On the differences between 2D and QG turbulence

Due to their mathematical tractability, two-dimensional (2D) fluid equations are often used by mathematicians as a model for quasi-geostrophic (QG) turbulence in the atmosphere, using Charney's 1971 paper as justification. Superficially, 2D and QG turbulence both satisfy the twin conservation of energy and enstrophy and thus are unlike 3D flows, which do not conserve enstrophy. Yet QG turbulence differs from 2D turbulence in fundamental ways, which are not generally known. Here we discuss ingredients missing in 2D turbulence formulations of large-scale atmospheric turbulence. We argue that there is no proof that energy cannot cascade downscale in QG turbulence. Indeed, observational evidence supports a downscale flux of both energy and enstrophy in the mesoscales. It is suggested that the observed atmospheric energy spectrum is explainable if there is a downscale energy cascade of QG turbulence, but is inconsistent with 2D turbulence theories, which require an upscale energy flux. A simple solved example is used to illustrate some of the ideas discussed.