Condensate, fluctuations and symmetries - a tale of 2D turbulence

Earths jet streams, Jupiters Great Red Spot and its zonal winds are all examples of persistent

large scale ows, whose dynamics is to a good approximation two-dimensional. These ows 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

I will describe a recent theoretical framework in which the prole of this coherent mean 

can be obtained, along with the mean momentum ux of the uctuations. 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 uctuations, but does not contribute at this order to the momentum 

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 rst-ever self-consistent theory of turbulence-ow interaction.