Coherent vortex versus chaotic state in two-dimensional turbulence

We examine analytically and numerically the state of a two-dimensional fluid in a finite box appearing as a result of the inverse cascade supported by a permanent pumping. We argue that there are two different states. One of the states is dominated by big coherent vortices with a well-defined mean profile. The other state is dominated by strong chaotic large-scale fluctuations. The character of the realized state depends on the ratio ν k_f² / α where ν is the kinematic viscosity coefficient, k_f is the characteristic wave vector of the pumping force and α the bottom friction coefficient. If the ratio νk_f² / α is large then one expects to observe the first state, whereas in the opposite state the second (chaotic) state is expected. To check the prediction we performed the direct numerical simulations of hydrodynamics of a weakly compressible two-dimensional fluid with no-slip boundary conditions. The pumping force is static and contains some spacial harmonics. The simulations are performed for different values of pumping and of the ratio ν k_f² / α . We introduce the criterion based on presence/absence of long time correlations to distinguish two above states. The numerical results confirm the analytical predictions.