  We present direct numerical simulations (DNS) of the mixing of the passive
scalar at modest Reynolds numbers (10 =< R_\lambda =< 42) and Schmidt numbers
larger than unity (2 =< Sc =< 32). The simulations resolve below the Batchelor
scale up to a factor of four. The advecting turbulence is homogeneous and
isotropic, and maintained stationary by stochastic forcing at low wavenumbers.
The passive scalar is rendered stationary by a mean scalar gradient in one
direction. The relation between geometrical and statistical properties of
scalar field and its gradients is examined. The Reynolds numbers and Schmidt
numbers are not large enough for either the Kolmogorov scaling or the Batchelor
scaling to develop and, not surprisingly, we find no fractal scaling of scalar
level sets, or isosurfaces, in the intermediate viscous range. The
area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of
the scalar fluctuations. The scalar flux across the isosurfaces, which is
determined by the conditional probability density function (PDF) of the scalar
gradient magnitude, has a stretched exponential distribution towards the tails.
The PDF of the scalar dissipation departs distinctly, for both small and large
amplitudes, from the lognormal distribution for all cases considered. The joint
statistics of the scalar and its dissipation rate, and the mean conditional
moment of the scalar dissipation, are studied as well. We examine the effects
of coarse-graining on the probability density to simulate the effects of poor
probe-resolution in measurements.
