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This analysis concerns ascertaining how different a magnetic cloud's field would be if its ideal helical field were confined to an elliptical cross-section while maintaining the same area as the "original" force free model of circular cross-section of radius Ro. This invariance of area will occur automatically, if the square of (Ro - b)/Ro is small, where b is the semi-minor axis of the perturbed flux rope. Below we express this field departure in terms of difference fields, normalized by the field along the axis of the cloud, Bo, in a Cartesian "magnetic cloud coordinate system." In this right-handed system the x'-axis is along the magnetic cloud axis and the y'-axis is along the observing spacecraft's closest approach vector. To give some examples of this effect the figures below display the normalized magnetic field departures, Delta-B'y/Bo and Delta-B'z/Bo, from the force free flux rope model to one having the elliptical cross- section ("new model"), presumably caused by some external perturbation in the solar wind, if the flux rope is a magnetic cloud. A measure of this shape distortion of the flux rope is given by the ratio b/Ro (Delta-psi = 1.0). The angle Delta-psi, measured counterclockwise around the flux rope axis (with the axis pointed toward) the observer) is the angle between the semi-minor axis of the new model's elliptical cross-section and the original model's closest-approach axis, the y'-axis. Yo/Ro is the ratio of closest approach distance (from the rope axis), Yo, to the radius of the original cross-sectional radius. For example, Yo/Ro = 0.0 indicates that the observing spacecraft crosses the axis of the flux rope. Note that Delta-B'x/Bo, the difference field along the rope axis, is exactly zero, because the elliptical distortion does not affect the B'x-component.
First set of figures: Difference-fields between the new (elliptical
cross-section) and original(circular) models normalized by the
field on the flux rope's axis, Bo, color-coded for various values
of Delta-psi of 0°(white), 45°(red), and 90°(green) for a fixed
closest approach distance from the flux rope axis of Yo/Ro = 0.50,
and for various values of b/Ro: 0.60, 0.70, 0.80, and 0.90..
Second set of figures: Difference-fields between the two models normalized by the field on the flux rope's axis, Bo, color-coded for various values Yo/Ro of
0.00 (white), 0.50, (red) and 0.90 (green) for a fixed value of Delta-psi = 0°,
and for various values of b/Ro: 0.60, 0.70, 0.80, and 0.90.
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