Abstract

How turbulence shapes a mean flow, and how it is altered by the mean flow in return, is a central problem in fluid mechanics. It is shared by a wide range of systems, from pipe flow to planetary scale winds and currents. Mean-flow-turbulence interactions take a special character in large-scale geophysical and astrophysical flows. Often constrained by rotation or stratification, the resulting flows are effectively two-dimensional. The mean flow can then spontaneously emerge in a process of self-organization, starting from small-scale fluctuations. The underlying mechanism is that of an inverse transfer of energy—- from small to large scales, leading to the accumulation of energy at large scales and the establishment of a system-size mean flow. A prominent example are the jets on Jupiter, which are powered by small scale turbulence and have remained remarkably unchanged for decades.

In this talk, I will discuss analytical and numerical results for turbulence-mean-flow-interactions in idealized two-dimensional systems. One of the difficulties in treating turbulence-mean-flow interactions is the non-local nature of incompressible fluid dynamics. I will focus on a model where this difficulty is eliminated, considering a rapidly rotating shallow layer of fluid in the limit where surface perturbations become localized. This is the so-called large-scale quasi-geostrophic (LQG) equation. I will show how low-order turbulent statistics are determined in this model, and how, depending on symmetries, they either shape the mean flow or are passively set by it. This emergent picture is consistent with previous results in two-dimensional incompressible Navier-Stokes, where the dynamics is non-local, pointing at its universality. On the other hand, I will also describe the dramatic influence of the local dynamics in LQG : the formation of regions with depleted fluxes towards small scales and a vanishing small-scale dissipation. Finally, I will comment on possible connections to mean-flow-turbulence interactions in three-dimensional flows, emphasizing commonalities and differences.

webinar: https://cassyni.com/events/Se4XRUaPhS4Pv7H9TTnptm?cb=functi