k = {1,2, \dots 8}$, thus $X$ has $8$ elements while $Y$ and $Z$ have $64$ and $512$ elements, respectively. Figure \[xyz\_demo\] shows examples of the chaotic temporal evolution of $X$, $Y$, and $Z$ obtained from directly solving Eqs. (\[eq:L1\])-(\[eq:L3\]). The examples demonstrate that $X$ has large amplitudes and slow variability; $Y$ has relatively small amplitudes, high-frequency variability, and intermittency; and $Z$ has small amplitudes and high-frequency variability. In the context of atmospheric circulation, the slow variable $X$ can represent the low-frequency variability of the large-scale circulation while the intermediate variable $Y$ and fast variable $Z$ can represent synoptic/baroclinic eddies and small-scale convection, respectively. Similar analogies in the context of ocean circulation and various other natural or engineering systems can be found, making this multi-scale Lorenz 96 system a useful prototype to focus on.

![Time evolution of (A) $X_k$ at grid point $k=1$, (B) $Y_{j,1}$ at grid point $j=1$, and (C) $Z_{i,1,1}$ at grid point $i=1$. The time series show chaotic behavior in $X$, which has large amplitude and slow variability; $Y$, which has relatively small amplitudes, high frequency variability, and intermittency; $Z$, which has small amplitudes and high frequency variability. The $x$-axis is in model time unit (MTU) which 