Asymmetry in the Earth’s magnetotail neutral sheet rotation due to IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document} sign?

Evidence suggests that a non-zero dawn–dusk interplanetary magnetic field (IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document}) can cause a rotation of the cross-tail current sheet/neutral sheet around its axis aligned with the Sun–Earth line in Earth’s magnetotail. We use Geotail, THEMIS and Cluster data to statistically investigate how the rotation of the neutral sheet depends on the sign and magnitude of IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document}. In our dataset, we find that in the tail range of -30<\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-30<$$\end{document} XGSM <-15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<-15$$\end{document}RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\mathrm{E}}$$\end{document}, the degree of the neutral sheet rotation is clearly smaller, there appears no significant rotation or even, the rotation is clearly to an unexpected direction for negative IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document}, compared to positive IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document}. Comparison to a model by Tsyganenko et al. (2015, doi:10.5194/angeo-33-1-2015) suggests that this asymmetry in the neutral sheet rotation between positive and negative IMF By\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_y$$\end{document} conditions is too large to be explained only by the currently known factors. The possible cause of the asymmetry remains unclear.


Introduction
The two magnetic hemispheres in the Earth's magnetotail are separated by a dawn-to-dusk-directed cross-tail current sheet. Within the current sheet, the boundary between the magnetic hemispheres is usually defined as a surface at which the X component (along the Sun-Earth line) of the tail magnetic field reverses ( B x = 0). It is called the neutral sheet (Ness 1965).
The position of the tail neutral sheet is often subjected to dynamical motion in the north-south Z direction termed as flapping (e.g., Speiser and Ness 1967;Lui et al. 1978;Sergeev et al. 1998Sergeev et al. , 2004Zhang et al. 2002Zhang et al. , 2005Gao et al. 2018). In general, the position of the neutral sheet is affected by three major causes: hinging, warping and twisting (or rotation) (Tsyganenko et al. 1998;Tsyganenko and Fairfield 2004;Tsyganenko et al. 2015;Xiao et al. 2016, and references therein). The hinging effect shifts the neutral sheet northward and southward with respect to the Sun-Earth line when the geomagnetic dipole tilt angle is positive (generally northern hemisphere summer) and negative (northern hemisphere winter), respectively. The warping effect is also dependent on the dipole tilt angle: for positive dipole tilt angle, the warped neutral sheet appears in the tail cross-sectional plane as a southward opening curve (see e.g., Tsyganenko et al. 2015, their Figure 9). For negative dipole tilt angle, the warped neutral sheet opens northward. The twisting or rotation effect depends strongly on the dawn-dusk component of the interplanetary magnetic field (IMF B y ). For duskward, IMF B y > 0 , the neutral sheet is rotated counter-clockwise in the tail cross-sectional plane when looking from Earth toward the tail. For dawnward, IMF B y < 0 , the rotation is clockwise.
In one scenario, the rotation of the neutral sheet is caused by the following sequence of events: a non-zero IMF B y drives an asymmetric magnetic reconnection on the dayside magnetopause. The magnetic tension force deflects the newly opened field lines (southward IMF) or the field lines remaining open (northward IMF) to opposite dawn-dusk directions in the two hemispheres, which leads to an asymmetric accumulation of magnetic flux in the tail lobes (e.g., Tenfjord et al. 2015;Tenfjord et al. 2018). This generates an excess of magnetic pressure on the dawnside of one and duskside of the other tail lobe so that a torque is exerted on the plasma sheet. The magnetotail then reaches a new equilibrium state by rotating the plasma sheet as well as the neutral sheet. Also, it has been suggested that the tangential stress on the tail magnetopause by the deflecting open field lines exerts a torque on the lobes, which can subsequently lead to the rotation of the neutral sheet (Cowley 1981).
Several statistical studies suggest that the neutral sheet rotation increases with tailward distance from Earth as well as with increasing IMF B y magnitude, and is generally stronger during northward compared to during southward IMF ( Tsyganenko et al. 1998;Tsyganenko and Fairfield 2004;Tsyganenko et al. 2015). The same IMF B y and B z dependence of the neutral sheet rotation appears also in MHD simulations (e.g., Kullen and Janhunen 2004, and references therein).
