Dynamics of potential vorticity staircase evolution and step mergers in a reduced model of beta-plane turbulence

Abstract

A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by, and regulates, a potential enstrophy field, ε. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analysed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of meta-stable quasi-periodic configurations, lasting for hundreds of time units, yet interspersed with abrupt (∆t 1) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Another types of stationary solutions are solitons and kinks in the PV gradient and ε -profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in inter-step coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PVflux which, however, abruptly drops to its global constant value once the merger is completed. I. INTRODUCTIONPattern and scale selection are omnipresent problems in the dynamics of fluids and related nonlinear continuum systems. In geophysical fluids, as described by the beta-plane [1] or quasi-geostrophic equations [2] -the mechanisms of formation and scale selection for arrays of jets or zonal flows [3] is of particular interest. The jets scale constitutes and emergent scale which often defines the extent of mixing, transport and other important physical phenomena. Beta-plane and quasi-geostrophic systems evolve by the Lagrangian conservation of potential vorticity (PV). The latter is an effective phase space-density which consists of the sum of planetary and fluid pieces. The question of scale selection then is inexorably wrapped up in the evolution of mixing of potential vorticity. Homogeneous mixing -predicted by the Prandtl-Batchelor theorem [1-3], leads to a uniform PV profile throughout the system with a sharp PV gradient at the boundary. Inhomogeneous mixing -linked to bistability of mixing, multi-scale PV patterns. One of these -a corrugated structure called the potential vorticity staircase -is of particular interest, as it is a long-lived, quasi-stationary pattern of jets. The struggle between homogenization and (homogeneous mixing) and corrugation (inhomogeneous mixing) of PV is central to the dynamics of staircase formation and evolution, which are the foci of this paper.The turbulent transport and structure formation phenomenon now commonly known as a arXiv:1811.12887v1 [physics.plasm-ph] 29 Nov 2018 'staircase', was first understood and described by Philips [4]. He considered a density...

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

“…Concerning the units, "a" is the tokamak minor radius, "c s " is the ion acoustic speed, and ρ is the dimensionless radial coordinate. plasma to generate staircase structures proves to be an established fact [3,4], it has been confirmed computationally using different numerical codes [11][12][13][14][15][16]; discussed theoretically, especially invoking flux-gradient time delay, flux landscape bistability or wave trapping [17][18][19][20][21][22]; and observed experimentally other than on Tore Supra also on DIII-D [16]; KSTAR [23]; and lately in HL-2A L-mode discharges [24].…”

Development and long-term evolution of density staircases in stirred stratified turbulence

“…These no-flux conditions imply that the total energy is changed only by dissipation and stirring. Models of the form (3)-( 4), and extensions thereof, have been applied to a number of different physical problems in which staircases have been observed, including double diffusive convection [22] and potential vorticity staircases [23]. Three-component models of a similar form have also been formulated to study E × B staircases in plasma drift-wave turbulence [24][25][26].…”

A self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

“…[1]−a rare, classic circumstance when the discovery is made au bout de sa plume, if the celebrated art phrase due to François Arago [10] is appropriate here. By the time this paper is being written, the natural tendency of L-mode plasma to generate staircase structures proves to be an established fact [3,4], it has been confirmed computationally using different numerical codes [11][12][13][14][15][16]; discussed theoretically−especially invoking flux-gradient time delay, flux landscape bistability or wave trapping [17][18][19][20][21][22]; and observed experimentally other than on Tore Supra also on DIII-D [16]; KSTAR [23]; and lately in HL-2A L-mode discharges [24].…”