Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
On Monte Carlo Operators for Studying Collisional Relaxation in Toroidal Plasmas
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns modelling of Coulomb collisions in toroidal plasma with Monte Carlo operators, which is important for many applications such as heating, current drive and collisional transport in fusion plasmas. Collisions relax the distribution functions towards local isotropic ones and transfer power to the background species when they are perturbed e.g. by wave-particle interactions or injected beams. The evolution of the distribution function in phase space, due to the Coulomb scattering on background ions and electrons and the interaction with RF waves, can be obtained by solving a Fokker-Planck equation.The coupling between spatial and velocity coordinates in toroidal plasmas correlates the spatial diffusion with the pitch angle scattering by Coulomb collisions.

In many applications the diffusion coefficients go to zero at the boundaries or in a part of the domain, which makes the SDE singular. To solve such SDEs or equivalent diffusion equations with Monte Carlo methods, we have proposed a new method, the hybrid method, as well as an adaptive method, which selects locally the faster method from the drift and diffusion coefficients. The proposed methods significantly reduce the computational efforts and improves the convergence.

The radial diffusion changes rapidly when crossing the trapped-passing boundary creating a boundary layer. To solve this problem two methods are proposed. The first one is to use a non-standard drift term in the Monte Carlo equation. The second is to symmetrize the flux across the trapped passing boundary. Because of the coupling between the spatial and velocity coordinates drift terms associated with radial gradients in density, temperature and fraction of the trapped particles appear. In addition an extra drift term has been included to relax the density profile to a prescribed one.

A simplified RF-operator in combination with the collision operator has been used to study the relaxation of a heated distribution function. Due to RF-heating the density of thermal ions is reduced by the formation of a high-energy tail in the distribution function. The Coulomb collisions tries to restore the density profile and thus generates an inward diffusion of thermal ions that results in a peaking of the total density profile of resonant ions.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , xi, 65 p.
Series
Trita-EES, 2013:014
Keyword [en]
Fusion plasma, thermonuclear fusion, tokamak, Coulomb collisions, stochastic differential equations with singular diffusion coefficients, Monte Carlo schemes, spatial diffusion, modelling, Fokker-Planck equation, RF-heating.
National Category
Fusion, Plasma and Space Physics
Identifiers
URN: urn:nbn:se:kth:diva-120590ISBN: 978-91-7501-709-9 (print)OAI: oai:DiVA.org:kth-120590DiVA: diva2:615851
Public defence
2013-05-13, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20130415

Available from: 2013-04-15 Created: 2013-04-12 Last updated: 2013-10-18Bibliographically approved
List of papers
1. On Solving Singular Diffusion Equations With Monte Carlo Methods
Open this publication in new window or tab >>On Solving Singular Diffusion Equations With Monte Carlo Methods
2010 (English)In: IEEE Transactions on Plasma Science, ISSN 0093-3813, E-ISSN 1939-9375, Vol. 38, no 9, 2185-2189 p.Article in journal (Refereed) Published
Abstract [en]

Diffusion equations in one, two, or three dimensions with inhomogeneous diffusion coefficients are usually solved with finite-difference or finite-element methods. For higher dimensional problems, Monte Carlo solutions to the corresponding stochastic differential equations can be more effective. The inhomogeneities of the diffusion constants restrict the time steps. When the coefficient in front of the highest derivative of the corresponding differential equation goes to zero, the equation is said to be singular. For a 1-D stochastic differential equation, this corresponds to the diffusion coefficient that goes to zero, making the coefficient strongly inhomogeneous, which, however, is a natural condition when the process is limited to a region in phase space. The standard methods to solve stochastic differential equations near the boundaries are to reduce the time step and to use reflection. The strong inhomogeneity at the boundary will strongly limit the time steps. To allow for longer time steps for Monte Carlo codes, higher order methods have been developed with better convergence in phase space. The aim of our investigation is to find operators producing converged results for large time steps for higher dimensional problems. Here, we compare new and standard algorithms with known steady-state solutions.

