Spectral theory of passive scalar with mean scalar gradient

A single-time two-point spectral closure is developed by approximation of the Lagrangian direct interaction approximation (LDIA) for a passive scalar in the presence of a mean scalar gradient in homogeneous isotropic turbulence. In the derivation of a single-time two-point spectral closure, the two assumptions, Markovianisation and the exponential form of Lagrangian velocity response function, are made for the LDIA, and angle dependence of the passive-scalar field is expressed by the second-order truncation of Legendre polynomials, in which such a truncation is justified by the linear theory. The resulting closure equations are derived in a straightforward way except for the above assumptions and further simplifications. The closures studied agree qualitatively with direct numerical simulation for one- and two-point statistics of a passive-scalar field in the case of unity Schmidt number. For both direct numerical simulation and closures, we show that the dependence of one-point passive-scalar statistics on the Péclet number based on scalar Taylor microscales collapses properly compared with that based on velocity microscales. We also propose universal scaling laws for second-order scalar structure functions and demonstrate their validity.