When the resolved and unresolved scales are broadly separated in space and time, it is reasonable to assume that a given configuration of the resolved flow is associated with a unique influence from the unresolved scales. Such a scale separation is implicit in traditional deterministic parameterisations. However, in the absence of a scale separation a given resolved flow may be consistent with a range of influences from the unresolved scales. In this situation (which is arguably the rule rather than the exception in GCMs) the subgrid scale parameterisations should include an element of randomness: that is, they should be stochastic. The explicit inclusion of stochasticity in parameterisation schemes can improve (typically underdispersive) ensemble spread and model biases (e.g. Jung et al. 2005). Furthermore, dynamical nonlinearities or state-dependence of the stochastic term can feed back and change the mean state of the climate (e.g. Penland 2003).
Two natural approaches to the implementation of such a parameterisation suggest themselves. First, the probability distribution of the upscale influence conditioned on the resolved flow can be determined (from first principles or empirically). At each timestep, this conditional distribution can be sampled (a number of times reflecting the degree of scale separation) and the sample mean value computed. Alternatively, models of subgrid scale variability (for example stochastic differential equations or cellular automata) can be run in parallel with the resolved flow as a ``superparameterisation''. While preliminary steps have been taken in the development of such parameterisations (e.g. Monahan 2007) and their implementation in operational models (e.g. Berner et al, in press) considerable theoretical and practical challenges remain.
REFS:
Berner J., G. Shutts, M. Leutbecher, and T.N. Palmer, 2008: A Spectral Stochastic Kinetic Energy Backscatter Scheme and its Impact on Flow-dependent Predictability in the ECMWF Ensemble Prediction System, in press, DOI: 10.1175/2008JAS2677.1
Jung, T., Palmer, T.N., and Shutts, G.J., 2005: Influence of a stochastic parameterization on the frequency of occurrence of North Pacific weather regimes in the ECMWF model", Geophys. Res. Lett., 32, doi:10.1029/2005GL024248.
Monahan, A.H., 2007: Empirical Models of the Probability Distribution of Sea Surface Wind Speeds , J. Climate , 20, 5798-5814.
Penland, C, 2003: Noise out of chaos and why it won't go away. Bull. Amer. Met. Soc., 84, 921-925.