  In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys.
Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented.
The detailed derivation of this closure is given here, relating it to the
Direct Interaction Approximation and Quasi-Normal types of closure. The
time-scale needed to obtain a self-consistent closure for both the energy
spectrum and the scalar variance spectrum is determined by evaluating the
correlation between the velocity and an advected displacement vector-field. The
relation between this latter correlation and the velocity-scalar correlation is
stressed, suggesting a simplified model of the latter. The resulting closed
equations are numerically integrated and results for the energy spectrum,
scalar fluctuation spectrum and velocity-displacement correlation spectrum are
presented for low, unity and high values of the Schmidt number.
