Convex Error Growth Patterns in a Global Weather Model

We investigate the error growth, that is, the growth in the distance $E$ between two typical solutions of a weather model. Typically $E$ grows until it reaches a saturation value $E_s$. We find two distinct broad log-linear regimes, one for $E$ below 2% of $E_s$ and the other for $E$ above. In each, $\log (E/E_s)$ grows as if satisfying a linear differential equation. When plotting $d\log (E)/dt$ vs $\log (E)$, the graph is convex. We argue this behavior is quite different from other dynamics problems with saturation values, which yield concave graphs.

Figure 1

Convex exponential error growth rate of the NCEP GFS model as a function of relative error $E/E_s$. The graph reveals two linear regimes. For $E<0.02E_s$, $E^{\prime }/E$ lies close to line $y_f={\lambda }_f\log (E/E_f)$ with ${\lambda }_f\approx -2.5$ and intercepts at $E_f\approx 2.1\%$ of the saturation level $E_s$. For $E>0.02E_S$, $E^{\prime }/E$ lies close to the line $y_s={\lambda }_s\log (E/E_s)$ with ${\lambda }_s\approx -0.17$. The doubling time of errors are $1/8$, $1/4$, 1, 2, and 3 days when $E$ is 0.23%, 0.7%, 1.6%, 13%, and 26% of the saturation level, respectively. Small errors grow in amplitude, moving from left to right, first along the left line and then along the right until they saturate at $E_s$. Each point plotted here is for an individual pair of initial conditions. The dashes represent the “superfast” regime for $E<(2\times {10}^{-3})E_s$ (see text).

Figure 2

Concave error growth rates, estimated by finite differences in (1), as functions of relative error $E/E_s$. (a) The error growth rate of the Lorenz-40 model fluctuates about $0.35$ for $E<{10}^{-3}{E}_s$ and decays in accordance to (2) where $\lambda =-0.22$ (see the line tangent to the growth rate) and $C=E_s$. (b) In the QG model, the growth rate fluctuates about $0.044$ when $E<{10}^{-3}{E}_s$ and decays asymptotically to (2) where $C=E_s$, with $\lambda =-0.035$ as $E>0.4E_s$. As time increases, errors move from left to right, asymptoting to $E_s$. Each point plotted here is averaged over $L=1000$ pairs of trajectories for the Lorenz-40 model and $L=100$ for the QG model.

Figure 3

Concave exponential growth rate, $E^{\prime }/E$ for (7) as a function of $E$ in logarithmic scale. We set $a=0.35$, $\lambda =-0.22$, $E(0)={10}^{-6}$, and $\Delta t=1$. When $E<{10}^{-3}$, we get $E^{\prime }/E\approx a=0.35$. As the error grows, the instantaneous doubling time $T_d$ increases. We show $T_d=2$, 3, and $4$ corresponding to exponential growth rates of $0.34$, $0.23$, and $0.17$, respectively. The line $y=\lambda \log (E)$ with $\lambda =-0.22$ is tangent to the $E^{\prime }/E$ curve at the asymptotic value $E=1$.