Section. Introduction

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Section. Dynamics of point vortices

A point vortex

Let us consider a fluid in a flat system, e.g., a lake with shallow waters. Let us also consider a vortex in the fluid that has a cylindrical shape and its diameter is very small compared to all other length scales in the system (such as the distance between two vortices).

In this case, the vorticity is confined to small areas which are called vortex filaments. (A vortex filament is a thin, swirling thread of fluid where rotation, or vorticity, is concentrated along a narrow tube across the thickness of the fluid.) In a shallow fluid (a film of fluid), the behaviour of the vortex filament follows the behaviour of the vortex that we can observe when looking on the surface of the fluid.

We can give the position of the vortex as a point $(x,y)$ on the plane. In this case, we have the approximation of a point vortex. (Helmholz 1858).

Two vortices

We now consider two point vortices,

• a vortex with vorticity $\gamma_1$ and position $\bm{r}_1=(x_1, y_1)$,
• a vortex with vorticity $\gamma_2$ and position $\bm{r}_2=(x_2, y_2)$.

Two vortices that are close to each other are interacting via the motion of the fluid in the region between them.

Their equations of motion are (Helmholtz 1858, Kirchhoff 1876)

$$ \begin{aligned} & \dot{x}_1 = -\gamma_2\,\frac{y_1-y_2}{|\bm{r}_1-\bm{r}_2|^2}, & \dot{x}_2 = -\gamma_1\,\frac{y_2-y_1}{|\bm{r}_1-\bm{r}_2|^2}, \\ & \dot{y}_1 = \gamma_2\,\frac{x_1-x_2}{|\bm{r}_1-\bm{r}_2|^2}, & \dot{y}_2 = \gamma_1\,\frac{x_2-x_1}{|\bm{r}_1-\bm{r}_2|^2}, \tag{1}\end{aligned} $$

where

$$ |\bm{r}_1-\bm{r}_2|\equiv\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} $$