2020/12/31 by Adrian van Kan, Alexandros Alexakis, Marc Brächet +1 · 1 citation
Engineering · Environmental Science · Mathematics · Physics and Astronomy · #Amplitude #Classical mechanics #Flow (mathematics) #Fluid Dynamics and Turbulent Flows #Intermittency #Logarithm #Mathematical analysis #Mathematics #Mechanics #Perturbation (astronomy) #Physics #Plant Water Relations and Carbon Dynamics #Quantum mechanics #Statistical physics #Theoretical and Computational Physics #Turbulence #Vortex #physics.flu-dyn
paper · pdf · doi:10.1103/physreve.103.053102
published as Phys. Rev. E 103, 053102 (2021)
arxiv created 2021/04/24 · openalex publication_date 2021/05/11 · arxiv updated 2021/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a regularized 2D point-vortex flow coupled to localized 3D perturbations ("ergophages"), such that ergophages can gain energy by altering vortex-vortex distances through an induced divergent velocity field, thus decreasing point-vortex energy. We investigate the model in three distinct stages of evolution: (i) The linear regime, where the amplitude of the ergophages grows or decays exponentially on average, with an instantaneous growth rate that fluctuates randomly in time. The instantaneous growth rate has a small auto-correlation time, and a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cutoff) depending on the point-vortex base flow. Consequently, the logarithm of the ergophage amplitude performs a Lévy flight. (ii) The passive-nonlinear regime of the model, where the 2D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2D flow. In this regime the system exhibits a new type of on-off intermittency that we name Lévy on-off intermittency, which we define and study in a companion paper [van Kan et al., Phys. Rev. E 103, 052115 (2021)1063-651X10.1103/PhysRevE.103.052115]. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude. (iii) Finally, we characterize the fully nonlinear regime, where ergophages feed back on the 2D flow, and study how the vortex temperature is altered by the interaction with ergophages. It is shown that when the amplitude of the ergophages is sufficiently large, the condensate is disrupted and the 2D flow saturates to a zero-temperature state. Given the limitations of existing theories, our model provides a new perspective on 3D instabilities growing on 2D flows, which will be useful in analyzing and understanding the much more complex results of DNS and potentially guide further theoretical developments.