Section. Introduction

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Section. Dynamics of point vortices

A point vortex

We consider a fluid in a flat system, e.g., a lake with shallow waters. Let us 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. 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 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 interaction via the motion of the fluid between them.

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

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

where

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