from the lower pitch angle of the equilibrium field at the high field side of the torus compared to the cylindrical case. If the perturbation coils would have been placed at the low field side, then the effective mode number would be smaller than the average value of 12. The total perturbation field shows a strong radial decay A ϕ r m 1. The plasma pressure, described by β pol, influences the position of the resonance surface via the Shafranov shift, which also changes the pitch angle of the field lines. The influence of the perturbation field on the field lines strongly depends on the angle between the field lines and the coils. As the pitch of the coils is fixed, the angle of the field lines is modified by changing β pol. The shift of the main resonance

47 44 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE to lower m-numbers results again from the lowered pitch angle at the HFS for increased β pol -values. The DED coils were laid out for a low value of β pol and for the high resonance pol =. pol =.4 pol = 1. H m m-mode number Figure 3.7: The absolute values of the perturbation spectrum for three different values of poloidal beta. with the field lines at q = 3. From figure 3.7 one sees, that the calculations confirm the initial design. The real spectrum of the perturbation field is well approximated by the sin(m m)θ c /(m m )θ c - dependence obtained from the cylindrical model. At β pol =., the 12/4 mode is indeed close to the maximum of the curve. At higher values of β pol, the maximum shifts to lower m-numbers and correspondingly to lower values of q. Therefore, the resonances are deeper inside the plasma. Because of the strong radial decay of the perturbation field, this would result in a reduction of the island widths. However, one can counter-react by shifting the plasma such that the optimum resonance remains at the same location. A spectrum like given in figure 3.7 has to be calculated for all relevant resonant surfaces. From the value of H m at that surface, the width of the individual islands is derived. Figure 3.8 gives the island widths for conditions used in 3.7. For higher beta poloidal the islands characterized by higher m-numbers are destroyed by overlapping with neighbors and do not exist anymore. The position of the island

48 3.3. THE PROGRAM ATLAS FOR THE TEXTOR-DED β pol =. β pol =.4 β pol = normalized island width normalized flux Figure 3.8: The width of the islands calculated for the same conditions as for figure 3.7. Numbers corresponds to the poloidal mode numbers. chains is changing with β pol due to the Shafranov shift. As mentioned before, one has many degrees of freedom to influence the island widths, e.g by plasma position or by I p. The superposition of the equilibrium and perturbation fields creates a threedimensional topology of the magnetic field in the plasma edge. The field lines are deflected from their regular trajectories in the vicinity of the DED coils. In the figure 3.9 two field lines are traced for one poloidal turn. The calculations are done for the plasma current, I plasma = 45kA and the poloidal beta β pol = 1.. The abscissa represents the poloidal angle and the ordinate the minor radius. The lines visualize trajectories projected on the poloidal section, at toroidal angle ϕ =. The poloidal extension of the DED coils is marked as a yellow rectangle. The black dots represent the unperturbed trajectory of the field line (I DED = ka) and the red and blue, the field lines under the influence of the perturbation field (I DED = 15 ka). The change in the radial coordinates for the unperturbed field line is due to choice of the major radius of the plasma, which is shifted by one centimeter to the high field side (HFS) from the geometrical center of the torus. The same shift is

49 46 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE Figure 3.9: Two field lines traced for one poloidal turn with Atlas codes for two cases: with and without the perturbation field. They are starting at the same radial and poloidal positions (r = 43.2 cm, θ = : (black dots) no perturbation; (blue dots) I c = 15 ka, ϕ = ; (red dots) I c = 15 ka, ϕ = 5. superimposed with the deflections coming from the DED field in the trajectories of the perturbed field lines. One sees that the trajectories of the perturbed field lines are deflected from their regular trajectories, only if their are in the vicinity of the DED coils, i.e. if the poloidal angle along the trajectory is in the range of the poloidal extension of the DED coils. If the field lines are outside of the poloidal extension of the DED coils, the field lines are unperturbed. In the figure 3.9 it is also shown, that the actual path of the field lines depends critically on the initial toroidal angle. Field lines with different starting points will undergo different orbits and reach different radial location. From this it follows that the structure of the ergodic and laminar zone becomes three dimensional. The deflection of the field line depicted as the red dots is completely different from the one depicted as the blue dots. The different starting toroidal angles in figure 3.9 are and 5. By

