REE14: Stochastic closures for weather and climate modelling
||Dr Wonjung Lee
|Team Leader(s):||Dr Chris Farmer
||Prof. Ibrahim Hoteit, KAUST
|Prof. Georgiy Stenchikov, KAUST|
|Dr Frank Kwasniok, University of Exeter
|Prof. Terry Lyons|
|Prof. Tim Palmer
|Dr Irene Moroz
|Dr Simon Cotter, University of Manchester|
In many applications of fluid dynamics, analytical solutions are out of the question, and computational resources are inadequate for a fully-resolved numerical solution. Therefore it is necessary to account for unresolved scales and processes in simulation by using some form of sub-grid modelling. This is usually referred to as a 'closure problem' in fluid dynamics and theoretical physics, and as a 'parameterisation problem' in meteorology and climate science . There are two basic problems involved in this project (i) how to derive or at least how to motivate the form or structure of the closure (ii) how to estimate the parameters in the closure once it has been postulated or derived. In motivating a closure the choice is between (a) simple postulation via axiomatic construction of the effective field theory represented by the closure and (b) a more systematic expansion method. Unfortunately in the general case there are no obvious expansion parameters.
Techniques and Challenges
To calibrate both stochastic and deterministic closures, it is necessary to solve high-dimensional parameter estimation and filtering problems. Such problems are set within a Bayesian framework. In the special case that observations (generated by a physical system or a fine resolution simulation) are separated by finite time intervals, between the observations the probability density evolves according to the Liouville equation or the Fokker-Planck equation . After each observation the probability density is updated using Bayes' rule. Our main task is to construct a Gaussian sum mixture approximation to the conditional probability density. The main computational difficulty is due to the inherent high-dimension in weather and climate dynamics.
A new approach – the 'ensemble Bayesian filter' or 'eBf' [12/47], based upon the previous work of Hoteit et al , and of Stordal et al  has been developed and in early tests appears quite promising. The current version of this method is for the case of a deterministic dynamical system. The eBf has the potential for a rigorous convergence proof and can solve problems such as the generalised Lorenz-96 model where extra, unknown parameters, are introduced. The figure shown on this page, using equations inspired by the work of DelSole and Yang [5,6], shows results after 250 time steps with 6 observations on the variables, and no observations on the parameters. Periodic boundary conditions are used. The solid line shows the 'truth' for the parameters and variables of an 80-dimensional Lorenz-96 example (40 variables and 40 parameters). The dotted line is the mean of an ensemble with 999 members, and the shaded region shows the mean plus or minus one standard deviation in the uncertainty. The details are given in [12/47].
Now that we have a promising method for stochastic parameter estimation, we will (i) develop the estimation method further to be applicable to systems with stochastic dynamics (ii) investigate the possible advantage of time dependent parameterisations compared to static parameterisations (iii) explore further methods for motivating particular closures.
[12/47] Farmer, C.L.: An ensemble Bayesian filter for state estimation
 Palmer, T., Williams P. (Editors): Stochastic Physics and Climate Modelling, Cambridge University Press, 2009
 Jazwinski A.H: Stochastic Processes and Filtering Theory, Dover, 1970
 Hoteit, I., Pham, D.-T., Triantafyllou, G. & Korres, G.: A new approximate solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography, Monthly Weather Review, 136, pp 317-334, 2008
 Stordal, A.S., Karlsen, H.A., Nævdal, G., Skaug, H.J., &Vallès, B.: Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter. Computational Geosciences, 15(2), pp 293-305, 2011
 Yang, X., and DelSole, T.: Using the ensemble Kalman filter to estimate multiplicative model parameters, Tellus, 61, pp 601-609, 2009
 DelSole, T. and Yang, X.: State and parameter estimation in stochastic dynamical models, Physica D, 239, pp 1781-1788, 2010