Instability driven relaxation of an anticyclone

We study the nonlinear evolution of the centrifugal instability appearing in a columnar anticyclone using a semi-linear approach to model the transient unsteady flow evolution in a self-consistent manner. For anticyclones in a homogeneous viscous flow, the fastest growing instability is without oscillation in time but with a finite axial wavenumber. Hence, the self-consistent model is developed around the spatially averaged time dependent meanflow and the fluctuation, which reduces the problem from 2D nonlinear to 1D semi-linear. The two linear meanflow and fluctuation equations are coupled via the Reynolds stress of the fluctuations. At a given rotation ratio between the vortex angular velocity and the background rotation, only the most linearly unstable mode is considered for Reynolds numbers $Re=800$ and $2000$ defined with the maximum angular velocity and the radius of the vortex. For both values of $Re$, the model predicts well the nonlinear evolution of the meanflow and the fluctuation amplitude. Higher harmonics are non-negligible only at the highest value of $Re$. The results show that the angular momentum of the meanflow is homogenized to a stable state via the action of the Reynolds stresses of the fluctuation.