Department of Mathematics

Introduction

Dr Cullen is a senior scientist at the Met Office and also holds a Visiting Professorship at the University of Reading. He is a long-established expert in atmospheric dynamics (particularly on large-scales) and numerical modelling of the atmosphere. He played the leading role in designing the numerical method used in the Met Office Unified Model from 1991-2002, and also carried out the initial design for the upgraded scheme used in the model since then. He has specialised in the mathematical analysis of limit equations which describe large-scale flows in the atmosphere and ocean, and has published a recent book and many articles on the subject. He has initiated collaborations between working meteorologists and several branches of rigorous mathematics. Many of these developed from the programme at the Isaac Newton Institute on "Atmosphere/Ocean Dynamics" held in 1996. These collaborations have resulted in the proving of a number of fundamental theorems showing that the equations describing large-scale flow can be solved. His current job within the Met Office is research into improving methods of data assimilation.

Course content:

The main purpose of the course is to describe the mathematical techniques which have been used to analyse the large-scale flow of the atmosphere. The derivation of suitable limit equations, which only describe large-scale flow, will be described. It turns out to be highly advantageous to derive these by Lagrangian averaging of the governing equations rather than Eulerian averaging. The limit equations are then naturally written in a Lagrangian form. Lagrangian analysis has had limited application for general fluid flows, because trajectories become highly tangled and it is impossible to take limits of approximate solutions. However, they are very powerful for the large-scale limit equations because it is possible to show that the pressure field has the same analytic properties as a convex function, which means that limits of both the pressure and pressure gradients can be taken.

The underlying conservation laws represented by the equations take a different from in Lagrangian coordinates. It is natural to express the mass conservation constraint using the mass transportation formulation introduced by Monge in 1780. The course will introduce the modern version of this theory, which has had many applications, such as in economics and image matching, as well as in meteorology. In this formulation the aim is to minimise a transport cost in which a given mass has to be transported between two locations. The problem sessions in the course will invite students to solve some simple mass transportation problems explicitly.

In large-scale meteorology, this cost represents the energy. The theory states that large-scale atmospheric flows evolve through a sequence of minimum energy states, where the energy is minimised with respect to a class of variations appropriate to large-scale flows. The course will review some of the key results that have been proved in this area, and outline some of the open questions which are currently being tackled.

References:

Cullen,M.J.P. (2006) A Mathematical Theory of Large-Scale Atmospheric Flow. Imperial College Press.

Majda, A. J. (2003) Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes, 9, American Math. Society.
Norbury,J. and Roulstone, I. eds. (2002) Large-Scale Atmosphere-Ocean Dynamics., vols.1 and 2. Cambridge University Press,

Villani, C. (2003) Topics in optimal transportation. Vol. 58 of Graduate Studies in Mathematics, Amer.Math. Soc., Providence, RI.

Programme (PROVISIONAL)

(ALL LECTURES IN ROOM 113 MATHS)

April 2

12-1 Lecture 1

[[Lunch break]]

Lagrangian form of the governing equations

2-3 Lecture 2

[[Coffee Break]]

Introduction to mass transportation

3:30-5 Problem session

Session 1: mass transportation problems

April 3

10-11 Lecture 3

[[Coffee break]]

Equations for large scale flow and solution as a sequence of energy minimising states

11:30-12:30 Lecture 4

[[Lunch break]]

Solution in a simple case

2-4 Problem session

[[Coffee Break]]

Session 2: semi-geostrophic problems

April 4

10-11 Lecture 5

[[Coffee break]]

Solution in further cases for uniform rotation

11:30-12:30 Lecture 6

[[Lunch break]]

Solution in spherical geometry and with additional physics

2-4 Problem session

[[Coffee Break]]

Accommodation

Registration

Further Information