Energy flows to large scales (k⁻⁵/³) · Enstrophy cascades forward (k⁻³) · Forced Navier-Stokes
2D Navier-Stokes: ∂ω/∂t + u·∇ω = ν∇²ω + f, where ω=curl(u) is vorticity.
Unlike 3D: vortex stretching absent → two conserved quadratic quantities: energy E=½⟨|u|²⟩ and enstrophy Z=½⟨ω²⟩.
Kraichnan 1967: dual cascade — energy flows to large scales (E(k)∝k⁻⁵/³ for k<k_f), enstrophy to small scales (E(k)∝k⁻³ for k>k_f).
The inverse cascade generates coherent large-scale vortices. Spectral simulation using pseudo-spectral method (simplified).