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 i d x i d t = ∂ H ∂ y i , k i d y i d t = − ∂ H ∂ x i , {\displaystyle k_{i}{\frac {dx_{i}}{dt}}={\frac {\partial H}{\partial y_{i}}},\qquad k_{i}{\frac {dy_{i}}{dt}}=-{\frac {\partial H}{\partial x_{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 ki is the circulation of the fluid around the ith 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 ki is the linear charge density of the ith 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
- Eyink, Gregory & Katepalli Sreenivasan (January 2006). "Onsager and the theory of hydrodynamic turbulence". Reviews of Modern Physics. 78 (1): 87–135. Bibcode:2006RvMP...78...87E. CiteSeerX 10.1.1.516.6219. doi:10.1103/RevModPhys.78.87.