A New Proof on Net Upscale Energy Cascade in 2D and QG Turbulence
  A general proof that more energy flows upscale than downscale in two-dimensional (2D) turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in Surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases. 
Eleftherios Gkioulekas and Ka Kit Tung