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.

中文翻译：

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使用半线性模型的离心不稳定性的非线性演化

我们使用半线性方法研究了在具有高斯角速度的柱状反气旋上发展的离心不稳定性的非线性演变。该模型由两个耦合方程组成：一个是最不稳定扰动对轴向平均平均流的线性演化，另一个是在由扰动引起的轴向平均雷诺应力的影响下平均流的演化。这种模型类似于 \cite{Vlado14} 的自洽模型，只是时间平均被空间平均取代。对于罗斯比数 $Ro=-4$ 和雷诺数的两个值，该半线性模型预测的平均流和扰动的非线性演变与 DNS 非常一致：$Re=800$ 和 $2000$（基于初始最大角速度和涡旋半径）。还考虑了考虑二次谐波扰动的改进模型。结果表明，通过波动的雷诺应力的作用，平均流的角动量朝着离心稳定的轮廓均匀化。由 \cite{Kloosterziel07} 在无粘性极限中预测的最终速度分布扩展到有限的高雷诺数。它与数值模拟非常吻合。由 \cite{Kloosterziel07} 在无粘性极限中预测的最终速度分布扩展到有限的高雷诺数。它与数值模拟非常吻合。由 \cite{Kloosterziel07} 在无粘性极限中预测的最终速度分布扩展到有限的高雷诺数。它与数值模拟非常吻合。