While semi-empirical neutral sheet models (e.g., Tsyganenko et al. 2015) assume that the degree of the rotation is independent of the sign of IMF B y , the statistical results by Kaymaz et al. (1994) and Owen et al. (1995) suggest a possibility that this would not be the case, although this is not explicitly discussed in their respective papers. Using IMP-8 magnetic field measurements in the range of −40 < XGSM < −25 R E (GSM, geocentric solar magnetospheric), Kaymaz et al. (1994) statistically inferred that the degree of the neutral sheet rotation in their dataset is slightly weaker for negative IMF B y than for positive IMF B y . Notably, the results by Kaymaz et al. (1994) showed signatures of nonlinear rotation, which weakens the inference of the rotation angle. Owen et al. (1995) investigated statistically the orientation of the edge of the plasma sheet boundary layer (PSBL) using particle measurements by ISEE-3 satellite from a large range of distances, between -240 and 0 R E XGSM. They found that the average PSBL edge tilt angle was smaller for negative than for positive IMF B y . Owen et al. (1995) concluded that the tail is generally more twisted around its axis for positive IMF B y . However, their results show very large scattering in the individual PSBL edge tilt angles and the method gives only indirect evidence of the neutral sheet rotation. Xiao et al. (2016), on the other hand, studied the average shape and position of the tail neutral sheet using data from multiple missions. They inferred that at low dipole tilt angles (absolute value less than 5 • ), the degree of the neutral sheet rotation was comparable for positive and negative IMF B y , when the range of the IMF B y magnitude was from 3 to 8 nT. Precisely taken, a slightly larger rotation was inferred for negative IMF B y . When separating the northward and southward IMF, the difference between positive and negative IMF B y in the rotation became clearer, but the characteristics of larger rotation for negative IMF B y remained in both cases.
In this letter, we investigate possible asymmetric responses in the tail neutral sheet rotation to different IMF directions. In the analysis approach, which is based on the identification of the measured neutral sheet crossings, we make use of the neutral sheet model by Tsyganenko et al. (2015). A major advance compared to the study by Kaymaz et al. (1994) is that we distinguish positive and negative IMF B z conditions. In addition, we distinguish small and large IMF B y magnitudes, which has been done neither by Kaymaz et al. (1994) nor Xiao et al. (2016).

Data
We use magnetotail data from the Geotail, Time History of Events and Macroscale Interactions during Substorms (THEMIS) and Cluster missions. Geotail data consist of the spin-averaged (3 s) magnetic field measurements from the MaGnetic Field experiment (MGF) (Kokubun et al. 1994) and the 12-s ion moments from the Low Energy Particle experiment (LEP) (Mukai et al. 1994)  For the solar wind data, we use the 1-min OMNI data propagated to the nominal bow shock nose (http:// omniw eb.gsfc.nasa.gov/) (King and Papitashvili 2005). In addition, we use 1-min SYM-H geomagnetic index data provided by the World Data Center for Geomagnetism, Kyoto, Japan, which are also retrieved through the OMNI database. The geocentric solar magnetospheric (GSM) coordinate system is used for all spacecraft data throughout the study. Pitkänen et al. Geosci. Lett. (2021)

Tsyganenko et al. (2015) neutral sheet model
In this study, we have utilized the semi-empirical neutral sheet model by Tsyganenko et al. (2015), hereafter denoted as the T15 neutral sheet model. Here, we briefly present the main features of the model. For a detailed description, the reader is referred to Tsyganenko et al. (2015).
The T15 model describes the global shape of the unperturbed magnetospheric equatorial current sheet (neutral sheet) at all local times as a function of geomagnetic dipole tilt angle ( ), solar wind dynamic pressure ( P dyn ), IMF B y and IMF B z . It has been derived from a vast data pool of magnetic field measurements from Polar, Cluster, Geotail and THEMIS missions between 1995 and 2013.
