  We consider the Navier-Stokes equations on the two-dimensional unit sphere
and study the linear stability of the two-jet Kolmogorov type flow which is a
stationary solution given by the zonal spherical harmonic function of degree
two. We prove the linear stability of the two-jet Kolmogorov type flow for an
arbitrary viscosity coefficient by showing the exponential decay of a solution
to the linearized equation towards an equilibrium which grows as the viscosity
coefficient tends to zero. The main result of this paper is the nonexistence of
nonzero eigenvalues of the perturbation operator appearing in the linearized
equation. By making use of the mixing property of the perturbation operator
which is expressed by a recurrence relation for the spherical harmonics, we
show that the perturbation operator does not have not only nonreal but also
nonzero real eigenvalues. As an application of this result, we get the enhanced
dissipation for the two-jet Kolmogorov type flow in the sense that a solution
to the linearized equation rescaled in time decays arbitrarily fast as the
viscosity coefficient tends to zero.
