Exact Magnus-Force Formula for Three-Dimensional Fluid-Core Vortices

Abstract

The two-dimensional Magnus formula ${\overrightarrow{\mathrm{f}}}_m=\rho ({\overrightarrow{\mathrm{v}}}_s-{\overrightarrow{\mathrm{V}}}_v)\times \overrightarrow{\kappa }$ can be extended to three-dimensional fluid-core vortices of arbitrary core structure by a precise redefinition of the stream velocity vector ${\overrightarrow{\mathrm{v}}}_s$, vortex velocity vector ${\overrightarrow{\mathrm{V}}}_v$, and circulation vector $\overrightarrow{\kappa }$. For example, ${\overrightarrow{\mathrm{V}}}_v$ becomes the velocity, in a plane slicing the vortex core, of the "center of mass" of the vorticity. The exact three-dimensional formula is derived by analyzing the motion of vorticity relative to the fluid particles - a relative motion caused by the action of nonconservative forces on the vortex core (e.g., the electric force on a charged vortex ring in superfluid helium). The analysis deals with the vorticity field $\overrightarrow{\omega }\equiv \overrightarrow{\nabla }\times \overrightarrow{\mathrm{v}}$ and, thus, can lead to simple formulas and insights in cases where the velocity field $\overrightarrow{\mathrm{v}}$ may be impossibly complicated. Examples introduced are (i) the concept of a "conserved vorticity current," (ii) a possible classical mechanism for the creation or destruction of vortex lines in superfluid helium, and (iii) a simple technique for analyzing the effect of viscosity on the structure of a vortex core.