  The statistical mechanical description of two-dimensional inviscid fluid
turbulence is reconsidered. Using this description, we make predictions about
turbulent flow in a rapidly rotating laboratory annulus. Measurements on the
continuously forced, weakly dissipative flow reveal coherent vortices in a mean
zonal flow. Statistical mechanics has two crucial requirements for equilibrium:
statistical independence of macro-cells (subsystems) and additivity of
invariants of macro-cells. We use additivity to select the appropriate Casimir
invariants from the infinite set available in vortex dynamics, and we do this
in such a way that the exchange of micro-cells within a macro-cell does not
alter an invariant of a macro-cell. A novel feature of the present study is our
choice of macro-cells, which are continuous phase space curves based on mean
values of the streamfunction. Quantities such as energy and enstrophy can be
defined on each curve, and these lead to a local canonical distribution that is
also defined on each curve. Our approach leads to the prediction that on a mean
streamfunction curve there should be a linear relation between the
ensemble-averaged potential vorticity and the time-averaged streamfunction, and
our laboratory data are in good accord with this prediction. Further, the
approach predicts that although the probability distribution function for
potential vorticity in the entire system is non-Gaussian, the distribution
function of micro-cells should be Gaussian on the macro-cells, i.e., for curves
defined by mean values of the streamfunction. This prediction is also supported
by the data. While the statistical mechanics approach used was motivated by and
applied to experiments on turbulence in a rotating annulus, the approach is
quite general and is applicable to a large class of Hamiltonian systems.
