The geometry of a vorticity model equation

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [22] describes the geodesic flow on the subgroup Diff∞1 (S) of orientation-preserving diffeomorphisms φ ∈ Diff∞(S) such that φ(1) = 1 with respect to right-invariant metric induced by the homogeneous Sobolev space Ḣ(S) and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class H with k ≥ 2.