REE14: Stochastic closures for weather and climate modelling

In many applications of fluid dynamics, analytical solutions are out of the question and computational resources are inadequate for a fully resolved numerical solution. It is thus necessary to account for unresolved scales and processes in simulation using some form of sub-grid modelling. This is usually referred to as a closure problem in fluid dynamics and theoretical physics, but as a 'parameterisation problem' in meteorology and climate science. In the past deterministic closures, relating average fluxes to resolved averages have been used. More recently, by generalising the closures to include stochastic parameters it has been found that improved forecasting skill for weather and climate can be achieved.

Aims

It is the aim of this project to investigate how the parameters in a stochastic closure can be estimated by solving the inverse problem of parameter estimation and data assimilation (E&A) using techniques founded upon clear statistical principles. This will require consideration of methods for solving the evolution equations for the conditional probability measures associated with such inverse problems. These problems evolve the joint probability measure of time varying dependent variables and time independent parameters.

Our principal aims are (i) to consolidate existing research advances into approximation methods that can be applied to problems in the same class as the high dimensional E&A problems encountered in climate research (ii) to show that the approximate probability measures converge to the exact measures of the evolution equation of the conditional measure (iii) to build intuition into the behaviour of the basic equations (for example the Kushner equation of estimation theory) in simple model examples (iv) to apply E&A methods to some models inspired by recent developments in stochastic closure theory by the Exeter group under Frank Kwasniok (v) to provide guidance to software applications developers as to the mathematical architecture required for software development and maintenance regarding principled methods for E&A.

Research Programme

At first some simple low dimensional examples using the simplest and most robust numerical methods, will be studied to build greater intuition and to provide some benchmark results for the more complicated algorithms likely to work in high dimensions. Then the canonical problem used in climate science - the 'Lorenz-96' equations - will be generalised to included deterministic and stochastic parameters. We hope to show that our newly developed E&A methods will work well on such problems. In developing an understanding of the difficulties likely to be encountered in applying the methods to real climate and weather problems the PDRA will make at least one extended visit to KAUST to work with Ibrahim Hoteit and Gera Stenchikov.