Profile of a Two-Dimensional Vortex Condensate Beyond the Universal Limit

It is well known that an inverse turbulent cascade in a finite ($2 \pi \times 2 \pi$) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length $l_f = 2\pi/k_f$ with $k_f$ ranging from $100$ to $12.5$. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior $\Omega(r) = (3 \epsilon/\alpha)^{1/2} r^{-1}$, where $r$ is the distance to the vortex center, $\alpha$ is the bottom friction coefficient, and $\epsilon$ is the inverse energy flux. Now we show that for a spatially homogeneous forcing with finite $k_f$ the vorticity profile becomes steeper, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. Qualitatively, this behaviour is related to a decrease in the effective pumping of the coherent vortex with distance from its center. To support this statement, we perform an additional simulation with spatially localized forcing, in which the effective pumping of the coherent vortex, on the contrary, increases with $r$ and show for the first time that in this case the vorticity profile can be flatter than the asymptotic limit.