Non-Oberbeck-Boussinesq Effects in the Ultimate State of Rapidly Rotating Rayleigh-Benard Convection

自然界中发生的许多湍流，例如在行星和恒星内部以及大气中，都是由对流驱动的，并受到快速旋转的强烈约束。旋转瑞利-贝纳德对流是一个很好的、数学上容易描述的模型系统。该模型包括一个液体或气体之间的温暖的底部边界和冷的顶部边界旋转的垂直轴旋转。但是，在瑞利-贝纳德对流的地面数值模拟和实验室实验中达到的湍流水平和相对旋转率（用控制参数瑞利数和埃克曼数表示）并不像自然环境中的参数那样极端。此外，大多数数值模拟和数学理论假设恒定的材料特性（例如粘度和热扩散率），与实际流体相反，它们随温度和压力而变化。因此，根据地球物理和天体物理学的流动来解释模拟和实验的结果有些问题。然而，湍流研究中有一个长期的原则，即如果流动变得足够湍流，即达到“极限状态”，则任何全局输运和宏观特征都变得独立于分子扩散率，特别是粘度和热扩散率。因此，至关重要的是，如果最终状态存在，从数值模拟和实验室实验到地球和天体物理系统的升级是可能的，尽管控制参数有许多数量级的差异。本研究的目的是检验快速旋转Rayleigh-Bénard对流最终状态下热量和动量输运的无扩散标度假设。尽管理论论证预测旋转系统比非旋转系统更容易达到最终状态，数值分辨率要求对于具有当今计算资源的蛮力方法是禁止的。为了减轻分辨率约束，我将考虑一个新的观点，通过采用不同的热扩散率和运动粘度在同一对流容器。材料性质的变化导致经典（非终极）瑞利-贝纳德问题中的上下对称性破缺。然而，在最终状态下，假设分子扩散率不再影响整体流动状态，人们可以预期这种对称性会恢复。这种对称性的恢复可用作达到最终状态的指标和量化措施，并可进行可靠的外推。此外，由于边界层是湍流热对流中热量和动量传输的关键参与者，我将比较无边界层的三重周期瑞利-贝纳德对流与实验室般的圆柱形设置，包括边界层的模拟。

Many of the turbulent flows occurring in nature, for example within planetary and stellar interiors, as well as atmospheres, are driven by convection and are strongly constrained by rapid rotation. An excellent and mathematically easily describable model system is rotating Rayleigh-Bénard convection. The model consists of a liquid or gas confined between a warm bottom boundary and a cold top boundary rotated around the vertical axis. But the level of turbulence and the relative rotation rates (expressed in terms of the control parameters Rayleigh and Ekman number) reached in earthbound numerical simulations and laboratory experiments of Rayleigh-Bénard convection, are not as extreme (yet) as the parameters in natural settings. Moreover, most numerical simulations and mathematical theories assume constant material properties (e.g. viscosity and thermal diffusivity), contrary to realistic fluids where they vary with temperature and pressure. Thus, interpreting results from simulations and experiments in the light of geophysical and astrophysical flows is somewhat problematic.However, there is a long-held tenet in turbulence research that if the flow only becomes turbulent enough, that is, reaches the "ultimate regime," any global transport and macroscopic features become independent of the molecular diffusivities, in particular, the viscosity and the thermal diffusivity. Hence, crucially, if the ultimate state exists, an upscaling from numerical simulations and laboratory experiments to geo- and astrophysical systems is possible despite many orders of magnitude difference in the control parameters. The objective of the proposed research is to test the hypothesis of a diffusion-free scaling of the heat and momentum transport in the ultimate state of rapidly rotating Rayleigh-Bénard convection.Even though theoretical arguments predict that the ultimate state is more easily accessible in rotating than in non-rotating systems, the numerical resolution requirements are prohibitive for a brute force approach with present-day computational resources.To alleviate the resolution constraints, I will consider a novel point of view by employing a varying thermal diffusivity and kinematic viscosity within the very same convection vessel. The variation of the material properties leads to a breaking of the top-bottom symmetry in the classical (non-ultimate) Rayleigh-Bénard problem. However, in the ultimate regime, one may expect that this symmetry gets restored, assuming that the molecular diffusivities do no longer affect the global flow state. The restoration of this symmetry can be used as an indicator and quantitative measure for reaching the ultimate regime and allows for reliable extrapolation. Further, as boundary layers are known to be key players in the transport of heat and momentum in turbulent thermal convection, I will compare simulations of boundary layer free triply periodic Rayleigh-Bénard convection with laboratory-like cylindrical set-ups that include boundary layers.