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On Formulations of Compressible Mantle Convection
Geophysical Journal International ( IF 2.8 ) Pub Date : 2020-02-13 , DOI: 10.1093/gji/ggaa078
Rene Gassmöller 1, 2 , Juliane Dannberg 1, 2 , Wolfgang Bangerth 3 , Timo Heister 4 , Robert Myhill 5
Affiliation  

SUMMARY
Mantle convection and long-term lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier-Stokes equations. Since on Earth-sized planets the influence of compressibility is not a dominant effect, density deviations from a reference profile are at most on the order of a few percent, and using the full governing equations poses numerical challenges, most modelling studies have simplified the governing equations. Common approximations assume a temporally constant, but depth-dependent reference profile for the density (the Anelastic Liquid Approximation), or drop compressibility altogether and use a constant reference density (the Boussinesq Approximation). In most previous studies of mantle convection and crustal dynamics, one can assume that the error introduced by these approximations was small compared to the errors that resulted from poorly constrained material behavior and limited numerical accuracy. However, as model parameterizations have become more realistic, and model resolution has improved, this may no longer be the case and the error due to using simplified conservation equations might no longer be negligible: while such approximations may be reasonable for models of mantle plumes or slabs traversing the whole mantle, they may be unsatisfactory for layered materials experiencing phase transitions or materials undergoing significant heating or cooling. For example at boundary layers or close to dynamically changing density gradients, the error arising from the use of the aforementioned compressibility approximations can be the dominant error source, and common approximations may fail to capture the physical behaviour of interest. In this paper, we discuss new formulations of the continuity equation that include dynamic density variations due to temperature, pressure, and composition without using a reference profile for the density. We quantify the improvement in accuracy relative to existing formulations in a number of benchmark models, and evaluate for which practical applications these effects are important. Finally, we consider numerical aspects of the new formulations. We implement and test these formulations in the freely available community software aspect, and use this code for our numerical experiments.


中文翻译:

关于可压缩地幔对流的公式

概要
地球和其他行星的地幔对流和长期岩石圈动力学可以看作是高粘度流体的缓慢变形,因此可以使用可压缩的Navier-Stokes方程来描述。由于对地球大小的行星而言,可压缩性不是主要影响因素,与参考曲线的密度偏差最多仅为百分之几,并且使用完整的控制方程会带来数值挑战,因此大多数建模研究都简化了控制过程。方程。通用逼近假定密度为时间常数,但与深度相关的参考曲线(非弹性液体逼近)或液滴可压缩性完全相同,并使用恒定的参考密度(Boussinesq逼近)。在先前有关地幔对流和地壳动力学的大多数研究中,可以假定,与由于不良约束的材料行为和有限的数值精度而导致的误差相比,由这些近似值引入的误差较小。但是,随着模型参数化变得更加现实,并且模型分辨率有所提高,情况可能不再如此,并且由于使用简化的守恒方程而产生的误差可能不再可以忽略不计:尽管这样的近似值对于地幔柱模型或地幔柱模型可能是合理的。平板横穿整个地幔,对于经历相变的层状材料或经历显着加热或冷却的材料,它们可能无法令人满意。例如,在边界层或接近动态变化的密度梯度时,使用上述可压缩性近似值引起的误差可能是主要的误差源,并且常见的近似值可能无法捕获感兴趣的身体行为。在本文中,我们讨论了连续性方程的新公式,其中包括由于温度,压力和成分而引起的动态密度变化,而没有使用密度的参考曲线。我们在许多基准模型中量化了相对于现有配方的准确性提高,并评估了这些影响在哪些实际应用中很重要。最后,我们考虑新配方的数值方面。我们在可免费获得的社区软件方面实施和测试这些公式,并将此代码用于我们的数值实验。我们讨论了连续性方程的新公式,其中包括由于温度,压力和成分引起的动态密度变化,而没有使用密度的参考曲线。我们在许多基准模型中量化了相对于现有配方的准确性提高,并评估了这些影响在哪些实际应用中很重要。最后,我们考虑新配方的数值方面。我们在可免费获得的社区软件方面实施和测试这些公式,并将此代码用于我们的数值实验。我们讨论了连续性方程的新公式,其中包括由于温度,压力和成分引起的动态密度变化,而没有使用密度的参考曲线。我们在许多基准模型中量化了相对于现有配方的准确性提高,并评估了这些影响在哪些实际应用中很重要。最后,我们考虑新配方的数值方面。我们在免费提供的社区软件方面实施和测试了这些公式,并将此代码用于我们的数值实验。并评估这些效果对于哪些实际应用很重要。最后,我们考虑新配方的数值方面。我们在可免费获得的社区软件方面实施和测试这些公式,并将此代码用于我们的数值实验。并评估这些效果对于哪些实际应用很重要。最后,我们考虑新配方的数值方面。我们在可免费获得的社区软件方面实施和测试这些公式,并将此代码用于我们的数值实验。
更新日期:2020-02-13
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