In the solar magnetic (SM) cylindrical coordinates, the position of the model neutral sheet in the Z axis, Z s , is determined by the form The first two terms describe the hinging and warping and the last third term determines the rotation of the neutral sheet. In the formula R H is the hinging distance, which depends on P dyn , IMF B z and the longitude φ for which tan φ = Y /X . is the dipole tilt angle, ρ = √ X 2 + Y 2 , a 0 and a 1 are Fourier coefficients, which depend on P dyn and IMF B z . The magnitude coefficient T depends on P dyn and the power index α depends on φ , P dyn and IMF B z . The power index β depends only on IMF B z . B y0 = 5 nT and ρ 0 = 10R E are numerical scaling factors.
The source code (Fortran) for the T15 neutral sheet model with input and output in GSM coordinates is provided at http://geo.phys.spbu.ru/~tsyganenko/modeling. html.

Methods
We have statistically investigated the rotation of the neutral sheet. Our approach is based on the identification of the neutral sheet positions and an investigation of their distribution. To identify the positions of the neutral sheet crossings in the magnetotail plasma sheet from the data, we have utilized the following procedure. First, the magnetospheric data were averaged to a 5-min time resolution. Small data gaps were allowed so that > 80% (= 4 min) of data must be available. For the data point to be included in the database, we required average plasma ion β ≥ 0.1 for the 5-min interval. The threshold for ion β , which is the ratio between ion thermal pressure and magnetic pressure, was used to exclude measurements from the low-β lobes. Then, we identified the neutral sheet (1) crossings in the tail region −30 < XGSM < −15R E and −25 < YGSM < +25R E as reversals in the magnetic field B x component in the 5-min data. The tail range of −30 < XGSM < −15R E was chosen to only include clearly taillike distances and it is limited in the far side by the apogees of the satellite trajectories. A crossing was accepted to the neutral sheet crossing dataset if the following conditions were fulfilled: (1) the ion speed V ⊥xy = V 2 ⊥x + V 2 ⊥y < 100 km/s in the XY plane for the 5-min data samples right before and after the crossing. This is expected to efficiently remove highspeed flows, which can perturb the neutral sheet, and tailward magnetosheath flows in the magnetotail flanks from the dataset. (2) The IMF magnitude B < 10 nT, the solar wind dynamic pressure 0.1 ≤ P dyn ≤ 10 nPa and −100 ≤ SYM-H ≤ 30 nT (Tsyganenko et al. 2015). Thus, the most extreme solar wind and magnetospheric conditions were excluded. IMF B was computed as a 40-min average over a time interval from 35 min prior to the neutral sheet crossing until 5 min after the crossing. This the same averaging window used by Tsyganenko et al. (2015). Notably, Case et al. (2018) suggest a time scale of 10-20 min for the response of the neutral sheet to the IMF B y reversals, which is within the averaging window of Tsyganenko et al. (2015) and the present study. P dyn and SYM-H were computed as averages over the 10-min period centered at the crossing. Tsyganenko et al. (2015) used a 5-min average and the instantaneous 1-min value for P dyn and SYM-H, respectively. We argue that 10-min averages describe better the general conditions around the crossing times. In the computation of IMF B and P dyn , total missing data up to 30% in the averaging time intervals were allowed. (3) The measured neutral sheet crossing position differs no more than 4 R E in the ZGSM direction from the model neutral sheet by Tsyganenko et al. (2015). This removes outliers caused by the most extreme neutral sheet perturbations from the dataset in the same way as has been done in the study by Tsyganenko et al. (2015).
In total, 2963 neutral sheet crossings were obtained for the dataset. Using the T15 neutral sheet model, we removed the contributions of the hinging and warping to the position for each neutral sheet crossing in this dataset. This was done by computing a new neutral sheet ZGSM position using the T15 model (equation (1)) without the rotation term, i.e., without the last term of equation (1), and subtracting the result from the observed ZGSM position for each data point of the dataset. As a result, one gets an estimate for the neutral sheet ZGSM position caused by the rotation effect only (referred to as dataset O). When using the T15 neutral sheet model in this study, we assume that it represents the nominal neutral sheet position in the absence of perturbations, such as north-south flappings. As input for the T15 model, IMF and P dyn values were used that are averaged in the same way as described above. For comparison, we also computed the T15 model neutral sheet ZGSM positions caused by the rotation effect only. This was done by subtracting the T15 model neutral sheet ZGSM position computed without rotation from the full T15 model, which is equivalent to computing the ZGSM position with the T15 rotation term only (referred to as dataset M). In the computation, the XGSM and YGSM positions and the IMF and P dyn that are associated with the actual measured neutral sheet crossings were used.
