Separating fast and slow modes in coupled chaotic systems

We test a simple technique based on breeding to separate fast and slow unstable modes in coupled systems with different time scales of evolution and variable amplitudes. The technique takes advantage of the earlier saturation of error growth rate of the fastest mode and of the lower value of the saturation amplitude of perturbation of either the fast or the slow modes. These properties of the coupled system allow a physically-based selection of the rescaling time interval and the amplitude of initial perturbations in the breeding of unstable modes (Toth and Kalnay, 1993, 1996, 1997; Aurell et a]., 1997; Boffetta et al., 1998) to isolate the desired mode. We perform tests in coupled models composed of fast and slow versions of the Lorenz (1963) model with different strengths of coupling. As examples we present first a coupled system which we denote weather with convection, with a slow, large amplitude model coupled with a fast, small amplitude model, second an ENSO system with a tropical atmosphere strongly coupled with a tropical ocean, and finally a triply coupled system denoted tropical-extratropical in which a fast model (representing the extratropical atmosphere) is loosely coupled to the ENSO system. We find that it is always possible to isolate the fast modes by taking the limit of small amplitudes and short rescaling intervals, in which case, as expected, the results are the same as the local Lyapunov growth obtained with the linear tangent model. In contrast, slow modes cannot be isolated with either Lyapunov or Singular vectors, since the linear tangent and adjoint models are dominated by the fast modes. Breeding is successful in isolating slow modes if rescaling intervals and amplitudes are chosen from physically appropriate scales.