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Modeling RF waves in hot plasmas using the finite element method and wavelet decomposition: Theory and applications for ion cyclotron resonance heating in toroidal plasmas
KTH, School of Electrical Engineering and Computer Science (EECS), Electrical Engineering, Fusion Plasma Physics.ORCID iD: 0000-0003-4343-6325
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Fusion energy has the potential to provide a sustainable solution for generating large quantities of clean energy for human societies. The tokamak fusion reactor is a toroidal device where the hot ionized fuel (plasma) is confined by magnetic fields. Several heating systems are used in order to reach fusion relevant temperatures. Ion cyclotron resonance heating (ICRH) is one of these systems, where the plasma is heated by injecting radio frequency (RF) waves from an antenna located outside the plasma.

This thesis concerns modeling of RF wave propagation and damping in hot tokamak plasmas. However, solving the wave equation is complicated because of spatial dispersion. This effect makes the wave equation an integro-differential equation that is difficult to solve using common numerical tools. The objective of this thesis is to develop numerical methods that can handle spatial dispersion and account for the geometric complexity outside the core plasma, such as the antenna and low-density regions (or SOL). The main results of this work is the development of the FEMIC code and the so-called iterative wavelet finite element scheme.

FEMIC is a 2D axisymmetric code based on the finite element method. Its main feature is the integration of the core plasma with the SOL and antenna regions, where arbitrary geometric complexity is allowed. Moreover, FEMIC can apply a dielectric response in the SOL and in the region between the SOL and the core plasma (i.e. the pedestal). The code can account for perpendicular spatial dispersion (or FLR effects) for the fast wave only, which is sufficient for modeling harmonic cyclotron damping and transit time magnetic pumping. FEMIC was used for studying the effect of poloidal phasing on the ICRH power deposition on JET and ITER, and was benchmarked against other ICRH modeling codes in the fusion community successfully.

The iterative wavelet finite element scheme was developed in order to account for spatial dispersion in a rigorous way. The method adds spatial dispersion effects to the wave equation by using a fixed point iteration scheme. Spatial dispersion effects are evaluated using a novel method based on Morlet wavelet decomposition. The method has been tested successfully for parallel and perpendicular spatial dispersion in one-dimensional models. The FEMIC1D code was developed in order to model ICRH and to study the properties of the numerical scheme. FEMIC1D was used to study second harmonic heating and mode conversion to ion-Bernstein waves (IBW), including a model for the SOL and pedestal. By studying the propagation and damping of the IBW, we verified that the scheme can account for FLR effects.

Abstract [sv]

Fusionsenergi har potentialen att erbjuda en hållbar lösning för storskalig energiproduktion för mänskligheten. Fördelarna med fusionsenergi inkluderar inga utsläpp av växthusgaser, inget långlivat radioaktivt avfall, pålitlig energiproduktion, hög säkerhet och stora bränslereserver på jorden.

Tokamaken är en fusionsreaktor med ringformad geometri, där det heta bränslet (eller plasmat) innesluts med starka magnetfält för att det inte ska få kontakt med t.ex. väggar och antenn. För att uppnå fusionsrelevanta temperaturer (ca. 100 miljoner grader) har tokamaker flera uppvärmningssystem. Joncyklotronresonansuppvärmning (ICRH) är ett system där plasmat värms upp med hjälp av radiovågor. ICRH kommer att ha en viktig roll på ITER, vilket är nästa generations tokamakexperiment som beräknas vara operativ mot slutet av 2020-talet.

Denna avhandling handlar om modellering och beräkningar av radiovågor i tokamakplasman för ICRH. Beräkningar av radiovågor görs genom att lösa Maxwells ekvationer. Att lösa Maxwells ekvationer är svårt p.g.a. fenomenet rumslig dispersion som finns i heta plasman. Denna effekt resulterar i integraloperatorer som är svåra att hantera med numeriska verktyg. Målet med detta arbete är att utveckla numeriska verktyg som kan hantera rumslig dispersion i Maxwells ekvationer och kunna hantera den geometriska komplexitet som finns utanför plasmat, t.ex. antennen och regionerna med låg plasmatäthet (SOL). Huvudresultaten av detta arbete är utvecklingen av den tvådimensionella FEMIC-koden och den så kallade "iterativa wavelet finita element" algoritmen.

