Subgrid-scale parametrization using cluster-weighted modeling

A method for stochastic subgrid modeling in atmosphere and ocean models is proposed. A probabilistic model of unresolved modes and processes is derived that is conditional on the state of the resolved variables. The parameters of this model are determined from observational or model data. The subgrid model is then added to the deterministic model for the resolved modes. The cluster weighted subgrid model consists of a small number of local models. Each of the local models has a Gaussian cluster in the space of the resolved variables and a regression model with Gaussian uncertainty into the space of unresolved variables. The regression model is a constant or linear model. The whole model is a state-dependent weighted mixture of the local models. The cluster centers and covariances as well as the parameters of the regression models are estimated from data according to maximum likelihood; only the number of clusters has to be prescribed. A well-established and robust algorithm is available for the parameter estimation.

Firstly, the method is exemplified on the Lorenz 1996 system as a conceptual atmospheric model. The scheme clearly outperforms simple statistical closure models.

Secondly, nonlinear stochastic low-order models of large scale atmospheric dynamics are derived. The dynamical framework is a three-level quasigeostrophic (QG) model with realistic climatological mean state and variance pattern as well as Pacific/North America and North Atlantic Oscillation teleconnection patterns. The reduced model is formulated in the space spanned by the leading empirical orthogonal functions (EOFs). Firstly, a bare truncation model is generated by projecting the equations of motion onto the resolved modes. Then a probabilistic model of the EOF tendency error of the bare truncation conditioned on the EOFs is fitted empirically from a large data set of the QG model using the technique of cluster-weighted modeling. The cluster-weighted model is added to the bare truncation to form the stochastic low-order model. It acts as a closure scheme to account for the influence of unresolved modes on the resolved modes, particularly the nonlinear backscattering from medium- and small-scale modes on the large-scale modes that has been found to play a role in forming the low-frequency variability of the QG model. A low-order model based on only 10-15 EOFs is capable of simulating self-consistently the long-term dynamics of these low-frequency modes. Monitored statistical quantities are the mean state, the standard deviation pattern and momentum fluxes as well as probability densities and autocorrelation functions.

The method has the potential to be used locally on a grid in a complex general circulation model.