In this paper we characterized isotropic random tangential vector fields on d-spheres for d?1 by the cross-covariance, and derived their Karhunen�Lo�ve 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?3. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows. � 2024 Elsevier B.V.