4.9 Variation of the ergodization
4.9.1 Variation of the plasma position
Due to technical constrains, the DED currents in the 12/4 regime were limited to 7 kA. In order to increase the level of the ergodization the plasma was shifted by 4 cm to the high-field side from the standard position R0 = 1.75 m to R0 = 1.71.
The shifting of the plasma towards the DED coils has a similar effect as increasing the plasma current, i.e. the resonant surfaces are closer to the DED coils. This effect was studied during discharge #92456, which was discussed in previous section. During the discharge the plasma position was changed by a series of horizontal shifts towards the high-field side. The heat flux density profile shows (see figure4.16) that during the change of the plasma position there is a change of the structure of the heat flux patterns. Each of the profile was taken for different plasma positions (indicated in the legend of the graph). As the plasma gets closer to the DED coils the profile becomes broader (blue curve - R0 = 1.715 m) and at R0 = 1.70 m (red
curve) instead of one maximum two can be seen. The strike points separate even more at R0 = 1.69 m (green curve). At the poloidal position s = 200 mm third
maximum appears, which suggest that the the heat influxes are strongly structured. Unfortunately, the equilibrium model (see appendix A) included in Atlas is only valid for plasma positions R0 > 1.74 m and we cannot analyze the structure of the
footprints for this discharge quantitatively. However, the separation of the power flux stripe when decreasing the distance between the resonant surfaces and the
4.9. VARIATION OF THE ERGODIZATION 101 1202 140 160 180 200 220 240 4 6 8 10 12 14 16x 10 4 s [mm] Q [W/m 2 ] R 0 = 1.725 m R 0 = 1.715 m R 0 = 1.704 m R 0 = 1.694 m #92456
Figure 4.16: The power flux profiles for one of the four stripes on the surface of the DED tiles during the discharge #92456. Different profiles are taken at different horizontal plasma positions.
perturbation coils is in agreement with the conclusions made in section 3.7 that at higher ergodization the stripes split up.
4.9.2
Variation of the plasma current
As discussed in section 3.7, the plasma current is a key parameter to influence the topology of the ergodic and laminar zones via the q-profile. If the plasma current is higher, the ergodization of the magnetic field lines in the edge is relatively stronger. For the footprint structure, the increase of the ergodization appears as the increase of the splitting of the magnetic footprint stripe. In figure4.17 two heat flux density profiles are presented for the same position, along the yellow rectangle from figure
4.9, for two different discharges. Both discharges were characterized by the same main parameters (R0 = 1.72 m, IDED = 7 kA, Bϕ = 1.9 T, additional heating
102 CHAPTER 4. THERMOGRAPHIC MEASUREMENTS
calculated for discharge #93573 with Ip = 350 kA, the red one for #93575 with
Ip = 400 kA. In the case of lower plasma current the power flux stripe has two
maxima, the distance between them is 28 mm. The value of the safety profile estimated with DIVA [39] is qa ≈ 2.8 at the plasma edge, which is almost m/n =
11/4. In the case of the red curve, we can clearly see the splitting of the power flux stripe. The value of the safety factor at the plasma edge estimated with DIVA is qa≈ 2.5, which corresponds to the flux surface m/n= 10/4. The distance between
220 240 260 280 300 320 340 2 3 4 5 6 7 8 9 10 11x 10 5 poloidal coordinate − s [mm]
heat flux density Q [W/m
2 ] I p = 350 kA I p = 400 kA #93573, #93575 D = 75 mm D = 28 mm
Figure 4.17: The power flux profiles for two discharges, with two different plasma currents.
the maxima is about 3 times larger then in the case of the blue curve, D≈75 mm. The asymmetry in the value of the power flux in both parts of the stripe are caused mainly by the imperfections in the geometry of the tiles. The power flux in the center of the heat flux tube is about three times higher than in the zone between. The result is consistent with the conclusions of section 3.7.2 that the ergodization of the magnetic field in the edge grows with the plasma current.
