We show that a variational implementation of probability density funct
ion (PDF) closures has the potential to make predictions of general tu
rbulence mean statistics for which a priori knowledge of the incorrect
ness is possible. This possibility exists because of realizability con
ditions on ''effective potential'' functions for general turbulence st
atistics. These potentials measure the cost For fluctuations to occur
away from the ensemble-mean value in empirical time-averages of the gi
ven variable, and their existence is a consequence of a refined ergodi
c hypothesis for the governing dynamical system (Navier-Stokes dynamic
s). Approximations of the effective potentials can be calculated withi
n PDF closures by an efficient Rayleigh-Ritz algorithm. The failure of
realizability within a closure for the approximate potential of any c
hosen statistic implies a priori that the closure prediction for that
statistic is not converged. The systematic use of these novel realizab
ility conditions within PDF closures is shown in a simple 3-mode syste
m of Lorenz to result in a statistically improved predictive ability.
In certain cases the variational method allows an a priori optimum cho
ice of free parameters in the closure to be made.