Energy flows upscale, enstrophy flows downscale — Kraichnan-Batchelor theory
In 2D turbulence, energy injected at scale k₀ cascades inversely to larger scales (E(k) ~ k⁻⁵/³) while enstrophy (∫ω²dA) cascades forward to smaller scales (E(k) ~ k⁻³). This is a consequence of having two inviscid invariants (energy and enstrophy). Kraichnan (1967) and Batchelor (1969) derived the dual cascade; vortex merging drives the inverse cascade in atmospheric and oceanic flows.