The culmination of an inverse cascade: mean flow and fluctuations.

Two dimensional turbulence has a remarkable tendency to self-organize into large, coherent structures, forming a mean flow. The purpose of this paper is to elucidate how these structures are sustained, and what determines them and the fluctuations around them. A recent theory for the mean flow will be reviewed. The theory assumes turbulence is excited by a forcing supported on small scales, and uses a linear shear model to relate the turbulent momentum flux to the mean shear rate. Extending the theory, it will be shown here that the relation between the momentum flux and mean shear is valid, and the momentum flux is non-zero, for both an isotropic and an anisotropic forcing, independent of the dissipation mechanism at small scales. This conclusion requires taking into account that the linear shear model is an approximation to the real system. The proportionality between the momentum flux and the inverse of the shear can then be inferred most simply on dimensional grounds. Moreover, for a homogeneous pumping, the proportionality constant can be determined by symmetry considerations, recovering the result of the original theory. The regime of applicability of the theory, its compatibility with observations from simulations, a formula for the momentum flux for an inhomogeneous pumping, and results for the statistics of fluctuations, will also be discussed.