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The dynamics of Alfvén eigenmodes excited by energetic ions in toroidal plasmas
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.ORCID iD: 0000-0002-3262-1958
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The future fusion power plants that are based on magnetic confinement will deal with plasmas that inevitably contain energetic (non-thermal) particles. These particles come, for instance, from fusion reactions or from external heating of the plasma. Ensembles of energetic ions can excite eigenmodes in the Alfvén frequency range to such an extent that the resulting wave fields redistribute the energetic ions, and potentially eject them from the plasma. The redistribution of ions may cause a substantial reduction of heating efficiency. Understanding the dynamics of such instabilities is necessary to optimise the operation of fusion experiments and of future fusion power plants.

Two models have been developed to simulate the interaction between energetic ions and Alfvén eigenmodes. One is a bump-on-tail model, of which two versions have been developed: one fully nonlinear and one quasilinear. The quasilinear version has a lower dimensionality of particle phase space than the nonlinear one. Unlike previous similar studies, the bump-on-tail model contains a decorrelation of the wave-particle phase in order to model stochasticity of the system. When the characteristic time scale for macroscopic phase decorrelation is similar to or shorter than the time scale of nonlinear wave-particle dynamics, the nonlinear and the quasilinear descriptions quantitatively agree. A finite phase decorrelation changes the growth rate and the saturation amplitude of the wave mode in systems with an inverted energy distribution around the wave-particle resonance. Analytical expressions for the correction of the growth rate and the saturation amplitude have been derived, which agree well with numerical simulations. A relatively weak phase decorrelation also diminishes frequency chirping events of the eigenmode.

The second model is called FOXTAIL, and it has a wider regime of validity than the bump-on-tail model. FOXTAIL is able to simulate systems with multiple eigenmodes, and it includes effects of different individual particle orbits relative to the wave fields. Simulations with FOXTAIL and the nonlinear bump-on-tail model have been compared in order to determine the regimes of validity of the bump-on-tail model quantitatively. Studies of two-mode scenarios confirmed the expected consequences of a fulfillment of the Chirikov criterion for resonance overlap. The influence of ICRH on the eigenmode-energetic ion system has also been studied, showing qualitatively similar effects as seen by the presence of phase decorrelation.

Another model, describing the efficiency of fast wave current drive, has been developed in order to study the influence of passive components close to the antenna, in which currents can be induced by the antenna generated wave field. It was found that the directivity of the launched wave, averaged over model parameters, was lowered by the presence of passive components in general, except for low values of the single pass damping of the wave, where the directivity was slightly increased, but reversed in the toroidal direction.

Abstract [sv]

De framtida fusionskraftverken baserade på magnetisk inneslutning kommer att hantera plasmor som oundvikligen innehåller energetiska (icke-termiska) partiklar. Dessa partiklar kommer exempelvis från fusionsreaktioner eller från externa uppvärmningsmekanismer av plasmat. Ensembler av energetiska joner kan excitera egenmoder i Alfvén-frekvensområdet i en sådan utsträckning att de resulterande vågfälten omfördelar de energetiska jonerna i rummet, och potentiellt slungar ut jonerna ur plasmat. Omfördelningen av joner kan orsaka en väsentligen minskad uppvärmningseffekt. Det är nödvändigt att förstå dynamiken hos denna typ av instabilitet för att kunna optimera verkningsgraden hos experiment och hos framtida fusionskraftverk.

Två modeller har utvecklats för att simulera interaktionen mellan energetiska joner och Alfvén-egenmoder. Den första är en bump-on-tail-modell, av vilken två versioner har utvecklats: en fullt icke-linjär och en kvasi-linjär. I den kvasi-linjära versionen har partiklarnas fasrum en lägre dimensionalitet än i den icke-linjära versionen. Till skillnad från tidigare liknande studier innehåller denna bump-on-tail-modell en dekorrelation av våg-partikelfasen för att modellera stokasticitet hos systemet. När den karakteristiska tidsskalan för makroskopisk fasdekorrelation är ungefär samma som eller kortare än tidsskalan för icke-linjär våg-partikeldynamik så stämmer den icke-linjära och den kvasi-linjära beskrivningen överens kvantitativt. En ändlig fasdekorrelation förändrar vågmodens tillväxthastighet och satureringsamplitud i system med en inverterad energifördelning omkring våg-partikelresonansen. Analytiska uttryck för korrektionen av tillväxthastigheten och satureringsamplituden har härletts, vilka stämmer väl överens med numeriska simuleringar. En relativt svag fasdekorrelation försvagar även "frequency chirping events" (snabba frekvensskiftningar i korttids-Fourier-transformen av egenmodens amplitudutveckling) hos egenmoden.

