A Statistical Closure of a Low-Order Barotropic Model

Abstract

In this paper, an attempt to close a low-order barotropic model for the neglected interactions in a perfect model setting is described. A barotropic T21 model with forcing and dissipation included is reformulated in terms of its EOFs. The EOFs are calculated from a long integration, and evolution equations are derived for the EOF amplitudes. A low-order EOF model is obtained by retaining only the 20 most dominant EOF structures and neglecting interactions with the remaining 211 EOFs. An attempt is made to describe the tendency error of the EOF model with a linear combination of resolved EOF amplitudes plus their quadratic combinations. The linear combination minimizes in a least squares sense the tendency error of the EOF model on the attractor of the full T21 model. It is found that, only if quadratic combinations of EOF amplitudes are taken into account, the closure reduces the tendency error substantially. The impact of the closure on the forecast skill of the EOF model is studied by making 100 3-week forecasts starting from independent initial conditions on the attractor of T21. The average useful forecast range increases from 12 days without closure to 18 days with closure. However, the method seems questionable in two aspects. First, the corrections to the coefficients of the evolution equations of the EOF amplitudes are as large as the coefficients themselves. Second, the closed EOF model could not simulate the climate. The closed model does not conserve energy in the absence of forcing and dissipation, and for this reason does not possess a stable attractor. Modifications to the proposed closure are mentioned to solve this problem. In conclusion, the proposed closure without the mentioned modifications leads to a statistical model that has an improved predictive skill but fails to simulate the climate of the original T21 barotropic model.