  We investigate both numerically and analytically the convective instability
of granular materials by two dimensional traffic equations. In the absence of
vibrations the traffic equations assume two distinctive classes of fixed bed
solutions with either a spatially uniform or nonuniform density profile. The
former one exists only when the function V(\rho) that monitors the relaxation
of grains assumes a cut off at the closed packed density, \rho_c, with
V(\rho_c)=0, while the latter one exists for any form of V. Since there is
little difference between the uniform and nonuniform solution deep inside the
bed, the convective instability of the bulk may be studied by focusing on the
stability of the uniform solution. In the presence of vibrations, we find that
the uniform solution bifurcates into a bouncing solution, which then undergoes
a supercritical bifurcation to the convective instability. We determine the
onset of convection as a function of control parameters and confirm this
picture by solving the traffic equations numerically, which reveals bouncing
solutions, two convective rolls, and four convective rolls. Further, convective
patterns change as the aspect ratio changes: in a vertically long container,
the rolls move toward the surface, and in a horizontally long container, the
rolls move toward the walls. We compare these results with those reported
previously with a different continuum model by Hayakawa, Yue and Hong[Phys.
Rev. Lett. 75,2328, 1995]. Finally, we also present a derivation of the traffic
equations from Enskoq equation.
