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We investigate the mixing properties of scalars stirred by spatially smooth,divergence-free flows and maintained by a steady source-sink distribution. Wefocus on the spatial variation of the scalar field, described by the {\itdissipation wavenumber}, $k_d$, that we define as a function of the meanvariance of the scalar and its gradient. We derive a set of upper bounds thatfor large P\'eclet number ($\Pe\gg1$) yield four distinct regimes for thescaling behavior of $k_d$, one of which corresponds to the Batchelor regime.The transition between these regimes is controlled by the value of $\Pe$ andthe ratio $\rho=\ell_u/\ell_s$, where $\ell_u$ and $\ell_s$ are respectively,the characteristic lengthscales of the velocity and source fields. A fifthregime is revealed by homogenization theory. These regimes reflect the balancebetween different processes: scalar injection, molecular diffusion, stirringand bulk transport from the sources to the sinks. We verify the relevance ofthese bounds by numerical simulations for a {two-dimensional, chaoticallymixing} example flow and discuss their relation to previous bounds. Finally, wenote some implications for three dimensional turbulent flows.

Bounding the scalar dissipation scale for mixing flows in the presence of sources