Spontaneous suppression of inverse energy cascade in instability-driven 2-D turbulence

Abstract

Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $\gamma$ , increases, the system undergoes two transitions. For $\gamma$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $\gamma$ , the condensate breaks down, and a gas of weakly interacting vortices with broken symmetry spontaneously emerges, characterised by preponderance of vortices of one sign only and suppressed inverse energy cascade. The latter transition is shown to depend on the damping mechanism. The number density of SVs in the broken symmetry state slowly increases via a random nucleation process. Bistability is observed between the condensate and mixed SV-condensate states. Our findings provide new evidence for a strong dependence of two-dimensional turbulence phenomenology on the forcing.

From a vortex gas to a vortex crystal in instability-driven two-dimensional turbulence

“…Starting from random, small-amplitude initial conditions, a short-lived inverse cascade is followed by the emergence of shielded vortices of both parities (σ t = 6.3 in figure 1). In a stochastic competition between the two species, one is eventually eliminated leading to spontaneous symmetry breaking, as discussed by van Kan et al (2022). This is clearly seen at σ t = 378 in figure 1.…”

“…In nearly inviscid, inertial fluid flows, the resulting shielded vortices typically evolve into tripoles (Carton, Flierl & Polvani 1989) consisting of a central vortex and two satellite vortices of opposite sign 180 • apart, as seen in both laboratory experiments (Van Heijst, Kloosterziel & Williams 1991) and direct numerical simulation (DNS) (Orlandi & van Heijst 1992). In instability-driven 2-D turbulence, the formation of such tripolar shielded vortices has been found to facilitate the spontaneous suppression of the inverse cascade (van Kan et al 2022).…”

“…Upon inspection of figure 1, the coherent vortices are seen to be tripolar, with an elliptical core and two satellites 180 • apart. As shown by van Kan et al (2022), these tripolar vortices are shielded, meaning that the circulation generated by any given vortex becomes small beyond a finite radius, located close to the edge of its satellites and comparable to the largest forcing scale.…”

“…This combination of two well-established forcing mechanisms, each of which has separately led to fundamental insights into turbulence, allows for an exploration of new aspects of non-universality in 2-D turbulence. With a cubic nonlinear damping term to saturate the linear instability, extensive DNS by van Kan et al (2022) revealed a number of transitions as the forcing function varies from stochastic to instability-like, from a large-scale condensate to a hybrid state consisting of large-scale circulation patterns with embedded mesoscale shielded vortices, and finally to a gas of shielded vortices characterised by a spontaneously broken symmetry, with all vorticity extrema in the core of the same sign at late times.…”

“…Owing to the significant numerical effort required to investigate the ultimate saturation of this process, the late-time limit of this slow evolution was not studied and remains an open problem. Here, we employ extensive DNS of this system with very long integration times to advance significantly beyond the results presented by van Kan et al (2022) and investigate in detail the late-time evolution of the broken symmetry shielded vortex gas state, revealing an approximately self-similar evolution towards a dense, statistically stationary state. This state is shown to persist over a wide range of forcing strengths, and to undergo a crystallisation transition at a critical parameter threshold.…”