Toth and Kalnay (1992, 1997) developed a simple method to create initial perturbations for ensemble forecasting, in which the difference between a perturbed and a control nonlinear integration of a model is rescaled periodically and added to the control. The result naturally “breeds” perturbations known as Bred vectors (BV), which are the finite amplitude, finite time generalization of leading Lyapunov vectors (LV), i.e., the fastest growing perturbations within the model's attractor. Leading Singular vectors (SV), by contrast, are the fastest growing perturbations outside the attractor, and derive their fast growth from the speed by which they return to the attractor.

In this talk we will present examples of the understanding that BVs can bring to the analysis of complex dynamical systems, since they are essentially the instabilities of a nonlinearly evolving background flow, and they dominate analysis and forecast errors. They only require two nonlinear model integrations, and the periodic rescaling of their difference. Like the leading Lyapunov vectors they are independent of the norm, unless the system has instabilities with different time scales, e.g., convective and baroclinic instabilities in the atmosphere, or weather and El Nino instabilities in a coupled ocean-atmosphere system. In that case, LV and SV can only isolate the fastest instability, but Pena and Kalnay (2003) showed that BV with amplitudes and rescaling times corresponding to the slow variables can isolate the slow instabilities as well as the fast ones. Corazza et al (2002) showed how BVs could be used to inexpensively account for “errors of the day” within a 3D-Var system. Cai et al (2003) and Yang et al (2005) showed how BVs of the coupled ocean-atmosphere ENSO instabilities. Vikhliaev et al (2006) showed that it is possible to obtain the leading instabilities of decadal variability by rescaling a coupled system every 10 years. Evans et al (2003) showed that BV growth is a predictor of regime change, and the duration of the new regime in the Lorenz (1963) model.