Geoeffective interplanetary magnetic field (IMF) from in situ data: realistic versus idealized spiral IMF

The geoeffective, southward IMF (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs) given in the GSM reference frame as nature presents is compared with that based on idealized, spiral IMF. We obtained \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs sorted by the IMF polarity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs fields) from in situ data at a high 16-second resolution. Idealized IMF is derived by omitting the fluctuation of the IMF in the GSEQ Z-direction. Results are: the absolute value of realistic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs is larger than the one from idealized IMF; realistic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs polarity fields exist in all seasons, while those from idealized IMF exist only around spring/fall when the IMF points toward/away from the Sun; idealized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs fields match the predictions of the Russell–McPherron (RM) model almost ideally. The present study has resolved the issue related to the patterns and absolute values of the observed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs fields and those from the RM model that assumes an idealized IMF. It confirms that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{z,\text {GSEQ}}$$\end{document}Bz,GSEQ plays a crucial role for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs. Finally, it paves a way to properly link the variations seen in geomagnetic activity with the pattern of the measured \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\text {s}$$\end{document}Bs fields.

www.nature.com/scientificreports/ with recent Parker Solar Probe observations (e.g. 25,26 and references therein). Some previous studies pointed out that IMF fluctuation about the spiral angle cannot be ignored in studies of semiannual, annual variations in geomagnetic activity (e.g. [27][28][29][30] ) and in study of B s fields (V.B. 2021). Thus, the deviations from the Parker spiral are an observable feature of the IMF which affect the solar wind coupling with the magnetosphere through B s and other coupling functions and should not be simply neglected when studying the relationship between B s and magnetospheric quantities. Nevertheless, to avoid that it further remains unexplained why the patterns and absolute values of the observed polarity fields and the absolute value of B s are not in line with the prediction of the RM model of B s , the present study aims to answer the following questions: (a) since the RM model does not describe the observations, is there any data set that the model can match, (b) if the latter is the case, what characteristics does that data set have?
Besides providing explicit evidence and explanation of the observed discrepancies between observed B s fields and RM model prediction, this study contributes to properly connect the variations seen in magnetospheric activity with the observed, measured B s fields.
The paper is organized as follows. The next section is devoted to data and method. In "Results" we present the obtained results. Characteristics of B s fields are presented in "Characteristics of B s fields: complete versus incomplete fields". Then follow the discussion and conclusion.

