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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 ofdirect numerical simulations of homogeneous incompressible flows (with andwithout mean gradient of the passive scalar) for $8 \leq Re_{\lambda} < 700$and of a reacting turbulent mixing layer. It is shown that the deterministicchaos in the passive scalar fluctuations at the small $Re_{\lambda}$ ischaracterized by exponential spatial (wavenumber) spectrum: $E(k) \propto\exp-(k/k_c)$, whereas the distributed chaos at turbulent $Re_{\lambda}$ ischaracterized by stretched exponential spectrum $E(k) \propto\exp-(k/k_{\beta})^{3/4}$. The Birkhoff-Saffman invariant related to themomentum conservation and, due to the Noether theorem, to the spatialhomogeneity has been used as a theoretical basis for this stretched exponentialspectrum. Although the $k_c$ and $k_{\beta}$ represent the large-scalestructures a relevance of the Batchelor scale $k_{bat}$ has been established aswell: the normalized values $k_c/k_{bat}$ and $k_{\beta}/k_{bat}$ exhibituniversality.

Chaotic and turbulent mixing of passive scalar