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.