The structure of Taylor's constraint in three dimensions
The structure of Taylor's constraint in three dimensions
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三维泰勒约束结构
DOI：
10.1098/rspa.2008.0091
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
A. Jackson
中科院分区：
文献类型：
--
作者：
P. Livermore;G. Ierley;A. Jackson
在1963年出版的《美国科学院院刊》中，Soc. A，J. B.泰勒（Taylor 1963 Proc. R. Soc. A 9，274-283）证明了在粘度和惯性可以忽略的快速旋转的导电流体中发电机作用的必要条件。他证明了洛伦兹力的方位角分量在任何地转等值线（即与旋转轴同轴的流体圆柱）上的平均值必须为零。由此产生的动态平衡，被称为泰勒状态，被认为是在地球的核心，因此对地球发电机中允许的场的类别施加限制。除了高度受限的例子之外，这种状态已被证明难以实现。特别是，还没有证明如何以一般的方式严格执行泰勒条件，似乎需要无限数量的约束。在这项工作中，我们在选择在原点规则的截断球谐基中扩展磁场后，推导出三维洛伦兹力的平均方位角分量的解析形式。由于该结果与一个适度次数的多项式成比例（简单地与谱展开的阶数有关），因此可以通过简单地使其每个系数等于零来使其在每个地转等值线上相同地消失。我们扩展的讨论，允许存在一个内核，它划分成三个不同的区域的地转轮廓。
In a 1963 edition of Proc. R. Soc. A, J. B. Taylor (Taylor 1963 Proc. R. Soc. A 9, 274–283) proved a necessary condition for dynamo action in a rapidly rotating electrically conducting fluid in which viscosity and inertia are negligible. He demonstrated that the azimuthal component of the Lorentz force must have zero average over any geostrophic contour (i.e. a fluid cylinder coaxial with the rotation axis). The resulting dynamical balance, termed a Taylor state, is believed to hold in the Earth's core, hence placing constraints on the class of permissible fields in the geodynamo. Such states have proven difficult to realize, apart from highly restricted examples. In particular, it has not yet been shown how to enforce the Taylor condition exactly in a general way, seeming to require an infinite number of constraints. In this work, we derive the analytic form for the averaged azimuthal component of the Lorentz force in three dimensions after expanding the magnetic field in a truncated spherical harmonic basis chosen to be regular at the origin. As the result is proportional to a polynomial of modest degree (simply related to the order of the spectral expansion), it can be made to vanish identically on every geostrophic contour by simply equating each of its coefficients to zero. We extend the discussion to allow for the presence of an inner core, which partitions the geostrophic contours into three distinct regions.