Vortex dynamics in R4
Vortex dynamics in R4
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DOI：
10.1063/1.3673800
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发表时间：
2012-01-01
影响因子：
1.3
通讯作者：
Shashikanth, Banavara N.
中科院分区：
文献类型：
--
作者：
Shashikanth, Banavara N.
研究了R-4中等密度流体流动时欧拉方程的涡动力学。本文主要讨论R-4中涡度二形式的奇异狄拉克分布。这些分布支持在称为膜的二维表面上，是R-3中的涡丝和R-2中的点涡的类似物。证明了膜的自激速度场是无界的，并使用局部感应近似正则化。正则化的自诱导速度场与膜的平均曲率矢量场成正比，但在法线平面上旋转90度。其次，提出了哈密顿膜模型。该模型的辛结构来源于Marsden和Weinstein提出的涡度分布的一般公式[《不可压缩流体的共伴轨道、涡和Clebsch变量》，《物理学D》7,305 -323(1983)]。最后，研究了四种形式的布尔和的动力学特性。结果表明，对于等密度情况，R-3中的Ertel涡量定理可以看作是这四种形式动力学的一种特殊情况。(C) 2012年美国物理研究所。(doi: 10.1063/1.3673800)
The vortex dynamics of Euler's equations for a constant density fluid flow in R-4 is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form omega in R-4. These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in R-3 and point vortices in R-2. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90 degrees in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein ["Coadjoint orbits, vortices and Clebsch variables for incompressible fluids," Physica D 7, 305-323 (1983)]. Finally, the dynamics of the four-form omega boolean AND omega is examined. It is shown that Ertel's vorticity theorem in R-3, for the constant density case, can be viewed as a special case of the dynamics of this four-form. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3673800]