Self-organization and transitions in anisotropic turbulence: from geophysical to bacterial scales

在地球物理和天体物理流动中，通常可以观察到大尺度的结构，如涡旋和喷流。飓风和急流是地球大气层的著名例子，但在气体巨星和恒星(如太阳)上也观察到了类似的状态。这种流动几乎总是高度湍流的，并显示出广泛的动态活跃的时间和空间尺度。此外，它们通常是高度各向异性的，容易受到行星快速自转、强烈层结、磁场或薄层几何结构的影响。在小得多的尺度上，生物系统，如细菌、精子或其他微泳者的悬浮液，会产生类似的流动模式，包括漩涡、喷流和涡旋晶格等强有序状态。这些系统属于活动物质的范畴，因为每个游泳者都会消耗能量来产生液体流动。因此，它们远远不是统计上的平衡。具体地说，我们谈到了活跃的湍流，尽管需要强调的是，这些流动不是经典意义上的湍流，其特点是雷诺数从小到中等。在非常不同的长度尺度上观察到的现象之间的相似性从形式上反映为两个系统的连续统描述导致相似的方程的事实。地球物理和天体物理湍流通常是由不稳定产生的，而不稳定明确地取决于不断演变的流动状态。在活性流体中，除了粘性应力外，还有活性应力，它也明确地(线性地)依赖于流场。在这个项目中，我们研究了复杂流动的自组织，不仅在行星尺度上，而且在微观尺度上。首先，我们考虑了由不稳定性驱动的强各向异性湍流的理想化模型。我们考虑二维流动，以及薄的三维流体层，其中流动是由依赖于速度的体力产生的。目标是扩展现有的二维情况下的结果，并测试它们对于三维扰动的稳健性。本项目的第二个主题是该系统的雷诺数依赖关系。我们改变雷诺数从小到大的值，从而弥合了模型微观的、活跃的实现与适用于地球物理和天体物理极限的高雷诺数流动之间的差距。最后，我们还将具体研究这些理想化结果在快速旋转的Rayleigh-Bénard对流中的应用，使用新的数值工具来研究大尺度结构的出现，即该系统中的涡旋和喷流。我们还利用最近引入的“Rainy-Bénard”模型，将这些研究扩展到由于水分和相变而产生的内部浮力源的情况。这个项目将对我们从根本上理解物理尺度上的自组织做出重大贡献。

Large-scale structures, such as vortices and jets are typically observed in geo- and astrophysical flows. Hurricanes and the Jet Stream are well known examples from Earth's atmosphere, but similar states are also observed on the gas giants and in stars, such as the Sun. Such flows are almost always highly turbulent, and display a wide range of dynamically active temporal and spatial scales. In addition, they are typically highly anisotropic, being subject to rapid planetary rotation, strong stratification, magnetic fields, or thin-layer geometries. On much smaller length scales, biological systems such as suspensions of bacteria, sperm cells, or other microswimmers, create similar flow patterns consisting of vortices, jets and strongly ordered states such as vortex lattices. These systems belong to the category of active matter, since individual swimmers consume energy to create fluid flow. Hence they are far from statistical equilibrium. Specifically, one speaks of active turbulence, although it is important to stress that these flows are not turbulent in the classical sense, being characterised by small to moderate Reynolds numbers. The similarity between the observed phenomena on very disparate length scales is formally reflected by the fact that the continuum descriptipn of both systems leads to similar equations. Geo- and astrophysical turbulence is typically born from instabilities, which explicitly depend on the evolving flow state. In active fluids, there are active stresses, in addition to viscious ones, which also depend explicitly (linearly) on the flow field. In this project, we study the self-organization of complex flows, at planetary, but also microscopic scales. First, we consider an idealized model of strongly anisotropic turbulence driven by instabilities. We consider two-dimensional flows, as well as thin three-dimensional fluid layers, where flow is created by a velocity-dependent body force. The goal is to extend existing results on the two-dimensional case, and to test their robustness with respect to three-dimensional perturbations. The second topic of this project is the Reynolds number dependence of this system. We vary the Reynolds number from small to large values, thus bridging the gap between the microscopic, active realization of the model, and the high-Reynolds number flow that applies to the geo- and astrophysical limit. Finally, we will also study specifically the application of these idealized results to rapidly rotating Rayleigh-Bénard convection, using novel numerically tools to study the emergence of large-scale structures, i.e. vortices and jets in this system. We also extend these investigations to the case of internal buoyancy sources due to moisture and phase changes, utilizing the recently introduced "Rainy-Bénard" model. This project will make a significant contribution to our fundamental understanding of self-organization across physical scales.