Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow

We have found that in two-dimensional Kolmogorov flow a spatially localized turbulent state (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of the control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmetry of the internal turbulence we introduce two coarse-grained variables characterizing the internal dynamics. These quantities follow the switching process reasonably. This suggests that the twin attracting invariant sets each of which corresponds to a one-way traveling SLT are embedded in the attractor of the moving SLT and the connection of the two sets is too complicated to be represented by a few degrees of freedom but the motion of an SLT is correlated with the internal turbulent dynamics.