  We develop a neutral vortex fluid theory on closed surfaces with zero genus.
The theory describes collective dynamics of many well-separated quantum
vortices in a superfluid confined on a closed surface. Comparing to the case on
a plane, the covariant vortex fluid equation on a curved surface contains an
additional term proportional to Gaussian curvature multiplying the circulation
quantum. This term describes the coupling between topological defects and
curvature in the macroscopic level. For a sphere, the simplest nontrivial
stationary vortex flow is obtained analytically and this flow is analogous to
the celebrated zonal Rossby-Haurwitz wave in classical fluids on a nonrotating
sphere. For this flow the difference between the coarse-grained vortex velocity
field and the fluid velocity field generated by vortices is solely driven by
curvature and vanishes in the corresponding vortex flow on a plane when the
radius of the sphere goes to infinity.
