(cpd). The zonal index features increasing power with decreasing frequency. At intraseasonal and shorter timescales, where the dominant balance of Equation 2b is between $i\omega Z$ and $M$, the power spectrum of zonal index can be interpreted, to the first order, as reddening of the power spectrum of eddy forcing (Figure 3b). The broad peak at synoptic timescale in the power spectrum of eddy forcing (Figure 3c) is an intrinsic characteristic of the mean-state-independent eddies (LH01). At timescales longer than around 50 days, a positive eddy-jet feedback is suggested to be responsible for the high power of both of the zonal index and eddy forcing, where the dominant balance of Equation 2b is between $Z/\tau$ and $M$. [A linear feedback model for]{} $M$ (e.g., Hasselmann 1976; LH01) can be [written as]{},

$$\renewcommand{\theequation}{3}
M = \tilde{M} + bZ,$$

where $\tilde{M}$ is the mean-state-independent eddy forcing, and $b$ is the strength of the eddy-jet feedback. In equilibrium, $b$ must be smaller than $1/\tau$ in both GCMs and the realistic atmosphere, otherwise the zonal index grows unboundedly. Plugging Equation 3 into Equation 2b returns,

$$\renewcommand{\theequation}{4}
i\omega Z = \tilde{M}  + (b - 1/{\tau})Z$$

If we consider $\tilde{M}$ as [white noise at low frequencies]{}, 