50 3.4. DESCRIPTION OF THE VISUALIZATION METHODS 47 this change, the field lines follow a path near the DED coils where they experience mainly the poloidal component of the perturbation field (therefore the radial shift is negligible) or the radial component (this generates the big deflection of the red curve). 3.4 Description of the visualization methods Poincaré plot The technique to visualize the mapping is the Poincaré plot. The principle of the Poincaré plots is explained in the section 2.2. For our application each point on the graph marks the intersection of a field line with the chosen poloidal section. The sketch illustrating these method is presented in figure 2.4. The tokamak torus is presented there with four poloidal sections, the distance between them being 9. Because of the toroidal symmetry of the 12/4 mode of the perturbation field, n = 4, one can use all those sections to mark the intersections of the field lines. This procedure reduces the computational time by factor of 4. The field line is traced either to maximum number of steps (e.g. 1 steps) or until it intersects the divertor target plates at r div = 47.7 cm. This latter method simulates the neutralization of the particles hitting the wall. An example of the Poincaré plot for the TEXTOR boundary with the DED in 12/4 mode is shown in figure 3.1. The parameters for the calculations are presented in table 3.2. All other calculations discussed in this subsection are performed with the same set of parameters. In figure 3.1a one sees the Poincaré plot in polar representation. The angle variable is the poloidal angle of the plasma; the radius corresponds to the minor radius. The ergodic and laminar regions cover only small fraction of the plasma minor radius (typically 5-7 cm), therefore the unfolded representation is better for the analysis of the topology of the magnetic field in the plasma edge. The transition is made by cutting the polar graph at θ = and subsequently unfolding. The result of the transformation is shown in figure 3.1b. The abscissa represents the geometrical poloidal angle θ from to 36 and the ordinate the minor radius of the plasma,

51 48 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE a) DED coils [m/n] minor radius - r [cm] edge fingers structure 12/4 Y q=3 1/4 Y q=2.5 19/8 9/4 17/8 25/12 core 8/4 b) Y q=2 7/4 poloidal angle - q [deg] Figure 3.1: A Poincare plot for the magnetic field lines in the plasma edge calculated with Atlas: a) in polar representation; b) in unfolded representation.

52 3.4. DESCRIPTION OF THE VISUALIZATION METHODS 49 plasma major radius of LCSF R = 1.74 m plasma minor radius a = 46.7 cm plasma current I plasma = 45 ka toroidal field at R B ϕ = 1.88 T poloidal beta β pol =. DED current distribution 12/4 DED current per coil I DED = 15 ka Table 3.2: Input parameters for Atlas used for creation of the Poincaré plot presented in figure 3.1. in the edge region expressed in centimeters. The poloidal extension of the DED coils is marked as a rectangular area in the top of the figure, which spans the angular range from θ min = (18 36), to θ max = ( ). The minor radius of the coils is r coils = cm. In this figure all characteristic features of the ergodized edge of the TEXTOR-DED can be seen. The visible island chains are marked on the right-hand side ordinate with the corresponding safety factor value, e.g. the m/n = 1/4 mode contains 1 islands. As discussed earlier, each island chain is generated by different modes. The number of the modes is related to the q-values of the resonant surfaces. Between the main island chains (q = 8/4) and (q = 9/4), the (q = 25/12) at r = 39.8 cm and (q = 17/8) at r = 4 cm islands exist. They are generated by higher order perturbation fields created by the modes with symmetry n = 8 and n = 12 respectively; however, their influence on global or local plasma parameters is insignificantly small. The island chains created at the q = 12/4 and q = 11/4 surface are completely destroyed, because they are very close to the DED coils. In the presented structure figure one can distinguish three different regions: 1. Good confinement zone r 4 cm: According to the KAM theorem (see section 2.2.3) the non-resonant surface, remains undestroyed for sufficiently small perturbation, and only slightly perturbed. The surfaces remain closed as long as the island chains surrounding

