The efficient computation of complex flows on the sphere, governed by the 2D shallow-water equations, is of acute importance in the modeling and forecasting of weather phenomenon on the earth. Some of the most powerful supercomputer clusters every built have been fully dedicated to this problem. In the years to come, increased performance in such clusters will be derived in large part from massive parallelization, to tens of thousands and even hundreds of thousands of computational nodes in the cluster. To facilitate such scalability, switchless interconnect systems coordinating the communication within the cluster are absolutely essential, as such systems eliminate an otherwise significant bottleneck (that is, the switch) impeding the communication between the nodes. The present work introduces a new switchless interconnect topology for supercomputer clusters which are dedicated specifically for computing such flows on the sphere. This topology is based on a class of Fullerenes (i.e., Buckyballs) with octahedral symmetry. In this topology, each node has direct send/receive capabilities with three neighboring nodes, and the cluster is itself physically connected in a spherical configuration. This natural correspondence between the interconnect network and the discretized physical model itself tends to keep most communication local (that is, between neighbors) during the flow simulation, thereby minimizing the density of packets being passed across the cluster and increasing dramatically the overall computational speed. One of the most communication-intensive steps of the flow simulation is related to solving the Poisson equation on the sphere; it is shown that the present topology is particularly well suited to this problem, leveraging multigrid acceleration with Red/Black Gauss-Seidel smoothing.