Development and Application of Turbulence Models in Numerical Geodynamo Simulations

地球液态铁核中的流体运动会产生磁场，保护地表环境免受太阳风和宇宙射线的有害影响。液态铁的低粘度(与液态水相当)确保了流体的运动具有高度的湍流和时间依赖性。流体运动和场产生的精确数值模拟超出了当前计算能力的限制，因此建模人员被迫使用流体粘度和其他物理参数的值，这些值与地球上的值相差很远。其结果是，动力学的性质被改变，我们解决重要科学问题的能力受到限制。通过强力提高分辨率，预计模型只会有适度的改进，因此迫切需要开发更好的模型来处理岩心的湍流。我们建议开发第二代地球发电机模型，从两个方面改进现有的模型。首先，新模型将采用一种更适合并行计算的求解方法。其次，新模型将包括对小尺度湍流的更复杂处理。为了实现这些目标，我们将使用有限元方法来求解控制方程，并实现一个自适应的、尺度相似的模型来考虑湍流的影响。有限元方法减少了并行计算机上的通信量，并允许我们局部地改进计算的分辨率，但只在需要的地方。尺度相似方法是一种成熟的技术，已经成功地在平面层发电机模型中实现，以解释小尺度上流动与磁场之间的复杂相互作用。我们保守地估计，在地球发电机模型中引入尺度相似模型将允许我们将流体粘度降低两个数量级，相对于目前的能力。这可能足以达到地球发电机被认为运行的动力机制(例如泰勒态)。新的模型还将允许我们探索发电机在低磁普朗特数Pm下的地球物理相关行为，Pm代表流体粘度与磁扩散系数的比率。目前最先进的数值模拟假设为PM 1，而地核的期望值为PM~1.0E-6。发电机在低PM下的运行提出了许多重要的科学问题，并构成了一个巨大的计算挑战。

Fluid motion in the liquid iron core of the Earth generates a magnetic field, which protects the surface environment from the harmful effects of solar wind and cosmic rays. The low viscosity of liquid iron (comparable to that of liquid water) ensures that the fluid motion is highly turbulent and time dependent. Accurate numerical simulations of the fluid motion and field generation are beyond the limit of current computational power, so modelers are forced to use values for the fluid viscosity and other physical parameters that are very far from Earth-like values. As a consequence, the nature of the dynamics is altered and our ability to address important scientific questions is limited. Only modest improvements in the models are expected through brute-force improvements in resolution, so there is a pressing need to develop better models for dealing with turbulence in the core.We propose to develop a second generation of geodynamo model that advances the current models in two ways. First, the new model will employ a method of solution that is better suited to parallel computing. Second, the new model will include a more sophisticated treatment of small-scale turbulence. To achieve these goals we will use the finite-element method to solve the governing equations and implement an adaptive, scale-similarity model to account for the effects of turbulence. The finite-element method reduces the amount of communication on a parallel computer and allows us to locally refine the resolution of the calculation, but only where it is needed. The scale-similarity method is a proven technique that has been successfully implemented in a plane-layer dynamo model to account for complex interactions between the flow and the magnetic field at small scales. We conservatively estimate that the introduction of the scale-similarity model in the geodynamo model will allow us to lower the fluid viscosity by two orders of magnitude, relative to current capabilities. This may be sufficient to reach the dynamical regime (e.g. a Taylor state) where the Earth's geodynamo is thought to operate. The new model will also allow us to probe the geophysically relevant behavior of dynamos at low magnetic Prandtl number Pm, which represents the ratio of fluid viscosity to magnetic diffusivity. Current state-of-the-art numerical simulations assume Pm 1, whereas the expected value for the Earth's core is Pm ~ 1.0e-6 . The operation of dynamos at low Pm raises a number of important scientific questions and poses a substantial computational challenge.