kth royal institute of technology Doctoral Thesis in Electrical Engineering Time Spectral Methods - Towards Plasma Turbulence Modelling KRISTOFFER LINDVALL Stockholm, Sweden 2021 Time Spectral Methods - Towards Plasma Turbulence Modelling KRISTOFFER LINDVALL Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Thursday, February 18, 2021, at 3:00 p.m. in F3, Lindstedsvägen 26, Stockholm. Doctoral Thesis in Electrical Engineering KTH Royal Institute of Technology Stockholm, Sweden 2021 © Kristoffer Lindvall © Jan Scheffel ISBN: 978-91-7873-759-8 TRITA-EECS-AVL-2021:7 Printed by: Universitetsservice US-AB, Sweden 2021 i Abstract Energy comes in two forms; potential energy and kinetic energy. Energy is stored as potential energy and released in the form of kinetic energy. This process of storage and release is the basic strategy of all energy alternatives in use today. This applies to solar, wind, fossil fuels, and the list goes on. Most of these come in diluted and scarce forms allowing only a portion of the energy to be used, which has prompted the quest for the original source, the Sun. As early as 1905 in the work by Albert Einstein on the connection between mass and energy, it has been seen theoretically that energy can be extracted from the process of fusing lighter elements into heavier elements. Later, this process of fusion was discovered to be the very source powering the Sun. Almost a century later, the work continues to make thermonuclear fusion energy a reality. Looking closer at the Sun, we see that it consists of a hot burning gas subject to electromagnetic fields, i.e. a plasma. The plasma in the Sun is contained by the massive gravitational force which allows for fusion to be created in a stable and continuous process. Taking inspiration from the Sun we see that a hot plasma and its containment are key to achieving fusion. The gravitational force is not present on Earth, and creating it artificially is, as of today, and insurmountable task. Fortunately, the plasma can be contained in another way; with magnetic fields. The challenges of making fusion a viable energy source are numerous and diverse. To deal with these challenges there are several fields of fusion re- search; engineering, physics, and numerical analysis. These of course overlap, but serve to illustrate the focus of different groups. This thesis work is focused on the latter two, physics and numerical analysis. The containment of the plasma in a fusion device is degraded by drift wave turbulence. The turbulence in the plasma occurs on the micro-scale, namely on the scale of particles travelling around the magnetic field lines. The physics behind turbulence and the drift waves responsible is a rich field with many future topics. Since the micro-turbulence can quickly grow and diffuse plasma through- out the device in a matter of micro-seconds, it becomes a difficult challenge to numerically resolve the turbulence over a longer span of time. The typical confinement times required in a fusion device is on the order of several sec- onds. Thus, the main focus of this thesis is on developing a numerical method that can effectively resolve the plasma physics over longer time-intervals. To this effect, a Time-Spectral method has been developed that utilizes the ad- vantageous properties of spectral methods to all domains, specifically the temporal domain. The numerical method has been implemented on com- pressible Navier-Stokes, ideal magnetohydrodynamics (MHD), and a toroidal two-fluid plasma turbulence model called the Weiland model. Keywords| Fusion, Turbulence, Micro-instabilities, Time-spectral, Navier-Stokes, MHD, Weiland model ii Sammanfattning Energi finns i tv˚aformer; potentiell energi och kinetisk energi. Energi lag- ras som potentiell energi och frig¨ors i form av kinetisk energi. Denna lagrings- och frig¨oringsprocess ¨ar den grundl¨aggande strategin f¨or alla energialternativ som anv¨ands idag. Detta g¨aller sol, vind, fossila br¨anslen och listan forts¨atter. De flesta av dessa kommer i utsp¨adda och knappa former som endast g¨or det m¨ojligt att anv¨anda en del av energin, vilket har lett till jakten p˚aden ur- sprungliga k¨allan, solen. Redan 1905 i Albert Einsteins arbete om sambandet mellan massa och energi har det teoretiskt sett att energi kan extraheras fr˚anprocessen att sl˚asamman l¨attare element till tyngre element. Senare uppt¨acktes att den- na fusionsprocess var sj¨alva k¨allan som driver solen. N¨astan ett sekel senare forts¨atter arbetet att g¨ora termonukle¨ar fusionsenergi till verklighet. N¨ar vi tittar n¨armare p˚asolen ser vi att den best˚arav en het br¨annande gas som uts¨atts f¨or elektromagnetiska f¨alt, dvs. ett plasma. Plasman i solen ¨ar innesluten av den massiva gravitationskraften som m¨ojligg¨or fusion att skapas i en stabil och kontinuerlig process. Med inspiration fr˚ansolen ser vi att en het plasma och dess inneslutning ¨ar nyckeln till att uppn˚afusion. Gravitationskraften finns inte p˚ajorden, och att skapa den artificiellt ¨ar fr˚an och med idag en o¨overstiglig uppgift. Lyckligtvis kan plasman inneslutas p˚a ett annat s¨att; med magnetf¨alt. Utmaningarna med att g¨ora fusion till en genomf¨orbar energik¨alla ¨ar m˚angaoch varierande. F¨or att hantera dessa utmaningar finns det flera f¨alt inom fusionsforskning; teknik, fysik och numerisk analys. Dessa ¨overlappar naturligtvis men tj¨anar till att illustrera olika gruppers fokus. Detta uppsats- arbete fokuserar p˚afysik och numerisk analys. Inneslutningen av plasma i en fusionsanordning f¨ors¨amras av drivv˚agsturb- ulens. Turbulensen i plasman sker p˚amikroskalan, n¨amligen p˚askalan av partiklar som r¨or sig runt magnetf¨altslinjerna. Fysiken bakom turbulens och drivv˚agornasom ¨ar ansvariga ¨ar ett rikt f¨alt med m˚angaframtida ¨amnen. Eftersom mikro-turbulensen snabbt kan v¨axa och sprida plasma genom hela enheten p˚an˚agrasekunder, blir det en sv˚arutmaning att numeriskt l¨osa turbulensen under en l¨angre tidsperiod. De typiska inneslutningstiderna som kr¨avs i en fusionsanordning ¨ar i storleksordningen flera sekunder. Av- handlingen fokuserar allts˚ap˚aatt utveckla en numerisk metod som effektivt kan l¨osa plasmafysiken ¨over l¨angre tidsintervall. F¨or detta ¨andam˚alhar en Tidsspektralmetod utvecklats som utnyttjar de f¨ordelaktiga egenskaperna hos spektrala metoder f¨or alla dom¨aner, s¨arskilt den temporala dom¨anen. Den nu- meriska metoden har implementerats p˚akomprimerbara Navier-Stokes, ideal magnetohydrodynamik (MHD) och en toroidformad tv˚av¨atske-plasmaturbul- ensmodell som kallas Weiland-modellen. List of Papers I A Time-Spectral Method for Initial-Value Problems Using a Novel Spatial Subdomain Scheme K. Lindvall and J. Scheffel Cogent Mathematics & Statistics, (2018) II Spectral Representation of Time and Physical Parameters in Numerical Weather Prediction K. Lindvall and J. Scheffel in Understanding of Atmospheric Systems with Efficient Numerical Methods for Ob- servation and Prediction, IntechOpen, (2018), DOI: 10.5772/intechopen.80351 III A Time-Spectral Approach to Numerical Weather Prediction J. Scheffel and K. Lindvall Computer Physics Communications, (2018) IV SIR|An efficient Solver for Systems of Equations J. Scheffel and K. Lindvall SoftwareX, (2018) V Optimizing Time-Spectral Solution of Initial-Value Problems J. Scheffel and K. Lindvall American Journal of Computational Mathematics, (2018) VI 2D Continuous Chebyshev-Galerkin Time-Spectral Method K. Lindvall and J. Scheffel Submitted to Journal of Scientific Computing, (2021) VII Temporal Smoothing - a Step Forward for Time-Spectral Methods J. Scheffel and K. Lindvall Submitted to Computer Physics Communications, (2021) Conference Contributions I Generalized Weighted Residual Method; Advancements and Current Studies J. Scheffel and K. Lindvall 58th Annual Meeting of the APS Division of Plasma Physics, (2016), San Jose, Cal- ifornia II A Time-Spectral Method for Fusion Physics Modelling K. Lindvall and J. Scheffel Annual Meeting of the Fusion Research Unit of Sweden, RUSA, (2016), Uppsala, Sweden iii iv LIST OF PAPERS III Can the Time-Spectral Method Compete in Fusion Transport Modelling? K. Lindvall and J. Scheffel 59th Annual Meeting of the APS Division of Plasma Physics, (2017), Milwaukee, Wisconsin IV A Time-Spectral Method Applied to Fusion Transport K. Lindvall and J. Scheffel 3rd Joint Nordic Fusion Conference, (2019), Denmark V 2D Drift Wave Turbulence Solved With a Time-Spectral Method K. Lindvall and J. Scheffel 61st Annual Meeting of the APS Division of Plasma Physics, (2019), Fort Lauderdale, Florida Acknowledgment I would like to extend my utmost appreciation and gratitude to my colleagues, past and present, at the division of fusion plasma physics. First, to Lorenzo and Thomas for being excellent teachers, and thanks to you Thomas for also being my co-supervisor. Also, a special thanks to Lorenzo for reviewing my work and giving me insightful comments and explanations. To my fellow PhDs; Pablo, Richard, Armin, Alvaro, Petter, Estera, and Emmi for welcoming me and showing me the ropes. I will miss our movie nights and discussions over lunch... well perhaps not the movies due to the them being absolutely horrible (by design). To my colleagues still in the trenches, Erik and Bj¨orn,may you finish with more grace I. To Hans Nordman for helping with the Weiland model and explaining the overall physics of anomalous transport. Your comments have been invaluable and I especially thank you for letting Jan and I to pick your brain on the subject of plasma turbulence. I am beyond grateful for the unending support from my wife, Jeniffer. If I start listing all the things you have done for me and the many times you have encouraged me, I'll never stop. Last, I would like to thank my supervisor Jan Scheffel.