Data and method
In this paper for the period 1998-2017 we used IMF components measured by Magnetometer (MAG 31 ) onboard the Advanced Composition Explorer (ACE 32 ) satellite, given in GSM at high 16-second resolution. First, we derive observed B s fields (thereafter complete observed fields) from B z,GSM and B y,GSEQ that we rectified at 16-seconds. The fields are defined as: B s =B z,GSM < 0 and undefined otherwise, B s =B s (B y,GSEQ < 0) for IMF pointing toward the Sun and B s =B s (B y,GSEQ > 0) for IMF pointing away from the Sun, respectively. In the next step, the IMF components given in GSM are transformed to GSEQ. Following the assumption that RM made to obtain their model, for each IMF vector we set B z,GSEQ to zero. In such a way obtained IMF vectors which have all components projected to the X-Y GSEQ plane are transformed back to GSM.
Generally, we have: so only the first term (denoted as I) in expression (1) remains. From such B z,GSM data set we derive B s fields, thereafter called incomplete observed B s fields. The angle α is the rotation angle between the GSEQ and GSM frames. Differences between complete and incomplete IMF vectors are schematically presented in Fig. 1. Complete and incomplete data sets in GSEQ (marked in blue) have the same X and Y components, but different Z components. By transforming from GSEQ to GSM, different IMF vectors of complete and incomplete data sets are obtained (marked in green and red).
To show both complete and incomplete observed B s fields as a function of day of the year (DOY), we calculate their means by averaging all 16-second values of all studied years within DOY intervals of 14-day and 1-month. To display data as pictograms in DOY-UT representation, we calculated means by averaging all 16-second values within the UT-interval of an hour for all days within the chosen DOY interval for all years. Further, for discussion purposes, we derived the hour-of-year means (365× 24 values within a year, thereafter HOY averages) by averaging all 16-second values of the same hours of all 20 considered years. These data are shown in the DOY-UT representation of 1-day × 1-hour. By averaging B s fields over many years (here 20 years), we reduced them to the (1) B z,GSM = B y,GSEQ sin α I + B z,GSEQ cos α II Figure 1. Schematic illustrating differences between complete and incomplete IMF vectors in GSEQ (marked in blue) and in GSM (marked in green for the complete data set and in red for the incomplete data set). www.nature.com/scientificreports/ epoch of 1 year. In the following they will be denoted as B s , B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) . We derive also the final means of B s fields by averaging all 16-second values within 20 years, named fin averages ( B s fin , B s (B y,GSEQ < 0) fin , B s (B y,GSEQ > 0) fin ). Finally, we obtain B s fields from the RM model following details about their calculations provided in V.B. 2021 which are based on assumptions made by RM. Let us briefly recall that according to the RM approach the constant IMF lies along the spiral angle, thus B z,GSEQ = 0. They calculated B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) . The B s was not derived directly, but as the average of the two polarity fields.
RM postulated B s to be: Since B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) do not overlap for any α , as commented in V.B. 2021, the above formula turns to: which we assign as B s from the RM model. The obtained characteristics of B s fields from complete and incomplete datasets, discussed in section "Characteristics of B s fields: complete versus incomplete fields" , have indicated how B s has to be expressed as function of the two polarity fields. It is shown that factor 1/2 in expression (4) has to be changed to 1 and that B s from the RM model shall be calculated using expression (8). B s derived using formula (8) will be denoted as B s predicted from the corrected RM model. Note that the temporal behavior of B s fields depends on angle α and is not affected by the initial IMF resolution. Since α exhibits annual and diurnal variations, we used hourly values of α which we find to be sufficient. Fig. 2b,d incomplete observed B s fields averaged on DOY-interval of 14-day and 1-month. The complete polarity fields exhibit the "pair of spectacles" pattern. They show enhancements in the favorable and reductions in unfavorable seasons of approximately the same amplitude. Amplitudes of complete B s are smaller than amplitudes that the complete polarity fields attain in their favorable seasons (black line in Fig. 2a,c is above the blue/red line in spring/fall). The B s fields oscillate around the average value (fin average) that for all of them amounts to ∼ −2.6 nT. On the other hand, the incomplete observed polarity fields do not exhibit the "pair of spectacles" pattern. Each of them lacks part of the pattern in unfavorable seasons: there is no B s (B y,GSEQ < 0) in fall and no B s (B y,GSEQ > 0) in spring. The fields oscillate around ∼ −0.5 nT, the value that is about five times smaller than the average about which the complete observed polarity fields oscillate. Also, their amplitudes are about twice the value of the amplitudes of the complete observed fields. The incomplete B s in spring and fall matches the values of B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) , respectively. Note that these results hold regardless of the resolution at which the fields are presented. Figure 3 shows: (a) incomplete observed fields along with the predictions of the RM model (calculated using expression (2) and expression (4)). Additionally, B s predicted from the corrected RM model (expression (8)) is depicted. (b) The contour plots of incomplete observed fields and (c) contour plots of the polarity fields from the RM model and that of B s from the corrected RM model. Plots related to the polarity fields show that they are in very good agreement with the predictions of the RM model. Both the patterns and the absolute values are in accordance. The B s (B y,GSEQ < 0) exists only around spring and B s (B y,GSEQ > 0) only around fall. Amplitude and absolute value of incomplete observed B s is not in accordance with B s predicted by the RM model, but matches well B s calculated using expression (8). Figure 4 shows the contour plots of the complete and incomplete polarity fields defined on the HOY scale.

