Nonlinear stochastic low-order models of atmospheric low-frequency variability using an empirical regime-weighted closure scheme

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. The total energy metric is used in the projection; the nonlinear terms in the reduced model then conserve total energy. Secondly, 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.

The cluster-weighted closure model consists of a small number (one to four) of local models. Each of the models has a Gaussian cluster in the space of the leading EOFs defining a regime in the large-scale circulation and a linear regression model with Gaussian uncertainty into the space of EOF tendency error. The whole model is a weighted mixture of the local models; hence the closure scheme amounts to locally linear but globally nonlinear deterministic corrections and multiplicative noise. The cluster centers and covariances as well as the linear models and their uncertainties are determined from the data according to maximum likelihood; only the number of clusters has to be prescribed. The algorithm is very robust as nonlinear correlations are treated in the form of state-dependent means and covariances. With only one cluster, the model degenerates to a globally linear closure scheme with only additive noise.

A low-order model based on only 10-15 EOFs is capable of simulating self-consistently with remarkable accuracy 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 that are known to be hard to capture in a low-order model.

The parameters of the proposed closure model are readily interpretable. Regimes in the space of the leading EOFs are defined with associated linear models. By introducing averages with respect to the posterior distribution of a particular cluster, regime-weighted quantities can be defined that may be used to characterize the dynamics of the model in certain regimes.

The present approach compares favorably with other recently proposed stochastic mode reduction schemes. Due to its partly empirical nature, it is applicable beyond the perfect-model scenario when the dynamical equations of the underlying system are not fully known or accessible. When deriving a reduced model from GCM or observational data there are not only unresolved modes but also unresolved processes that could be accounted for by the present closure scheme.