The objective of this project is to understand the predictability of behaviour of the earth’s atmosphere to explain the success of operational weather forecasts based on space-time averaged models of the real system. An investigation will be made of the relationship between solutions of the full Navier-Stokes equations and their reduction to the semi-geostrophic equations, valid on large scales where the flow is dominated by the effects of the earth’s rotation. Understanding the high predictability of the real system through this research could then be exploited to improve operational systems.

David Gilbert (Postgraduate Associate, Reading)
Beatrice Pelloni (Academic supervisor, Reading)
Mike Cullen (Industrial supervisor, Met Office)
Melvin Brown (Technology Translator, Industrial Mathematics KTN)

This project is being carried out at Reading, in conjunction with Met Office. It is supported by an EPSRC industrial CASE award. Start date: January 2009; duration: 3.5 years.

The behaviour of the atmosphere is governed by the compressible Navier-Stokes equations and the laws of thermodynamics, together with equations describing phase changes, source terms and boundary fluxes. These equations describe everything that happens in the atmosphere down to the smallest scales. A direct numerical solution of these equations would require computers 30 orders of magnitude more powerful than those currently available. Operational weather forecasts therefore have to use a space-time averaged form of the equations. Despite this, forecasts are regularly successful in predicting the general weather pattern for about a week ahead, though the detail of day to day variations can only be predicted for a few days.

This success implies that the weather patterns are largely controlled by large-scale dynamics, so that the ‘butterfly’ effect must be limited. This can be explained mathematically by deriving a reduction of the Navier-Stokes equations valid on large scales. The semi-geostrophic equations are such a reduction, valid on scales where the flow is dominated by the effect of the Earth’s rotation. The high predictability of the real system could be explained by showing that the semi-geostrophic equations can be solved independently, that the solutions behave in a predictable fashion, and that there is a solution of the Navier-Stokes equations close to the semi-geostrophic solution in a suitable asymptotic limit. If these steps can be carried out, they will give an understanding of the predictability of the system which could be exploited in making further improvements to operational systems.