GRILLIX: A 3D turbulence code for magnetic fusion devices based on a field line map Andreas Korbinian Stegmeir TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Max-Planck-Institut f¨urPlasmaphysik GRILLIX: A 3D turbulence code for magnetic fusion devices based on a field line map Andreas Korbinian Stegmeir Vollst¨andigerAbdruck der von der Fakult¨atf¨urPhysik der Technischen Universit¨atM¨unchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Stephan Paul Pr¨uferder Dissertation: 1. Hon.-Prof. Dr. Sibylle G¨unter 2. Univ.-Prof. Dr. Katharina Krischer 3. Univ.-Prof. Dr. Karl-Heinz Spatschek (i.R.) Heinrich-Heine-Universit¨atD¨usseldorf Die Dissertation wurde am 13.10.2014 bei der Technischen Universit¨atM¨unchen eingereicht und durch die Fakult¨atf¨urPhysik am 13.01.2015 angenommen. Abstract The main contribution to the (anomalous) cross field transport in tokamaks is known to be due to turbulence and numerical codes are essential tools in order to predict transport levels and understand physical mechanisms. Whereas for the interior closed field line region sophisticated turbulence codes are already quite advanced, the outer region of a tokamak, i.e. the edge and scrape-off layer (SOL), still lacks such tools to a large extent. The presence of many spatio- temporal scales and the complex geometry in diverted machines pose a huge challenge for the modelling of the edge/SOL. In this work the newly developed code GRILLIX is presented, which is aimed to set a first milestone in the development of a 3D turbulence code for the edge/SOL. GRILLIX uses a simpli- fied physical model (Hasegawa-Wakatani), but is capable to treat the complex geometry across the separatrix. The usually employed field aligned coordinate systems are avoided by using a cylindrical grid (R; Z; ') which is Cartesian within poloidal planes. The discretisation of per- pendicular (w.r.t. the magnetic field) operators is straight forward and parallel operators are discretised with a field line map procedure, i.e. field line tracing from plane to plane and in- terpolation. Via a grid-sparsification in the toroidal direction the flute mode character of the solutions can be exploited computationally. Ultimately, tokamak geometries with an arbitrary poloidal cross section, including a separatrix, can be treated with GRILLIX. In non-field-aligned grids numerical diffusion, i.e. a spurious perpendicular coupling depending on parallel dynamics, arises unavoidably. This numerical diffusion can be fatal for codes, since the parallel dynamics is usually orders of magnitude faster than perpendicular dynamics in tokamaks. A new numerical scheme is developed and applied in GRILLIX which maintains the self-adjointness property of the parallel diffusion operator on the discrete level and reduces numerical diffusion drastically. Many benchmarks in several geometries are presented to validate the field line map approach in general and GRILLIX in special. First effects of the geometry in diverted machines on drift wave turbulence were studied with GRILLIX. Field aligned structures get strongly distorted as they enter the X-point region. Their perpendicular spatial extent decreases thereby drastically towards the X-point and are thus subject to enhanced dissipation. Since ultimately close to the X-point fluctuations die out, the X-point constitutes a kind of barrier for fluctuations. This mechanism is similar to the previously found resistive X-point mode. i Zusammenfassung Radialer (anormaler) Transport in Tokamaks wird haupts¨achlich durch turbulente Prozesse getra- gen und numerische Simulationsprogramme sind heutzutage ein unverzichtbares Werkzeug, um Vorhersagen ¨uber das Transportlevel zu treffen und um physikalische Mechanismen zu verstehen. W¨ahrendf¨urden inneren Bereich geschlossener Feldlinien, hochentwickelte Programme bereits zur Verf¨ugungstehen, gibt es f¨urden ¨außerenBereich (Rand und Absch¨alschicht) von Tokamaks kaum Ans¨atze.Das Vorhandensein vieler raumzeitlicher Skalen und eine komplexe Geometrie in Divertormaschinen stellen eine grosse Herausforderung beim Modellieren des Randbereiches dar. In dieser Arbeit wurde das Simulationsprogramm GRILLIX entwickelt, welches einen er- sten Meilenstein bei der Entwicklung eines 3D Turbulenzprogrammes f¨urden Rand und die Absch¨alschicht setzt. GRILLIX basiert noch auf einem vereinfachten physikalischen Model (Hasegawa-Wakatani), aber kann daf¨urauf die komplexe Geometrie angewandt werden, z.B. sind Simulationen ¨uber die Separatrix hinweg m¨oglich. Durch das Verwenden eines zylindrischen numerischen Gitters (R; Z; '), welches kartesisch innerhalb poloidaler Ebenen ist, werden die ¨ublicherweise verwendeten Feldlinien-angepassten Koordinaten umgangen. Zur Diskretisierung senkrechter (im Bezug auf die Magnetfeldlinien) Operatoren k¨onnendamit Standardmethoden herangezogen werden. Die Diskretisierung paralleler Operatoren erfolgt mittels Feldlinienabbil- dung, d.h. Feldlinien werden von Ebene zu Ebene verfolgt und Werte an den entsprechenden Stellen interpoliert. Strukturen sind ¨ublicherweise stark elongiert