This paper develops a fractional stochastic partial differential equation
(SPDE) to model the evolution of a random tangent vector field on the unit
sphere. The SPDE is governed by a fractional diffusion operator to model the
L\'{e}vy-type behaviour of the spatial solution, a fractional derivative in
time to depict the intermittency of its temporal solution, and is driven by
vector-valued fractional Brownian motion on the unit sphere to characterize its
temporal long-range dependence. The solution to the SPDE is presented in the
form of the Karhunen-Lo\`{e}ve expansion in terms of vector spherical
harmonics. Its covariance matrix function is established as a tensor field on
the unit sphere that is an expansion of Legendre tensor kernels. The variance
of the increments and approximations to the solutions are studied and
convergence rates of the approximation errors are given. It is demonstrated how
these convergence rates depend on the decay of the power spectrum and variances
of the fractional Brownian motion.Comment: 20 page