Anisotropic decay of the energy spectrum in two-dimensional dense granular flows

We study anisotropic collective motions of two-dimensional granular particles under simple shear deformations. Employing molecular-dynamics simulations of large system sizes, we find that anisotropic fluidized bands develop in the system yielding under quasistatic deformations, where the spectrum of nonaffine velocities, which is associated with the energy spectrum for turbulent flows, exhibits a quadrupole structure. To explain theoretically the anisotropic spectrum, we derive hydrodynamic modes from a continuum model of dense granular materials, where we find that fluidized bands are caused by long-lived hydrodynamic fluctuations characterized by compressibility.