  We construct different equivalent non-equilibrium statistical ensembles in a
simple yet instructive $N$-degrees of freedom model of atmospheric turbulence,
introduced by Lorenz in 1996. The vector field can be decomposed into an
energy-conserving, time-reversible part, plus a non-time reversible part,
including forcing and dissipation. We construct a modified version of the model
where viscosity varies with time, in such a way that energy is conserved, and
the resulting dynamics is fully time-reversible. For each value of the forcing,
the statistical properties of the irreversible and reversible model are in
excellent agreement, if in the latter the energy is kept constant at a value
equal to the time-average realized with the irreversible model. In particular,
the average contraction rate of the phase space of the time-reversible model
agrees with that of the irreversible model, where instead it is constant by
construction. We also show that the phase space contraction rate obeys the
fluctuation relation, and we relate its finite time corrections to the
characteristic time scales of the system. A local version of the fluctuation
relation is explored and successfully checked. The equivalence between the two
non-equilibrium ensembles extends to dynamical properties such as the Lyapunov
exponents, which are shown to obey to a good degree of approximation a pairing
rule. These results have relevance in motivating the importance of the chaotic
hypothesis. in explaining that we have the freedom to model non-equilibrium
systems using different but equivalent approaches, and, in particular, that
using a model of a fluid where viscosity is kept constant is just one option,
and not necessarily the only option, for describing accurately its statistical
and dynamical properties.
