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Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics.
Journal of Nonlinear Science ( IF 3 ) Pub Date : 2017-03-16 , DOI: 10.1007/s00332-017-9367-4 Darryl D Holm 1 , Henry O Jacobs 1
Journal of Nonlinear Science ( IF 3 ) Pub Date : 2017-03-16 , DOI: 10.1007/s00332-017-9367-4 Darryl D Holm 1 , Henry O Jacobs 1
Affiliation
Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.
中文翻译:
多极涡流斑点(MVB):辛几何和动力学。
涡流斑点法通常以正则化长度标度为特征,在该长度标度下,对于孤立的斑点而言,动力学是微不足道的。在本文中,我们观察到,如果愿意将Dirac delta泛函的分布导数视为有效涡度分布,则动力学并不一定非同小可。更具体地说,提出了一种新的奇异涡旋理论,用于平面上理想不可压缩流的正则欧拉流体方程。我们确定条件,在这种条件下,这种正规化的Euler流体方程可以接受比delta函数(例如delta函数的导数)更强的涡度奇点。我们还描述了与这些增强涡旋结构相关的辛几何,并将动力学描述为哈密顿量。还讨论了在类似于漩涡斑点方法的数值方法设计中的应用。这些发现阐明了在正则化长度尺度以下发生的丰富动态,并启发了我们对正则化流体模型捕获多尺度现象的潜力的看法。
更新日期:2017-03-16
中文翻译:
多极涡流斑点(MVB):辛几何和动力学。
涡流斑点法通常以正则化长度标度为特征,在该长度标度下,对于孤立的斑点而言,动力学是微不足道的。在本文中,我们观察到,如果愿意将Dirac delta泛函的分布导数视为有效涡度分布,则动力学并不一定非同小可。更具体地说,提出了一种新的奇异涡旋理论,用于平面上理想不可压缩流的正则欧拉流体方程。我们确定条件,在这种条件下,这种正规化的Euler流体方程可以接受比delta函数(例如delta函数的导数)更强的涡度奇点。我们还描述了与这些增强涡旋结构相关的辛几何,并将动力学描述为哈密顿量。还讨论了在类似于漩涡斑点方法的数值方法设计中的应用。这些发现阐明了在正则化长度尺度以下发生的丰富动态,并启发了我们对正则化流体模型捕获多尺度现象的潜力的看法。