Statistics of Macroturbulence from Flow Equations

Probability distribution functions of stochastically-driven and frictionally-damped fluids are governed by a linear framework that resembles quantum many-body theory. Besides the Fokker-Planck approach, there is a closely related Hopf functional method\footnote{Ookie Ma and J. B. Marston, J. Stat. Phys. Th. Exp. P10007 (2005).}; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we generalize the flow equation approach\footnote{F. Wegner, Ann. Phys. {\bf 3}, 77 (1994).} (also known as the method of continuous unitary transformations\footnote{S. D. Glazek and K. G. Wilson, Phys. Rev. D {\bf 48}, 5863 (1993); Phys. Rev. D {\bf 49}, 4214 (1994).}) to find the zero mode. We test the approach using a prototypical model of geophysical and astrophysical flows on a rotating sphere that spontaneously organizes into a coherent jet. Good agreement is found with low-order equal-time statistics accumulated by direct numerical simulation, the traditional method. Different choices for the generators of the continuous transformations, and for closure approximations of the operator algebra, are discussed.