Two-dimensional point vortex gas

The two-dimensional point vortex gas is a discrete particle model used to study turbulence in two-dimensional ideal fluids. The two-dimensional guiding-center plasma is a completely equivalent model used in plasma physics.

General setup

The model is a Hamiltonian system of "N" points in the two-dimensional plane executing the motion:$k\_ifrac\{dx\_i\}\{dt\}\; =\; frac\{partial\; H\}\{partial\; y\_i\},qquad\; k\_ifrac\{dy\_i\}\{dt\}\; =\; -frac\{partial\; H\}\{partial\; x\_i\},$

where the "k"_"i" are constant and:

where "r"_"ij" is the distance between the "i"th and "j"th points. [Eyink and Sreenivasan p.90; scaling constants have been omitted]

The "x"_"i" and "y"_"i" are not quite canonically conjugate quantities, due to the "k"_"i" in the equations of motion. The conjugate quantities are instead:$k\_i^\{1/2\}x\_i,quad\; k\_i^\{1/2\}y\_i$

(In the confined version of the problem, the logarithmic potential is modified.)

Interpretations

In the point-vortex gas interpretation, the particles represent either point vortices in a two-dimensional fluid, or parallel line vortices in a three-dimensional fluid. The constant "k"_"i" is the circulation of the fluid around the "i"th vortex. The Hamiltonian "H" is the interaction term of the fluid's integrated kinetic energy; it may be either positive or negative. The equations of motion simply reflect the drift of each vortex's position in the velocity field of the other vortices.

In the guiding-center plasma interpretation, the particles represent long filaments of charge parallel to some external magnetic field. The constant "k"_"i" is the linear charge density of the "i"th filament. The Hamiltonian "H" is just the two-dimensional Coulomb potential between lines. The equations of motion reflect the guiding center drift of the charge filaments, hence the name.

Notes

References

*cite journal |author=Eyink, Gregory and Katepalli Sreenivasan |title=Onsager and the theory of hydrodynamic turbulence |journal=Reviews of Modern Physics |volume=78 |year=2006 |month=January |pages=87–135 |doi=10.1103/RevModPhys.78.87