Characteristics of B s fields: complete versus incomplete fields
In this section we focus on the existence of B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) within the year, explanation of the observed features and on the relationship between B s fields.
Complete B s fields. According to Fig. 2a,c complete B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) can exist at the same point in time. Figure 4a confirms that this is valid for every HOY and that the results are not influenced by www.nature.com/scientificreports/ averaging the 16-second field values on DOY interval of 14-day and 1-month. As noted in V.B. 2021, the B s will exist at some point in time as long as the following condition is satisfied: Since the signs of B y,GSEQ and B z,GSEQ vary randomly through the years and thus they are not seasonal dependent, on averaging over many years both complete B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) fields exist in all seasons (favorable and in unfavorable seasons). This explains why these fields can exist at any HOY within the year (as observed in Fig. 4a). Further, from Fig. 2a,c it follows that B s is not a simple average of B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) , but can be expressed as a function of the two fields as follows: Incomplete B s fields. Fig. 3a shows that incomplete observed B s (B y,GSEQ < 0) exists around spring and B s (B y,GSEQ > 0) around fall. In summer and winter these fields overlap. Figure 3b suggests that these are independent on UT in fall and spring, but dependent on UT in summer and winter. Thus, there is an indication that the fields in all seasons do not exist at the same point in time. This issue solves Fig. 4b by clearly revealing that at each single HOY the field is either B s (B y,GSEQ < 0) or B s (B y,GSEQ > 0) regardless of the season. In this way we have shown that the overlap of the polarity fields in summer and winter seen in Fig. 3a is caused by their averaging on DOY-interval of 14-day which does not enable to resolve the UT dependence.
Adopting the assumption that B z,GSEQ equals zero to obtain incomplete fields, the expression (5) reduces to: That indicates that f 1 and f 2 are constants (equal to 1) in this case and that expression (6) turns to: RM model. B s fields both from the RM model and from the incomplete dataset are based on the same assumption that B z,GSEQ is zero. Because of that B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) predicted by the RM model (expression 2) have the same characteristics as incomplete polarity fields: they are mutually exclusive for each HOY. For the model to be consistent with the incomplete dataset this feature must be taken into account when calculating B s from the polarity fields. Thus, B s shall be calculated using expression (8), which represents the corrected RM model, instead of expression (4). This is clearly seen in Fig. 3a

