Equilibrium Design for the COMPASS-U Tokamak

WDS'18 Proceedings of Contributed Papers — Physics, 99–104, 2018. ISBN 978-80-7378-374-7 © MATFYZPRESS Equilibrium Design for the COMPASS-U Tokamak L. Kripner, M. Peterka, M. Imr´ıˇsek,and T. Markoviˇc Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, Czech Academy of Sciences, Institute of Plasma Physics, Prague, Czech Republic. J. Urban, J. Havl´ıˇcek,M. Hron, F. Jaulmes, R. Pánek,D. Sesták,ˇ V. Weinzettl, V. Yanovskiy, and the COMPASS team Czech Academy of Sciences, Institute of Plasma Physics, Prague, Czech Republic Abstract. The COMPASS-U tokamak (R = 0:86 m, a = 0:28 m, 20 −3 n ≈ (1:5{2:5) × 10 m , B0 = 5 T) is being designed at the Institute of Plasma Physics of Czech Academy of Sciences in Prague. A wide range of plasma shapes (elongation κ up to 1.8 and triangularity up to 0.6) will be explored. This paper presents methods used for calculation of plasma equilibria. Furthermore, it is concerned with the design of poloidal field coils for the COMPASS-U. It explains the reasons for moving of the poloidal field coils position inside the toroidal field coils with respect to the original COMPASS-U design, and it proposes a method to evaluate heating stress of the coils. The accessibility of the planned high-performance scenario is shown in the current design of poloidal field coils. Introduction The COMPASS-U tokamak is going to be built at the Institute of Plasma Physics of Czech Academy of Sciences in Prague (IPP Prague), where it is going to replace the existing COMPASS tokamak. COMPASS-U [Panek et al., 2017] is designed with major radius 0.89 m and minor radius 0.28 m, which makes it 1.6 times larger according to the major radius than the current COMPASS tokamak. COMPASS-U will operate at high magnetic field, up to 5 T on the magnetic axis, high plasma density (1:5{2:5)×1020 m−3, and with up to 8 MW of additional heating (NBI, ECRH). These parameters yield a heat flux of 15{20 MW=m2 to the divertor due to the short power decay length (λq ≈ 1 mm) according to the Eich scaling [Eich et al., 2013]. Tokamak COMPASS-U aims to explore wide range of different plasma configurations in- cluding single null and double null divertor configuration, with high values of shape parameters (elongation κ up to 1.8 and triangularity up to 0.6), since these parameters are supposed to be important for plasma stability [Fontana et al., 2018] and for possibility to suppress edge localized modes. An essential feature of the tokamak is its ability to move the strike points, which enables an extensive study of divertor power exhaust. In this paper we focus on the modelling of high-performance equilibria with the current design of poloidal field (PF) coils. Various available tools are used to find the coils' possibilities and limitations and to improve the design. The studied scenario has high triangularity κ = 0:6, high elongation δ = 1:77, high centre density n ≈ 2:5 × 1020 m−3 and high plasma current Ip = 2 MA. In the second section, we introduce the theoretical background of the calculations and the available tools. Tokamak coil design together with its possibilities is described in the third section. In the fourth section, we demonstrate the possibility to model a whole discharge as a set of static plasma equilibria to calculate the change of coil parameter during the discharge. Plasma equilibrium To successfully operate a tokamak, the plasma has to be in a stable equilibrium. Although the plasma stability is crucial, in this article we focus on the plasma equilibrium, which is the necessary condition. Having the plasma in equilibrium means that a force balance is established 99 KRIPNER ET AL.: EQUILIBRIUM DESIGN FOR COMPASS-U between the current and pressure inside the plasma. For the tokamak toroidal symmetry Grad{ Shafranov (GS) equitation can be derived [Grad and Rubin, 1958; Shafranov, 1958] as dp fdf ∆∗ = −µ R2 − ; (1) 0 d d where ∆∗ is Stokes operator [Zohm, 2015] @ 1 @ @2 ∆∗ = R + ; (2) @R R @R @Z2 p is pressure and f = RBφ. Both p and f are functions of magnetic poloidal flux , and together contain information about the profile of toroidal plasma current and toroidal magnetic field Bθ. The solution of GS equation is the combination of functions (R; Z), p0( ) and ff 0( ), which provides the complete information about magnetic field B~ and plasma current ~j. The final equilibrium is a combination of vacuum magnetic field obtained from the coils and the magnetic field generated by plasma. In our approach, we are investigating combinations of currents in the set of PF and central solenoid (CS) coils needed to form the plasma equilibrium with required shape of plasma boundary, plasma current Ip and value of pressure p in the plasma centre. This requires