Inertial Ranges in Two-Dimensional Turbulence

TLDR

Two‑dimensional turbulence conserves kinetic energy and mean‑square vorticity, giving rise to two inertial ranges with spectra E(k)∼ε2/3k−5/3 and E(k)∼η2/3k−3. The study conjectures that constant‑rate energy injection at a narrow wavenumber band ∼ki in high‑Reynolds‑number 2D turbulence produces a quasi‑steady state with a −5/3 range for k≪ki and a −3 range for k≫ki up to the viscous cutoff. The authors model energy injection at a fixed rate into a narrow wavenumber band ∼ki and analyze the resulting cascade dynamics in the high‑Reynolds‑number limit. The analysis shows that the −5/3 range drives a backward energy cascade with zero vorticity, while the −3 range produces an upward vorticity flux with zero energy transfer; the apparent paradox is resolved by the triangular nature of wavenumber interactions, and the formal −3 range requires logarithmic corrections, leading to a steadily increasing kinetic energy and vanishing viscous dissipation as viscosity tends to zero.

Abstract

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.