Extended Abstract (684K)6.8
Are Bred Vectors the same as Lyapunov vectors?
Eugenia Kalnay, University of Maryland, College Park, MD; and M. Corazza and M. Cai
The importance of local atmospheric predictability (in space and time) has become very apparent through the operational use of ensemble forecasting. Spaghetti plots used to display ensemble forecasts frequently show simultaneous high predictability in some regions and low predictability in others, associated with the stability of the evolving basic flow. Stability properties have been studied with Lyapunov vectors, singular vectors and bred vectors. A major difference between Lyapunov vectors (LVs) and bred vectors (BVs) is that the LVs are computed using global orthogonalization, whereas the BVs are never orthogonalized (e.g., Toth and Kalnay, 1993, 1997). Since the global atmosphere is large compared to the horizontal scales of individual weather systems (which have a finite lifetime), at any given time there usually are several independent weather systems in the globe growing and decaying independently from each other. This suggests that their predictability may be essentially independent from each other, and therefore that local (BV) and global (LV) methods may yield different results.
By construction, LVs and BVs are closely related. In fact, for infinitely long times and infinitely small amplitudes, BVs are identical to the leading LVs. As a result, they share some of their important properties, in particular that the direction of the BV is independent of the norm. In practical applications, however, BV are different from LVs in two important ways: a) BVs are never orthogonalized, and are intrinsically local in space and time, and b) they are finite amplitude, finite time vectors.
In this paper we compare the properties of BVs with LVs for a quasi-geostrophic model. We find that it is enough to use 5-15 BVs to represent the unstable growth of perturbations. By contrast, there are well over a hundred growing globally orthogonalized LVs (with positive Lyapunov exponents). Another difference is that, and beyond the first LV (identical to a bred vector), the LVs do not represent well the shape of the errors in the data assimilation system. Finally, we present heuristic arguments to explain why the number of BVs required to represent the unstable growth directions is much smaller than the number of LVs.
Session 6, Ensembles and data assimilation
Thursday, 17 January 2002, 8:45 AM-1:30 PM
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