Intermittency of three-dimensional perturbations in a point-vortex model

Abstract

Three-dimensional (3-D) instabilities on a (potentially turbulent) two-dimensional (2-D) flow are still incompletely understood, despite recent progress. Here we propose a simple energy-conserving model of a regularised 2-D point-vortex flow coupled to localised 3-D 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 cut-off) depending on the background state of the point-vortex flow. Consequently, the logarithm of the ergophage amplitude performs a free Lévy flight. (ii) The passive-nonlinear regime of the model, where the 2-D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2-D 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 [1]. 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 characterise the the fully nonlinear regime, where ergophages feed back on the 2-D 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 2-D flow saturates to a zero-temperature state. Given the limitations of existing theories, our model provides a new perspective on 3-D instabilities growing on 2-D flows, which will be useful in analysing and understanding the much more complex results of DNS and potentially guide further theoretical developments.