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We construct different equivalent non-equilibrium statistical ensembles in asimple yet instructive $N$-degrees of freedom model of atmospheric turbulence,introduced by Lorenz in 1996. The vector field can be decomposed into anenergy-conserving, time-reversible part, plus a non-time reversible part,including forcing and dissipation. We construct a modified version of the modelwhere viscosity varies with time, in such a way that energy is conserved, andthe resulting dynamics is fully time-reversible. For each value of the forcing,the statistical properties of the irreversible and reversible model are inexcellent agreement, if in the latter the energy is kept constant at a valueequal to the time-average realized with the irreversible model. In particular,the average contraction rate of the phase space of the time-reversible modelagrees with that of the irreversible model, where instead it is constant byconstruction. We also show that the phase space contraction rate obeys thefluctuation relation, and we relate its finite time corrections to thecharacteristic time scales of the system. A local version of the fluctuationrelation is explored and successfully checked. The equivalence between the twonon-equilibrium ensembles extends to dynamical properties such as the Lyapunovexponents, which are shown to obey to a good degree of approximation a pairingrule. These results have relevance in motivating the importance of the chaotichypothesis. in explaining that we have the freedom to model non-equilibriumsystems using different but equivalent approaches, and, in particular, thatusing a model of a fluid where viscosity is kept constant is just one option,and not necessarily the only option, for describing accurately its statisticaland dynamical properties.

Equivalence of Non-Equilibrium Ensembles and Representation of Friction in Turbulent Flows: The Lorenz 96 Model