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 A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...

of ''N'' points in the two-dimensional plane executing the motion :

k_i\frac = \frac,\qquad k_i\frac = -\frac,

(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 fluid dynamics, a vortex (: vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...

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 In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...

of the ''i''th filament. The Hamiltonian ''H'' is just the two-dimensional

Coulomb potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...

between lines. The equations of motion reflect the

guiding center In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relat ...

drift of the charge filaments, hence the name.

Notes

References

*{{cite journal , author=Eyink, Gregory , author2=Katepalli Sreenivasan , name-list-style=amp , title=Onsager and the theory of hydrodynamic turbulence , journal=Reviews of Modern Physics , volume=78 , issue=1 , date=January 2006 , pages=87–135 , doi=10.1103/RevModPhys.78.87 , bibcode=2006RvMP...78...87E, citeseerx=10.1.1.516.6219 Turbulence models Plasma theory and modeling