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 multiplied 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). In fact, to Helmholtz the points of vorticity were the intersections with a fixed plane of parallel lines of vorticity in 3 dimensions.

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

Mathematical formulation

For $N\geq 2$, 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

$H(r_{1},\dots ,r_{N})=-{\frac {1}{4\pi }}\sum _{j\neq k}\Gamma _{j}\Gamma _{k}\ln(\rho _{jk}),$

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

${\dot {z}}_{j}={\frac {1}{2\pi i}}\sum _{k\neq j}\Gamma _{k}{\frac {1}{{\overline {z}}_{k}-{\overline {z}}_{j}}}.$

And the Hamiltonian version is, for the same Hamiltonian as above,

$\Gamma _{j}{\dot {z}}_{j}={\frac {1}{2\pi i}}\,{\frac {\partial H}{\partial {\overline {z_{j}}}}}\qquad ({\text{for }}j=1,\dots N).$

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 ‘’rotational impulse’’ (or “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

$P+iQ=\sum _{j=1}^{N}\Gamma _{j}z_{j}.$

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 gravity, the vector

$\mathbf {c} ={\frac {1}{\Lambda }}{\begin{pmatrix}P\\Q\end{pmatrix}}\in \mathbb {R} ^{2}$

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 $\Gamma _{1}+\Gamma _{2}\neq 0$ 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 and 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 mutual distance. [check ??? - or is there a 4π?]

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. A summary can be found in the paper of Aref (1979).

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 18?? Kelvin analyzed the equations of motion near such a ring and showed that for $N\leq 6$ the ring is linearly stable, while if $N\geq 8$ 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 ring (with $N=7$) 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 first written by Bogomolov (1977), with important early work by Y. Kimura (1999) on surfaces of constant curvature.

More recent work by Boatto and Dritschel (20??) and by Rodgrigues?? shows the relevance of the Greens function for the Laplacian and the Robin function for surfaces of non-constant curvature.

Diffeomorphism group

Based on foundational ideas of V.I. Arnold (1966), 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 a surface.

References

• Aref, H. (1979), "Motion of three vortices", Physics of Fluids, 22: 393–400
• Aref, H (2007), "Point vortex dynamics: A classical mathematics playground", J. Math. Phys., 48: 065401
• Arnold, V. I. (1966), "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits", Ann. Inst. Fourier Grenoble, 16: 319–361
• Bogomolov, V.A. (1977), "Dynamics of vorticity at a sphere", Fluid Dynamics, 6: 863–870
• Kimura, Y (1999), "Vortex motion on surfaces with constant curvature", Proc. R. Soc. Lond. A, 455: 245–259
• Marsden, J.E.; Weinstein, A. (1983), "Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids", Physica D, 7: 305–323
• Newton, Paul (2001). The N vortex Problem (2nd ed.). Springer.
• Schmidt, D. (2004), "The stability of the Thomson heptagon", Regular and Chaotic Dynamics, 9: 519–528