Spatio-temporal correlation functions in scalar turbulence from functional renormalization group

We provide the leading behavior at large wavenumbers of the two-point correlation function of a scalar field passively advected by a turbulent flow. We first consider the Kraichnan model, in which the turbulent carrier flow is modeled by a stochastic vector field with a Gaussian distribution, and then a scalar advected by a homogeneous and isotropic turbulent flow described by the Navier-Stokes equation, under the assumption that the scalar is passive, i.e. that it does not affect the carrier flow. We show that at large wavenumbers, the two-point correlation function of the scalar in the Kraichnan model decays as an exponential in the time delay, in both the inertial and dissipation ranges. We establish the expression, both from a perturbative and from a nonperturbative calculation, of the prefactor, which is found to be always proportional to $k^2$. For a real scalar, the decay is Gaussian in $t$ at small time delays, and it crosses over to an exponential only at large $t$. The assumption of delta-correlation in time of the stochastic velocity field in the Kraichnan model hence significantly alters the statistical temporal behavior of the scalar at small times.