Holm Darryl D, Jacobs Henry O
Department of Mathematics, Imperial College, London, SW7 2AZ UK.
J Nonlinear Sci. 2017;27(3):973-1006. doi: 10.1007/s00332-017-9367-4. Epub 2017 Mar 16.
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.
涡旋斑点方法通常以一个正则化长度尺度为特征，在该尺度以下，对于孤立斑点而言动力学是平凡的。在本文中，我们观察到，如果愿意将狄拉克δ泛函的分布导数视为有效的涡度分布，那么动力学未必是平凡的。更具体地说，针对平面上理想不可压缩流的正则化欧拉流体方程，提出了一种新的奇异涡旋理论。我们确定了这样的正则化欧拉流体方程可能允许比δ函数更强的涡度奇点（例如，δ函数的导数）的条件。我们还描述了与这些增强涡旋结构相关的辛几何，并将动力学表征为哈密顿量。还讨论了在设计类似于涡旋斑点方法的数值方法方面的应用。这些发现揭示了在正则化长度尺度以下发生的丰富动力学，并启发了我们对于正则化流体模型捕捉多尺度现象潜力的认识。
