Draft:Point Vortices

In Mathematics, and in particular Fluid Mechanics, the point vortex system is a well-studied dynamical system consisting of a number of points on the plane (or other surface) moving according to a law of interaction that derives from fluid motion. It was first introduced by Hermann von Helmholtz in 1858, as part of his investigation into the motion of vortex filaments in 3 dimensions. It was first written in Hamiltonian form by Gustav Kirchhoff a few years later.

A single vortex in a fluid is like a whirlpool, with the fluid rotating about a central point (the point vortex). The speed of the fluid is inversely proportional to the distance of a fluid element to the point vortex. The constant of proportionality (or it divided by 2π) is called the vorticity or vortex strength of the point vortex. This is positive if the fluid rotates anticlockwise (like a cyclone in the Northern hemisphere) or negative if it rotates clockwise (anticyclone in the N. hemisphere), and has a constant value for each point vortex.

It was realized by Helmholtz that if there are several point vortices in an ideal fluid, then the motion of each depends only on the positions and strengths of the others, thereby giving rise to a system of ordinary differential equations whose variables are the coordinates of the point vortices (as functions of time).

The study of point vortices has been called a Classical Mathematics Playground by the fluid dynamicist Hassan Aref (2007), due to the many areas of classical mathematics that can be brought to bear for the analysis of this dynamical system.

Mathematical formulation
For $$N\geq2$$, consider $$N$$ points in the plane, with coordinates $$(x_j,y_j)$$ (for $$j=1,\dots,N$$) and vortex strengths $$\Gamma_1,\dots,\Gamma_N$$. Then Helmholtz’s equations of motion of the points are


 * $$\begin{array}{rcl}

\dot{x}_j &=& \frac{1}{2\pi}\sum_{k\neq j} \Gamma_k \frac{y_k-y_j}{\rho_{jk}} \\ \dot{y}_j &=& -\frac{1}{2\pi}\sum_{k\neq j} \Gamma_k \frac{x_k-x_j}{\rho_{jk}} \end{array}$$

where $$\rho_{jk}=(x_j-x_k)^2+(y_j-y_k)^2$$ (the square of the distance between vortex $$j$$ and vortex $$k$$).

Hamiltonian version
Consider the function (called the Kirchhoff-Routh Hamiltonian) given by
 * $$H(r_1,\dots,r_N) = -\frac1{4\pi}\sum_{j\neq k}\Gamma_j\Gamma_k\ln(\rho_{jk}),$$

where $$\rho_{jk}$$ is given above.

The equations of motion are then given by
 * $$\Gamma_j\dot x_j = \frac{\partial H}{\partial y_j},\quad \Gamma_j\dot y_j = -\frac{\partial H}{\partial x_j}\qquad(\text{for } j=1,\dots N).$$

Note: if we put, for example, $$q_j=x_j$$ and $$p_j=\Gamma_j y_j$$ then these become Hamilton’s usual canonical equations.

Complex notation
Authors often use complex numbers to denote the positions of the vortices, with $$z_j=x_j+iy_j$$. Then $$\rho_{jk}=|z_j-z_k|^2$$ and Helmholtz’s equations become


 * $$\dot{z}_j = \frac1{2\pi i}\sum_{k\neq j}\Gamma_k \frac1{\overline{z}_k-\overline{z}_j}.$$

And the Hamiltonian version is, for the same Hamiltonian as above,
 * $$\Gamma_j\dot{z}_j = \frac1{2\pi i}\, \frac{\partial H}{\partial \overline{z_j}}\qquad(\text{for } j=1,\dots N).$$

Conserved quantities
In addition to the Hamiltonian, there are three (real) conserved quantities. Two of these form the moment of vorticity and the third is the angular impulse:


 * $$P=\sum_{j=1}^N \Gamma_j x_j,\quad \text{and}\quad Q =\sum_{j=1}^N \Gamma_j y_j$$

In complex notation, this becomes
 * $$P+iQ = \sum_{j=1}^N\Gamma_jz_j.$$

The third is
 * $$I= \sum_{j=1}^N \Gamma_j\left(x_j^2+y_j^2\right) = \sum_{j=1}^N\Gamma_j|z_j|^2.$$

In the case that the total vorticity (or total circulation) $$\Lambda=\sum_j\Gamma_j$$ is non-zero, then by analogy with the centre of mass, the vector
 * $$\mathbf{c} =\frac1{\Lambda} \begin{pmatrix}P\\Q\end{pmatrix}\in\mathbb{R}^2$$

is called the centre of vorticity.

These conserved quantities are related to the Euclidean symmetries of the system via Noether’s theorem.

Two point vortices
This simplest case can be solved exactly, and was already described by Helmholtz. Since the Hamiltonian is conserved it follows that the distance between the two vortices is constant.

If $$\Gamma_1+\Gamma_2\neq0$$ then the centre of vorticity $$\mathbf{c}$$ is fixed and it follows from a short calculation that each of the two vortices moves in circles with centre $$\mathbf{c}$$. The rate of rotation is constant, with period equal to $$2\pi d^2/(\Gamma_1+\Gamma_2)$$, where d is the separation of the vortices: this is easily derived from the equations of motion.

If $$\Gamma_1+\Gamma_2=0$$ then the two vortices move at constant velocity in a direction perpendicular to the line joining the vortices, with speed equal to $$|\Gamma_1|/2\pi d$$ where d is the separation of the vortices.

Three point vortices
The case of 3 point vortices is completely integrable so is not chaotic. The first study of this case was made by Gröbli in his 1877 thesis. Further work was done by J.L. Synge. A summary and further investigation can be found in the paper of Aref (1979).

One property is there are two types of relative equilibrium, which are motions where the 3 vortices move while maintaining their separations. The two types are firstly where the 3 vortices are collinear, and secondly where they form the vertices of an equilateral triangle. This is similar to the analogous property of 3 attracting bodies in celestial mechanics, especially their central configurations.

Rings of point vortices
If N identical point vortices are placed at the vertices of a regular polygon, the configuration is known as a ring. It was known to Kelvin that given such a configuration the vortices rotate at a constant rate (depending on the common vorticity and the radius) about the centre of vorticity, which lies at the centre of the polygon; he also knew that for three identical vortices this motion was (linearly) stable.

In his Adams Prize essay of 1843, J.J. Thomson (later famous for discovering the electron) analyzed the equations of motion near such a ring and showed that for $$N\leq 6$$ the ring is linearly stable, while for $$N\geq8$$ it is unstable. The $$N=7$$ case is known as Thomson’s heptagon and is degenerate (more eigenvalues of the linear approximation are zero than would be expected from the conservation laws). The nonlinear stability of Thomson’s heptagon was finally settled by Dieter Schmidt (2004) who showed, using normal form theory, that it is indeed stable.

Point vortices on non-planar surfaces
There has been much interest in recent years on adapting the point vortex system to surfaces other than the plane. The first example was the system on the sphere, where the equations of motion were written down by Bogomolov (1977), with important early work by Kimura (1999) on general surfaces of constant curvature.

More recent work by Boatto and Dritschel (2015) and others shows the importance of the Green's function for the Laplacian and the Robin function for surfaces of non-constant curvature.

Diffeomorphism group
Based on foundational ideas of V.I. Arnold, Marsden and Weinstein (1983) showed that the point vortex system could be modelled as a Hamiltonian system on a natural finite dimensional coadjoint orbit of the diffeomorphism group of any orientable surface.