Shock-like focusing of inertial waves - the localized generation of turbulence

这个项目的目的是发展一种关于初始轴对称旋转流的演化的理论，该旋转流包含从振动的圆环中涌出并汇聚在焦点上的锥形惯性波。对这一原型流的理解提出了与其拓扑和动力学有关的几个问题。我们将通过将对称群理论和数值模拟相结合来解决这些问题，从简单的线性模型发展到更复杂的湍流模型。首先，对于发生线性波传播的低强迫振幅，流的对称性将被详尽地理解。然后，这将允许在增加波幅的情况下处理弱非线性区域，其中在惯性波的焦点处发生局部激波现象。这引发了波与波之间复杂的能量转移，但也引发了一种尚未解释的大规模运动转移。分析的基础将是一个从欧拉方程导出的方程，该方程在高转速极限下使用奇异渐近性，并对大波幅有效。结构对称性破缺将由稳定性理论和群论相结合来解释，我们计划将其与从惯性波的三元相互作用的统计分析中得出的动力学论点联系起来。除了这种系统方法，我们还将研究局部现象，例如将非线性的核心限制在流动中的局部区域的机制。在全面研究了波-湍流型之后，我们将考虑完全湍流型，在这种全湍流型中，非线性很强，使得流动中的传递既由惯性波交换又由经典湍流交换来调节，从而产生更复杂的耦合。我们最初的方法将是执行两点统计方程的对称性分析，并将其与有螺旋度和无螺旋度的直接数值模拟中的各向同性破坏联系起来。在螺旋线的情况下，必须考虑附加的不变量。具体几何形状的作用也将通过对有约束和无约束的锥形惯性波的参数研究来评估，也可以通过直接数值模拟来评估。因此，我们项目最具原创性的方面是将基于旋转湍流对称性的新理论和尺度间各向异性传递的动力学观点整合到一个单独的研究中。

This project aims at developing a theory on the evolution of an initially axisymmetric rotating flow containing conical inertial waves that emerge from a vibrating torus and meet in a focal point. The understanding of this archetypal flow raises several questions pertaining to its topology and dynamics. We will address these questions by combining symmetry group theory and numerical simulations, progressing from the simpler linear model to the more complex turbulent one. First, the symmetry properties of the flow will be exhaustively understood for low forcing amplitudes for which linear wave propagation occurs. Then, this will permit to tackle the weakly nonlinear regime at increasing wave amplitude, where a local shock-like phenomenon occurs at the focal point of the inertial waves. This triggers complex energy transfers between waves but also a yet to be explained transfer to large-scale motion. The analytic basis will be an equation derived from the Euler equation using singular asymptotics in the limit of high rotation rates and valid for large wave amplitudes. The structural symmetry breaking will be explained by a combination of stability theory and group theory, which we plan to relate to dynamical arguments drawn from a statistical analysis of triadic interactions of inertial waves. In addition to this system approach, we will study local phenomena, such as the mechanism limiting the core of non-linearity to a localized region in the flow. After the comprehensive study of the wave-turbulence regime, we will consider the fully turbulent regime in which nonlinearities are strong so that transfers in the flow are mediated by both inertial waves exchanges and by classical turbulent ones, thus producing more complex couplings. Our original approach will be to perform the symmetry analysis of two-point statistical equations, and to relate this to the isotropy-breaking in Direct Numerical Simulations with and without helicity. In the helical case, additional invariants have to be considered. The role of the specific geometry will also be evaluated by a parametric investigation of the cone-shaped inertial waves with and without confinement, also by Direct Numerical Simulations. The most original aspect of our project is thus to integrate in a single study a new theory based on the symmetries of rotating turbulent flows, and a dynamical point of view for anisotropic transfers between scales.