低罗斯贝数和大普朗特数极限下的无惯性对流发电机模型
Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number
摘要：
成分对流被认为是行星内部产生磁场的重要能源。普朗特数，$Pr$，表征成分对流是显着大于单位，这表明，惯性力可能不是重要的对流小尺度上，只要浮力不是太强。我们建立了小罗斯贝数和大普朗特数的渐近发电机模型，其中对流尺度上没有惯性。该极限的相关扩散率参数是组成罗伯茨数，$q = D/η$，它是组成和磁扩散率的比率。发电机模型开发的一阶$q$和更相关的低$q$限制。在这两种情况下，磁能密度与动能密度之比$M$是渐近大的，反映了阿尔文波已经从动力学中滤除的事实。随着以前的调查的渐近发电机模型$Pr=O(1)$，我们的研究结果表明，比例$M$是不是一个有用的指示器的主导力量平衡的动量方程，因为许多不同的渐近极限$M$可以得到不改变主导秩序的地转平衡。此外，本模型表明，惯性不是驱动低$q$，大规模发电机的要求。
英文摘要：
Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, $Pr$, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is not too strong. We develop asymptotic dynamo models for the case of small Rossby number and large Prandtl number in which inertia is absent on the convective scale. The relevant diffusivity parameter for this limit is the compositional Roberts number, $q = D/η$, which is the ratio of compositional and magnetic diffusivities. Dynamo models are developed for both order one $q$ and the more geophysically relevant low $q$ limit. For both cases the ratio of magnetic to kinetic energy densities, $M$, is asymptotically large and reflects the fact that Alfvén waves have been filtered from the dynamics. Along with previous investigations of asymptotic dynamo models for $Pr=O(1)$, our results show that the ratio $M$ is not a useful indicator of dominant force balances in the momentum equation since many different asymptotic limits of $M$ can be obtained without changing the leading order geostrophic balance. Furthermore, the present models show that inertia is not a requirement for driving low $q$, large-scale dynamos.
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