Fluctuations in two-dimensional reversibly damped turbulence

Gallavotti proposed an equivalence principle in hydrodynamics, which states that forced-damped fluids can be equally well represented by means of the Navier-Stokes equations and by means of time-reversible dynamical systems called GNS. In the GNS systems, the usual viscosity is replaced by a state-dependent dissipation term which fixes one global quantity. The principle states that the mean values of properly chosen observables are the same for both representations of the fluid. In the same paper, the chaotic hypothesis of Gallavotti and Cohen is applied to hydrodynamics, leading to the conjecture that entropy fluctuations in the GNS system verify a relation first observed in non-equilibrium molecular dynamics. We tested these ideas in the case of two-dimensional fluids. We examined the fluctuations of global quadratic quantities in the statistically stationary state of (a) the Navier-Stokes equations and (b) the GNS equations. Our results are consistent with the Validity of the fluctuation relation, and of the equivalence principle, indicating possible extensions thereof. Moreover, in these results the difference between the Gallavotti-Cohen fluctuation theorem and the Evans-Searles identity is evident.