Energy extremals and Nonlinear Stability in a Variational theory of Barotropic Flow on a Rotating Sphere

The main result is that at any rate of spin Ω and relative enstrophy Qrel, the unique global energy maximizer for fixed relative enstrophy corresponds to solid-body rotation, w0 Max(Qrel) = √ Qrelψ10 in the direction of Ω. Another solution, the counter-rotating steadystate w0 min(Qrel) = − √ Qrelψ10, is a constrained energy minimum provided the relative enstrophy is small enough, i.e., Qrel < Ω 2C2 where C = || cos θ ||2. If Ω2C2 < Qrel, then w0 min(Qrel) is a saddle point. For all energy H below a threshold value Hc which depends on the relative enstrophy Qrel and spin Ω, the constrained energy extremals consist of only minimizers and saddles in the form of counter-rotating states, −Qrelψ10. Only when the energy exceeds this threshold value Hc, can pro-rotating states, √ Qrelψ10 arise as global maximizers. Given the conditions for w0 min(Qrel) to be a local constrained minima, the solid-body rotation opposite to spin w0 min(Qrel) is nonlinearly stable. The global constrained maximizer w0 Max(Qrel) corresponding to solid-body rotation in the direction of spin, is always nonlinearly stable.