An upper bound for passive scalar diffusion in shear flows

This study is concerned with the diffusion of a passive scalar $\Theta(\r,t)$ advected by general $n$-dimensional shear flows $\u=u(y,z,...,t)\hat{x}$ having finite mean-square velocity gradients. The unidirectionality of the incompressible flows conserves the stream-wise scalar gradient, $\partial_x\Theta$, allowing only the cross-stream components to be amplified by shearing effects. This amplification is relatively weak because an important contributing factor, $\partial_x\Theta$, is conserved, effectively rendering a slow diffusion process. It is found that the decay of the scalar variance $<\Theta^2>$ satisfies $d<\Theta^2>/dt\ge -C\kappa^{1/3}$, where $C>0$ is a constant, depending on the fluid velocity gradients and initial distribution of $\Theta$, and $\kappa$ is the molecular diffusivity. This result generalizes to axisymmetric flows on the plane and on the sphere having finite mean-square angular velocity gradients.