The 2-dimensional Navier-Stokes equations, and several related models such as the Charney equation or the quasi-geostrophic equations, possess a second inviscid invariant in addition to the kinetic energy. This second inviscid invariant, usually called the enstrophy, fundamentally changes the dynamics compared to 3 dimensions. In particular, it is responsible for the inverse cascade phenomenon in 2 dimensions which provides a natural mechanism for the self-organization of large scale coherent structures such as vortices or jets from small scale random fluctuations. In this talk I will briefly review the classical theory of 2-dimensional classical and quasi-geostrophic turbulence. I will then discuss more recent questions pertaining to the role played by the large scale boundary conditions in determining the coherent flow which emerges at large scales and how the presence of a large scale coherent flow affects the small scale turbulent fluctuations.
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