On the Closure Problem in the Reduction of Complex Atmospheric Models by PIPs and EOFs: A Comparison for the Case of a Two-Layer Model with Zonally Symmetric Forcing

Abstract

The reduction of complex systems to their essential degrees of freedom (e.g., patterns spanning that part of state space that the system’s trajectory passes in time) and development of reduced models based on these might be one tool for improving basic understanding of simulation results obtained from current, increasingly more complex, meteorological circulation models. Some successful work on the reduction of complex systems by principal interaction patterns (PIPs) or empirical orthogonal functions (EOFs) has already been done. However, the parameterization of the influence of unresolved modes onto the selected patterns, called closure in this context, is still an outstanding problem. Nonlinear closure schemes, although greatly improving prediction on shorter timescales, have so far been observed to lead to absolute instability of the reduced model, that is, explosive unbounded energy growth after some integration time. In this work a method is outlined for circumventing this problem. Energy conservation constraints are formulated that can be used in the extraction of a useful empirical closure from synthetic model data by minimizing the error between tendencies in the full and the reduced model with the closure parameters as variables. In the present context the computational size of the associated minimization calculations could be reduced by utilizing the zonal symmetry in the full model’s forcing and boundary conditions. So it could be shown that each EOF or PIP must be an element of the subspace of a single zonal wavenumber. Coupling conditions for the closure coefficients are derived that further decrease the dimensionality of the problem. The method is tested by reducing multiple baroclinic wave life cycles in a quasigeostrophic two-layer model on the basis of both EOFs and PIPs. It is shown that the aforementioned stability problems connected with the nonlinearity of a closure are indeed avoided by the method. Furthermore, the closure greatly improves the simulation capabilities of the reduced model both on short and long timescales. In contrast to previous results on linear closure schemes, the authors find that in nonlinear closure schemes also the linear terms have to be handled carefully in order to ensure realistic behavior of the reduced model on longer timescales. In a comparison between reductions based on EOFs and PIPs substantial superiority of the latter in effectively extracting the essential degrees of freedom is demonstrated.