# Nonpositive Curvature of the quantomorphism group and quasigeostrophic motion

## TL;DR

This paper calculates the sectional curvature of the quantomorphism group related to the quasi-geostrophic equation, revealing how physical parameters like Froude and Rossby numbers influence flow stability.

## Contribution

It provides an explicit curvature formula for the quantomorphism group and analyzes the effects of physical parameters on flow stability in geophysical fluid dynamics.

## Key findings

- Nonpositive curvature criterion derived
- Froude and Rossby numbers stabilize flows
- Explicit Green's function used for calculations

## 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.