53 5 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE them do not overlap. The trajectories of the field lines are distorted in the vicinity of the DED coils, i.e. on the HFS, as it was presented in section The field lines remain always on the same flux surface, and after infinite number of poloidal turns fill the flux surface. Between the non-resonant surfaces island chains exist, created on the q = 2 and q = 7/4 flux surfaces. It is assumed that the island chains for q 2 can be tolerated as long as they do not overlap. If they overlap, they might cause disruptions as was found in Tore Supra. The boundary of the good confinement zone is between the island chains at q = 25/12 and q = 17/8 at a minor radius of r 4 cm. 2. Ergodic region 4 cm r 44 cm: The typical picture of the ergodic zone in the TEXTOR-DED experiment is the volume with mixture of the areas with completely destroyed flux surfaces and some remnants of islands. The trajectories of the field lines in the ergodic region are irregular, i.e. it is practically impossible to predict the trajectory over long distances (long as compared to the Kolmogorov length). Initially neighboring field lines will deviate from each other substantially and in an unpredictable way. This is in contrast to field lines in the good confinement zone and in islands. There, the field line trajectories remain inside the island chain and are separated from the stochastic areas. They still keep a regular character. Because a limiting wall (divertor target plates) is placed inside the ergodic zone, the character of the field lines is changed to an open chaotic system. Even though the field lines may remain for very long path inside the ergodic zone, they will finally hit the wall. It was found that the field lines leave the ergodic zone in a well ordered way, namely along the so called fingers. They are rather thin ( θ < 5 ) and surround the laminar zone. 3. Laminar zone r 43.5 cm: Close to the divertor coils, the effects of the near field from the divertor coils is so significant, that the field lines are very strongly deflected towards the wall; thus these field lines have short connection lengths. The laminar zone

54 3.4. DESCRIPTION OF THE VISUALIZATION METHODS 51 is visible in the Poincaré plot as the white regions between and outside the fingers. The Poincaré points of the previous (q = 11/4) surface and outwards have practically vanished in the laminar zone. In order to visualize a laminar zone the new imaging technique was developed, called a laminar plot [34]. The principles of the laminar plot are explained in the following section A method to characterize the laminar zone A perturbing coil system, apart from the ergodic structures, forms also a helical near field system, where the field lines have relatively short connection lengths. To visualize the field line topology, we also use a mapping technique. We trace many starting points in both directions until they intersect the wall. The total length between the intersections with the wall are decoded in color such that the laminar plot gives the distribution of the connection lengths. A connection length corresponding to a single poloidal turn, is equivalent to that of the scape off layer of a conventional poloidal divertor. For the laminar plot only those connection lengths are of interest which are smaller than the Kolmogorov length, which corresponds to about 3 poloidal turns. For the mapping we have still the freedom to choose an intersection plane of the Poincaré points. It looks natural to select a plane in which the laminar plot looks highly symmetric. The underlying physics of the laminar plot is the wish to give information of the expected plasma flow in the region of open field lines. The point of highest symmetry of a flow in the SOL is obviously its stagnation point, i.e. the point with equal distance to the intersections with the wall. Provided that the sources for the SOL flows are uniformity distributed. For a poloidal divertor, this stagnation point is poloidally just apposite of the X-point. If we accept this picture for the DED, we would choose as a reference plane a radial cut along the outer equatorial midplane. Indeed, as expected we find there the stagnation points for those magnetic field lines which go once poloidally around the torus. We find even more stagnation points namely for all field lines with a connection length corresponding to an odd number of poloidal turns.

