Two-dimensional turbulence in inviscid fluids or guiding center plasmas

Analytic theory for two-dimensional turbulent equilibria for the inviscid Navier–Stokes equation (or the electrostatic guiding center plasma) is tested numerically. A good fit is demonstrated for the approach to a predicted energy per Fourier mode obtained from the two-temperature canonical ensemble of Kraichnan: 〈‖u(k) ‖2〉 = (α+βk2)−1, where k is the wavenumber and α and β are reciprocal energy and enstrophy temperatures. Negative as well as positive temperature regimes are explored. Fluctuations about the mean energy per mode also compare well with theory. In the regime α0, β≳0, with the minimum value of α+βk2 near zero, contour plots of the stream function reveal macroscopic vortex structures similar to those seen previously in discrete vortex simulations by Joyce and Montgomery. Kraichnan’s assertion that thermodynamic limits exist for the negative temperature states is questioned. Eulerian direct interaction equations, which can be used to follow the approach to inviscid equilibrium, are derived.