Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations
Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations
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DOI：
10.1007/s00205-018-1312-7
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发表时间：
2019-03-01
影响因子：
2.5
通讯作者：
Renault, Coralie
中科院分区：
文献类型：
--
作者：
Dritschel, David Gerard;Hmidi, Taoufik;Renault, Coralie
本文研究了准地转浅水（QGSW）方程单连通旋转涡斑平衡点分岔图的解析和数值解。QGSW方程是欧拉方程的推广，并包含一个附加参数，Rossby变形长度λ-1，它进入流函数和（位）涡之间的关系。欧拉方程在极限ε 0下恢复。我们证明，接近圆形（朗肯）涡，任意Rossby变形长度的分歧图的持久性。然而，我们表明，两重分支，对应于基尔霍夫椭圆的欧拉方程，从来没有连接，甚至为小值，并确实分裂成一个可数集的不相交的连接分支。精确的数值计算的全局结构的分岔图和极限平衡状态也提出了补充的数学分析。
We study analytical and numerical aspects of the bifurcation diagram of simply connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length epsilon-1, which enters into the relation between the stream function and (potential) vorticity. The Euler equations are recovered in the limit epsilon 0. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the two-fold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values epsilon, and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.