entlang Magnetfeldlinien und diese Eigenschaft wird ausgenutzt durch eine Ausd¨unnung des Rechengitters in toroidaler Rich- tung. Tokamak Geometrien mit beliebigem poloidalen Querschnitt, einschließlich einer Separa- trix, k¨onnenmit GRILLIX behandelt werden. In Rechengittern, die nicht Feldlinien angepasst sind, tritt numerische Diffusion auf, d.h. eine f¨alschliche numerische senkrechte Kopplung, die von der parallelen Dynamik abh¨angt.Diese nu- merische Diffusion kann fatal sein f¨urSimulationsprogramme, da die parallele Dynamik ¨ublicher- weise Gr¨oßenordnungen schneller ist als die senkrechte. Ein neues numerisches Schema wurde daher entwickelt und in GRILLIX angewendet, welches die Selbstadjungiertheit das paralle- len Diffusionsoperators auf der diskreten Ebene erh¨altund die numerische Diffusion drastisch reduziert. Viele Tests in verschiedenen Geometrien werden pr¨asentiert, um das Konzept der Feldlinienabbildung im Allgemeinen und GRILLIX im Speziellen zu verifizieren. Erste Geometrieeffekte in Maschinen mit Divertor auf Drift-Wellen Turbulenz wurden mit GRILLIX untersucht. Feldlininen ausgerichtete Strukturen werden stark deformiert in der N¨ahe des X-Punktes. Deren senkrechte Ausdehnung nimmt zum X-Punkt hin stark ab und Dissipation wird dominant. Der X-Punkt stellt letztlich eine Art Barriere f¨urFluktuation dar, da diese in der N¨ahedes X-Punktes praktisch verenden. Der Mechanismus ¨ahneltdamit der bereits zuvor gefundenen resistiven X-Punkt Mode. iii Acknowledgments Many ideas and most strategic decisions for this work were performed in meetings among David Coster, Karl Lackner, Klaus Hallatschek and me. Hence, I want to cordially thank all three of them equally for supervising me. Karl Lackner motivated not only this work but also myself during some difficult phases. With his overview and belief in the whole project we were always able to make the right strategic decisions. I want to thank David Coster for always having an open door for me and keeping me motivated at any time with many good words (and cookies on Friday). At many problems he could help with his often pragmatic view on things from distance. As well, I was also always able to discuss the smallest details with him. I want to thank Klaus Hallatschek for his brilliant ideas contributing to this work. His detailed and critical view on results was often very helpful for the next steps and I enjoyed working with him very much. Furthermore, Omar Maj helped me a lot at the development of the numerical scheme and his door was still open on many Fridays late. I also want to thank Matthias H¨olzlfor some help with the numerical scheme and his advice with the solver, Hans-Joachim Klingshirn, who helped me several times with computational issues, Michele Martone for his advice with the LIBRSB library, Andreas Kammel for exchanging his experience about the Hasegawa-Wakatani model with me and Emanuele Poli for his advice during the final stage of this work. A very special thanks goes to my office mate and good friend Johannes Grießhammer. Not only he helped me several times with frustrating bugs in the code and other computational issues, but also we were able to answer most questions and solve most problems in our discussions. Working with him was a perfect symbiosis (Maybe, I was even slightly parasitic). Finally, I thank my significant other Daniela Gl¨aserfor supporting me in every aspect. Not only once, she encouraged me during difficult phases. I want to thank my family for supporting and ultimately making all this possible for me. v Contents Abstract i Zusammenfassung iii Acknowledgments v 1. Introduction 1 1.1. Tokamaks . 2 1.2. Stability and transport . 3 1.3. Turbulence basics . 4 1.4. Edge and SOL . 6 1.5. Motivation and outline . 7 2. The Hasegawa-Wakatani model 11 2.1. Braginskii equations . 11 2.2. Derivation . 13 2.2.1. Robust simplifications . 13 2.2.2. Drift approximation . 13 2.2.3. Strong approximations . 16 2.2.4. The equations . 16 2.3. Linear dynamics . 19 2.3.1. Drift waves . 19 2.3.2. Ideal interchange instability . 21 2.4. Fluctuation free energy theorem . 22 2.5. Turbulence in a box . 23 3. Field line map 29 3.1. Field and flux aligned coordinates . 29 3.2. Field line map approach . 31 3.3. Field line tracing . 33 3.4. Parallel gradient . 35 3.4.1. Finite difference method . 35 3.4.2. Coordinate free representation . 35 3.5. Parallel diffusion . 36 3.5.1. Naive discretisation . 36 3.5.2. Support operator method . 36 3.6. A simple model problem . 39 3.6.1. Stencil . 39 3.6.2. Numerical analysis . 40 3.7. Benchmarks in 3D . 42 3.7.1. Basic test q = ................................ 43 3.7.2. q = ......................................1 43 6 1 vii Contents 3.8. Map distortion . 46 3.8.1. Remedy 1 . 47 3.8.2. Remedy 2 . 48 3.9. Conclusions . 50 4. GRILLIX: A field line map based 3D turbulence code 51 4.1. Environment and data flow . 51 4.2. main@GRILLIX . 53 4.2.1. Paralellisation scheme . 53 4.2.2. Initialisation phase . 53 4.2.3. Time stepping phase . 56 4.3. Computational resources and efficiency . 58 4.3.1. MPI scaling . 58 4.3.2.