Staircases of active scalar concentration in cellular flow

Staircases are quasi-periodic layered structures characterized by regions of enhanced mixing (‘steps’) separated by microbarriers (‘jumps’) that impede transport and locally steepen gradients. These staircases emerge in many physical systems, including magnetically confined plasmas, where interspersed regions of localized shear are thought to form through E ✕ B feedback. Prior studies have demonstrated that the simplest type of staircase arises in passive-scalar advection driven by the interplay of two disparate timescales, that of fast cell turnover (τH) and slow diffusion across cell boundaries. This study replaces the passive scalar with an active scalar and examines its staircase evolution and transport within a fluctuating vortex array model. The active scalar field, the magnetic potential, interacts with vortex dynamics through Lorentz forces, as in 2D MHD. Resistive diffusion introduces non-ideal MHD phenomena such as flux expulsion (kinematic) and cell disruption (dynamic). Numerical experiments focusing on a broad range of excitations typical of modest levels of turbulence in magnetic confinement experiments (i.e., Kubo number ≤ 1) reveal that magnetic potential staircases form in both kinematic and dynamic regimes. In the kinematic regime, where cells expel magnetic field lines to cell boundaries, the background magnetic field enhances the elasticity of the vortex array against cell scattering and suppresses the transport of magnetic potential across cell boundaries, as measured by turbulent resistivity (ηT). Inhomogeneous mixing occurs when τH << τηT. In the dynamic regime, although the magnetic field disrupts the cell structure, some cells remain unaffected, unexpectedly allowing the layered structure to persist. Notably, without external magnetic forcing, the magnetic field eventually decays in 2D. Thus, we also examine the effects of external stochastic magnetic potential forcing on the persistence of active scalar staircases. Finally, we revisit the passive scalar case and introduce a unified framework that identifies shared mechanisms underlying staircase formation in both passive and active scalar transport, offering new physical insight into barrier formation and resilience in cellular flows.