Rotating fluids frequently show nonlinear wave interactions and turbulence. This is true in particular for non-uniformly rotating systems. One example of such a non-uniform rotating object is the Earth. Due to its fast rotation it is not exactly spherical. As a result of the interaction with the Sun and Moon the non-spherical Earth cannot rotate uniformly but shows precession and libration. This has consequences for the fluid enclosed in the outer Earth core. Due to the forcing it might become turbulent, one of the key factors in the present theories explaining the generation of the geomagnetic field. In the present paper we show experimental results from a system that is simpler than classical precession experiments but still shows very similar wave interactions and a collapse to turbulence. This system consists of a partly filled rotating annulus that rotates about its symmetry axis slightly tilted with respect to the gravity vector and allows us to explore the Ekman numbers ranging from $7.9\times {10}^{-6}$ to $3.2\times {10}^{-5}$. In analogy to the more classical precession experiments we also find a resonant collapse when the forcing frequency corresponds with a resonant frequency of the rotating tank. The forced mode and two free Kelvin modes give rise to triadic resonance. Beside the parametric triadic resonance we further observed a shear type instability of the nonlinearly excited geostrophic flow. This instability give rise to a barotropic mode that interacts with the forced mode and generates secondary modes. We also observed a dependency of the mode frequencies on the Ekman number, which can, at least partly, be explained by a Doppler shift due to the mean flow. Finally, we try to connect our data to a low-order dynamical system that describes the main features of single triad interaction in precession experiments. Although this model is originally not designed for the multiple triads we observe, it is still useful for a qualitative understanding of mode interactions, e.g. for the mechanism of geostrophic mode excitation.

中文翻译：

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具有自由表面的旋转倾斜圆柱环的惯性模式相互作用

旋转流体经常显示出非线性波相互作用和湍流。对于非均匀旋转的系统尤其如此。这样的非均匀旋转物体的一个例子是地球。由于其快速旋转，它并非完全是球形的。由于与太阳和月球的相互作用，非球形地球不能均匀旋转，而是表现出进动和自由。这对于封闭在外部地心中的流体有影响。由于强迫作用，它可能会变得湍流，这是解释地磁场产生的当前理论中的关键因素之一。在本文中，我们显示了比经典进动实验更简单的系统的实验结果，但仍显示出非常相似的波相互作用和湍流塌陷。$7.9\times {10}^{-6}$ 至 $3.2\times {10}^{-5}$。与更经典的岁差实验类似，当强迫频率与旋转坦克的共振频率相对应时，我们也发现了共振崩溃。强制模式和两个自由开尔文模式会引起三重共振。除了参数三重共振，我们还观察到了非线性激发的地转流的剪切型不稳定性。这种不稳定性产生了与强制模式相互作用并产生次级模式的正压模式。我们还观察到模式频率对Ekman数的依赖性，这可以至少部分地由平均流量引起的多普勒频移来解释。最后，我们尝试将我们的数据连接到描述进动实验中单个三合会互动的主要特征的低阶动力学系统。