Jeffrey B. Parker (Lawrence Livermore National Laboratory)
A quasilinear foothold to gaining insight into zonal flows
Zonal flows, which consist of a regular spatially alternating pattern of flows forming spontaneously in fluid systems, are of fundamental physical interest. They occur in diverse systems such as planetary atmospheres, with the banded flows of Jupiter being a prominent example, and in magnetically confined plasmas. In laboratory plasmas that are aimed at fusion-energy production, zonal flows have been linked to the suppression of detrimental turbulent transport, and have therefore attracted significant attention. The study of zonal flows in both plasma physics and in geophysics are by a happy coincidence closely linked: each field has produced idealized models for studying fluid flow that take the same mathematical form. These are the Hasegawa-Mima equations for a magnetized plasma slab and the barotropic vorticity equation on a beta-plane. While simulations are essential for quantitative studies of realistic devices, developing a mathematical theory of zonal flow is important for gaining physical insight. Recent work based on combining a quasilinear approach with a statistical formulation has brought a fresh look at zonal flow. In this theory, zonal flows grow as a coherent structure due to the symmetry-breaking zonostrophic instability---the broken symmetry being statistical homogeneity. Just beyond the stability threshold, the system obeys the Real Ginzburg-Landau equation, linking the study of zonal flow to the wider field of pattern formation. These analytic findings have been confirmed with numerics. From this quasilinear approach, a conceptual framework involving the growth and saturation of zonal flow has emerged that can inform future studies.