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

该项目旨在发展一种关于最初轴对称旋转流的演化理论，该旋转流包含从振动环面出现并在焦点处相遇的锥形惯性波。对这种原型流的理解提出了几个与其拓扑结构和动力学有关的问题。我们将结合对称群理论和数值模拟来解决这些问题，从简单的线性模型发展到更复杂的湍流模型。首先，对于线性波传播发生时的低强迫振幅，将彻底理解流动的对称性。然后，这将允许在增加的波幅下处理弱非线性状态，其中在惯性波的焦点处发生局部冲击状现象。这引发了波之间复杂的能量转移，但也引发了尚未解释的大规模运动转移。分析基础将是从欧拉方程导出的方程，该方程在高旋转速率的极限下使用奇异渐近性，并且对大波幅有效。结构对称性破缺将由稳定性理论和群论的结合来解释，我们计划将其与来自惯性波三元相互作用的统计分析的动力学参数相关联。除了这个系统的方法，我们将研究本地的现象，如机制限制的核心非线性流动中的局部区域。在对波动-湍流状态进行全面研究之后，我们将考虑完全湍流状态，在该状态下，非线性很强，因此流中的传输由惯性波交换和经典湍流交换介导，从而产生更复杂的耦合。我们最初的方法是进行两点统计方程的对称性分析，并将其与有螺旋性和无螺旋性的直接数值模拟中的各向同性破缺联系起来。在螺旋的情况下，必须考虑额外的不变量。具体的几何形状的作用也将进行评估的锥形惯性波的参数调查，并没有限制，也通过直接数值模拟。因此，我们的项目的最原始的方面是在一个单一的研究中整合一个新的理论的基础上旋转湍流的对称性，和尺度之间的各向异性传输的动力学观点。

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.