Scalar variance decay in chaotic advection and Batchelor-regime turbulence

The decay of the variance of a diffusive scalar in chaotic advection flow (or equivalently Batchelor-regime turbulence) is analyzed using a model in which the advection is represented by an inhomogeneous baker’s map on the unit square. The variance decays exponentially at large times, with a rate that has a finite limit as the diffusivity κ tends to zero and is determined by the action of the inhomogeneous map on the gravest Fourier modes in the scalar field. The decay rate predicted by recent theoretical work that follows scalar evolution in linear flow and then averages over all stretching histories is shown to be incorrect. The exponentially decaying scalar field is shown to have a spatial power spectrum of the form $P\left(k\right)\mathrm{}\sim k^{-\sigma }$ at wave numbers small enough for diffusion to be neglected, with $\sigma <1.\mathrm{}$