  In this article we study the induced geodesic distance of fractional order
Sobolev metrics on the groups of (volume preserving) diffeomorphisms and
symplectomorphisms. The interest in these geometries is fueled by the
observation that they allow for a geometric interpretation for prominent
partial differential equations in the field of fluid dynamics. These include in
particular the modified Constantin-Lax-Majda and surface quasi-geostrophic
equations. The main result of this article shows that both of these equations
stem from a Riemannian metric with vanishing geodesic distance.
