## MATH3459 Astrophysical Fluid Dynamics

### 15 creditsClass Size: 35

Module manager: Prof David Hughes
Email: d.w.hughes@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2019/20

### Pre-requisites

 MATH3620 Fluid Dynamics 2

### This module is mutually exclusive with

This module is not approved as a discovery module

### Module summary

This module introduces some of the most important ideas in Astrophysical Fluid Dynamics. Gases in astrophysics (e.g. in stellar interiors) are typically electrically conducting, and hence can support a magnetic field, and also compressible. The module will first expand the standard equations of incompressible fluid dynamics to incorporate these effects. It will then look at some of the key properties introduced by a magnetic field. Three specific topics of widespread astrophysical relevance will be covered in detail:- Wave motions in astrophysical fluids.- Dynamo theory. How can a magnetic field be maintained?- Strong flows onto and away from stars (stellar accretion and winds).

### Objectives

At the end of the module, students should be able to:
- State the equations of compressible fluid dynamics.
- Derive Bernoulli's equation
- Derive the induction equation from Maxwell's equations and Ohm's law
- Describe the Lorentz force in terms of a magnetic pressure and tension
- Describe magnetic fields in terms of field lines and flux functions
- Give basic properties of the induction equation at low and high magnetic Reynolds number (Rm)
- Derive the equations of the different linear waves in magnetohydrodynamics.
- Obtain the solution for dynamo waves.
- Derive the solutions for spherical accretion and winds.

Learning outcomes
At the end of the module, students should be able to:
- State the equations of compressible fluid dynamics.
- Derive Bernoulli's equation
- Derive the induction equation from Maxwell's equations and Ohm's law
- Describe the Lorentz force in terms of a magnetic pressure and tension
- Describe magnetic fields in terms of field lines and flux functions
- Give basic properties of the induction equation at low and high magnetic Reynolds number (Rm)
- Derive the equations of the different linear waves in magnetohydrodynamics.
- Obtain the solution for dynamo waves.
- Derive the solutions for spherical accretion and winds.

### Syllabus

- Extending the equations of fluid dynamics to incorporate compressibility. Simple considerations of thermodynamics.
- The equations of magnetohydrodynamics (MHD). Deriving the magnetic induction equation. Incorporating the Lorentz force into the Navier-Stokes equation.
- The induction equation. The magnetic Reynolds number Rm. The low Rm limit. The perfectly conducting limit and the Cauchy solution. Simple solutions with advection and diffusion.
- The Lorentz force. Magnetic pressure and tension. Potential and force-free fields.
- Waves: Alfvén waves, magnetoacoustic waves, internal gravity waves, inertial waves.
- Introduction to dynamo theory. Anti-dynamo theorems. Dynamo waves.
- Spherically symmetric stellar accretion, winds and braking.

### Teaching methods

 Delivery type Number Length hours Student hours Lectures 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course materials. Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular example sheets

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated