Ensemble Dynamics and Bred Vectors

We propose a variant of the Bred Vector (BV) algorithm, orginally introduced by Z. Toth and E. Kalnay (Bulletin of the American Meteorological Society 74:2317 2330, 1993) to assess the sensitivity of model outputs to changes in initial conditions for weather forecasting. The new algorithm, which we call the Ensemble Bred Vector or EBV, is based on collective dynamics in an essential way. As such, it features distinctive qualities compared to the classical breeding algorithm. By construction, the EBV produces one or more dominant vectors, and is less prone to spurious results than the BV algorithm. It retains the attractive features of the BV with regard to being able to handle legacy codes with minimal additional coding. We investigate the performance of EBV, comparing it to the BV algorithm as well as the finite-time Lyapunov Vectors. With the help of a continuous-time adaptation of the algorithms we give a theoretical justification to the observed fact that the vectors produced by BV, EBV, and the finite-time Lyapunov vectors are similar to each other for small amplitudes. The continuum theory is also used to establish the relationship between the BV and general directional derivatives. Numerical comparisons of BV and EBV for the 3-equation Lorenz model and for a forced, dissipative partial differential equation that arises in the thermohaline circulation demonstrate that the EBV yields a size-ordered description of the perturbation field, and is more robust than the BV in the higher nonlinear regime. The EBV yields insight into the fractal structure of the Lorenz attractor, and of the inertial manifold for the Cahn-Hilliard-type partial differential equation.