  Predicting the long time or late time states of two-dimensional
incompressible, high Reynolds number, slowly decaying turbulence has been one
of the long-standing problems. Using ``point vortices'' as ``inviscid''
building blocks, which do not respect incompressibility, statistical mechanical
models conserving only total energy and zero total circulation result in the
well-known sinh-Poisson relation between vorticity and stream function. On the
other hand, statistical mechanics of ``inviscid patch'' vortices, which
respects incompressibility by conserving regions of zero and nonzero vorticity,
predicts a generalized relaxed state, which has never been systematically
compared with direct numerical simulations (DNS). In this study, starting from
highly packed regions of nonzero initial vorticity, we demonstrate using high
resolution, high Reynolds number DNS that the late time states agree with
predictions from patch vortex models. As total circulation is reduced or
diluted, we show that late time states of our DNS systematically and
unambiguously lead to the sinh-Poisson relationship between vorticity and
stream function. We believe that our quantitative findings solve one of the
long-standing problems in two-dimensional turbulence.
