Isotropic random tangential vector fields on spheres

In this paper we characterized isotropic random tangential vector fields on $d$-spheres for $d\ge 1$ by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a $d$-sphere for $d\ge 3$. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.