Den andra modellen, kallad FOXTAIL, har ett mycket bredare giltighetsområde än bump-on-tail-modellen. FOXTAIL kan simulera system med flera egenmoder, och den inkluderar effekter av olika enskilda partikelbanor relativt vågfälten. Simuleringar med FOXTAIL och med bump-on-tail-modellen har jämförts för att kvantitativt bestämma bump-on-tail-modellens giltighetsområde. Studier av scenarier med två egenmoder bekräftar de förväntade effekterna av när Chirikov-kriteriet för resonansöverlapp uppfylls. Även inflytandet av ICRH på dynamiken mellan egenmoder och energetiska joner har studerats, vilket har visat kvalitativt liknande effekter som har observerats i närvaron av fasdekorrelation.

En annan modell, vilken beskriver effektiviteten hos "fast wave current drive" (strömdrivning med snabba magnetosoniska vågor), har utvecklats för att studera inflytandet av passiva komponenter nära antennen, i vilka strömmar kan induceras av vågfälten som genereras av antennen. Det visades att den utskickade vågens direktivitet, medelvärdesbildat över modellparametrar, generellt sett minskade vid närvaron av passiva komponenter, förutom vid låg "sinlge pass damping" (dämpning av vågen vid propagering genom hela plasmat), då direktiviteten istället ökade något, men bytte tecken i toroidal riktning.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , 106 p.
Series
TRITA-EE, ISSN 1653-5146 ; 2016:141
Keyword [en]
Fusion plasma physics, Tokamak, Wave–particle interactions, Toroidal Alfvén eigenmodes, Bump-on-tail instabilities, Magnetohydrodynamics, Nonlinear dynamics, Quasilinear dynamics, Hamiltonian mechanics, Monte Carlo method, ICRH, FWCD
National Category
Fusion, Plasma and Space Physics
Research subject
Electrical Engineering
Identifiers
URN: urn:nbn:se:kth:diva-193029ISBN: 978-91-7729-122-0 (print)OAI: oai:DiVA.org:kth-193029DiVA: diva2:974450
Public defence
2016-10-28, D2, Lindstedtsvägen 5, Stockholm, 09:15 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 621-2011-5387
Note

QC 20160927

Available from: 2016-09-27 Created: 2016-09-26 Last updated: 2017-06-08Bibliographically approved
List of papers
1. Monte-Carlo model for nonlinear interactions of Alfvén eigenmodes with energetic ions
Open this publication in new window or tab >>Monte-Carlo model for nonlinear interactions of Alfvén eigenmodes with energetic ions
2012 (English)In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 401, no 1, 012024- p.Article in journal (Refereed) Published
Abstract [en]

A Monte-Carlo model for interactions between a single Alfvén eigenmode and energetic ions in a tokamak is presented. A phenomenological decorrelation of the wave-particle phase is introduced to mimic decorrelation by collisions or by other waves. Analysis is dedicated to how the strength of the phase decorrelation affects the nonlinear wave-particle interactions. Several of the phenomena that have been observed in some earlier models describing the nonlinear dynamics of Alfvén eigenmodes have been verified, such as the growth and saturation of the wave mode amplitude giving rise to a localized flattening of the distribution function, as well as the generation of coherent structures in the distribution function. The degree of phase decorrelation is shown to have a strong effect on the dynamics of the Alfvén eigenmode excitation.

