  Combinatorial Calabi flows are introduced by Ge in his Ph.D. thesis
(Combinatorial methods and geometric equations, Peking University, Beijing,
2012), and have been studied extensively in Euclidean and hyperbolic background
geometry. In this paper, we introduce the combinatorial Calabi flow in
spherical background geometry for finding ideal circle patterns with prescribed
total geodesic curvatures. We prove that the solution of combinatorial Calabi
flow exists for all time and converges if and only if there exists an ideal
circle pattern with prescribed total geodesic curvatures. We also show that if
it converges, it will converge exponentially fast to the desired metric, which
provides an effective algorithm to find certain ideal circle patterns. To our
knowledge, it is the first combinatorial Calabi flow in spherical background
geometry.
