A generalized poloidal-toroidal decomposition and an absolute measure of helicity
A generalized poloidal-toroidal decomposition and an absolute measure of helicity
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DOI：
10.1088/1751-8121/aaea88
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发表时间：
2018-12-07
影响因子：
2.1
通讯作者：
Hornig, G.
中科院分区：
文献类型：
--
作者：
Berger, M. A.;Hornig, G.
在流体力学和磁流体力学中，将矢量场分解为极向和环向分量是很有用的。在球面几何中，极向分量包含了场的所有径向部分，而环向分量的旋度包含了所有径向电流。本文探讨了它们如何在更一般的几何中工作，其中空间由嵌套的单连通曲面分叶。向量场仍然可以分为极向和环向分量，但在缺乏球对称的几何中，将极向场进一步划分为标准部和“形状”项是有意义的，其本身的行为就像环向场，并由曲率的变化产生。广义的P-T分解导致螺旋度的一个简单定义，它不依赖于减去一个潜在参考场的螺旋度。相反，螺旋度测量的是标准极向场与环向场以及新形状场的网络连接。这种螺旋度与球面和平面几何中的相对螺旋度是一致的。它的时间导数由于场线在一个表面上的运动有一个简单和直观的令人愉快的形式。
In fluid mechanics and magneto-hydrodynamics it is often useful to decompose a vector field into poloidal and toroidal components. In a spherical geometry, the poloidal component contains all of the radial part of the field, while the curl of the toroidal component contains all of the radial current. This paper explores how they work in more general geometries, where space is foliated by nested simply connected surfaces. Vector fields can still be divided into poloidal and toroidal components, but in geometries lacking spherical symmetry it makes sense to further divide the poloidal field into a standard part and a 'shape' term, which in itself behaves like a toroidal field and arises from variations in curvature.The generalised P-T decomposition leads to a simple definition of helicity which does not rely on subtracting the helicity of a potential reference field. Instead, the helicity measures the net linking of the standard poloidal field with the toroidal field as well as the new shape field. This helicity is consistent with the relative helicity in spherical and planar geometries. Its time derivative due to motion of field lines in a surface has a simple and intuitively pleasing form.