Vorticity field and dual cascade in 2D Navier-Stokes turbulence
Vorticity ω(x,y)
Energy Spectrum E(k)
t=0 | E=0 | Z=0
In 2D turbulence (Kraichnan 1967, Batchelor 1969), two conserved quantities—energy and enstrophy Z=½⟨ω²⟩—lead to a dual cascade.
Energy cascades upscale (inverse) with spectrum E(k) ~ k−5/3, while enstrophy cascades downscale with E(k) ~ k−3.
Forcing at intermediate scale k_f drives both cascades simultaneously. Large coherent vortices emerge via the inverse cascade.
This is the opposite of 3D turbulence and explains large-scale coherent structures in planetary atmospheres.