4.9. VARIATION OF THE ERGODIZATION 103
4.9.3
Variation of the poloidal beta
The value of the poloidal beta is dependent on the energy content of the plasma. Therefore additional heating (e.g. with the neutral beam) changes βpol. For the
ohmic plasmas, where the energy content of the plasma is P = IpUv ≈ 0.3 MW,
the poloidal beta is of order of 0.25, while with 1 MW of additional heating power βpol is of order of 0.6. Hence, additional heating power should decrease the level
of ergodization (see section 3.7.1). During the experiments discussed in this thesis there was no measurement of the poloidal beta. Thus all values ofβpol are calculated
using the scaling for the previous experimental campaigns [21].
During discharge #93100 additional heating power started to be injected about half a second after the DED was switched on (see figure 4.1). The discharge pa- rameters are as follows: R0 = 1.72 m, Bϕ = 1.9 T, Ip = 380 kA, Uv = 0.8 V ; the
DED with the current amplitude of IDED = 7 kA was operated from t = 1.7 s to
2.7 s and the additional heating powerPN BI = 0.7 MW was injected sincet= 2.1 s.
The poloidal beta in ohmic phase was about 0.3 and in the NBI-phase about 0.5. The power flux profile for two different phases of the discharge: ohmic and with the neutral beam are presented in figure 4.18. The blue curve present the heat flux density profile for the ohmic phase and the red one for the neutral beam phase. Both profiles are normalized to their maximum value. Even if the change of poloidal beta is relatively small ∆βpol = 0.2 there is measurable change of the separation of the
power flux parts. In the ohmic phase (βpol ≈0.3) the separation isDohmic = 59 mm;
in the phase with the additional heating (βpol = 0.5) the separation decreases by
about 8 mm – DNBI = 51 mm. This behavior of the flux tubes is in agreement
with the tendencies shown for the topology of the magnetic footprints discussed in section3.7.1. One should also note that the separation of the parts of the flux tubes are in between the cases discussed in section 4.9.2. As the plasma current during the discharge #93100 is in between the values of the plasma currents during the discharges discussed there. However, the comparison cannot be made directly, as the poloidal beta differs in these cases.
104 CHAPTER 4. THERMOGRAPHIC MEASUREMENTS 240 260 280 300 320 340 0 0.2 0.4 0.6 0.8 1 poloidal coordinate − s [mm]
normalized power flux −
Q norm ohmic NBI−1 44 kV D ohmic = 59 mm D NBI = 51 mm #93100
Figure 4.18: The power flux profiles for two phases of the #93100 discharge: (blue curve) – the ohmic phase; (red curve) – the neutral beam phase.
Chapter 5
Summary
In the thesis the topological properties of the magnetic field in the TEXTOR-DED plasma boundary are considered. By the DED the magnetic equilibrium field is per- turbed by external currents. Sixteen perturbation coils produce a helical multipolar magnetic field. The linear superposition of the equilibrium and the perturbation fields creates a region in the plasma edge, where the field lines possess stochas- tic properties. The axial symmetry of the magnetic field topology is broken and the magnetic flux surfaces are destroyed. As a result an open chaotic system is formed. Thus, one can distinguish two different areas in the plasma edge: The ergodic region, where the field lines have connection lengths much longer than the Kolmogorov length, and the outermost laminar zone, where the connection lengths are smaller than the Kolmogorov length. Both laminar and ergodic zones influence the transport properties of the plasma. In this thesis it the comprehensive studies of the topological properties of the magnetic field in the edge of the TEXTOR-DED plasma have been performed.
The 12/4 configuration of the perturbation field is analyzed using a cylindrical model. The resulting perturbation field (centered around the m0 = 12 poloidal
mode) consists of several modes, with amplitudes showing a strong radial decay Amn ∝rm. In the toroidal model the poloidal spectrum is localized near the central
mode m0 = 2πn/θc ≈ 20 instead of 12. The reason for the increased exponent m0
relative to the cylinder approximation results from a lower pitch angle of the equi- 105
106 CHAPTER 5. SUMMARY
librium field at the high field side of the torus. The perturbation field is localized radially within a few centimeters on the high-field side of the tokamak.
