  Spatio-temporal deterministic chaos at small Taylor-Reynolds numbers
$Re_{\lambda} \lesssim 40$ and distributed chaos at turbulent $Re_{\lambda}
\gtrsim 40$ in passive scalar dynamics have been studied using results of
direct numerical simulations of homogeneous incompressible flows (with and
without mean gradient of the passive scalar) for $8 \leq Re_{\lambda} < 700$
and of a reacting turbulent mixing layer. It is shown that the deterministic
chaos in the passive scalar fluctuations at the small $Re_{\lambda}$ is
characterized by exponential spatial (wavenumber) spectrum: $E(k) \propto
\exp-(k/k_c)$, whereas the distributed chaos at turbulent $Re_{\lambda}$ is
characterized by stretched exponential spectrum $E(k) \propto
\exp-(k/k_{\beta})^{3/4}$. The Birkhoff-Saffman invariant related to the
momentum conservation and, due to the Noether theorem, to the spatial
homogeneity has been used as a theoretical basis for this stretched exponential
spectrum. Although the $k_c$ and $k_{\beta}$ represent the large-scale
structures a relevance of the Batchelor scale $k_{bat}$ has been established as
well: the normalized values $k_c/k_{bat}$ and $k_{\beta}/k_{bat}$ exhibit
universality.
