Abstract
We present a theoretical framework to analyze the 3D coronal vector magnetic-field structure. We assume that the vector magnetic field exists and is a priori smooth. We introduce a generalized connectivity phase space associated with the vector magnetic field in which the basic elements are the field line and its linearized variation: the Spatial Propagator. We provide a direct formulation of these elements in terms of the vector magnetic field and its spatial derivatives, constructed with respect to general curvilinear coordinates and the equivalence class of general affine parameterizations. The Spatial Propagator describes the geometric organization of the local bundle of field lines, equivalent to the kinematic deformation of a propagated volume tied to the bundle. The Spatial Propagator’s geometric properties are characterized by dilation, anisotropic stretch, and rotation. Extreme singular values of the Spatial Propagator describe quasi-separatrix layers (QSLs), while true separatrix surfaces and separator lines are identified by the vanishing of one and two singular values, respectively. Finally, we show that, among other possible applications, the squashing factor [Q] is easily constructed from an analysis of particular sub-matrices of the Spatial Propagator.
Highlights
Geometry describes the measurable lengths and angles associated with a system’s configuration, whereas topology is concerned with those properties preserved under continuous 76 Page 2 of 41J.K
We introduce a generalized connectivity phase space associated with the vector magnetic field in which the basic elements are the field line and its linearized variation: the Spatial Propagator
In this article we introduce a generalized field-line connectivity phase space associated with the vector magnetic field in which the geometric and topological features of the system are made explicit
Summary
This is an extremely popular approach to numerical/computational solar- and space-plasma physics, but arguably leads to spurious effects (see, e.g., Edmondson, 2012, for a discussion); we offer no judgment regarding the veracity of these approaches This perspective, requires the development of new mathematical tools/description/framework to analyze the geometric organization of the magnetic-field connectivity map. The purpose of this article is the introduction, development, and presentation of a generalized connectivity phase space of field-line geometry associated to the vector magnetic field, in which the geometric structure and topological constraints are made explicit This formalism does not alter the physics of Maxwell’s equations, or MHD, but it is a fundamentally different framework that describes and analyzes vector magnetic fields. We discuss the various representations of the Spatial Propagator (Section 2.4): covariance with respect to local curvilinear
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