UC San Diego

In this dissertation, we establish a unified theoretical foundation for staircase formation and persistence using a hierarchy of reduced physical models. The objective is to identify the minimal physical mechanisms responsible for staircase formation, determine the conditions that allow these structures to persist, and assess their relevance to transport regulation in magnetized plasmas. In Chapter 1, we study staircase formation in passive scalar advection by stationary and fluctuating cellular flows. We show that layering emerges from the separation between rapid intra-cell mixing and slow inter-cell transport. Staircases remain robust to global transverse shear and weak vortex scattering when the diffusive timescale exceeds the turnover timescale and profile curvature remains sufficiently large. Persistence is quantified using the cellular Peclet number and profile curvature. In Chapter 2, we extend this model to active scalar transport by studying staircase formation in the magnetic potential of a two-dimensional magnetohydrodynamic system coupled to a fluctuating vortex array. Magnetic feedback suppresses transport across cell boundaries, increasing the disparity between intra-cell mixing and inter-cell transport. This produces a positive feedback loop in which weak magnetic fields reinforce barriers and promote sharper staircases. Magnetic staircases persist in both flux-expulsion and vortex-disruption regimes and can be sustained against resistive decay by stochastic forcing. In Chapter 3, we combine the passive and active scalar results into a common framework for staircase formation and persistence. Staircase emergence is shown to arise from inhomogeneous transport generated by rapid local mixing, slow barrier-crossing transport, and feedback. We identify curvature and Peclet number as robust measures of resiliency and clarify how fluctuations, disorder, and magnetic feedback modify staircase morphology and lifetime. In Chapter 4, we investigate staircase formation in reduced flux-driven models for the coupled evolution of temperature, turbulent kinetic energy, and mean flow. Unlike classical models based on a local layer-forming instability, persistent staircases arise through a metastable near-marginal regime with sustained transport contrast. Persistence is governed by the competition between transport redistribution, turbulent damping, and mean-flow shear. These results connect staircase dynamics to plasma transport self-organization and identify a distinct barrier-formation mechanism near marginal stability.