The spectrum of decaying scalar in a flow

After discussing the effect of the flow on the average concentration, we turn to the analysis of the statistics of the concentration fluctuations in such decay-type reaction systems. Even in the case of linear decay, where the average concentration is not affected by mixing, the fluctuations depend on the advecting flow. The distribution of the fluctuations over different length scales can be characterized by the power spectral density of the concentration field (taken along a one-dimensional section C(x), for simplicity) defined as 

The distribution of a decaying scalar field advected by a turbulent flow was studied by Corrsin (1961) who generalized the Obukhov-Corrsin theory of passive scalar turbulence for the linear decay problem F(C) = S(x) — bC. As in the case of the passive non-decaying scalar field, depending on the length scales considered, one can identify inertial-convective and viscous-convective regimes with qualitatively different characteristics. 

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