of $Z$ for the $t \sim 4\,{\rm Gyr}$ snapshot of the Model 1 simulation. We see that while the $m=1$ mode is clearly leading, the $m=2$ term is trailing, and the $m=3$ term is a combination of both leading and trailing. Furthermore, the $m=1$ mode is the dominant mode beyond a radius of $\sim 15 \, {\rm kpc}$.

![Time evolution of $|Z_1(R)|$. The upper and lower panels show Model 1 and Model 2, respectively. Both models show a general trend of increasing strength propagating outward from intermediate radii to the outer disc over time. \[fig:PS\_evolution\]](fig5.pdf){width="\columnwidth"}

We explore how this dominant mode in the outer disc forms and evolves in Fig. \[fig:PS\_evolution\] by showing $|Z_1(R)|$ for both models at various snapshots. Both models show a trend of increasing $|Z_{1}(R)|$ from intermediate radii to the outer disc with time. This is most evident in Model 1 where there is a factor of $\sim 2-3$ decrease in $|Z_1|$ at $R \sim 10 - 15 \, {\rm kpc}$ and an approximately equal increase in the outer disc after $t \simeq 1$ Gyr. We attribute the large suppression of $|Z_1|$ at intermediate radii to the bar, which forms at $\sim 2.5$ Gyr. This trend, which 