Analytical Verification of the Normal Vector Determination Methods for Planar Magnetic Structures

Abstract Minimum variance analysis of magnetic fields (MVAB) is conventionally implemented to determine the normal vector of a planar magnetic structure (PMS) for which the average of the normal magnetic field component vanishes (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfenced close="〉" open="〈"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> = 0). A more suitable normal vector determination method for single-spacecraft measurements, based on the minimum variance concept, is recommended for PMS studies, namely the MVAB method associated with the constraint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfenced close="〉" open="〈"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> = 0 (MVABC). An analytical PMS model is employed to verify the performance of the MVAB and MVABC methods for three different types of PMSs, under the circumstances that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> values are highly fluctuated but the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfenced close="〉" open="〈"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> remains zero. The results show that MVABC outperforms MVAB in estimating the PMS normal vector. By analyzing a large number of test data sets randomly selected from the original data set, the conclusion remains unchanged. The normal of a PMS discontinuity, which is associated with significant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> fluctuations, is accurately predicted by MVABC and the cross-product method, as compared to MVAB. Moreover, the normal of the discontinuity aligns with the PMS normal, as expected. Two PMS events observed by the Advanced Composition Explorer spacecraft in the sheath of an interplanetary coronal mass ejection are presented. From the MVAB perspective, these two events are not classified as PMSs. The estimated normals of the in situ PMS discontinuities from the cross-product method form angles between 10° and 20° with the predicted PMS normals from MVABC. Discussions are given regarding the failure of the MVAB method in PMS identification.