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Magnetic Field Magnitude Modification for a Force-free Magnetic Cloud Model

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Abstract

A scheme was developed by Lepping, Berdichevsky, and Wu (Solar Phys. 292, 27, 2017) [called the LBW article here] to approximate the average magnetic field magnitude (\(B\)-) profile of a typical magnetic cloud (MC) at/near 1 AU. It was based on actual Wind MC data, taken over 21 years, that were used to modify a time-shifted Bessel function (force-free) magnetic field, where shifted refers to a field that was adjusted for typical MC self-similar expansion. This was developed in the context of the Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 11957, 1990) [called LJB here] MC parameter fitting model and should provide more realistic future representations of the MC \(B\)-profile in most cases. In the LBW article, we showed through testing that in about 80% of the MC cases (but this varies according to the actual closest approach distances of the spacecraft) on average, the MC \(B\)-profile of the modified model is expected to significantly improve when this scheme is used. We describe how this scheme can be employed practically to modify the LJB MC fitting model, and we test a new and slightly better (and less unwieldy) version of the scheme, the non-shifted (of Bessel functions) version, which is indeed used in the LJB model modification. The new scheme is based on slightly more accurate modification formulae compared to the old scheme, and it is expected to improve the \(B\)-profile in approximately 83% of the cases on average. The schemes are applicable for use with data originating only at/near 1 AU, since the magnetic field and plasma data used in the development of the associated formulae were taken only from the Wind spacecraft, which was and is at 1 AU.

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Acknowledgements

We thank the Wind MFI and SWE teams for the care they employ in producing the magnetic field and plasma data used in this study. C. Kay’s research was supported by an appointment to the NASA Postdoctoral Program at NASA GSFC, administered by the Universities Space Research Association under contract with NASA, and by the Dept. of Physics, The Catholic University of America, Washington DC 20064, USA. This study was partially supported by the Chief of Naval Research, and NASA LWS program, Grant No. 80HQTR18T0023 (CCW).

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Appendices

Appendix A: Magnetic Cloud (Cl) Coordinate System

In the Cl system the \(\mathbf{X}_{\mathrm{Cl}}\)-axis is along the MC axis, positive in the direction of the positive polarity of the axial magnetic field, the \(\mathbf{Z}_{\mathrm{Cl}}\)-axis passes through the MC axis and is aligned with the projection of the trajectory of the spacecraft (relative to the MC velocity, which is approximately along the \(\mathbf{X}_{\mathrm{GSE}}\)-axis) onto the cross-section of the MC, and \(\mathbf{Y}_{\mathrm{Cl}} = \mathbf{Z}_{\mathrm{Cl}} \times \mathbf{X}_{\mathrm{Cl}}\). See Figure 3, which shows the circular cross-section of an ideal MC and the projection of the spacecraft path onto the cross-section in Cl coordinates, and see the Wind/MFI website https://wind.gsfc.nasa.gov/mfi/ecliptic.html for further discussion and derivation of the coordinate transformation from GSE coordinates to Cl coordinates.

Appendix B: Criteria for Estimating the Quality of the Magnetic Cloud Fit

Here we quantify the quality (\(Q _{0}\)) of the model parameter fit of a given magnetic cloud (MC) into three possibilities, \(Q _{0} = 1, 2, 3\), for excellent, good/fair, and poor, respectively, given below in terms of magnetic field quantities resulting from use of the MC model (Lepping, Jones, and Burlaga 1990). However, for the sake of compactness, we often refer to Quality as a measure of the MC per se, where it is mainly the quality of the MC parameter fit that is being estimated.

We first describe the characteristics of those MCs that fall into the \(Q _{0} = 3\) (poor) category. This category arises from satisfying any one of the following \(Q _{0} = 3\) criteria: \(|\mbox{check}| \geq55\%\), \(|\mbox{CA}| \geq97\%\), \(\langle B_{\mathrm{X}}\rangle_{\mathrm{Cl}} \leq -1.5~\mbox{nT}\), either the f-flag or the F-flag = NOT OK, \(\mbox{diameter} \geq 0.45~\mbox{AU}\), \(\mbox{asf} \geq 40\%\), cone angle \(( \beta_{\mathrm{CA}}) \leq 25^{\circ}\) or \(\beta _{\mathrm{CA}} \geq155^{\circ}\), and \(\chi _{\mathrm{R}} \geq 0.215\). Note that \(\chi _{\mathrm{R}} = 0.215\) corresponds to an MC field noise level \(\nu\) of 4.0 nT, according to Lepping, Berdichevsky, and Ferguson (2003, 2004), and this is the highest MC noise level that they found acceptable. The remaining cases, comprising designated “\(Q _{0} = 1\) or 2,” are next examined to distinguish the “good” cases (\(Q _{0} = 1\)) from the “fair” (\(Q _{0} = 2\)) ones. The \(Q _{0} = 1\) cases must satisfy all of the following criteria: \(|\mbox{check}| \leq 20\%\), \(|\langle B _{\mathrm{Y}}\rangle_{\mathrm{Cl}}| \leq 3.0~\mbox{nT}\), \(\mbox{asf} \leq 30\%\), \(45^{\circ} \leq \beta _{\mathrm{CA}} \leq 135^{\circ}\), and \(\chi _{\mathrm{R}} \leq 0.165\). These are the “\(Q _{0} = 1\) set.” Note that \(\chi _{\mathrm{R}} = 0.165\) corresponds to an MC field noise level \(\nu\) of 3.0 nT, according to Lepping, Berdichevsky, and Ferguson (2003, 2004). The remaining cases within set 1, 2, i.e. those not satisfying the \(Q _{0} = 1\) criteria, are placed in category \(Q _{0} = 2\).

There are many ways that an MC can achieve a \(Q _{0} = 3\) quality, so there is no typical \(Q _{0} = 3~\mbox{MC}\). However, \(\chi _{\mathrm{R}}\) and asf are usually the two most important parameters in judging MC quality. The quality criteria (meaning for all \(Q _{0} = 1, 2, 3\)) were derived from our experience in the application of the Lepping, Jones, and Burlaga (1990) model and partly from a desire to be consistent with the results of the error study by Lepping, Berdichevsky, and Ferguson (2003, 2004). It should be stressed that by our criteria, an MC may well satisfy the original Burlaga et al. (1981) definition of an MC and still not have a good flux rope structure by the Lepping, Jones, and Burlaga (1990) model and therefore not qualify for a \(Q _{0}\) of 1 or 2.

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Lepping, R.P., Wu, CC., Berdichevsky, D.B. et al. Magnetic Field Magnitude Modification for a Force-free Magnetic Cloud Model. Sol Phys 293, 162 (2018). https://doi.org/10.1007/s11207-018-1383-5

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