En av FEMIC-kodens viktigaste egenskaper är att den kan beskriva vågfysiken både i det heta inre plasmat och det omgivande SOL-området, där godtycklig geometrisk komplexitet är tillåten för att beskriva SOL, väggar och antenn. Dessutom tillämpar FEMIC en dielektricitetsmodell i SOL-området och i området mellan plasmat och SOL (som kallas för pedestalen). Koden kan beskriva vinkelrät rumslig dispersion (FLR-effekter) för den snabba vågen enbart, vilket är tillräckligt för att beskriva viktiga mekanismer som t.ex. harmonisk dämpning och magnetisk pumpning. FEMIC har används för att studera effekten av poloidal fasning i tokamakerna JET och ITER, samt validerats emot andra ICRH-koder framgångsrikt.

Den iterativa wavelet finita element algoritmen utvecklades för att behandla rumsligt dispersiva effekter på ett rigoröst sätt. I denna algoritm adderas rumsligt dispersiva effekter till vågekvationen med hjälp av iterationer. För att evaluera rumslig dispersion har en ny metod baserad på Morlet wavelets tillämpats. Algoritmen har testats framgångsrikt för vinkelrät och parallell dispersion i endimensionella modeller. Koden FEMIC1D utvecklades för att studera algoritmens egenskaper och för att simulera ICRH, inklusive FLR-effekter. Koden har tillämpats på ett fall för att studera harmonisk dämpning och modkonvertering till så kallade jon-Bernsteinvågor. I denna studie verifierades att algoritmen kan ta hänsyn till FLR-effekter genom att studera jon-Bernsteinvågens egenskaper.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2019. , p. 77
Series
TRITA-EECS-AVL ; 2020:4
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Electrical Engineering
Identifiers
URN: urn:nbn:se:kth:diva-265016ISBN: 978-91-7873-397-2 (print)OAI: oai:DiVA.org:kth-265016DiVA, id: diva2:1377064
Public defence
2020-01-17, F3, Lindstedtsvägen 26, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20191211

Available from: 2019-12-11 Created: 2019-12-10 Last updated: 2019-12-19Bibliographically approved
List of papers
1. An iterative method to include spatial dispersion for waves in nonuniform plasmas using wavelet decomposition
Open this publication in new window or tab >>An iterative method to include spatial dispersion for waves in nonuniform plasmas using wavelet decomposition
2016 (English)In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 775, no 1, article id 012016Article in journal (Refereed) Published
Abstract [en]

A novel method for solving wave equations with spatial dispersion is presented, suitable for applications to ion cyclotron resonance heating. The method splits the wave operator into a dispersive and a non-dispersive part. The latter can be inverted with e.g. finite element methods. The spatial dispersion is evaluated using a wavelet representation of the dielectric kernel and added by means of iteration. The method has been successfully tested on a low frequency kinetic Alfvén wave with second order Larmor radius effects in a nonuniform plasma slab.

Keywords
Cyclotron resonance, Finite element method, Fusion reactions, Iterative methods, Wave equations, Wavelet decomposition, Ion cyclotron resonance heating, Larmor radius effects, Nonuniform plasma, Second orders, Solving wave equations, Spatial dispersion, Wave operators, Wavelet representation, Dispersion (waves)
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-201866 (URN)10.1088/1742-6596/775/1/012016 (DOI)000409406300016 ()2-s2.0-85009809630 (Scopus ID)
Conference
29 August 2016 through 2 September 2016
Note

QC 20170308

Available from: 2017-03-08 Created: 2017-03-08 Last updated: 2019-12-10Bibliographically approved
2. Modeling RF waves in spatially dispersive inhomogeneus plasma using an iterative wavelet spectral method
Open this publication in new window or tab >>Modeling RF waves in spatially dispersive inhomogeneus plasma using an iterative wavelet spectral method
2017 (English)In: EPJ Web of Conferences, EDP Sciences, 2017, Vol. 157, article id 03059Conference paper (Refereed)
Abstract [en]

The wave equation for a spatially dispersive inhomogeneous magnetized plasma is given by an integro-differential equation. The effects caused by spatial dispersion in the directions perpendicular and parallel to the magnetic field are quite different. In this study, we show how to solve the wave equation using a newly developed iterative wavelet spectral method for two cases. In the first case, the method is applied to a propagating kinetic Alfvén wave in the perpendicular direction and solved to all orders in FLR. To conserve the kinetic energy flux, first order corrections in equilibrium gradients are used in the dielectric response tensor. In the second case, we verify the method for a fast wave minority heating scenario and study the up-and downshift in the parallel wave number.

