Numerical simulations of the geodynamo (and other planetary dynamos) have made significant progress in recent years. As computing power has advanced, some new models claim to be ever more appropriate for understanding Earth's core dynamics. One measure of the success of such models is the ability to replicate the expected balance between forces operating within the core; Coriolis and Lorentz forces are predicted to be most important. The picture is complicated for an incompressible flow by the existence of the pressure gradient force which renders the gradient parts of all other forces dynamically unimportant. This can confuse the situation, especially when the scale dependence of forces are considered. In this work we investigate force balances through the alternative approach of eliminating gradient parts of each force to form `solenoidal force balances'. We perform a lengthscale dependent analysis for several spherical simulations and find that removal of gradient parts offers an alternative picture of the force balance compared to looking at traditional forces alone. Solenoidal force balances provide some agreement with the results of previous studies but also significant differences. They offer a cleaner overall picture of the dynamics and introduce differences at smaller scales. This has implications for geodynamo models purporting to have reached Earth-like regimes: in order to achieve a meaningful comparison of forces, only the solenoidal part of forces should be considered.
Force balance in numerical geodynamo simulations: a systematic study
地球外核中的发电机作用预计将由压力、科里奥利力、浮力和洛伦兹力之间的平衡控制，惯性力和粘性力的作用很小。然而，由于计算的局限性，目前对地球发电机的数值模拟在大得多的惯性和粘性下进行。这使人们对这些模型的物理相关性产生了一些怀疑。我们的工作的目的是找到发电机模型在一个温和的计算制度，重现领先的顺序力平衡的地球。通过对埃克曼数在$10^{-6} \leq E \leq 10^{-4}$范围内进行系统的参数空间测量，研究了力平衡在改变强迫（Rayleigh数，$Ra$）和粘性扩散率与磁扩散率之比（磁普朗特数，$Pm$）时的变化。对于偶极占主导地位的发电机，我们观察到，在整个调查的参数空间的力的平衡结构是强大的，表现出准地转（QG）的平衡（科里奥利和压力之间的平衡）在零阶，其次是一阶MAC之间的非地转科里奥利，浮力和洛伦兹力的平衡。在第二阶，这种平衡受到惯性力和粘性力的影响。具有不同的力序列的发电机，其中惯性和/或粘性在一阶力平衡中取代了洛伦兹力，只能在接近发电机作用的开始和多极状态中找到。我们的研究表明，大多数经典的数值发电机控制的QG-MAC平衡，而粘度和惯性起主导作用的情况下，是例外，而不是规范。
