Vertical velocities from proper motions of red clump giants
  We derive the vertical velocities of disk stars in the range of
Galactocentric radii of R=5-16 kpc within 2 kpc in height from the Galactic
plane. This kinematic information is connected to dynamical aspects in the
formation and evolution of the Milky Way, such as the passage of satellites and
vertical resonance and determines whether the warp is a long-lived or a
transient feature.
  We used the proper motions of the PPMXL survey, correcting of systematic
errors with the reference of quasars. From the color-magnitude diagram K versus
(J-K) we selected the standard candles corresponding to red clump giants and
used the information of their proper motions to build a map of the vertical
motions of our Galaxy. We derived the kinematics of the warp both analytically
and through a particle simulation to fit these data. Complementarily, we also
carried out the same analysis with red clump giants spectroscopically selected
with APOGEE data, and we predict the improvements in accuracy that will be
reached with future Gaia data.
  A simple model of warp with the height of the disk z_w(R,phi)=gamma (R-R_sun)
sin(phi-phi_w) fits the vertical motions if d(gamma)/dt/gamma=-34+/-17
Gyr^{-1}; the contribution to d(gamma)/dt comes from the southern warp and is
negligible in the north. The vertical motion in the warp apparently indicates
that the main S-shaped structure of the warp is a long-lived feature, whereas
the perturbation that produces an irregularity in the southern part is most
likely a transient phenomenon. With the use of the Gaia end-of-mission products
together with spectroscopically classified red clump giants, the precision in
vertical motions can be increased by an order of magnitude at least.
