REE14: Stochastic closures for weather and climate modelling
| Researcher: |
Dr Wonjung Lee |
| Team Leader(s): | Dr Chris Farmer |
| Collaborators: |
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 |
Background
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 [1]. 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 [2]. 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.
Results
A new approach – the 'ensemble Bayesian filter'
or 'eBf' [12/47], based upon the
previous work of Hoteit et al [3],
and of Stordal et al [4] 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].
The Future
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.
References
[12/47] Farmer, C.L.: An ensemble Bayesian filter for state estimation
[1] Palmer, T., Williams P. (Editors): Stochastic Physics and Climate Modelling, Cambridge University Press, 2009
[2] Jazwinski A.H: Stochastic Processes and Filtering Theory, Dover, 1970
[3] 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
[4] 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
[5] Yang, X., and DelSole, T.: Using the ensemble Kalman filter to estimate multiplicative model parameters, Tellus, 61, pp 601-609, 2009
[6] DelSole, T. and Yang, X.: State and parameter estimation in stochastic dynamical models, Physica D, 239, pp 1781-1788, 2010