Place, publisher, year, edition, pages
EDP Sciences, 2017
Series
EPJ Web of Conferences, ISSN 2101-6275 ; 157
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-217484 (URN)10.1051/epjconf/201715703059 (DOI)2-s2.0-85032624870 (Scopus ID)
Conference
22nd Topical Conference on Radio-Frequency Power in Plasmas 2017, Centre de Congres, Aix en Provence, France, 30 May 2017 through 2 June 2017
Note

QC 20171113

Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2019-12-10Bibliographically approved
3. A numerical tool based on FEM and wavelets to account for spatial dispersion in ICRH simulations
Open this publication in new window or tab >>A numerical tool based on FEM and wavelets to account for spatial dispersion in ICRH simulations
2018 (English)In: Journal of Physics: Conference Series, Institute of Physics Publishing , 2018, no 1Conference paper, Published paper (Refereed)
Abstract [en]

Modeling of Ion Cyclotron Resonance Heating (ICRH) is difficult because of spatial dispersion. Numerical methods based on finite element or finite difference have difficulties in handling spatial dispersive effects, because the response is non-local. Fourier spectral methods can handle spatial dispersion, however, these methods have difficulties in handling the complex geometries outside the plasma domain and tend to produce dense matrices that are time consuming to invert. In this study, we investigate the potential of a new numerical method for solving the spatially dispersive wave equation based on FEM and wavelets. The spatially dispersive terms in the wave equation are evaluated using wavelets, and its contribution is represented as an induced current density in the wave equation. The wave equation is then solved using a finite element scheme, where the induced current density is represented as an inhomogeneous term and added using a fixed point iteration scheme. The method is applied to a case of one dimensional fast wave minority heating, including the up- and downshift in the parallel wave number, where we show that convergence can be obtained in a few iterations.

Place, publisher, year, edition, pages
Institute of Physics Publishing, 2018
Keywords
Cyclotron resonance, Finite element method, Fusion reactions, Iterative methods, Numerical methods, Wave equations, Complex geometries, Dispersive waves, Finite element schemes, Fixed point iteration, Ion cyclotron resonance heating, Numerical tools, Spatial dispersion, Spectral methods, Dispersion (waves)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-247044 (URN)10.1088/1742-6596/1125/1/012020 (DOI)2-s2.0-85058276573 (Scopus ID)
Conference
2018 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas, 27 August 2018 through 31 August 2018
Note

QC 20190625

Available from: 2019-06-25 Created: 2019-06-25 Last updated: 2019-12-10Bibliographically approved
4. Effect of poloidal phasing on ion cyclotron resonance heating power absorption
Open this publication in new window or tab >>Effect of poloidal phasing on ion cyclotron resonance heating power absorption
Show others...
2019 (English)In: Nuclear Fusion, ISSN 0029-5515, E-ISSN 1741-4326, Vol. 59, no 7, article id 076022Article in journal (Refereed) Published
Abstract [en]

Two ion cyclotron resonance heating (ICRH) systems are planned for ITER, each system containing 24 antennas distributed as a two by four array of poloidal triplets. The ITER antennas are designed to operate at a poloidal phase difference between the upper and lower triplet of Δθpol = -90° in the antenna currents. Since current tokamak experiments normally operate at Δθpol = 0°, experience from ICRH schemes with Δθpol °= 0 is lacking. In this paper, the effects of poloidal phasing on ICRH power absorption and coupling are studied using the novel code FEMIC, which is described here. Simulations of the ITER antenna and the JET ITER-like antenna show that increasing the poloidal phase difference increases the destructive interference of the fast magnetosonic wave near the equatorial plane. This causes a degradation of the on-axis heating performance and reduces the total coupled power to the plasma. Best on-axis heating was obtained for Δθpol = 0°, resulting in peaked profiles. By increasing the poloidal phase difference the absorption profiles tend to become less peaked or hollow on-axis. The effect is localized and occurs for °pol ° 0.1, i.e. near the magnetic axis. The total coupled power was found to be asymmetric around Δθpol = 0° due to the plasma gyrotropy, where the maximum coupled power occurs within ?33° ° Δθpol ° ?22° on ITER and JET. The exact location of the maximum depends on the width of the pedestal. The strength of the asymmetry increases with the pedestal width.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2019
Keywords
FEMIC, ICRH, ITER, Poloidal phasing
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-262632 (URN)10.1088/1741-4326/ab1ab7 (DOI)000470808100004 ()2-s2.0-85069038864 (Scopus ID)
Note

