# Mathematics

## ECMM731 - Magnetic Fields and Fluid Flows (2018)

MODULE TITLE CREDIT VALUE Magnetic Fields and Fluid Flows 15 ECMM731 Dr Joanne Mason (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 26
DESCRIPTION - summary of the module content

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.

Prerequisite module: ECM3707 or equivalent

AIMS - intentions of the module

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 ECM3707 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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.

SYLLABUS PLAN - summary of the structure and academic content of the module

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.

- force-free fields, including potential fields and field-parallel currents;
- pressure balanced configurations and plasma confinement;

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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 27 Lectures Scheduled learning and teaching activities 6 Examples classes Guided independent study 30 Problem sheets Guided independent study 87 Reading, revision, preparation

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Problem sheets containing a mixture of short answer questions and more comprehensive exercises 6 hours/4 questions per problem sheet (5 sheets) 1-6 Examples class, solutions and general comments uploaded to ELE

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 100 2 hours - Summer Exam Period 1-4 Oral on request

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

Referred and deferred assessment will be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: http://vle.exeter.ac.uk