Keyword
Coherent structure, De correlations, Eigen modes, Energetic ion, Monte-carlo models, Nonlinear interactions, Wave modes, Wave-particle interactions
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-118913 (URN)10.1088/1742-6596/401/1/012024 (DOI)000312549900024 ()2-s2.0-84874163425 (Scopus ID)
Conference
13th Joint Varenna-Lausanne International Workshop on Theory of Fusion Plasmas; Varenna; Italy; 27 August 2012 through 31 August 2012
Note

QC 20130304

Available from: 2013-03-04 Created: 2013-03-04 Last updated: 2017-12-06Bibliographically approved
2. Modeling the dynamics of toroidal Alfvén eigenmodes
Open this publication in new window or tab >>Modeling the dynamics of toroidal Alfvén eigenmodes
2013 (English)Conference paper, Published paper (Other academic)
Abstract [en]

A model describing nonlinear dynamics of a single Alfvén eigenmode excited by an inverted energy distribution of energetic ions is presented, suitable for drift orbit averaged Monte Carlo codes. The nonlinear dynamics of the wave mode is modeled with a complex wave amplitude, and is characterized by the formation of coherent structures in phase space, caused by wave-particle interaction. The transition to a quasilinear regime is modeled with a phenomenological decorrelation of the wave-particle phase. As the decorrelation is increased the coherent phase-space structures diminishes, and frequency chirping events in the marginal stability region is limited. The strength of the decorrelation modifies the saturation level and saturation time of the eigenmode amplitude.

Place, publisher, year, edition, pages
International Atomic Energy Agency, 2013
Keyword
Monte Carlo model, wave-particle interactions, bump-on-tail instabilities, toroidal Alfvén eigenmodes, nonlinear dynamics
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-161926 (URN)
Conference
13th IAEA Technical Meeting on “Energetic Particles in Magnetic Confinement Systems”, Beijing, China, September 17 - 20, 2013
Note

QC 20150319

Available from: 2015-03-19 Created: 2015-03-19 Last updated: 2016-09-27Bibliographically approved
3. Comparisons of the nonlinear and the quasilinear model for the bump-on-tail instability with phase decorrelation
Open this publication in new window or tab >>Comparisons of the nonlinear and the quasilinear model for the bump-on-tail instability with phase decorrelation
2014 (English)Conference paper, Published paper (Refereed)
Abstract [en]

The dynamics of discrete global modes in a toroidal plasma interacting with an energetic particle distribution is studied, and in particular when the dynamics of the system using the nonlinear and quasilinear descriptions are macroscopically similar. The dynamics can be described with a nonlinear bump-on-tail model in a two-dimensional phase space of particles. A Monte Carlo framework is developed for this model with an included decorrelation of the wave-particle phase, which is used to model extrinsic stochastisation of the wave-particle interactions. From this description, a quasilinear version of the model is also developed, which is described by a diffusive process in energy space due to the added phase decorrelation. Due to the reduced dimensionality of phase space, the quasilinear description is typically less computationally demanding than the nonlinear description. The purpose of the studies is to find conditions when a quasilinear model sufficiently describes the same phenomena of the wave-plasma interactions as a nonlinear model does. Via numerical and theoretical parameter studies, regimes where the two models overlap macroscopically are found. These regimes exist above a given threshold of the strength of the decorrelation, where coherent phase space structures are destroyed on time scales shorter than characteristic time scales of nonlinear particle motion in phase space close to the wave-particle resonance. Specifically for the quasilinear model, a theoretical value of the time scale of quasilinear flattening is derived and numerically verified.

Series
Journal of Physics: Conference Series, ISSN 1742-6588
Keyword
Dynamics, Time measurement, Wave plasma interactions
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-159056 (URN)10.1088/1742-6596/561/1/012019 (DOI)000346423600019 ()2-s2.0-84919338077 (Scopus ID)
Conference
Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas, SEP 01-05, 2014, Varenna, ITALY
Note

QC 20150122

Available from: 2015-01-22 Created: 2015-01-20 Last updated: 2017-03-24Bibliographically approved
4. The effects of phase decorrelation on the dynamics of the bump-on-tail instability
Open this publication in new window or tab >>The effects of phase decorrelation on the dynamics of the bump-on-tail instability
2015 (English)In: Physics of Plasmas, ISSN 1070-664X, E-ISSN 1089-7674, Vol. 22, no 8, 082106Article in journal (Refereed) Published
Abstract [en]

The dynamics of the bump-on-tail instability has been studied. The novelty of the work is the analysis of how the bump-on-tail dynamics is affected by an extrinsic stochastisation of the phase of the wave-particle interaction; here referred to as phase decorrelation. For this purpose, a nonlinear Monte Carlo model has been developed. When the characteristic time scale for macroscopic phase decorrelation becomes shorter than time scales of nonlinear wave-particle dynamics, the system may be described quasilinearly, with the phase decorrelation being replaced by a quasilinear diffusion coefficient in particle energy. A purely quasilinear Monte Carlo model, which is typically less computationally demanding than the fully nonlinear description due to the reduced dimensionality of phase space, has been developed for comparison. In this paper, parameter regimes, where the nonlinear and the quasilinear descriptions quantitatively agree on a macroscopic level, have been investigated, using combined theoretical and numerical analyses. Qualitative effects on the macroscopic dynamics by the presence of phase decorrelation and/or by structures of the energy distribution function in the proximity of the wave-particle resonance are also studied.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2015
Keyword
Monte Carlo model, wave-particle interactions, bump-on-tail instabilities, nonlinear dynamics, quasilinear dynamics, toroidal Alfvén eigenmodes
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-161935 (URN)10.1063/1.4928094 (DOI)000360647600010 ()2-s2.0-84938790389 (Scopus ID)
Funder
Swedish Research Council, 621-2011-5387
Note

QC 20151005. Updated from submitted to published.

Available from: 2015-03-27 Created: 2015-03-19 Last updated: 2017-12-04Bibliographically approved
5. A bump-on-tail model for Alfvén eigenmodes in toroidal plasmas
Open this publication in new window or tab >>A bump-on-tail model for Alfvén eigenmodes in toroidal plasmas
2015 (English)Conference paper, Published paper (Other academic)
Abstract [en]

Presented is a numerical model for solving the nonlinear dynamics of Alfvén eigenmodes and energetic ions self-consistently. The model is an extension of a previous bump-on-tail model [1,2], taking into account particle orbits and wave fields in realistic toroidal geometries. The model can be used in conjunction with an orbit averaged Monte Carlo code that handles heating and current drive (similar to e.g. the SELFO code), which enables modeling of the effects of MHD activity on plasma heating. For rapid particle tracing, the unperturbed guiding center orbits are described with canonical action-angle coordinates [3], and the perturbed Hamiltonian for wave-particle interaction is included as Fourier components in the same angles [4]. This allows the numerical integrator to take time steps over several transit periods, which efficiently resolves the relevant time scales for nonlinear wave-particle dynamics. The wave field is modeled by a static eigenfunction and a dynamic complex amplitude driven by the interactions with resonant and non-resonant particles.

[1] E. Tholerus, T. Hellsten and T. Johnson, Phys. Plasmas 22, 082106 (2015)

[2] S. Tholerus, T. Hellsten and T. Johnson, J. Phys.: Conf. Ser. 561, 012019 (2014)

[3] A. N. Kaufman, Phys. Fluids 15, 1063 (1972)

[4] H. L. Berk, B. N. Breizman and M. S. Pekker, Nucl. Fusion 35, 1713 (1995)

Place, publisher, year, edition, pages
International Atomic Energy Agency, 2015
Keyword
Monte Carlo model, wave-particle interactions, bump-on-tail instabilities, toroidal Alfvén eigenmodes, nonlinear dynamics
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192940 (URN)
Conference
14th IAEA Technical Meeting on “Energetic Particles in Magnetic Confinement Systems”, Vienna, Austria, September 1 - 4, 2015
Funder
Swedish Research Council, 621-2011-5387
Note

QC 20160927

Available from: 2016-09-23 Created: 2016-09-23 Last updated: 2016-09-27Bibliographically approved
6. FOXTAIL: Modeling the nonlinear interaction between Alfvén eigenmodes and energetic particles in tokamaks
Open this publication in new window or tab >>FOXTAIL: Modeling the nonlinear interaction between Alfvén eigenmodes and energetic particles in tokamaks
2016 (English)In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944Article in journal (Refereed) Submitted
Abstract [en]

FOXTAIL is a new hybrid magnetohydrodynamic-kinetic code used to describe interactions between energetic particles and Alfvén eigenmodes in tokamaks with realistic geometries. The code simulates the nonlinear dynamics of the amplitudes of individual eigenmodes and of a set of discrete markers in five-dimensional phase space representing the energetic particle distribution. Action-angle coordinates of the equilibrium system are used for efficient tracing of energetic particles, and the particle acceleration by the wave fields of the eigenmodes is Fourier decomposed in the same angles. The eigenmodes are described using temporally constant eigenfunctions with dynamic complex amplitudes. Possible applications of the code are presented, e.g., making a quantitative validity evaluation of the one-dimensional bump-on-tail approximation of the system. Expected effects of the fulfillment of the Chirikov criterion in two-mode scenarios have also been verified.

