Wood, Andrew Maclean
(1999)
*Lattice Boltzmann Magnetohydrodynamics.*
PhD thesis, University of Glasgow.

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## Abstract

Understanding the dynamics of plasmas presents a formidable challenge in theoretical physics. Being governed by a complicated set of non-linear equations, analytic descriptions of their behaviour are only possible in the simplest of cases and therefore numerical methods are essential to understand any realistic situation. This thesis presents an application of the lattice Boltzmann (LB) method to the solution of the magnetohydrodynamic (MHD) equations, which model the low frequency motions of plasmas. The lattice Boltzmann method, which has been developed over the past decade or so, is a kinetic model of fluid like systems, derived from the statistical mechanics of lattice gas cellular automata. In chapter 1, after a brief derivation of the equations to be modelled and discussion of standard numerical methods, the basic ideas of cellular automata (CA) are reviewed along with some examples. Special attention is given to lattice gas models of hydrodynamics and magnetohy- drodynamics (MHD), with a discussion of the particular problem that an MHD model faces, namely the representation of the essentially non-local Lorentz force, and how this was overcome. In chapter 2 the lattice Boltzmann method is discussed in some detail. The LB models of two dimensional hydrodynamics and magnetohydrodynamics are explained and the Navier-Stokes and MHD equations are derived from these models. The derivation is standard in the literature and bears important similarities to the theory discussed in chapter 1 despite, in the case of the MHD model, the fundamentally different means by which the interactions between the particles and magnetic field are represented. An improvement of the Mill) model is proposed and a linear stability analysis is carried out. Alternative methods of discretising the lattice Boltzmann equation are also discussed. Various tests are presented in chapter 3. The simulations of Hartmann flow confirm previously published results, although we also model the evolution of the flow towards a steady state in the case of an unmagnetised fluid. Damped Alfven waves are also modelled. Both of these linear problems show good agreement between the numerical lattice Boltzmann solutions and the analytic solutions. Simulations of a non-linear reconnection problem are also presented, namely the coalescence of magnetic islands. The simulations reproduce correctly the qualitative features of island coalescence found in the literature. The lattice Boltzmann method is applied to a practical problem in chapter 4, namely the shedding of vortices in the wake of an obstacle. This problem is relevant to the dynamics of solar active regions, in which the photosphere is either stirred by or drags along an erupting magnetic flux tube. The observed vorticity in such regions is greater than can be accounted for by the action of the Coriolis force on the upwelling or downwelling fluid. The effect of a magnetic field on the vortex shedding process is investigated, and it is found that if the magnetic field is strong enough, then Alfven waves transport vorticity sufficiently fast to supress the vortex shedding process. In the case of a perpendicular magnetic field, reconnection is also observed in the wake. Generalisations of the lattice Boltzmann MHD model are proposed in chapter T). A thermal MHD model and a three dimensional model are presented, and the thermal model is tested by simulating magnetosonic waves, which show good agreement with the analytic solutions. Conclusions and suggestions for future work are discussed in chapter 6. The computer code for the numerical simulations is contained in the appendix. The original work for this thesis is the modification of the lattice Boltzmann MHD model in section 2.4.4, the stability analysis of section 2.4.5 and the work which appears in chapters 3, 4 and 5.

Item Type: | Thesis (PhD) |
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Qualification Level: | Doctoral |

Additional Information: | Adviser: Declan Diver |

Keywords: | Theoretical physics, Plasma physics |

Date of Award: | 1999 |

Depositing User: | Enlighten Team |

Unique ID: | glathesis:1999-75905 |

Copyright: | Copyright of this thesis is held by the author. |

Date Deposited: | 19 Nov 2019 17:38 |

Last Modified: | 19 Nov 2019 17:38 |

URI: | https://theses.gla.ac.uk/id/eprint/75905 |

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