- v_{\rm cir})\sin{i},\\
 && \Delta v_{\xi}   \equiv -v_{R} \sin{i} + (v_{\phi} - v_{\rm cir})\cos{i},\end{aligned}$$ where $i$ is the pitch angle of the spiral arm, and $v_{\rm cir}$ is the circular velocity determined by the azimuthally averaged gravitational field. For this definition, inflow to the arm corresponds to $\Delta v_{\eta} > 0$ for $\eta < 0$, and $\Delta v_{\eta} < 0$ for $\eta > 0$. In the amplification phase ($T_{\rm rot} \lesssim 12.2$), we see a clear inflow motion to the spiral arm along both sides of the chosen strip (solid lines). In the initial stages of the damping phase ($T_{\rm rot} \simeq 12.3$), the streaming velocity around the spiral arm gradually transits from inflow to outflow. At the end of the damping phase ($T_{\rm rot} \simeq 12.5$), it is clear that the stars in the spiral arm move away from the arm along both its sides (dot-dashed lines). During the dynamical evolution, the parallel component of the velocity, $\Delta v_{\xi}$, does not change its sign or show a decrease its magnitude. The left panels of Figure \[fig:VelocityProfile\] show a schematic view of the time evolution of the non-circular velocity field associated with the non-steady spiral arm.

The middle right panels 