We sorted the neutral sheet crossings in eight different categories: those observed during northward and southward IMF conditions and further, those with positive and negative IMF B y with large ( > 3 nT) and small ( 0 < IMF B y < 3 nT) magnitudes. Then we investigated the degree of the rotation in each of the categories in the two datasets M (model) and O (observations). The locations of the neutral sheet crossings in each category projected to the GSM XY plane are shown in Fig. 1. The relative numbers of the removed outliers in each IMF category varied between 1 and 9% of the numbers of the included neutral sheet crossings. Figure 2 displays the results for the northward IMF conditions (IMF B z > 0 ), for dataset M (T15 model) on the left panels and for dataset O (Data) on the right panels. Shown are the positions of the neutral sheet crossings in the GSM YZ plane for each IMF category (when looking from Earth tailward). In each category, we have fitted a line of best fit of linear least squares sense to the data points to statistically estimate the neutral sheet rotation. The rotation is measured by the angle α between the YGSM axis and the regression line and it is increasing anti-clockwise from the positive YGSM axis. The angle α is computed as a gradient of the best-fitted line. The given error estimate for α is the standard error. Note that in the T15 model, the curve describing the neutral sheet in each tail cross-sectional plane due to the rotation, is precisely taken nonlinear. This is because the rotation term in the T15 model is dependent on the sinus of the longitude φ , i.e., the rotation in one tail cross-section increases towards the flanks (see equation (1) and Tsyganenko et al. 2015, their Figure 10). A linear fit is, however, sufficient for our purpose, since we are interested in quantifying the degree of the rotation at general level. By intercomparing the model results (dataset M, Fig. 2a,c,e,g), we see that the degree of the rotation ( |α| ) is clearly a few degrees larger for large than for small IMF B y magnitudes. This is expected, as the rotation angle in the T15 model depends, among other parameters, on the magnitude of IMF B y . The magnitude of the angle α is also slightly larger for positive IMF B y compared to negative IMF B y . But in general, the degrees of the rotation are comparable between the two IMF B y signs. Since the T15 neutral sheet model does not distinguish between positive and negative IMF B y in the degree of the rotation, this means that any differences seen in the model dataset M for positive and negative IMF B y can only come from the differences in the distributions of the model input parameters and the XY positions of the neutral sheet crossings.

Results
The magnitude of the rotation angle increases with the distance in the X direction in the T15 neutral sheet model (Tsyganenko et al. 2015). However, the scatter of the modeled data points from the regression line in each category are small, as also indicated by the high r-squared values. Therefore, we conclude that the distributions of the crossings in XGSM (Fig. 1) do not affect the determination of the rotation angle.
When looking at the observations (dataset O, Fig. 2b, d, f, h), the scattering of the data points with respect to the regression line is significantly larger if compared to the model. This is understandable, because other effects such as flapping contribute to the measured positions of the neutral sheet. In the model, these effects are absent. Generally, the magnitudes of the rotation angles are smaller for the data than for the model, except for low-magnitude positive IMF B y (Fig. 2f ).
