Reversible Navier-Stokes equation on logarithmic lattices

The three-dimensional reversible Navier-Stokes (RNS) equations are a modification of the dissipative Navier-Stokes (NS) equations, first introduced by Gallavotti [Phys. Lett. A 223, 91 (1996)0375-960110.1016/S0375-9601(96)00729-3], in which the energy or the enstrophy is kept constant by adjusting the viscosity over time. Spectral direct numerical simulations of this model were performed by Shukla et al. [Phys. Rev. E 100, 043104 (2019)2470-004510.1103/PhysRevE.100.043104] and Margazoglou et al. [Phys. Rev. E 105, 065110 (2022)10.1103/PhysRevE.105.065110]. Here we consider a linear, forced reversible system obtained by projecting RNS equations on a log lattice rather than on a linearly spaced grid in Fourier space, as is done in regular spectral numerical simulations. We perform numerical simulations of the system at extremely large resolutions, allowing us to explore regimes of parameters that were out of reach of the direct numerical simulations of Shukla et al. Using the nondimensionalized forcing as a control parameter, and the square root of enstrophy as the order parameter, we confirm the existence of a second-order phase transition well described by a mean-field Landau theory. The log-lattice projection allows us to probe the impact of the resolution, highlighting an imperfect transition at small resolutions with exponents differing from the mean-field predictions. Our findings are in qualitative agreement with predictions of a 1D nonlinear diffusive model, the reversible Leith model of turbulence. We then compare the statistics of the solutions of RNS and NS, in order to shed light on an adaptation of the Gallavotti conjecture, in which there is equivalence of statistics between the reversible and irreversible models, to the case where our reversible model conserves either the enstrophy or the energy. We deduce the conditions in which the two are equivalent. Our results support the validity of the conjecture and represent an instance of nonequilibrium system where ensemble equivalence holds for mean quantities.