Logarithmic bred vectors in spatiotemporal chaos: Structure and growth

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growth rates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of its amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz 1996 model.

Figure 1

(Color online) Upper panel: surfaces associated with the LV and with the log-BV with the smallest value of ${\epsilon }_0$ used ${\epsilon }_0^{\text{min}}=e^{-20}$ (Lorenz 1996 model with $F=8.0$ and $L=1024$). Lower panel: snapshot of the difference fields ${\eta }_{{\epsilon }_0}=h_{{\epsilon }_0}\left(x,t\right)-h_{LV}\left(x,t\right)$ (curves are arbitrarily shifted to improve their visibility).

Figure 2

(Color online) Data collapse of ${\lambda }_{{\epsilon }_0}\left(L\right)$, defined in Eqs. (11, 13), for different values of ${\epsilon }_0$ and system size $L$ $\left(F=8\right)$. Points fall approximately on a line, the scaling function $g$ in Eq. (17). Inset: the value of the LE $\lambda \left(L=\infty \right)$ was determined by extrapolating the largest LE from finite systems using Eq. (14).

Figure 3

(Color online) The lines with symbols are structure factors $S\left(k\right)$ obtained for log-BVs with different perturbation amplitudes for $F=8.0$ and $L=1024$. The straight lines without symbols indicate the method of estimating the values $k_c\left({\epsilon }_0\right)$ which are determined from the intersection of a horizontal line at $S\left(k_{\text{min}}=2\pi /L\right)$ and a power law $S\left(k\right)\propto k^{-2.02}$ obtained from fitting the structure factor for $\text{min}\left(\text{ln}{\epsilon }_0\right)$.

Figure 4

(Color online) Collapse of the structure factors using the values $k_c\left({\epsilon }_0\right)$ determined from the structure factors $S\left(k\right)$ in Fig. 3.

Figure 5

(Color online) The lines with symbols indicate structure factors $S\left(k\right)$ obtained for bred vectors with different perturbation amplitudes with $F=6,8,14,20$ and $L=1024$. The values $k_c\left({\epsilon }_0\right)$ are determined like in Fig. 3.

Figure 6

(Color online) Lyapunov spectra (solid black lines) and bred exponents (symbols). Size $L=1024$ and $F=6,7,\dots ,20$.

Figure 7

(Color online) The colored lines with symbols indicate structure factors $S\left(k\right)$ obtained for log-BVs with different perturbation amplitudes for $F=6,8,14,20$ and $L=128$. The values $k_c\left({\epsilon }_0\right)$ are determined like in Fig. 3.

Figure 8

(Color online) Lyapunov spectra (solid black lines) and bred exponents (symbols). Size $L=128$ and $F=6,7,\dots ,20$.