The striking difference between the data (dataset O) and the model (dataset M) is the strong rotation asymmetry between positive and negative IMF B y in dataset O. For the model (dataset M), |α| for positive and negative IMF B y are comparable, both in the case of large ( Fig. 2a and c) and small IMF B y magnitudes ( Fig. 2e and g). For the data (dataset O), for large IMF B y magnitudes ( Fig. 2b and d), |α| is clearly smaller for negative IMF B y , 1.80 ± 0.39 • , compared to 5.48 ± 0.30 • for positive IMF B y . For small IMF B y magnitudes, a clear rotation can be deduced for positive IMF B y ( 4.17 ± 0.25 • , Fig. 2f ), but no significant rotation for negative IMF B y ( 0.25 ± 0.29 • , Fig. 2h). In fact, in the latter case the sense of the rotation is in an unexpected direction ( α > 0 ), but the rotation angle stays within the error limits. Figure 3 displays the results for the southward IMF (IMF B z < 0 ). From the model results (dataset M), we see that similar to the northward IMF, the degree of the rotation is a couple of degrees larger for large than for small IMF B y magnitudes (Fig. 3a,c,e,g). For large IMF B y magnitudes, |α| is almost the same for positive and negative IMF B y (Fig. 3a and c). For small IMF B y magnitudes, |α| is slightly larger for positive IMF B y but still comparable ( Fig. 3e and g).
When looking at the observations (Fig. 3b, d, f, h), we again observe large differences in |α| between positive and negative IMF B y as in the case of the northward IMF. At large IMF B y magnitudes, a clear rotation can be seen for positive IMF B y ( 2.85 ± 0.30 • , Fig. 3b), but no significant rotation for negative IMF B y ( 0.27 ± 0.39 • , Fig. 3d). In the latter case, the rotation angle is slightly positive, i.e., in the unexpected direction, but within the error limits. At small IMF B y magnitudes, similarly, a clear rotation appears for positive IMF B y ( 2.68 ± 0.21 • , Fig. 3f ) (which notably has larger α than the model, Fig. 3e). But contrary to what are expected, the rotation angle for negative IMF B y is clearly opposite, positive ( 1.26 ± 0.28 • , Fig. 3h). Generally, the degree of the neutral sheet rotation is larger for northward than southward IMF conditions, which is consistent with previous studies (e.g., Owen et al. 1995;Tsyganenko et al. 2015).

Discussion
The observational results (dataset O) presented in Figs. 2 and 3 indicate differences in the degree of the neutral sheet rotation between positive and negative IMF B y conditions. Comparison to the model (dataset M) implies that these differences are so large that they cannot be explained by the distributions of model input parameters and spatial XY positions of the neutral sheet crossings.
The tail neutral sheet crossings are generally measured when the neutral sheet moves fast, i.e., flaps in the northsouth direction and passes the satellite position rather than the satellite moves across a static neutral sheet. From the large scattering of the crossings in dataset O, it is clear that the flapping affects the estimates of the rotation angle. However, it has been deduced that the typical amplitudes of the flapping motion are on the order of 1-2 R E (Sergeev et al. 2006). Because we obtain a clear positive rotation angle α for positive IMF B y in all categories, we argue that the scattering caused by the flapping cannot explain (i) the small magnitude of negative α in Fig. 2d, (ii) the very small opposite rotation in Figs. 2h and 3d, and (iii) the clear opposite rotation direction in Fig. 3h, for negative IMF B y (the rotation is counterclockwise as for positive IMF B y while clockwise rotation is expected for negative IMF B y ). It might explain the difference, if flapping would be more significant for negative IMF B y , but this is fully speculative.
Another cause for the scattering could arise from the response of the neutral sheet to IMF B y reversals (Case et al. 2018). Assume that a neutral sheet crossing is detected in the dusk and ZGSM > 0 quadrant in the YZ plane when IMF B y > 0 has been prevailing for a longer time. The crossing is detected either as the neutral sheet flaps or the positive rotation increases. The crossing is thus observed in the expected quadrant in the corresponding IMF B y > 0 category (the expected quadrant is the quadrant one expects the neutral sheet to rotate into, based on the prevailing IMF B y conditions). Then IMF B y suddenly reverses to IMF B y < 0, and a second crossing is detected if the neutral sheet responds to the IMF B y reversal and rotates to a new (opposite) angle corresponding to the IMF B y < 0 condition. If this would happen, this second crossing would still be observed in the quadrant, which is in accordance with the IMF B y > 0 condition (dusk and ZGSM > 0) because the satellite position practically remains unchanged. If the separation of the crossing times were long enough, the averaged IMF B y value would be negative and the crossing would appear in the unexpected quadrant for IMF B y < 0. This would subsequently increase the scattering in the corresponding IMF B y < 0 category.
