Influence of Ekman friction on the velocity profile of a coherent vortex in a three-dimensional rotating turbulent flow

Physics of Fluids. 2021. Vol. 33. No. 11. Article 115128.

Parfenyev V., Vergeles S. S.

In the presence of strong background rotation, the velocity field tends to become quasi-two-dimensional, which leads to the inverse energy cascade. If the damping is small enough, then the energy is accumulated at the largest scales of the system, forming coherent columnar vortex structures known as condensates. Recently, it was found that the radial velocity profiles of axisymmetric cyclones and anticyclones are described by the dependence $U^{\varphi}_{\scriptscriptstyle G}(r) = \pm \sqrt{\epsilon/\nu} \, r \ln (R/r)$, where $\epsilon$ is statistically stationary turbulent forcing power per unit mass, $\nu$ is the kinematic viscosity of a fluid, and $R$ is the transverse size of the vortex. However, the corresponding theory did not take into account the boundary effects and therefore was mainly applicable to numerical simulations with periodic boundary conditions. Here we demonstrate that for typical experimental conditions the damping of the condensate far enough from the symmetry axis is determined by the linear Ekman friction $\alpha = 2 \Omega_0 E^{1/2}$ associated with the no-slip conditions at the lower and upper boundaries of the system, where $\Omega_0$ is the angular velocity of the background rotation and $E$ is the Ekman number. In this case, the azimuthal velocity of the coherent vortex does not depend on the distance to the vortex center and is determined by the expression $U^{\varphi}_{\scriptscriptstyle G} = \pm \sqrt{3 \epsilon/ \alpha}$. We discuss the structure of the coherent vortex in this case and compare the results with velocity profiles of condensates in two-dimensional systems.

Keywords: турбулентность, вихревые течения, turbulence, vortex flows, rotating flows, cyclones and anticyclones, циклоны и антициклоны