Staircase resiliency in a fluctuating cellular array

Abstract

Inhomogeneous mixing by stationary convective cells set in a fixed array is a particularly simple route to layering. Layered profile structures, or staircases, have been observed in many systems, including drift-wave turbulence in magnetic confinement devices. The simplest type of staircase occurs in passive-scalar advection, due to the existence and interplay of two disparate timescales, the cell turn-over (τ H ), and the cell diffusion (τ D ) time. In this simple system, we study the resiliency of the staircase structure in the presence of global transverse shear and weak vortex scattering. The fixed cellular array is then generalized to a fluctuating vortex array in a series of numerical experiments. The focus is on regimes of low-modest effective Reynolds numbers, as found in magnetic fusion devices. By systematically perturbing the elements of the vortex array, we learn that staircases form and are resilient (although steps become less regular, due to cell mergers) over a broad range of Reynolds numbers. The criteria for resiliency are (a) τ D >>τ H and (b) a sufficiently high profile curvature (κ ≥ 1.5). We learn that scalar concentration travels along regions of shear, thus staircase barriers form first, and scalar concentration "homogenizes" in vortices later. The scattering of vortices induces a lower effective speed of scalar concentration front propagation. The paths are those of the least time. We observe that if background diffusion is kept fixed, the cell geometric properties can be used to derive an approximation for the effective diffusivity of the scalar. Furthermore, the effective diffusivity of the fluctuating vortex array does not deviate significantly from that of the fixed cellular array.