  We consider the vorticity form of the Navier-Stokes equations on the
two-dimensional unit sphere and study the nonlinear stability of the two-jet
Kolmogorov type flow which is a stationary solution given by the zonal
spherical harmonic function of degree two. In particular, we assume that a
perturbation contains a nondissipative part given by a linear combination of
the spherical harmonics of degree one and investigate the effect of the
nondissipative part on the long-time behavior of the perturbation through the
convection term. We show that the nondissipative part of a weak solution to the
nonlinear stability problem is preserved in time for all initial data.
Moreover, we prove that the dissipative part of the weak solution converges
exponentially in time towards an equilibrium which is expressed explicitly in
terms of the nondissipative part of the initial data and does not vanish in
general. In particular, it turns out that the asymptotic behavior of the weak
solution is finally determined by a system of linear ordinary differential
equations. To prove these results, we make use of properties of Killing vector
fields on a manifold. We also consider the case of a rotating sphere.
