Collapse of generalized Euler and surface quasigeostrophic point vortices.
Collapse of generalized Euler and surface quasigeostrophic point vortices.
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DOI：
10.1103/physreve.98.023110
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发表时间：
2018-05
期刊：
影响因子：
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通讯作者：
G. Badin;A. Barry
中科院分区：
文献类型：
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作者：
G. Badin;A. Barry
提出了广义欧拉方程的点涡模型，该方程具有活动标量与流函数之间的分数阶拉普拉斯关系。特别关注曲面拟等转（SQG）方程的情况，对于这些方程，有限时间奇点的存在性仍然是一个争论的问题。点涡轨迹用Nambu动力学表示。该公式基于非正则括号，并允许将轨迹作为哈密顿和卡西米尔水平集的交集的几何解释。在这个设置中，我们关注三点涡模型解的坍缩。特别地，我们证明了对于SQG，坍缩可以是自相似的，也可以是非自相似的。自相似只在哈密顿量为零时出现，而非自相似在哈密顿量为零时出现。对于这两种情况，在允许的间隔内，任何选择的循环都允许崩溃。这些结果明显不同于经典的点涡模型，在点涡模型中坍缩对任何哈密顿量都是自相似的，但涡环流必须满足严格的关系。结果也可能揭示SQG偏微分方程中奇点的形成，其中奇点被认为只能以自相似的方式达到。
Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.