Effective equations and the inverse cascade theory for Kolmogorov flows

A detailed study is presented of the inverse cascade process in the Kolmogorov flows in the limit as the forcing wave number goes to infinity. Extensive numerical results reveal that energy is transferred from high wave numbers to low ones through two distinctive stages. At the early stage, the low wave-number spectrum evolves to a universal k^-4 decay law. At the same time, enough scales are generated between the forcing scale (small) and the size of the system (large scale) so that there is no longer separation of scales in the system. In the next stage the flow undergoes a transition to a continuum of scales. This transition proceeds through the formation of a hierarchy of layers of elongated vortices at increasingly larger scales with alternating orientation. To explain these phenomena, effective equations governing the evolution of the large-scale quantities are derived. Strong numerical evidence is presented that even with smooth initial data, the solution to the effective equation develops a k^-4 type singularity at a finite time. The effective equation also exhibits a weak instability which suggest that gradients in the direction of the forcing will grow much quicker than gradients in other directions.