  We numerically compute the flow induced in a spherical shell by fixing the
outer sphere and rotating the inner one. The aspect ratio
$\epsilon=(r_o-r_i)/r_i$ is set at 0.04 and 0.02, and in each case the Reynolds
number measuring the inner sphere's rotation rate is increased to $\sim10\%$
beyond the first bifurcation from the basic state flow. For $\epsilon =0.04$
the initial bifurcations are the same as in previous numerical work at
$\epsilon=0.154$, and result in steady one- and two-vortex states. Further
bifurcations yield travelling wave solutions similar to previous analytic
results valid in the $\epsilon\to0$ limit. For $\epsilon=0.02$ the steady
one-vortex state no longer exists, and the first bifurcation is directly to
these travelling wave solutions, consisting of pulse trains of Taylor vortices
travelling toward the equator from both hemispheres, and annihilating there in
distinct phase-slip events. We explore these time-dependent solutions in
detail, and find that they can be both equatorially symmetric and asymmetric,
as well as periodic or quasi-periodic in time.
