Self-organization and transition to turbulence in isotropic fluid motion driven by negative damping at low wavenumbers

We observe a symmetry-breaking transition from a turbulent to a self-organized state in direct numerical simulation of the Navier-Stokes equation at very low Reynolds number. In this self-organised state the kinetic energy is contained only in modes at the lowest resolved wavenumber, the skewness vanishes, and visualization of the flows shows a lack of small-scale structure, with the vorticity and velocity vectors becoming aligned (a Beltrami flow).我们在极低雷诺数（Reynolds number）下的纳维-斯托克斯方程（Navier-Stokes equation）直接数值模拟中，观测到了从湍流态到自组织态的对称破缺转变。在该自组织态下，动能仅集中于最低分辨波数对应的模态中，偏度消失；流动可视化结果显示其缺乏小尺度结构，且涡量与速度矢量趋于对齐，此即为贝尔特拉米流（Beltrami flow）。