  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.
