Aims

To introduce linear stability theory in the context of dynamical systems. To apply these ideas to mathematical models of real-world systems, taking examples from fluid mechanics and climate modelling.

Module Summary
Why can you hang an umbrella from a hook but not stand it on its point? Why do some fluid flows remain smooth while others become turbulent? Is the Earth's climate approaching a tipping point? Linear stability theory provides a mathematical framework to answer such questions.

Many real-world systems can be described using mathematical models, but the governing equations are often complicated and cannot be solved in general. Nevertheless, steady states can often be found, and linear stability theory then used to determine whether such states are robust against small perturbations. The theory also provides physical insight into the fundamental behaviour of the system.

Outline Of Syllabus

Developing mathematical models:
- Dimensionless variables and parameters
- Systems of ODEs and PDEs
- Models for the Earth’s ocean and atmosphere
- Linear stability theory:
- Linearization around steady state
- Normal modes
- Deriving stability criteria
- Advanced examples:
- A coupled “daisyworld” climate-vegetation model
- Kelvin-Helmholtz instability
- Rayleigh-Benard thermal convection

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work.

Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one way for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem-solving techniques, to practice the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.