Nonpositive curvature of the quantomorphism group and quasigeostrophic motion

Abstract

In this paper, we compute the sectional curvature of the group whose Euler-Arnold equation is the quasi-geostrophic (QG) equation in geophysics and oceanography, or the Hasegawa-Mima equation in plasma physics: this group is a central extension of the quantomorphism group Dq(M). We consider the case where the underlying manifold M is rotationally symmetric, and the fluid flows with a radial stream function. Using an explicit formula for the curvature, we will also derive a criterion for the curvature operator to be nonpositive and discuss the role of the Froude number and the Rossby number on curvature. The main technique to obtain a usable formula is a simplification of Arnold's general curvature formula in the case where a vector field is close to a Killing field, and then use the Green's function explicitly together with a criterion for nonnegativity of a general bilinear form. We show that nonzero Froude number and Rossby numbers typically both tend to stabilize flows in the Lagrangian sense, although there are counterexamples in general.