Magnetic Fields and Fluid Flows - 2024 entry

This module deals with the motion of electrically conducting fluids in the presence of magnetic fields, a subject known as magnetohydrodynamics (MHD). MHD flows play a crucial role in the dynamics of a variety of astrophysical systems (including stars, planets, accretion discs and galaxies). MHD flows are also studied in the laboratory with a view towards engineering applications (e.g. electromagnetic stirring and fusion plasmas). In this module, you will see how the mutual interaction of the fluid flow and the electromagnetic field reveals a variety of new and interesting phenomena. You will learn how to formulate a real physical problem in terms of a system of partial differential equations. We will solve these using a variety of techniques of applied mathematics.

Pre-requisite module: MTH3007 Fluid Dynamics, or equivalent

The aim of this module is to give you an introduction to the subject of electrically conducting fluid dynamics. This module can be seen as an extension of the third year module MTH3007 on viscous fluids. You will learn how the equations of fluid dynamics are modified when electromagnetic effects are taken in account. The mathematical theory will be illustrated with examples from astrophysics, geophysics and laboratory plasma physics.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Define the concepts and equations governing the interaction of magnetic fields and electrically conducting flows;

2 Prove some important theorems and analyse the solutions of the MHD equations using a variety of techniques of applied mathematics;

Discipline Specific Skills and Knowledge:

3 Formulate a real physical problem mathematically;

4 Explain mathematical solutions in terms of physical effects;

Personal and Key Transferable / Employment Skills and Knowledge:

5 Develop communication skills via in-class discussions;

6 Practice time management skills in order to meet coursework deadlines.

Fundamentals and Simple Examples:

- Motivation for studying magnetic fields and fluid flow;

- A brief introduction to electrodynamics: Maxwell’s equations;

- The equations of electrically conducting fluid dynamics;

- Basic properties of the induction equation, including the diffusive limit, the perfectly conducting limit and Alfvén’s theorem of flux freezing;

Magnetohydrostatics and Steady MHD Flows:

- Force-free fields, including potential fields and field-parallel currents;

- Pressure balanced configurations and plasma confinement;

- Steady unidirectional flow;

MHD Waves:

- Alfvén waves in an infinite, inviscid, incompressible fluid of infinite electrical conductivity;

- Phase velocity and group velocity;

- Damped Alfvén waves;

An Introduction to Dynamo Theory:

- Observational evidence for astrophysical magnetic fields;

- Formulation of the kinematic dynamo problem;

- Poloidal and toroidal fields;

- Antidynamo theorems;

- Qualitative picture of large-scale dynamo action, involving differential rotation and helical turbulence;

- Mean-field electrodynamics;

- Parker’s dynamo waves.

*Please refer to reassessment notes for details on deferral vs. referral reassessment

ELE: https://ele.exeter.ac.uk

Reading list for this module: