  We study the nonlinear evolution of the centrifugal instability developing on
a columnar anticyclone with a Gaussian angular velocity using a semi-linear
approach. The model consists in two coupled equations: one for the linear
evolution of the most unstable perturbation on the axially averaged mean flow
and another for the evolution of the mean flow under the effect of the axially
averaged Reynolds stresses due to the perturbation. Such model is similar to
the self-consistent model of \cite{Vlado14} except that the time averaging is
replaced by a spatial averaging. The non-linear evolutions of the mean flow and
the perturbations predicted by this semi-linear model are in very good
agreement with DNS for the Rossby number $Ro=-4$ and both values of the
Reynolds numbers investigated: $Re=800$ and $2000$ (based on the initial
maximum angular velocity and radius of the vortex). An improved model taking
into account the second harmonic perturbations is also considered. The results
show that the angular momentum of the mean flow is homogenized towards a
centrifugally stable profile via the action of the Reynolds stresses of the
fluctuations. The final velocity profile predicted by \cite{Kloosterziel07} in
the inviscid limit is extended to finite high Reynolds numbers. It is in good
agreement with the numerical simulations.