55 52 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE For the field lines with an even number of poloidal turns, the stagnation points are at the high field side of the torus. Therefore it is meaningful to use two laminar plots, one cut at the LFS and the other one at the HFS. tokamak vessel planes for laminar plots at = 45 = 9 = 45 plane for footprints plot planes for laminar plots at = equatorial plane Figure 3.11: Sketch presenting the planes used for visualization of the structure of the magnetic field lines in the laminar zone and their footprints on the divertor wall. The set-up of the DED is highly symmetric. Toroidally, the DED for the 12/4 mode is four-fold symmetric and in addition it is helically symmetric. We expect that this symmetry is also reflected in the shape of the areas forming a stagnation point. This is indeed true as one can see in figure Because of the four-fold toroidal symmetry it is sufficient to use a toroidal angle of 9 only. The experience shows that one can select two toroidal angles separated by 45 as origin of the laminar plot in which the symmetry is highest, i.e. in both the stagnation points for the flux tubes appear. Instead of using as reference plane the equatorial cut we are also using poloidal cuts. The toroidal angle of these cuts is given by the prescription specified below where the stagnation points are symmetric. In total, one can define by this procedure four poloidal cuts of highest symmetry, two at the LFS, two at the HFS (each cut separated toroidally by 45 ). These planes are presented in figure The laminar

56 3.4. DESCRIPTION OF THE VISUALIZATION METHODS 53 zone has a symmetry of the perturbation current. The information of the equatorial cut and of the poloidal cuts are equivalent, if the poloidal cut is limited to one mode which is 3 in intrinsic coordinates for the 12/4 mode (i.e. 1/12 of the full poloidal circumference). If one uses however the full poloidal cut one can see how e.g. the flux bundle of the stagnation area develops until it finally hits the divertor target plates. II I Figure 3.12: Sketch showing the stagnation planes for field lines. The field lines are plotted as projections onto the poloidal section. The blue line has a connection length of one poloidal turn, the length of the field line expressed in meters would be L c 1 2πqr; it has a stagnation point at the area marked with I. The green line has a connection length equal to two poloidal turns (L c 2 2πqr); its stagnation point is at the area marked with II. Dark blue lines represent the field lines with very short connection length L c.5 2πqr. The DED coils and tiles, as well as the tokamak vessel, are also shown. The flux tubes intersecting the divertor wall form specific structures of the magnetic footprints. These structures define the heat and particle deposition patterns [11]. The reference section for the footprints is not shown in figure 3.12, as it is in the (θ ϕ)-plane. The four-fold symmetry of the magnetic structures in the TEXTOR-

57 54 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE DED boundary allows to limit the size of the reference area to (θ min = 18 36, θ max = , ϕ min =, ϕ max = 9 ). The θ limits are defined by the poloidal extension of the coils; toroidally they cover one full period of the perturbation in the toroidal direction. 3.5 Structure of the laminar zone Based on the imaging technique described in section the laminar and magnetic footprints plots are created. The laminar plots for the reference regions defined in section are presented in figure The structures are calculated for the parameters taken from the table 3.2. For these parameters the level of ergodization is quite high, so the laminar zone is well developed. The dark red areas represent the ergodic region. Here, it is assumed that if a field line has the connection length equal to or greater then five poloidal turns, then it belongs already to the ergodic zone. To confirm this choice the Poincaré plot (Fig. 3.13a) is plotted for the same area. One can see that the ergodic region from the Poincaré plot corresponds to the dark red region in the laminar plot (Fig. 3.13b). The fingers from the Poincaré plot appear as a very thin structures surrounded by the flux tubes belonging to the laminar zone (green and blue areas). Their size depends on the level of the ergodization (see section 3.7.2). In principle, all the field lines intersect the divertor wall, therefore also the fingers are attached to the DED target plates. The reason, why the finger-structure from the right-hand side does not exist in the Poincaré plot, is that the field lines, which are used to obtain the Poincaré plot are traced only in one direction, whereas in case of the laminar zone the field lines are traced co- and counter-clockwise. Additionally to the laminar plots, the histograms are calculated to count the distribution of the different flux tubes in the laminar zone (see Fig. 3.14). The histograms are calculated for the same conditions as in figure 3.13 (see table 3.2). The size of the reference section is defined to cover one mode (i.e. one poloidal

