  In geophysical fluid dynamics, the screened Poisson equation appears in the
shallow-water, quasi geostrophic equations. Recently, many attempts have been
made to solve those equations on the sphere using different numerical methods.
These include vortex methods, which solve a Poisson equation to compute the
stream-function from the (relative) vorticity. Alternatively, the
stream-function can be computed directly from potential vorticity (PV), which
would offer the possibility of constructing more attractive vortex methods
because PV is conserved along material trajectories in the inviscid case. On
the spherical shell, however, the screened Poisson equation does not admit a
known Green's function, which limits the extension of such approaches to the
case of a sphere. In this paper, we derive an expression of Green's function
for the screened Poisson equation on the spherical shell in series form and in
integral form. A proof of convergence of the series representation is then
given. As the series is slowly convergent, a robust and efficient approximation
is obtained using a split form which isolates the singular behavior. The
solutions are illustrated and analyzed for different values of the screening
constant.
