  In [12], the existence of ideal circle patterns in Euclidean or hyperbolic
background geometry under the combinatorial conditions was proved using flow
approaches. It remains as an open problem for the spherical case. In this
paper, we introduce a combinatorial geodesic curvature flow in spherical
background geometry, which is analogous to the combinatorial Ricci flow of Chow
and Luo in [4]. We characterize the sufficient and necessary condition for the
convergence of the flow. That is, the prescribed geodesic curvature satisfies
certain geometric and combinatorial condition if and only if for any initial
data the flow converges exponentially fast to a circle pattern with given total
geodesic curvature on each circle. Our result could be regarded as a resolution
of the problem in the spherical case. As far as we know, this is the first
combinatorial curvature flow in spherical background geometry with fine
properties, and it provides an algorithm to find the desired ideal circle
pattern.
