  Numerical and analytical studies of "final states" of two-dimensional (2D)
decaying turbulence are reported. The first part of this work is trying to give
a definition for final states of 2D decaying turbulence. Although the
functional relation of $\omega-\psi$ is frequently used as the characterization
of those "final states," it is just a sufficient but not necessary condition so
it is not proper to be used as the definition. It is found the way through the
value of the effective area S covered by the scatter $\omega-\psi$ plot, which
is initially suggested by Read, is more general, and more suitable for the
definition. Based on this concept, we gave out a definition that can cover all
existing results in late states of decaying 2D flows, including some weird
double-valued $\omega-\psi$ scatter plots that can not be explained before. The
rest part of the paper is trying to further investigate 2D decaying turbulence
with the assistance of our new definition. Some new numerical results, which
lead to "bar" final states and further verify the predictive ability of
statistical mechanics [2], are reported. It is realized that some simulations
with narrow-band energy spectral initial conditions, which can be called
"turbulence" doubtfully, lead to some final states that can not be very well
explained by the statistical theory (in the meanwhile, they are still in the
scope of our new definition of the "final state"). For those simulations with
initial conditions of broadband energy spectra that lead to the famous dipole,
we give out a mathematical re-interpreting for the so-called sin-hyperbolic
("sinh") $\omega-\psi$ scatter plot in final states. We suggest the term "sinh"
here should be replaced by "sinh-like." The corresponding physical meaning of
this re-interpreting will also be discussed.
