Reduced models of extratropical low-frequency variability

Low-frequency variability (LFV) of the atmosphere refers to its behavior on time scales of 10-100 days, longer than the life cycle of a mid-latitude cyclone but shorter than a season. This behavior is still poorly understood and hard to predict. It has been helpful in gaining understanding that might improve prediction to use various simplified models.; The present study compares and contrasts various mode reduction strategies that help derive systematically such simplified models of LFV. Three major strategies have been applied to reduce a fairly realistic, high-dimensional, quasi-geostrophic, 3-level (QG3) atmospheric model to lower dimensions: (i) a purely empirical, multi-level regression procedure, which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression; (ii) an empirical-dynamical method, which retains only a few components in the projection of the full QG3 model equations onto a specified basis (the so-called bare truncation), and finds the linear deterministic and additive stochastic corrections empirically; and (iii) a dynamics-based technique, employing the stochastic mode reduction strategy of Majda et al. (2001; MTV). Subject to the assumption of significant time-scale separation in the physical system under consideration, MTV derives the form of the reduced model and finds its coefficients with minimal statistical fitting. The empirical-dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi-level regression models; the extended Kalman filter (EKF) was used for the parameter estimation.; In constructing the reduced models, the choice of basis functions is also important. We considered as basis functions a set of empirical orthogonal functions (EOFs). These EOFs were computed using (a) an energy norm; and (b) a potential-enstrophy norm. We also devised a method, using singular value decomposition of the full-model's linearized propagator, to rotate the EOF basis in a way that emphasizes the low-frequency modes.; The reduced model's performance for a given basis was judged by how well that model reproduces statistical properties of the leading LFV modes of the full model, in the same basis. The statistical properties considered included one dimensional probability density functions (PDFs) and autocorrelation functions. We also used Gaussian mixtures to estimate a multi-dimensional PDF in a subspace of leading EOFs to study the regime behavior of the full and reduced models considered.; Overall, the reduced models perform better when more statistical information is used in model construction. Thus, the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full QG3 model's LFV, i.e. its autocorrelations and spectra, as well as the nonlinear properties, i.e. the persistent flow regimes that induce non-Gaussian features in the model's PDF. The empirical-dynamical models capture the basic statistical properties of the full model's LFV, such as the variance and integral correlation time scales of the leading LFV modes, as well as some of the regime-behavior features, but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes. Dynamical models that use data-assimilation corrections do capture the linear statistics to a degree comparable with that of empirical-dynamical models, but do much less well on the full QG3 model's nonlinear dynamics.; There is no appreciable difference between the performance of the models constructed using the energy norm or the potential-vorticity norm in defining the EOFs. The rotated bases were shown to emphasize the system's linear dynamics, which makes them unsuitable for studies of nonlinear behavior of mid-latitude LFV.; The analysis of and comparisons between all the reduced models showed that the cubic nonlinear corrections sugges...