# Mathematics / Applied Mathematics MSc

# 5M: Magnetohydrodynamics MATHS5050

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Either Semester 1 or Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

Magnetohydrodynamics (MHD for short) is the study of the flow of an electrically conducting fluid in the presence of a magnetic field. It is the basis of our understanding of a wealth of phenomena, for example: the generation of magnetic fields in planetary cores, the behaviour of solar flares and other phenomena in the atmosphere of the Sun, and the magnetic control of plasmas or liquid metals.

#### Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

#### Requirements of Entry

Mandatory Entry Requirements

Recommended Entry Requirements

#### Assessment

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

This course aims to give an introduction to the subject of Magnetohydrodynamics (the study of the flow of an electrically conducting fluid in the presence of a magnetic field) and in particular to present the basic theory and apply it to a variety of physical problems.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

1. State Maxwell's equations, make the magnetohydrodynamic approximation and derive the magnetic induction equation;

2. State the equation of mass continuity and the Navier-Stokes equations for an electrically conducting fluid;

3. Define the magnetic Reynolds number Rm and describe magnetic field evolution in the limits Rm → 0 and Rm → ∞ (where the field lines are "frozen into the fluid");

4. Solve the induction equation in the limit of infinite Rm via the Cauchy solution;

5. Sketch field lines for a given magnetic field B and deduce the directions of the magnetic pressure and tension forces;

6. Solve the governing equations [those discussed in (1.) and (2.)] for steady unidirectional flows ("Hartmann flow") and interpret the solution in the limits of small and large Hartmann number M;

7. Solve the governing equations for simple hydrostatic "pinch" configurations and discuss the role of such field configurations in plasma containment;

8. Discuss qualitatively the stability of pinch fields;

9. Linearise the governing equations and obtain the dispersion relations for Alfv´en waves and magneto-acoustic waves;

10. Define the Rayleigh number and solve the governing equations for the onset of buoyancy driven flow in a non-magnetic system, and describe how this is modified in a rotating system and/or when a magnetic field is present;

11. Discuss the application of MHD to Dynamo Theory and solve straightforward examples in this application.

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.