Empirical Low-Order Models of Barotropic Flow

Empirical low-order models are constructed in the framework of a quasigeostrophic barotropic spectral model, truncated at T21. The spectral model is projected onto a linear subspace spanned by only a limited number of spatial structures called principal interaction patterns. The expansion coefficients of these modes are assumed to be governed by a forced, dissipative dynamical system with a quadratic nonlinearity that conserves turbulent kinetic energy and turbulent enstrophy. A simple empirical scheme for modeling the energy and enstrophy cascade assuming localization of the nonlinear interactions with respect to spatial scale is applied. The optimal low-order model, that is, the optimal basis functions and the optimal interaction coefficients, is determined by minimizing the mean-squared error between trajectories of the reduced model and trajectories of the T21 model. A model with 40 degrees of freedom succeeds in well capturing, in a long-term integration, the behavior of the T21 model monitored by the mean state, the variance pattern, autocorrelation functions, and probability distributions of the streamfunction. The model based on principal interaction patterns offers a substantial improvement on a model based on empirical orthogonal functions with the same number of degrees of freedom in capturing the autocorrelation function, the probability distribution, and the response of the system to a change in the forcing.