Earths jet streams, Jupiters Great Red Spot and its zonal winds are all examples of persistent large scale flows, whose dynamics is to a good approximation two-dimensional. These flows are also highly turbulent, and the interaction between the turbulence and these coherent structures remains poorly understood. Apart from its geophysical relevance, 2D turbulence is a rich and beautiful fundamental system—where turbulence takes a counter-intuitive role. Indeed, in 2D, energy is transferred to progressively larger scales, which can terminate in the self organization of the turbulence into a large scale coherent structure, a so called condensate, on top of small scale fluctuations.
I will describe a recent theoretical framework in which the profile of this coherent mean flow can be obtained, along with the mean momentum flux of the fluctuations. I will explain how and when the relation between the two can be deduced from dimensional analysis and symmetry considerations, and how it can be derived. Finally, I will show that, to leading order, the velocity two-point correlation function solves a scale invariant advection equation. The solution determines the average energy of the fluctuations, but does not contribute at this order to the momentum flux, due to parity + time reversal symmetry. Using analytic expressions for the solutions, matched to data from extensive numerical simulations, it is then possible to determine the main characteristics of the average energy. This is the first-ever self-consistent theory of turbulence-flow interaction.
