Transitions between turbulent states in a two-dimensional shear flow

Abstract: We study the bifurcations of the large scale jets in the turbulent regime of a forced shear flow using direct numerical simulations of the Navier-Stokes equations. The bifurcations are seen in the probability density function (PDF) of the largest scale mode with the control parameter being the Reynolds number based on the friction coefficient denoted as $Rh$. As one increases $Rh$ in the turbulent regime, the PDF of the large scale mode first bifurcates from a Gaussian to a bimodal behaviour, signifying the emergence of reversals of the large scale flow where the flow fluctuates between two distinct turbulent states. Further increase in $Rh$ leads to a bifurcation from bimodal to unimodal PDF which denotes the disappearance of the reversals of the largest scale mode. We attribute the latter transition to the long-time memory that the large scale flow exhibits related to low frequency $1/f^\alpha$ type of noise with $0 < \alpha < 2$. We also demonstrate that a minimal model with 15 modes, obtained from the truncated Euler equation, is able to capture the bifurcations of the large scale jets exhibited by the Navier-Stokes equations.