We have investigated the possible influence of IMF B y reversals occurring prior to the crossings on the results. We computed ∼ 1-h average (60 min prior to until 5 min after a crossing) of IMF and compared this 1-h IMF B y direction to the assigned (35 min prior to until 5 min after a crossing) IMF B y direction. One can assume that if these are collinear, no significant reversal took place. We find that for the categories of large IMF B y magnitudes, there appear only 0-2 neutral sheet crossings in each category for which the 1-h IMF B y direction is opposite to the assigned IMF B y direction. Thus, IMF B y reversals do not affect the results in these categories. In the case of small magnitude IMF B y categories, the relative number of the IMF B y reversal crossings vary from 5 to 12% and the effects to the neutral sheet rotation angle are small, 4-5% , except for northward IMF B z and −3 < IMF B y < 0 nT-category (Fig. 2h). While in this category, the contribution of the neutral sheet crossings associated with the IMF B y reversal leads to a relatively high increase of the rotation angle to the unexpected direction, in percentage ∼733%, the absolute increase in the angle is only 0.22 • (from 0.03 • to 0.25 • ). All the deviations in the neutral sheet rotation angle caused by the IMF B y reversals are within the error limits. Therefore, we argue that IMF B y reversals cannot explain the large differences in the degree of the rotation.
The distributions of the IMF B y magnitudes inside each category could also affect the results. If the distribution of the IMF B y magnitudes would be biased  et al. Geosci. Lett. (2021) 8:3 toward larger magnitudes for positive and toward smaller magnitudes for negative IMF B y , that might explain at least partly the larger degree of the rotation for positive IMF B y . However, by comparing the distributions for positive and negative IMF B y , we do not find any significant differences between them and thus exclude that as the potential cause. The mean and median of IMF B y (and IMF B z ) in each IMF category are shown in Table 1.
The observational results that the degree of the neutral sheet rotation is larger for positive IMF B y are in accordance with the results by Kaymaz et al. (1994). Notably, Kaymaz et al. (1994) found in their dataset a quite clear rotation also for negative IMF B y , whereas in our dataset the rotation is clear and in the expected direction for negative IMF B y only for northward IMF and IMF B y < −3 nT (Fig. 2d). In their dataset, Kaymaz et al. (1994) used only data for which IMF |B y | > |B z | and did not distinguish between northward and southward IMF. We have checked subsets of data using the same condition, but the differences in the results are relatively small. The rotation angles (without error estimates, which have the similar magnitudes as in Figs. 2 and 3 Owen et al. (1995) found that the average PSBL edge tilt angle was smaller for negative than positive IMF B y . Owen et al. (1995) further distinguished northward and southward IMF B z and found that the magnitudes of the PSBL tilt angles were clearly larger for northward than southward IMF. If one assumes that the PSBL tilt angle reflects the rotation of the neutral sheet, our results agree with this aspect with the results by Owen et al. (1995). Also, the T15 neutral sheet model gives larger rotation for northward than for southward IMF, but it does not distinguish between positive and negative IMF B y (Tsyganenko et al. 2015).
The results that the rotation of the neutral sheet is weaker for negative than for positive IMF B y raises a question of a possible intrinsic degree of the rotation when IMF B y = 0 . In a hypothetical scenario, the tail neutral sheet is always slightly rotated with a small positive α under zero IMF B y condition. Therefore, one would need a small negative IMF B y to rotate α to zero. However, the results by Xiao et al. (2016) do not support this scenario. Xiao et al. (2016) also studied the neutral sheet rotation at low dipole tilt angles for −1 < IMF B y < 1 nT, so approximately for zero IMF B y . They found no indication of intrinsic rotation. Furthermore, Xiao et al. (2016) found a slightly larger rotation for clearly negative IMF B y ( −8 < IMF B y < −3 nT) than for clearly positive IMF B y ( 3 < IMF B y < 8 nT). This is not in accordance with the results of the present study and with the results by Kaymaz et al. (1994) and Owen et al. (1995).
