Exact spherical vortex-type equilibrium flows in fluids and plasmas

Jason M. Keller, Alexei F. Cheviakov
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Abstract

The famous Hill’s solution describing a spherical vortex with nested toroidal pressure surfaces, bounded by a sphere, propelling itself in an ideal Eulerian fluid, is re-derived using Galilei symmetry and the Bragg–Hawthorne equations in spherical coordinates. The correspondence between equilibrium Euler equations of fluid dynamics and static magnetohydrodynamic equations is used to derive a generalized vortex type solution that corresponds to dynamic fluid equilibria and static plasma equilibria with a nonzero azimuthal vector field component, satisfying physical boundary conditions. Separation of variables in Bragg–Hawthorne equation in spherical coordinates is used to construct further new fluid and plasma equilibria with nested toroidal flux surfaces, featuring respectively boundary vorticity sheets and current sheets. Finally, the instability of the original Hill’s vortex with respect to certain radial perturbations of the spherical flux surface is proven analytically and illustrated numerically.

流体和等离子体中的精确球形旋涡型平衡流动
利用伽利略对称性和球面坐标中的布拉格-霍桑方程,重新推导出了著名的希尔解,该解描述了在理想欧拉流体中,以球面为界,具有嵌套环形压力面的球形涡旋的推进过程。利用流体动力学平衡欧拉方程和静态磁流体动力学方程之间的对应关系,推导出一种广义旋涡型解法,该解法对应于动态流体平衡和静态等离子体平衡,具有非零方位矢量场分量,满足物理边界条件。利用布拉格-霍桑方程在球面坐标下的变量分离,进一步构建了具有嵌套环形通量面的新流体和等离子体平衡,分别以边界涡流片和电流片为特征。最后,分析和数值说明了原始希尔旋涡在球形通量面的某些径向扰动下的不稳定性。
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