Aims

To introduce linear stability theory, and demonstrate how it can be used to understand the behaviour of mathematical models representing real-world systems, particularly in the field of 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 current climate stable? Linear stability theory provides a mathematical framework to answer such questions.

The time-evolution of innumerable real-world systems can be described using mathematical models, but the resulting equations can be complicated and nonlinear. Often there are no general solutions. Nonetheless, linear stability theory provides a way to determine whether a particular steady state of the system is stable against small perturbations. The theory also provides insight into the nature of the systems of equations themselves, and highlights profound connections between the theory of differential equations and linear algebra.

Outline Of Syllabus

Developing mathematical models:
•       Dimensionless variables and parameters
•       Equations of motion: ODEs and PDEs
•       Derive Energy Balance Climate Models
Introduction to linear stability theory:
•       Linearization around steady state
•       Normal modes
•       Classification of solutions: stability criteria
Advanced examples:
•       A coupled “daisyworld” climate-vegetation model, with ice-albedo feedback
•       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 ways 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 practise 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.