On the Magnetic Squashing Factor and the Lie Transport of Tangents
On the Magnetic Squashing Factor and the Lie Transport of Tangents
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DOI：
10.3847/1538-4357/aa8a64
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发表时间：
2017-10
期刊：
影响因子：
--
通讯作者：
R. Scott;D. Pontin;G. Hornig
中科院分区：
文献类型：
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作者：
R. Scott;D. Pontin;G. Hornig
矢量场的挤压因子（或挤压程度）是两个表面之间映射的场线变形的定量度量。在太阳磁场的背景下，它通常用于识别不同通量域之间的基本磁通量管映射中的梯度。映射中这些梯度较大的区域称为准分界层 (QSL)，并且是分离器和分界面的连续延伸。这些 QSL 被观察到是形成强电流的潜在位点，因此对于三维磁重联的研究非常重要。由于挤压因子 Q 是根据场线映射的雅可比行列式定义的，因此最常通过首先确定两个表面（或其某种近似值）之间的映射，然后进行数值微分来计算。 Tassev 和 Savcheva 引入了另一种方法，其中他们参数化相邻场线之间间隔的变化，然后沿着各个场线积分以获得雅可比行列式的估计，而不需要对映射本身进行数值微分。但是，虽然他们的方法提供了一定的计算优势，但它是在场线轨迹的微扰描述上制定的，并且该方法的准确性尚不完全清楚。在这里，我们通过另一种推导表明，这个积分公式原则上是精确的。然后，我们展示了线性 3D 磁零点情况下的结果，这样可以进行精确的分析描述并与数值估计进行直接比较。
The squashing factor (or squashing degree) of a vector field is a quantitative measure of the deformation of the field line mapping between two surfaces. In the context of solar magnetic fields, it is often used to identify gradients in the mapping of elementary magnetic flux tubes between various flux domains. Regions where these gradients in the mapping are large are referred to as quasi-separatrix layers (QSLs), and are a continuous extension of separators and separatrix surfaces. These QSLs are observed to be potential sites for the formation of strong electric currents, and are therefore important for the study of magnetic reconnection in three dimensions. Since the squashing factor, Q, is defined in terms of the Jacobian of the field line mapping, it is most often calculated by first determining the mapping between two surfaces (or some approximation of it) and then numerically differentiating. Tassev & Savcheva have introduced an alternative method, in which they parameterize the change in separation between adjacent field lines, and then integrate along individual field lines to get an estimate of the Jacobian without the need to numerically differentiate the mapping itself. But while their method offers certain computational advantages, it is formulated on a perturbative description of the field line trajectory, and the accuracy of this method is not entirely clear. Here we show, through an alternative derivation, that this integral formulation is, in principle, exact. We then demonstrate the result in the case of a linear, 3D magnetic null, which allows for an exact analytical description and direct comparison to numerical estimates.