The investigation of the topological properties of the ergodized magnetic field is performed with the “Atlas” program, which has been developed during this thesis. Atlas is based on the fast, symplectic mapping scheme [2, 4] and allows to visualize various structures of the 3D structures of the ergodized magnetic field, as well as the identification of different flux tubes in the scrape-off layer. Using the Atlas, we have carried out the analysis of the general features of the ergodic and the laminar regions:
– Typically the ergodic region is composed of a mixture of a stochastic region and island chains. The connection lengths of the field lines in the ergodic region are of the order of hundreds of poloidal turns. They penetrate the whole available space. Eventually, the field lines are deflected towards the DED coils and leave the plasma volume via so-called fingers.
– The laminar zone has some similarities with the scrape-off layer of the poloidal divertor, however, it has a more complicated three dimensional structure. Ac- cording to the Monte Carlo modelling [26], the main channels of heat and particle exhaust are the flux tubes with connection lengths of one and two poloidal turns. The single turn flux tube has a high poloidal symmetry: Its stagnation point of the plasma flows is positioned at the outer equatorial mid- plane. It seems to be a general feature of all investigated cases, that the flows from the stagnation point of the single turn flux tube towards the DED wall are deposited in the frame of one magnetic footprint stripe. Due to a four-fold toroidal symmetry of the 12/4 perturbation, there exist four helical magnetic footprint structures.
The key plasma parameters, which define the topology of the magnetic field in the plasma edge, have been identified:
– The plasma current, which determines the safety factor profile and thus defines the position of the resonant magnetic flux surfaces.
107 – The plasma pressure, which influences the position of the resonant flux surfaces via the Shafranov shift and thus the interaction of the perturbation field and the plasma.
The dependence of the width and the structure of the perturbed volume on the plasma current and the plasma pressure expressed via the poloidal beta has been studied. By changing the plasma current and/or the poloidal beta one can modify the structure of the perturbed volume in such a way that either the ergodic or the laminar zone dominates the perturbed volume. The size of the perturbed volume changes nonlinearly with the plasma current, however, the variation is not large (about 13%, WP ∼ 7−8 cm). At lower ergodization level the “sink” areas within
a footprint stripe are connected together; while at the higher level of ergodization they split up forming a private flux zone in between them.
The power deposition pattern is defined mainly by the magnetic footprint struc- ture. Therefore, the modelled structures were compared with thermographic mea- surements of the temperature distribution on the DED target plates. To measure the surface temperature a fast infrared system was designed and implemented. The thermographical measurements confirm the plasma boundary structures cal- culated by Atlas. One observes four helical strike zones, which are parallel to the divertor coils. The structure of the single power flux stripe is consistent with the topology of the flux tubes; namely, each of the power flux stripe consists of two parts with different direction of incoming heat and particle fluxes. The direct comparison of the power flux profile and connection length profile are in good agreement. How- ever, the equilibrium model applied in the calculations needs further improvements. The variation of the structure of the heat flux density deposition pattern with the plasma current and poloidal beta has been measured. For increasing level of the ergodization the strike zone broadens and at some point splits up as it is predicted by Atlas.
This work forms the basis for the experimental investigations of the three dimen- sional structures of the magnetic field induced by the Dynamic Ergodic Divertor in the plasma edge. For the future investigations it is planned to include the numeri- cally calculated spectrum of the perturbation and the magnetic equilibrium.
Acknowledgements
I would like to thank all those who have helped me during my PhD work:
First of all, I would like to express my thanks to my mentor Dr. Karl Heinz Finken for giving me the opportunity to do my PhD thesis in J¨ulich and for his constant support during the PhD work. Furthermore, I am very thankful to Dr. S.S. Abdullaev, who patiently introduced me to the world of chaos and always had time to answer my questions.
I am deeply grateful to my supervisor Prof. Robert Wolf for his inestimable help and valuable advices, which made this thesis better. I want to thank Dr. Michael Lehnen and Dr. Masahiro Kobayashi for fruitful discussions during last three years and Dr. G¨unther Mank for helping me at the beginning of my stay in J¨ulich.
My deepest thanks go to my wife Katarzyna for her support and understanding during all these years.
I would like to thank my family and all those people, who helped me to reach this point. Thank you all very much...
Appendix A
The model of magnetic
equilibrium field used in the Atlas
code
In the Hamiltonian definition (see equation 2.14) the part representing the unper- turbed part depends only on the toroidal flux ψ, which for the equilibrium case is defined as[4]: ψ = 1 2πBtR20(a) Z r 0 r0dr0 Z 2π 0 Bϕ(r0, θ)dθ, (A.1)
where Bt is a toroidal magnetic field on the plasma magnetic axis and it can be
expressed via the current flowing through the wire along the Z-axisIϕ:
Bt=
µ0Iϕ
2πR0(a)
.
