Inertial Ranges in Two-Dimensional Turbulence

Two-dimensional turbulence has both kinetic energy and mean-square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k)∼ε2/3k−5/3 and E(k)∼η2/3k−3, where ε is the rate of cascade of kinetic energy per unit mass, η is the rate of cascade of mean-square vorticity, and the kinetic energy per unit mass is ∫0∞E(k) dk. The −53 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero-vorticity flow. The −3 range gives an upward vorticity flow and zero-energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber interactions. The formal −3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers ∼ki and the Reynolds number is large, it is conjectured that a quasi-steady-state results with a −53 range for k « ki and a −3 range for k » ki, up to the viscous cutoff. The total kinetic energy increases steadily with time as the −53 range pushes to ever-lower k, until scales the size of the entire fluid are strongly excited. The rate of energy dissipation by viscosity decreases to zero if kinematic viscosity is decreased to zero with other parameters unchanged.