QC 20191016

Available from: 2019-10-16 Created: 2019-10-16 Last updated: 2019-12-10Bibliographically approved
5. Iterative addition of finite Larmor radius effects to finite element models using wavelet decomposition
Open this publication in new window or tab >>Iterative addition of finite Larmor radius effects to finite element models using wavelet decomposition
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Modeling the propagation and damping of electromagnetic waves in a hot magnetized plasma is difficult due to spatial dispersion. In such media, the dielectric response becomes non-local and the wave equation an integro-differential equation. In the application of RF heating and current drive in tokamak plasmas, the finite Larmor radius (FLR) causes spatial dispersion, which gives rise to physical phenomena such as higher harmonic ion cyclotron damping and mode conversion to electrostatic waves. In this paper, a new numerical method based on an iterative wavelet finite element scheme is presented, which is suitable for adding non-local effects to the wave equation by iterations. To verify the method, we apply it to a case of one-dimensional fast wave heating at the second harmonic ion cyclotron resonance, and study mode conversion to ion Bernstein waves in a toroidal plasma. Comparison with a local (truncated FLR) model showed good agreement in general. The observed difference is in the damping of the ion Bernstein wave, where the proposed method predicts stronger damping on the ion Bernstein wave.

Keywords
Morlet wavelets, finite element method, ion cyclotron resonance heating, mode conversion, ion-Bernstein waves
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Electrical Engineering
Identifiers
urn:nbn:se:kth:diva-265018 (URN)
Available from: 2020-02-03 Created: 2019-12-10 Last updated: 2019-12-18Bibliographically approved
6. The effects of electron cyclotron heating and current drive on toroidal Alfven eigenmodes in tokamak plasmas
Open this publication in new window or tab >>The effects of electron cyclotron heating and current drive on toroidal Alfven eigenmodes in tokamak plasmas
Show others...
2018 (English)In: Plasma Physics and Controlled Fusion, ISSN 0741-3335, E-ISSN 1361-6587, Vol. 60, no 1, article id 014026Article in journal (Refereed) Published
Abstract [en]

Dedicated studies performed for toroidal Alfven eigenmodes (TAEs) in ASDEX-Upgrade (AUG) discharges with monotonic q-profiles have shown that electron cyclotron resonance heating (ECRH) can make TAEs more unstable. In these AUG discharges, energetic ions driving TAEs were obtained by ion cyclotron resonance heating (ICRH). It was found that off-axis ECRH facilitated TAE instability, with TAEs appearing and disappearing on timescales of a few milliseconds when the ECRH power was switched on and off. On-axis ECRH had a much weaker effect on TAEs, and in AUG discharges performed with co- and counter-current electron cyclotron current drive (ECCD), the effects of ECCD were found to be similar to those of ECRH. Fast ion distributions produced by ICRH were computed with the PION and SELFO codes. A significant increase in T-e caused by ECRH applied off-axis is found to increase the fast ion slowing-down time and fast ion pressure causing a significant increase in the TAE drive by ICRH-accelerated ions. TAE stability calculations show that the rise in T-e causes also an increase in TAE radiative damping and thermal ion Landau damping, but to a lesser extent than the fast ion drive. As a result of the competition between larger drive and damping effects caused by ECRH, TAEs become more unstable. It is concluded, that although ECRH effects on AE stability in present-day experiments may be quite significant, they are determined by the changes in the plasma profiles and are not particularly ECRH specific.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD, 2018
Keywords
energetic particles, Alfven eigenmodes, ECRH, ECCD, ICRH
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-217926 (URN)10.1088/1361-6587/aa90ee (DOI)000414369100001 ()
Note

QC 20171121

Available from: 2017-11-21 Created: 2017-11-21 Last updated: 2019-12-10Bibliographically approved
7. 3D Finite Element Modelling of ICRH in WEST
Open this publication in new window or tab >>3D Finite Element Modelling of ICRH in WEST
2019 (English)In: Proceedings 46th EPS Conference on Plasma Physics, 2019, article id P4.1082Conference paper, Published paper (Other academic)
Abstract [en]

The Ion Cyclotron Resonance Heating (ICRH) antenna in WEST has been modelled with the finite element method in 3D. A detailed geometry was used along with a hot plasma model in the plasma region. The convergence of the total absorbed power and the electron power partition was studied by varying different mesh parameters. To obtain a better resolved solution and a wave field without reflections, it is estimated that 1 TB of RAM is required. The coupled power spectrum was also studied using a two-dimensional Fourier decomposition of the electromagnetic fields.

Keywords
ICRH, FEM
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Electrical Engineering
Identifiers
urn:nbn:se:kth:diva-265017 (URN)2-s2.0-85074275287 (Scopus ID)
Conference
46th EPS Conference on Plasma Physics, July 8-12, 2019, Milan, Italy
Note

QC 20191219

Available from: 2019-12-10 Created: 2019-12-10 Last updated: 2019-12-19Bibliographically approved

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