The differences in the results, specifically between the present study (larger rotation for positive IMF B y ) and the study by Xiao et al. (2016) (larger rotation for negative IMF B y ) are difficult to explain. The two studies use data from approximately similar time intervals for almost the same satellites, although Xiao et al. (2016) are using more data with one additional mission, TC-1 .
The difference could arise from the different methods used. In our method, the individual neutral sheet crossings are identified and the T15 neutral sheet model is used to remove the hinging and warping effects from the neutral sheet positions. Then a line is fitted to the datapoints in the YZ plane to extract the rotation. In the Xiao et al. (2016) method, tail B x measurements are binned to squares in the YZ plane and the majority of the B x measurement samples in a bin determines whether the bin is assigned by positive or negative B x bin. A line is then fitted to the position points that separate the positive and negative B x regions to get the rotation angle. This is done for the data with low dipole tilt angles (absolute value less than 5 • ) to minimize the hinging and warping. We have checked the neutral sheet rotation in each IMF category for a subset of our dataset using the same range for the low dipole tilt angle as Xiao et al. (2016). While the number of neutral sheet crossings in each category are much smaller indicating weaker statistics, the asymmetry between the positive and negative IMF B y remains (data not shown).
Our method is likely to have larger scattering of the datapoints that construct the fit, because all the datapoints in the particular IMF category are included in the fit. While not explicitly stated by Xiao et al. (2016), we assume that in the method by Xiao et al. (2016), only the position data assigned by white color (in their Figures 10  and 11) are included in the fit. This means that for each spatial Y bin, only one position is taken to the fitting process, and the scattering of these datapoints is generally much smaller. However, generally, the scattering from the fitted line in our method does not vary significantly between the IMF B y categories (Figs. 2 and 3).
The difference in the degree of the neutral sheet rotation could be caused by differences in the asymmetric accumulation of magnetic flux into the tail lobes in case the accumulation would not be an exact mirror process for positive and negative IMF B y . It is also possible that there exists another still unidentified mechanism that is more efficient under one IMF B y direction. These questions require further investigation. As a future work, the influence of IMF B y on the neutral sheet rotation should be studied using numerical global magnetosphere models. It should be tested whether the present models produce the difference in the degree of the rotation between positive and negative IMF B y , and if yes, then investigated the physical process(es) behind the asymmetry.

Summary
We have used Geotail, THEMIS and Cluster data from 1995 to 2009 to statistically investigate the rotation of the neutral sheet under the influence of non-zero IMF B y in the Earth's magnetotail. With help of T15 neutral sheet model (Tsyganenko et al. 2015), we find in the tail range of −30 < XGSM < −15 R E , the degree of the neutral sheet rotation is clearly smaller, there appears no significant rotation or the rotation is even clearly in an opposite direction to what is expected for negative IMF B y compared to positive IMF B y . For positive IMF B y , the inferred rotation angle varies between 2.68 • and 5.48 • and for negative IMF B y , the rotation angle gets values from −1.80 • to 1.26 • . A comparison to the T15 model suggests that this asymmetry in the neutral sheet rotation between positive and negative IMF B y conditions is too large to be explained only by an uneven distribution of the neutral sheet crossings or other solar wind conditions at the observed neutral sheet crossings, which have been deduced to contribute to the position of the neutral sheet.
The possible physical mechanism remains unclear. The discrepancy between the results from different studies indicate that more research is needed to understand the IMF B y influence on the rotation of the neutral sheet. Different approaches are desired to find out all related aspects. While numerical modeling, such as global magnetosphere models, can be used in investigations, they cannot replace observational studies. Based on the results of the present study, we suggest that in the future, magnetospheric models such as semi-empirical neutral sheet models, should be parameterized so that asymmetric effects due to the IMF B y direction are allowed for.
The asymmetries related to the IMF B y sign are not limited to the neutral sheet rotation. Recent studies have found IMF B y sign-related asymmetries for instance in the high-latitude geomagnetic activity (Holappa and Mursula 2018) and in the polar cap size (Reistad et al. 2020).