the so-called inverse approach to the GS solution, where the coil currents are taken as the output of the calculation. This approach presents an ill-posed problem, where, in principle, exist infinite number of coil current combinations which similarly satisfy the shape constrains . As a consequence, additional constraints should be inserted into the task to find a sufficient solution. Beside the points of the requested plasma boundary, which are required to have the same value of , and the position of the x-point, mathematical regularization is generally used to reduce the differences in currents between neighbouring coils at the cost of a slightly relaxed match of the plasma boundary. The importance of a constrain can be stressed by change of its weight. To solve this problem we are using free boundary codes with given constraints. Since the request for a given plasma shape is only approximate (and not necessarily achievable), a fixed- boundary code cannot be used. Functions ff 0 and p0 are inserted to the equation as an input. These functions are chosen to satisfy requirements on total plasma current, plasma pressure on the magnetic axis and zero value of f and p on the plasma boundary. Since the resulted plasma boundary is not strongly dependent on the shape of the ff 0 and p0 functions, linear ansatz is generally used, which results in the quadratic profile of p and f. Calculated reconstruction of the and magnetic field of the high-performance case is shown in Fig. 1. There exist numerous codes to solve GS equation with free boundary. The codes differ in used calculation methods, mesh generation, possible free parameters, and optimizations. Comparison of results of tested codes Freebie [Artaud and Kim, 2012], FreeGS [Dudson, 2018] and Fiesta [Cunningham, 2013] with comparable setting of free parameters is shown in Fig. 2a. All tested codes were able to produce equilibrium with the requested plasma shape with similar values of coil currents for the Fiesta and Freebie code, while the FreeGS code ended with the slightly different set of coil currents which lead to the shift of the upper strike point. The difference might be caused by limited possibility to set value of 0 on the plasma boundary and though the equilibrium in different part of the discharge may be calculated in the FreeGS code (a non-zero d =dt throughout the discharge provides the loop voltage for ohmic heating and generation of plasma current). Fiesta will be used in the ongoing analysis due to the flexibility, reliability and speed of the code. Final plasma shape and magnitude of coil currents is compromise in precise shape of pre- scribe boundary and level of regularization which, generally speaking, redistribute the current in coils to reduce current differences trying to fulfil the other constrain. A scan of various regularization weights was performed with the Fiesta code as shown in Fig. 2b. Level of regularization 100 KRIPNER ET AL.: EQUILIBRIUM DESIGN FOR COMPASS-U Figure 1. High-performance equilibrium calculated by Fiesta code. The poloidal flux , absolute value of the poloidal magnetic field and absolute value the toroidal field are shown from left to right. It can be seen that with the given profile of f the plasma is slightly paramagnetic and the dependence of the toroidal field is not strictly / 1=R. (a) (b) Figure 2. (a) Benchmark of the available codes for the same shape of the plasma (value of on the boundary has been set identical only for Freebie and Fiesta code). Angular axis of the polar graphs represents the angular position of the coil with respect to the geometrical centre; the radial axis represents current per turn in the coil. (b) Calculated coil currents with different level of regularization weight performed by Fiesta code (changes in plasma shape were negligible). were selected not to violate the boundary and still keeping reasonable level of coil currents. Re- distribution of the currents within the coils helps to reduce the load on stressed coils for the cost of slight change of the strike point position. This also shows that the small difference between the coil currents in the results from Fiesta and Freebie can easily be explained by a different regularizing method. Poloidal coils design The elementary coil design of any tokamak contains the toroidal field (TF) coils, PF coils and the CS coils forming the central solenoid. The TF coils generate the main toroidal field of the tokamak. Axial asymmetry caused by the finite number of TF coils is causing a so-called toroidal ripple, which is however neglected in our axisymmetric calculations. The primary purpose of the CS coils is to inductively drive the current through the plasma by d =dt.

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