spiral arms is not sufficiently large to erase all the spiral features. The curves in Figure \[fig:rotcurve\]c indicate that the value of $Q$ increases from an initial value of $1.2$ to a final value of $1.4$ around $R \approx 2 R_{\rm sd}$ at $T_{\rm rot} = 15$. However, the spirals are not completely erased due to dynamical heating [see @Fujii+2011]. It is to be noted that heating effect along the vertical direction to the disk plane is negligibly small (bottom panels in Figure \[fig:snapshotsMS\]), which is consistent with the results of previous studies [e.g., @BinneyLacey1988; @JenkinsBinney1990].

Growth and Damping Phases of Stellar Spiral Arms {#sec:SpiralDynamics}
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In this section, we investigate the dynamical evolution of co-rotating spiral arms in detail. Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\] show the time evolution of a spiral arm in the co-rotating frame. A weak density enhancement can be observed around $\phi \simeq 240-300^{\circ}$ and $R \simeq 7-10$ kpc for $T_{\rm rot} = 11.7$. This density enhancement causes the growth of a prominent spiral arm until $T_{\rm rot} = 12.0$, and the arm has a maximum amplitude around $T_{\rm rot} \sim 12.2-12.3$. Beyond $T_{\rm rot} \simeq 12.3$, this spiral arm rapidly fades out ($T_{\rm rot} = 12.3-12.5$). 