{SUMMARY}
A new numerical method for solving the axisymmetric unsteady incompressible
Navier-Stokes equations using vorticity-velocity variables on staggered grid
is presented. The solution is advanced in time with an explicit two-stage
Runge-Kutta method. At each stage a vector Poisson equation for velocity is
solved. Some important aspects of staggering of the variable location,
divergence-free correction to the velocity field by means of a suitably
chosen scalar potential and numerical treatment of the vorticity boundary
condition are examined.  Axisymmetric spherical Couette flow between two
concentric differentially rotating spheres is computed as an initial-value
problem. The comparison of the computational results by using staggered grid
with those by using non-staggered grid shows that the staggered grid is 
superior to the non-staggered grid. The computed scenario of  the transition
from 0-vortex to 2-vortex at moderate Reynolds number agrees to that simulated
by using a pseudo-spectral method, thus validating the temporal accuracy of our
method.