Stochastic Partial Differential Equations with applications in Modelling of Oceans and Atmosphere

The human impact on oceans and atmosphere is a pressing concern. It is a problem that has been rapidly evolving which makes clear not only the need for action, but the reality of the level of uncertainty that we face. For example, one sees the effects of our CO2 emissions and plastic waste in the oceans. In particular the upper ocean feels that effect with the high proportion of marine life towards the surface. The upper ocean responds to these effects by interacting with the air and thus influencing the weather and climate. The project aims to contribute to the modelling side of fluid dynamics. A rigorous mathematical analysis of the various models available for the upper layer of the ocean will underpin any further rigorous prediction methods for rising sea levels, heat uptake, changes in pH and more in order to combat climate change. In particular, the project will analyse the use of stochasticity in fluid dynamics models. This is needed as high-resolution data will is not be always predictable in deterministic simulations. The derivation and analysis of stochastic mathematical models will then be validated through the numerical work and so that they cope with uncertainty in both observation and simulation. As a specific objective we will look at well-posedness results for stochastic partial differential equation with boundaries (the ocean has shores!). The new class of stochastic equations considered as viable fluid dynamics models has added uncertainty in the transport of fluid parcels to reflect the unresolved scales. These models afford proper energy circulation and conservation dynamics and preserve fundamental properties of their deterministic counterparts. Their analysis is still an open problem and is the focus of the current project. Specific examples included the Euler and Rotating Shallow Water Equations. The plan is to follow a procedure involving an approximating sequence of solutions via a truncated equation and recover a solution through relative-compactness arguments. A treatment of the boundary in this new stochastic framework is uncharted territory and hinges on the interplay between long established deterministic results and this novel stochastic methodology. Well-posedness of the stochastic 2D Euler equation with a smooth boundary represents an exciting first aim for the project, with subsequent aims to examine more delicate and relevant scenarios. Synergistically with this we could explore a general result about the potential regularising effect adding stochasticity could have, which will constitute a massive development in tackling the new class of equations given the extent to which their deterministic counterparts have been studied. It is a concept which has been theorised and attempted with varying success sofar. The outlined plan firmly aligns with the EPSRC's Strategic Theme of Mathematical Sciences, and Research Area of Mathematical Analysis.