Characterizing the metastable balance between chaos and diffusion

We examine some new diagnostics for the behavior of a field ρ evolving in an advective–diffusive system. One of these diagnostics is approximately the Fourier second moment (denoted as χ2) and the other is the linear entropy or field intensity S, the latter being significantly easier to compute or measure. We establish that as a result of chaos the increasing structure in ρ is accompanied by χ increasing exponentially rapidly in time at a rate given by ρ-dependent Lyapunov exponents Λi and dominated by the largest one Λmax. Noise or diffusive coarse-graining of ρ causes χ to decrease as χ2≈14Dt, where D is a measure of the diffusion. When both effects are present the competition between the processes leads to metastability for χ followed by a final diffusive tail. The initial stages may be chaotic or diffusive depending upon the value of Λ−1max2Dχ2(0) but the metastable value of χ2 is given by χ2∗=∑iΛi/2D irrespective. Since Ṡ=−2Dχ2, similar analysis applies to S, and in particular there exists a metastable decay rate for S given by Ṡ∗=∑iΛi. These arguments are verified for a simple case, the Arnol’d Cat Map with added diffusive noise.