A nonlinear dynamical perspective on model error: A proposal for non‐local stochastic‐dynamic parametrization in weather and climate prediction models

Abstract

S u MMARYConventional parametrization schemes in weather and climate prediction models describe the effects of subgrid-scale processes by deterministic bulk formulae which depend on local resolved-scale variables and a number of adjustable parameters. Despite the unquestionable success of such models for weather and climate prediction, it is impossible to justify the use of such formulae from first principles. Using low-order dynamicalsystems models, and elementary results from dynamical-systems and turbulence theory, it is shown that even if unresolved scales only describe a small fraction of the total variance of the system, neglecting their variability can, in some circumstances, lead to gross errors in the climatology of the dominant scales. It is suggested that some of the remaining errors in weather and climate prediction models may have their origin in the neglect of subgrid-scale variability, and that such variability should be parametrized by non-local dynamically based stochastic parametrization schemes. Results from existing schemes are described, and mechanisms which might account for the impact of random parametrization error on planetary-scale motions are discussed. Proposals for the development of non-local stochastic-dynamic parametrization schemes are outlined, based on potential-vorticity diagnosis, singular-vector analysis and a simple stochastic cellular automaton model.

Extremes and Recurrence in Dynamical Systems

“…The basic results are that one can approximately treat the effect of fast modes on the slow dynamics by adding suitably defined deterministic and stochastic (with white spectrum) forcing terms in the evolution equations of the slow variables [327,328]. While the use of deterministic, mean field parametrizations for processes like convection, which cannot yet be captured by the relatively coarse grids of most weather and climate models is a standard practise in geophysical fluid dynamical modelling [329,330,24], currently weather and climate modelling centres are moving in the direction of introducing stochastic parametrizations [331,332], as mounting evidences suggest that they are more effective than usual deterministic methods [333,334]. It is indeed not clear to what extent such methods, which aim at being able to describe the typical behaviour of the slow variables, perform in terms of providing a good representation of extreme events.…”

A comparative method to evaluate and validate stochastic parametrizations

“…This may be similar to the upscale energy cascades that are simulated by some current stochastic parametrizations (Palmer, 2001;Shutts, 2005). An average correlation of less than −0.5 was found between the value of adv in a T 95 grid box and the subgrid-scale interquartile range of the T 799 temperature tendencies contained in that grid box.…”

“…This is likely to be because these phenomena depend on small-scale organization, which simply is not resolved, nor properly represented by existing deterministic parametrizations. Introducing structured but random noise has the potential to help NWP models evolve more realistically (Palmer, 2001). Williams et al (2003) showed experimentally and later in a model (Williams et al, 2004) that adding noise can cause a rotating annulus experiment to transition from wave number two to wave number one.…”

Sensitivity of a double-gyre ocean model to details of stochastic forcing

“…After all, stochastic processes are the macroscopic manifestation of unresolved nonlinear interactions. As stochastic parameterizations become popular (e.g., Buizza et al, 1999;Palmer, 2001), the accuracy of probabilistic forecasts will increasingly depend on how appropriate the stochastic representation is to the physical process it is meant to represent, despite the fact that ensemble weather forecasting does take an average of the ensemble members to improve the forecast. The dynamical form the CLT (Papanicolaou and Kohler, 1974;Sardeshmukh et al, 2001) offers some guidance for doing this.…”