The magnetic equilibrium field of the TEXTOR tokamak plasma consists of nested, circular flux surfaces. The centers of the magnetic surfaces R0(r) due to
effects of the plasma pressure and electric current are shifted with respect to the geometrical plasma center. This shift called Shafranov shift is given by [4]:
∆(r) = R0(r)−R0(a) = R20(a) + (Λ + 1)(a2−r2)1/2 −R0(a), (A.2) where Λ =βpol+li/2−1. 111
112 APPENDIX A. MAGNETIC EQUILIBRIUM FIELD
Here, li denotes the plasma internal inductance, which typically for the TEXTOR
tokamak has a value of li ≈1.2.
The toroidal component of the magnetic field equilibrium equals to: Bϕ(r, θ) = Bt
R0(a)
R0(r)
1
1 + Rr0 cosθ, (A.3)
The poloidal component depends on the current density profile, but at the edge it can be approximate by the expression:
Bθ(r, θ) = µ0Ip 2πr 1 + Λ r R0 cosθ (A.4) .
According to equation A.1 one can obtain the relationship between the normal- ized toroidal flux ψ and the radius of the magnetic flux surface r:
ψn= R0(r) R(a) " 1− 1− r 2 R2 0(r) 1/2# , (A.5) r= a2−R 2 0−R20(a) Λ + 1 1/2 , R0(r) = R0(a)[a1ψ+ (1 +a2ψ2n+a3)1/2] a1 =−Λ−1, a2 = Λ2+ 3Λ + 2, a3 = (Λ + 1)a2/R20(a)
If the value of Shafranov shift is small enough equation A.5 can be simplified to a following form: ψn = 1− 1− r 2 R2 0(a) 1/2 , r≈R0(a)[2ψn−ψn2]1/2. (A.6)
Appendix B
The THEODOR code
The measured time evolution of the surface temperature can serve as an input for the heat flux evaluation. In the thesis we have performed those calculations with the THEODOR code [44]. The THEODOR code performs two dimensional heat flux calculation taking into account the temperature dependent material parameters. It
Figure B.1: The sketch illustrating the geometry of the tiles in the THEODOR code
is based on direct numerical solution of the heat conduction equation for the whole body of a tile. The measured top-surface temperature serves as a time-dependent boundary condition.
114 APPENDIX B. THE THEODOR CODE
The fundamental equations are [43] qs =−
∂U ∂x,
∂U
∂t =D∆U (B.1)
where D is the temperature-dependent heat diffusion coefficient [7]: D(T(U)) = λ
ρκ. (B.2)
Here, ρstands for the density,κ– the heat capacitance andλ– the heat conduction coefficient for the material, which tiles are made of. The transformation B.1 is of computational advantage if λ is fitted in such a way that U(T) becomes a simple analytic function. The calculations are made for an assumption that the heat does not penetrate along the power stripes produced by the DED.
The geometry of the tile and the edge conditions are shown in figure B.1. Tiles are made of IGU-110 (ρ= 1780 kg·m−3,κ= 837 J·kg−1·K−1) the heat conduction
coefficient for different temperatures is presented in table B.1. T [◦C] 0 500 1000
λ[ W
m·K] 94 63 47
Table B.1: The heat conduction coefficient of the IGU-110 for different temperatures of the material [59]
At the top surface there is imposed a thin layer (marked as a red rectangle in figure B.1), which can simulate the changed heat transfer of the surface due to surface modification (e.g. impurity implantation or melting). In the THEODOR code it is equivalent to a heat transmission edge condition:
qs =αtop(Tmeasured−T(0)), (B.3)
where Tmeasured is surface temperature measured by the infrared camera,T(0) – the
temperature of the tile αtop = λlayer/d is a heat transmission coefficient. For the
results obtained during the first experimental campaign of the TEXTOR-DED we have assumed that the surface of the divertor target plates was not modified. Thus, we setα= 1.44·1010kWm−2K−1, which is equivalent to neglecting the top layer. The
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