Variational principles for stochastic parameterisations in geophysical fluid dynamics

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Our proposal is inspired by the clear and present need for understanding statistical variability of weather and climate. Dynamical weather prediction stems from the deterministic laws of mechanics and thermodynamics, established by the mid-19th century. With the advent of digital computers in the second half of the 20th century, these ideas led to operational Numerical Weather Prediction (NWP) and shortly thereafter, with the advent of satellite observations, to numerical experiments that explored the atmosphere's general circulation. The new type of scientific exploration via numerical simulations soon raised the issue of limits of predictability of atmospheric dynamics, due to uncertainty in the initial state, unresolved scales of motion, and the extreme sensitivity of the numerical output to these uncertainties. This sensitivity was famously popularised as the Butterfly Effect. The recognition of the loss of predictability for NWP summoned research into a stochastic approach in designing simulators for NWP. NWP cannot be entirely deterministic, but must also involve a form of randomness, or noise. A new approach to NWP arose, which coupled randomness and probability with determinism. Parallel processing methods in the early 1990's and improved operational forecasting systems, in both simulator physics and data assimilation methods, have led to more reliable forecasts produced by modern operational stochastic dynamic Ensemble Prediction Systems (EPS) now used at ECMWF, and the UK Met Office. Yet it still remains to determine the most appropriate way to introduce stochastic dynamics into the simulator, so as to couple data assimilation with ensemble forecasting and to determine the number of samples in the ensemble sufficient for a required reliability. Current work continues to explore these avenues with great vigour.This project addresses the remaining challenge of Stochastic Dynamics for NWP, by taking an integrated approach to data-driven mathematical modelling, compatible numerics and model-driven data assimilation. The mathematical modelling uses an optimal, systematic method of introducing stochasticity into Geophysical Fluid Dynamics (GFD). The method is based on a stochastic version of the family of variational principles whose critical points yield the entire sequence of deterministic equations of motion for ideal GFD at each level of approximation. The levels of approximation are obtained from asymptotic expansion of the unapproximated variational principle that yields the fundamental Euler equations for a rotating, stratified, incompressible fluid. Stochasticity is introduced into the variational principle by using resolved spatial correlations of data obtained from observations of fluctuating tracer paths. In turn, the stochastic variational principle generates the equations of motion for the fluid flow carrying these tracers along their fluctuating paths. The proposed mathematical research on these new equations of motion will be integrated with numerical simulations and data assimilation methods, aiming to create an implementable modelling approach of significance for the mathematical foundations of NWP, climate science, and other highly unstable fluid dynamics applications. For this, we adopt a Bayesian perspective in blending the newly developed SPDEs with data completely integrated with its modelling and simulation efforts with connections as shown in Figure 1. Likewise, the numerical algorithms will be informed by the mathematical analysis. Once the numerical simulations are developed and performed, the subsequent data assimilation will produce the posterior distribution of the current state of the model via particle filtering methods.