Instability-driven two-dimensional turbulence*

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. The system exhibits four distinct branches of statistically stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two symmetry-broken states composed of like SVs, a dense vortex gas and a hexagonal vortex crystal. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. As the forcing strength decreases the vortex gas undergoes a sharp transition to a vortex crystal, and the vortex diffusivity drops to zero. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of forcing details in studies of two-dimensional turbulence.