A Low-Dimensional Description of the Vortex Merger Process Driven by Hyperbolic Instability of 2D Taylor-Green Vortex

Two-dimensional fluid dynamics is investigated numerically using the Taylor-Green vortex as the initial condition. The merging of same-signed columnar vortices is driven by the nonlinear growth of hyperbolic instability, which gives rise to the inverse energy cascade. The nonlinear saturation of this unstable mode essentially corresponds to the vortex merger process, leading to a quasi-steady, stable, and large-scale state in which most of the energy remains undissipated. Since hyperbolic instability is an oscillatory mode, the final state is uniquely determined by the phase angle at the moment of saturation—similar to a roulette wheel, although the probability distribution is nontrivial. This transition from an unstable to a stable state is well captured by a low-dimensional dynamical system, as the forward energy cascade is not dominant. A five-dimensional model is derived from the 2D incompressible Navier-Stokes equations, and its solutions and conservation laws are compared with those from direct numerical simulations.