Persistent patterns in deterministic mixing flows

Abstract. – We present a theoretical approach to the description of persistent passive scalar patterns observed in recent experiments with non-turbulent time-periodic two-dimensional fluid flows. The behaviour of the passive scalar is described in terms of eigenmodes of the evolution operator which coincides with the Frobenius-Perron operator for the corresponding Lagrangian dynamics with small noise; the latter represents the molecular diffusion. The asymptotic behaviour is dominated by the eigenmode with the slowest decay rate, which is shown to be localized in the non-mixing region of the flow. Recently, in a number of experiments [1–3] it has been demonstrated that during chaotic advection in a fluid, a dye can form long-living, persistent structures. From the theoretical point of view, the problem is of asymptoticbehaviour of solutions of linear equations for a passive scalar. Because of the linearity, the description in terms of eigenmodes is appropriate [4]. However, the determination of these modes for chaotically mixing flows is non-trivial (cf. a similar approach to the dynamo problem [5]). In this letter we demonstrate that qualitative features of the persistent structures in chaotic fluid mixing observed in [1] can be modeled by flows having both chaotic and regular components (mixed phase space). Because there are non-mixing regions in the phase space, the long-time evolution toward the uniform concentration of the passive scalar density depends on the molecular diffusion constant. We study the spectrum of the eigenmodes for models of not globally chaotic flows and investigate the dependence of the decay rates of the passive scalar density on the parameters and the diffusion constant. It is worth noting that the same problem appears in the description of the correlations decay for chaos, the role of molecular diffusion being taken by additive noise; we extensively use this analogy below. The evolution of a passive scalar density φ(x, y, t) in a two-dimensional incompressible flow V (x, y, t) is described by the equation ∂φ + V (x,t) ·∇φ = D∆φ, (1