Physics of Dynamic Profile Staircases*

Staircases are quasi-periodic layered states of inhomogeneous mixing zones interspersed by microbarriers, which impede transport and so locally steepen gradients. Staircases and layering are observed in many physical systems, including, but not limited to, magnetically confined plasmas. There, the microbarriers are thought to be due to E ✕ B shear layer feedback. However, we show that staircase formation is a much simpler and more general phenomenon. In fact, inhomogeneous mixing will occur when the turbulence field is organized in an array of nearly overlapping convective cells. Such a configuration is typical of systems near marginal stability, which occur frequently, including in confined plasmas. In such a cellular array, passive scalar staircases form due to the interplay of two disparate time scales, that of (fast) cell turn-over and (slow) diffusion across cell boundaries. To address the important features of cellular variability and jitter, we study staircase formation in a fluctuating cellular array. There we see that scalar staircases remain resilient over a broad range of excitations typical of the modest levels of turbulence in magnetic confinement experiments (i.e., Kubo number ≤ 1). Staircase curvature and cellular Peclet number are identified as figures-of-merit for the resiliency of dynamic staircases. We also observe that as long  as cell streamlines are maintained, effective diffusion across cell boundaries does not deviate significantly from that for the fixed cellular array [D*∝ √(D₀ D_cell)]. In addition, we examine the active scalar staircase. The dynamics of the active scalar are analogous to that of the magnetic potential in 2D MHD, where fields are expelled to cell boundaries and so stabilize the staircase cells. Here, turbulent resistivity measures cellular array elasticity. The active scalar system exhibits a novel feedback mechanism that reinforces the global staircase structure and promotes self-organization.