  This paper presents a general form of the covariance matrix structure for a
vector random field that is axially symmetric and mean square continuous on the
sphere and provides a series representation for a longitudinally reversible
one. The series representation is somehow an imitator of the covariance matrix
function, and both of them have simpler forms than those proposed in the
literature in terms of the associated Legendre functions and are useful for
modeling and simulation. Also, a general form of the covariance matrix
structure is derived for a spatio-temporal vector random field that is axially
symmetric and mean square continuous over the sphere, and a series
representation is given for a longitudinally reversible one.
