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In this paper, we compute the sectional curvature of the quantomorphism group$\mathcal{D}_q(M)$ whose geodesic equation is the quasi-geostrophic (QG)equation in geophysics and oceanography, for flows with a stream functiondepending on only one variable. Using this explicit formula, we will alsoderive a criterion for the curvature operator to be nonpositive and discuss therole of the Froude number and the Rossby number on curvature. The maintechnique to obtain a usable formula is a simplification of Arnold's generalformula in the case where a vector field is close to a Killing field, and thenuse the Green's function explicitly. We show that nonzero Froude number andRossby numbers both tend to stabilize flows in the Lagrangian sense.

Nonpositive Curvature of the quantomorphism group and quasigeostrophic motion