Place, publisher, year, edition, pages
Elsevier, 2016
Keyword
Magnetohydrodynamic waves, Tokamaks, Fast particle effects, Nonlinear dynamics, Hybrid plasma simulation methods, Lagrangian and Hamiltonian mechanics
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192923 (URN)
Funder
Swedish Research Council, 621-2011-5387
Note

QC 20160927

Available from: 2016-09-22 Created: 2016-09-22 Last updated: 2017-06-08Bibliographically approved
7. Modelling the Dynamics of Energetic Ions and MHD Modes Influenced by ICRH
Open this publication in new window or tab >>Modelling the Dynamics of Energetic Ions and MHD Modes Influenced by ICRH
2016 (English)Report (Other academic)
Abstract [en]

FOXTAIL is a code used to describe the nonlinear interactions between toroidal Alfvén eigenmodes and an ensemble of resonant energetic particles in tokamaks with realistic geometries. This report introduces an extension of the code, including effects from ion cyclotron resonance heating (ICRH) of energetic ions using a quasilinear diffusion operator in adiabatic invariant space. First results of the effects of ICRH diffusion on the system consisting of a single Alfvén eigenmode linearly excited by resonant ions are presented. It is shown that the presence of ICRH diffusion allows for the mode amplitude to grow larger than in the case of nonlinear saturation in the absence of sources and sinks. Gradually increasing the strength of ICRH diffusion also decreases the linear growth rate of the mode. Both these phenomena are previously observed also for the case of a finite phase decorrelation operator in bump-on-tail systems with a single eigenmode.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2016
Series
TRITA-EE, ISSN 1653-5146 ; 2016:142
Keyword
Monte Carlo model, wave-particle interactions, bump-on-tail instabilities, nonlinear dynamics, toroidal Alfvén eigenmodes, ion cyclotron resonance heating
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-192924 (URN)
Funder
Swedish Research Council, 621-2011-5387
Note

QC 20160927

Available from: 2016-09-22 Created: 2016-09-22 Last updated: 2016-09-27Bibliographically approved
8. On the Coupling of Waves for FWCD
Open this publication in new window or tab >>On the Coupling of Waves for FWCD
2014 (English)In: Radiofrequency power in plasmas, 2014, Vol. 1580, 330-333 p.Conference paper, Published paper (Refereed)
Abstract [en]

Coupling of an ICRF antenna to fast magnetosonic waves in a toroidal plasma is studied, considering incomplete damping of waves and passive conducting elements, e.g. conducting limiters, near the antenna. The system is characterized by coupling to a broad spectrum of partially overlapping resonant modes, whose frequency widths are due to finite damping. Small variations of plasma parameters, in particular the plasma density, can change the coupling to individual modes significantly. Therefore, a statistical analysis of the coupling is required. Currents in passive conductors near the antenna, which are induced, in particular by resonant modes, redistribute the coupling to on- and off-resonant modes. A statistical analysis is made to study how coupling to a large set of modes varies with respect to continuous variation of plasma parameters. At low single pass damping it is found that the coupling spectrum can be significantly modified by passive conducting components. This affects the directivity of the launched spectrum, which is important for fast wave current drive (FWCD).

Series
AIP Conference Proceedings, ISSN 0094-243X
Keyword
Fast Wave Current Drive, ICRF, Modeling
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-149997 (URN)10.1063/1.4864555 (DOI)000339626400060 ()2-s2.0-84906330004 (Scopus ID)978-0-7354-1210-1 (ISBN)
Conference
20th Topical Conference on Radio Frequency Power in Plasmas, JUN 25-28, 2013, Sorrento, ITALY
Note

QC 20140829

Available from: 2014-08-29 Created: 2014-08-29 Last updated: 2016-09-27Bibliographically approved

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