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arXiv · 1907.10256

Nonpositive Curvature of the quantomorphism group and quasigeostrophic motion

Jae Min Lee·Stephen C. Preston
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Abstract

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 function depending on only one variable. Using this explicit formula, 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 formula in the case where a vector field is close to a Killing field, and then use the Green's function explicitly. We show that nonzero Froude number and Rossby numbers both tend to stabilize flows in the Lagrangian sense.

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Jae Min Lee, Stephen C. Preston. 2019-07-24. Nonpositive Curvature of the quantomorphism group and quasigeostrophic motion. https://arxiv.org/abs/1907.10256

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