Discussion
Results have shown that complete B s fields oscillate around the mean value (fin average) which is 5 times higher than the average about which the incomplete fields oscillate. We attribute that to the larger contribution of the second term in expression (1) to B s . For the examined period (1998-2017), the fin average of B y,GSEQ sorted by IMF polarity (± 3.40 nT) is greater than the fin average of B z,GSEQ sorted by IMF polarity (± 2.45 nT), which would indicate a larger contribution of B y,GSEQ component to B s fields. But, since IMF that is ordered in GSEQ contributes to B z,GSM in combination with α which is in the range ± 37 • , the second term in expression (1) www.nature.com/scientificreports/ indeed dominates. The importance of the second term has already been noticed by 27 and 28 and discussed in more detail in V.B. 2021. Further, we showed that the amplitude of the incomplete B s is too large compared to the complete B s . With the corrected relationship between the B s fields (expression 8), compared to the one proposed by RM, the absolute value of the incomplete field is matched. Nevertheless even this value, nor the one postulated by RM, is not in accordance with observations. Since for the incomplete observed B s fields each HOY in all seasons is characterized with one of the two polarities, it follows that at some HOY in all 20 years the polarity of the field is always the same. This further means that polarity in this idealized IMF situation is seasonal dependent. Further, the incomplete observed B s is obtained by merging both polarity fields (expression 8) rather than be an average of both (expression 3). This is caused by unreal polarity separation as explained above. In this context, in the present study we have made progress by deriving the general relationship between the fields (expression 6) which shows that B s can not be expressed as a simple average of B s ordered by IMF polarity, as assumed by RM. The analysis and conclusions are based on realistic situation in which for specific HOY, the fields can randomly be of any polarity (toward or away from the Sun). Therefore, by averaging over many years (here 20 years) both B s (B y,GSEQ < 0) and B s (B y,GSEQ > 0) can appear in favorable and in unfavorable seasons. The averaging retains the information about the existence of both field polarity.
Crooker and Siscoe 33 already in 1986 pointed out (on pages 209-210): "... although the polarity effect itself is an outstanding feature in data sets separated according to polarity, the net effect of mixed polarities makes only a small contribution to the semiannual variation. When a realistic distribution of the north-south component of the IMF is used in a model of the polarity effect, the annual variation of geomagnetic activity for a given polarity is not at nearly zero level for half of the year, as it would be for an idealized spiral IMF, but instead varies gradually in a sinusoidal-like way. Consequently, the net effect of these two annual variations of opposite phase is a semiannual wave of amplitude considerably smaller than that predicted on the basis of an idealized spiral IMF". These very advanced notices that are in line with results obtained in the present study have been unfortunately forgotten and not considered. In our view, probably because in the interpretation of semiannual, annual and diurnal variations of magnetospheric quantities, the observed B s fields have not been considered and the RM model based on idealized spiral IMF was adopted. Consequently, what has been shown by us here that the semiannual amplitude of realistic (observed) B s is small, and much smaller than the one from the RM model, along with confirmation that the pattern of polarity fields is the "pair of spectacles" pattern (two annual sinusoidal-like variations of opposite phase) should really be taken into account when B s is considered to be the causative agent of geomagnetic activity.
The present study has clearly shown why the RM model based on the idealized IMF cannot match the observed fields by providing the data set (incomplete data set) that this model accommodates. The discrepancy between the B s from complete and from incomplete data sets becomes especially noticeable in the cases when the polarities are considered separately.
Note that we did not analyze variations in any magnetospheric quantity. Nevertheless, based on the obtained results, in the following we provide some possible explanations why studies that considered different mechanisms responsible for variations in geomagnetic activity commented that the contribution of B s is small (e.g. 34 ). First, if noticed that B s has little influence on the semiannual variation it may not necessarily be because B s is not important in confront to other effects, but just because the amplitude of the semiannual variation of observed B s is low. Further, if one finds in magnetospheric quantity separated according to IMF polarity an enhancement in the favorable season and reduction in the unfavorable season, but not zero activity in unfavorable seasons, it could be a sign of the influence of polarity fields. Then, this indicates that B s fields do contribute to the variations seen in magnetospheric quantity. In particular when geomagnetic indices are sorted by IMF polarity, the impact of the complete polarity fields, and not the impact of incomplete ones which the RM model well described, becomes clearly evident. For instance 27,28 and 35 obtained enhancements in the favorable and reductions in unfavorable seasons (a pattern similar to the "pair of spectacles" pattern) when geomagnetic indices AL, am and AE, and am are ordered by the IMF polarity respectively. The obtained variations in these geomagnetic indices reveal the pattern of the complete B s fields shown in our Fig. 2a,c. Further, a recent study by 36 has shown how important it is to use the complete, observed pattern of B s as input when modeling geomagnetic indices Dst and Kp sorted by IMF polarity. This work used the information from coronal holes on the Sun that are of a strictly defined polarity. As the prior function (input for the model) they employed the sinusoidal function, the form of a realistic B s polarity field. In this way the seasonal variations in the geomagnetic activity were well reproduced. If as a prior function the patterns of incomplete B s polarity fields were chosen, then the method would not lead to meaningful results.
There are solar wind-magnetosphere coupling functions which are combinations of different measured interplanetary parameters (for details about different coupling functions the reader is referred to the study by 37 ). They are used to quantitatively predict magnetospheric activity. Most of them contain IMF orientation factor F(θ ) via sin(θ/2) on some exponents, where θ is the clock angle defined as tan(θ ) = |B y |/B z in GSM. These coupling functions sorted by the IMF polarity are not zero in unfavorable seasons. They exhibit enhancements and reductions within the year (e.g. see Figure 12b in 35 ), similar to the pattern of observed, complete B s polarity fields. The reason for that is that sin ( θ/2) allows stronger coupling during southward and weaker coupling during northward pointing IMF B z,GSM component. This indicates that much of the IMF dependence reflected in geomagnetic activity originates from the southward component of the IMF given in GSM, further confirming the importance to clarify the real, observed pattern of B s fields.
To summarize, in light of the obtained results and the above discussion, the observed B s fields are those that can contribute to the magnetospheric activity and not the incomplete fields. We note that discussion related to the imprint of B s fields in geomagnetic quantities does not rule out other parameters and mechanisms that besides B s can affect seasonal variations in geomagnetic activity.

Conclusion
Although recent studies have already provided evidence that the RM model of B s does not match the observations, the present work has explicitly proved that and has provided explanations. We have derived incomplete observed B s fields and have demonstrated that it is exactly this data set that the RM model can describe. Comparison of the B s fields obtained from the observed data set, incomplete observed data set and those predicted with the RM model allows us to explicitly deduce where the differences between the observations and model predictions come from. The results have confirmed that B z,GSEQ plays a significant role and in combination with angle α it becomes crucial to obtain B s fields as nature presents. In summary, the present study has resolved the issue related to the pattern and absolute value of the observed B s fields and those obtained with the RM model. The results have pointed out that it is very important to consider the pattern of observed B s fields when interpreting semiannual (annual) variations in magnetospheric quantities and moreover when modeling geomagnetic indices. Finally, it has shown that the new model of the B s fields which will take into account the fluctuation of IMF about the spiral direction, the most probable IMF orientation, and in that way be in accordance with observations is needed. This is the subject of our work in progress.

Data availability
The interplanetary magnetic field data analysed during the current study are available at https:// izw1. calte ch. edu/ ACE/ ASC/ level2/ lvl2D ATA_ MAG. html.