Numerical Study of Granular Turbulence and the appearance of $k^{-5/3}$   energy spectrum without flow
  The vibrated bed of powder, a vessel that mono disperse glass beads fill and a loud speaker shakes, is investigated numerically with the distinct element method, a kind of molecular dynamics. When the bed is heavily shaken, the displacement vectors of powder have the power spectrum with the dependence upon the wave number $k$ as $k^{-5/3}$. The origin of this spectrum is suggested to be the balance between the injected and dissipative energy, analogous to the proposal by Kolmogorov to explain the $k^{-5/3}$ energy spectrum observed in the fluid turbulence. Furthermore, the same spectrum still appears even without flows of powder. Thus Kolmogorov's argument is more universal than believed before. 
Y-h. Taguchi (Department of Physics, Tokyo Institute of   Technology,Oh-okayama, Meguro-ku, Tokyo 152 Japan)