Coherent vortex in two-dimensional turbulence: Interplay of viscosity and bottom friction

We examine coherent vortices appearing as a result of the inverse cascade of two-dimensional turbulence in a finite box in the case of pumping with arbitrary correlation time in the quasilinear regime. We demonstrate that the existence of the vortices depends on the ratio between the values of the bottom friction coefficient $\alpha$ and the viscous damping of the flow fluctuations at the pumping scale $\nu k_f^2$ ($\nu$ is the kinematic viscosity coefficient and $k_f$ is the characteristic wave vector at the pumping scale). The coherent vortices appear if $\nu k_f^2\gg \alpha$ and cannot exist if $\nu k_f^2\ll \alpha$. Therefore there is a border value $\alpha \sim \nu k_f^2$ separating the regions. In numerical simulations, $\nu k_f^2/\alpha$ can be arbitrary, whereas in a laboratory experiment $\nu k_f^2/\alpha \lesssim 1$ and the coherent vortices can be observed solely near the border value of $\nu k_f^2/\alpha$.