  We investigate the occurrence of spontaneous convection in a coaxial cylinder
highly filled with mono-disperse spheres.
  To analyze the flow field non-invasively, initial pulses consisting of
colored particles are placed at equal intervals.
  By analyzing the spatio-temporal distribution of these pulses, we obtained
axial velocity profiles for both the surface and subsurface regions.
  Our advection-diffusion equations with steady advection terms incorporate
experimentally obtained axial velocity profiles in the surface layer, while the
rest of the components are estimated using azimuthal symmetry and volume
conservation.
  The validity of our model is confirmed by comparing experimental data with
numerical solutions for both the spatio-temporal distribution and
cross-sectional profile of the colored particles.