Keyword
Diffusion equations, Monte Carlo methods, simulation, stochastic differential equations
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-27684 (URN)10.1109/TPS.2010.2057259 (DOI)000283252500014 ()2-s2.0-77956617100 (Scopus ID)
Note
QC 20101221Available from: 2010-12-21 Created: 2010-12-20 Last updated: 2017-12-11Bibliographically approved
2. On Monte Carlo operators describing Coulomb collisions in toroidal plasmas
Open this publication in new window or tab >>On Monte Carlo operators describing Coulomb collisions in toroidal plasmas
2011 (English)Conference paper, Published paper (Other academic)
Series
38th EPS Conference on Plasma Physics 2011, EPS 2011 - Europhysics Conference Abstracts
National Category
Natural Sciences
Identifiers
urn:nbn:se:kth:diva-81183 (URN)2-s2.0-84867638897 (Scopus ID)978-161839593-1 (ISBN)
Conference
38th EPS Conference on Plasma Physics 2011, EPS 2011; Strasbourgh, France, 27 June - 1 July 2011
Note

QC 20140908

Available from: 2012-02-10 Created: 2012-02-10 Last updated: 2014-10-02Bibliographically approved
3. On modelling Coulomb collisions in toroidal plasmas with orbit averaged Monte Carlo operators
Open this publication in new window or tab >>On modelling Coulomb collisions in toroidal plasmas with orbit averaged Monte Carlo operators
(English)Article in journal (Other academic) Submitted
National Category
Natural Sciences
Identifiers
urn:nbn:se:kth:diva-120606 (URN)
Note

QS 2013

Available from: 2013-04-15 Created: 2013-04-15 Last updated: 2013-04-15Bibliographically approved
4. A model Monte Carlo collision operator for toroidal plasmas
Open this publication in new window or tab >>A model Monte Carlo collision operator for toroidal plasmas
2013 (English)In: Plasma Physics and Controlled Fusion, ISSN 0741-3335, E-ISSN 1361-6587, Vol. 55, no 10, 105002- p.Article in journal (Refereed) Published
Abstract [en]

In order to simulate radio refquency (RF)-heating in toroidal plasmas in the banana regime a model collision operator has been developed, which relaxes the distribution function towards a prescribed local Maxwellian either determined by experiments or transport codes. The pitch angle scattering by Coulomb collisions gives rise to spatial diffusion in toroidal plasmas because of the coupling between spatial and velocity coordinates. The coupling between the spatial and velocity components results in drift terms in the Monte Carlo formulation of the Fokker-Planck equation due to spatial derivatives of the Jacobian, the fraction of the trapped particles, the density and the temperature profiles. A simple RF operator is used to test the collision operator in conjunction with RF heating. The formation of a high-energy tail on the distribution function during RF heating leads to reduction of the density of the thermal ions as the tail builds up. For central heating this reduction can lead to hollow density profiles of thermal ions. The spatial diffusion caused by the relaxation of the thermal ions towards a prescribed density profile then gives rise to an increase of the density of resonant ions in regions with strong heating where the thermal ions diffuse towards higher energies.

Keyword
Diffusion, Tokamaks, Waves, Equations
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-120607 (URN)10.1088/0741-3335/55/10/105002 (DOI)000324625600003 ()2-s2.0-84885131727 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20131018. Updated from submitted to published.

Correction in: Plasma Physics and Controlled Fusion, vol. 55, issue 11. article nr. 119601, doi: 10.1088/0741-3335/55/11/119601, wos: 000326242200016, ScopusID: 2-s2.0-84886896267

Available from: 2013-04-15 Created: 2013-04-15 Last updated: 2017-12-06Bibliographically approved

Open Access in DiVA

Qaisar_Thesis(2173 kB)358 downloads
File information
File name FULLTEXT02.pdfFile size 2173 kBChecksum SHA-512
593a9dcd69fc01987d82f8351f2fc9c1244e537d749e5d3ecd36257645744e72f864438e3fcc9ebc958c815e02df1225ce77490c268a356d884d8ef260209c4b
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Mukhtar, Qaisar
By organisation
Fusion Plasma Physics
Fusion, Plasma and Space Physics

Search outside of DiVA

GoogleGoogle Scholar
Total: 358 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 188 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf