  The on-going vertical phase mixing, manifesting itself as a snail shell in
the $Z-V_{Z}$ phase space, has been discovered with the Gaia DR2 data. To
better understand the origin and properties of the phase mixing process, we
study the vertical phase-mixing signatures in arches (including the classical
``moving groups'') of the $V_{R}-V_{\phi}$ phase space near the Solar circle.
Interestingly, the phase space snail shell exists only in the arches with
$|V_{\phi} - V_{\rm LSR}| \lesssim 30$ km/s, i.e., stars on dynamically
``colder'' orbits. The snail shell becomes much weaker and eventually
disappears for increasingly larger radial action ($J_{R}$), quantifying the
``hotness'' of orbits. Thus one should pay closer attention to the colder
orbits in future phase mixing studies. We also confirm that the Hercules stream
has two branches (at fast and slow $V_{\phi}$), which may not be explained by a
single mechanism, since only the fast branch shows the prominent snail shell
feature. The hotter orbits may have phase-wrapped away already due to the much
larger dynamical range in radial variation to facilitate faster phase mixing.
To explain the lack of a well-defined snail shell in the hotter orbits, the
disk should have been perturbed at least $500$ Myr ago. Our results offer more
support to the recent satellite-disk encounter scenario than the internal bar
buckling perturbation scenario as the origin of the phase space mixing. Origin
of the more prominent snail shell in the $V_{\phi}$ color-coded phase space is
also discussed.
