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 3-D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a 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 ergophage amplitude grows or decays exponentially on average, with a randomly fluctuating instantaneous growth rate. The growth rate has a small auto-correlation time, and follows a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cut-off), depending on the point-vortex base flow. Consequently, the logarithmic ergophage amplitude performs a free L\'evy 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 L\'evy on-off intermittency, and which we study in a companion paper. 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 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 2D flow saturates to a zero-temperature state. Given the limitations of existing theories ...
Intermittency of three-dimensional perturbations in a point-vortex model