  In this paper, we establish three Arnold-type stability theorems for steady
or rotating solutions of the incompressible Euler equation on a sphere.
Specifically, we prove that if the stream function of a flow solves a
semilinear elliptic equation with a monotone nonlinearity, then, under
appropriate conditions, the flow is stable or orbitally stable in the Lyapunov
sense. In particular, our theorems apply to degree-2 Rossby-Haurwitz waves.
These results are achieved via a variational approach, with the key ingredient
being to show that the flows under consideration satisfy the conditions of two
Burton-type stability criteria which are established in this paper. As
byproducts, we obtain some sharp rigidity results for solutions of semilinear
elliptic equations on a sphere.
