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 ${\mathcal{D}}_q(M)$. 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.

在本文中，我们计算了Euler-Arnold方程是地球物理和海洋学中的准地转（QG）方程或等离子物理学中的Hasegawa-Mima方程组的截面曲率：该组是拟同构的中心扩展组 ${\mathcal{d}}_q（\mathrm{中号}）$。我们考虑下面的歧管M旋转对称且流体以径向流函数流动的情况。使用显式的曲率公式，我们还将导出曲率算符为非正的准则，并讨论Froude数和Rossby数对曲率的作用。获得可用公式的主要技术是在向量场接近Killing场的情况下简化Arnold的一般曲率公式，然后显式地使用Green函数以及一般双线性形式的非负性准则。我们显示，尽管通常存在反例，但非零弗洛德数和Rossby数通常都趋向于拉格朗日意义上的稳定流量。