58 3.5. STRUCTURE OF THE LAMINAR ZONE private flux zone minor radius [cm] minor radius [cm] a) poloidal angle θ [deg] b) poloidal angle θ [deg.] minor radius [cm] minor radius [cm] c) poloidal angle θ [deg.] d) poloidal angle θ [deg.] Figure 3.13: The laminar plot visualizes the topology of the field lines in the laminar zone. The connection lengths of the field lines are presented as a contour plot. The color dots mark interconnected areas belonging to the same flux tube, e.g. red dot indicates areas belonging to the single turn flux tube. The strike points of these flux tubes are presented in figure 3.18 in section 3.6. a) Poincaré plot plotted for the laminar zone. b) Laminar plot for the reference section in inner equatorial midplane. c) Laminar plot for the reference section in the outer equatorial midplane. d) Laminar plot for the reference section in the outer equatorial midplane shifted by 45 degrees in toroidal direction.

59 56 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE Y i /Σ(Y i ) 1% Y i /Σ(Y i ) 1% a) connection length [poloidal turn] b) connection length [poloidal turn] Figure 3.14: Histograms presenting contribution of the field lines with different connection lengths: a) calculated for the reference section at the high-field side; b) calculated for the reference section at the low-field side. period of the perturbation field in the intrinsic coordinates) in the laminar zone, i.e. r 43 cm, 16 θ 18 on the high-field side and r 43 cm, 28 θ 28 on the low-field side. The dominating field lines in the laminar zone are those with the connection length equal to one poloidal turn L c = 1; the contribution in the overall number of the field lines is about 5%. The field lines with L c <.5 also constitute a large fraction of the total number of the field lines in the laminar zone, i.e. 16% on the HFS and 28% on the LFS. The contribution of these field lines at the low-field side is higher, because the field lines, which leave the plasma before they reach the reference section at the inner equatorial midplane, are counted as well. They intersect the divertor wall at θ < 15 or θ > 18 ; they contribute to other modes at the HFS. Therefore, the proportional contribution of the field lines with connection length longer than one poloidal turn is higher in figure 3.14a then in figure 3.14b. The percentage of the flux tubes with L c 2 is small compared to the the shorter ones; they should not influence the transport properties significantly. It is predicted by the modelling [9, 26, 13] that the particle and heat fluxes coming from the ergodic region will be diffused from the fingers to the laminar region and reach the wall mainly via the flux tubes with a connection length of one poloidal turn.

60 3.5. STRUCTURE OF THE LAMINAR ZONE 57 In the laminar zone the field lines form the flux tubes, i.e. there are relatively large areas in which the field lines have the same connection lengths. As proof the structure of the laminar zone is presented in figure 3.15, without changing the numbers of the connection lengths to integer numbers as it is done in the contour plot. One sees that all the field lines within an area, which belongs to flux tube, have similar connection length. The histograms (fig. 3.14) show that the contribution of radius r [cm] poloidal angle θ [deg] Figure 3.15: The laminar plot for the same input parameters as in figure 3.13 shown without any mathematical processing. The range of poloidal angle covers full period. The red dashed line depicts the poloidal extension of the ergodization coils. One should notice that the colors do not correspond to the other laminar plots. the field lines with growing connection length is decreasing. Therefore the size of the flux tubes characterized by the longer connection lengths is decreasing as well. In the laminar zone the transport is mainly parallel (see section 3.2). The particles are able to follow the flux tube if the average diameter of the tube is larger then the characteristic diffusion length, which is comparable to the size of the scrape-off layer of the poloidal divertor, i.e. about 1 cm. Only the flux tubes with connection length up to 2 turns are of this size. They should influence significantly the transport of heat and particles in the plasma edge, what is confirmed by the modelling with the EMC3D code [26]. In the examples presented in figures 3.13b-d there exist one