the amplitude of $Z$ is inversely proportional to the difference between $1/\tau$ and $b$ at the low-frequency limit (i.e., neglecting the left hand side of Equation 4). The stronger the eddy feedback is (i.e., the closer $b$ is to $1/\tau$), the higher power $Z$ has at intraseasonal and longer timescales. Note that if $b=0$, the amplitude of $Z$ is inversely proportional to $1/\tau$ at the low-frequency limit, and at intraseasonal to interannual timescales the zonal index will still have increasing power with decreasing frequency [@Hasselmann1976], although the annular mode will be less persistent than that with a positive eddy feedback.

![image](g02.pdf){width="\textwidth"}

The autocorrelation function of the zonal index decreases more slowly with lag time than that of the eddy forcing (Figure 3cd). The negative autocorrelations of eddy forcing at small lag time indicates the quasi-oscillatory nature of the eddies (Figure 3d), which is consistent with the broad maximum in the power spectrum at 7-15 days. The cross-correlation of $m$ and $z$ peaks at around 0.53, when the zonal index lags eddy forcing by 1-2 days as the zonal index is driven by the eddy forcing (Figure 4). Negative cross-correlations when the zonal index leads eddy forcing by a fews days 