A normalized statistical study of Earth's cusp region based on nine‐years of Cluster measurements

Nine years (2001–2009) of data from the Cluster spacecraft are analyzed in this study of the Earth's mid‐ and high‐altitude (2–9R E) cusp. Properties of the cusp region, and its location and size in the Solar Magnetic coordinate system, are studied statistically. The survey shows that (1) the relationships between X and Z are nearly linear for the poleward, equatorward boundaries and the center of the cusp; (2) the relationship between cusp width in the X direction and Z can be expressed by a quadratic function; (3) the cusp region is almost dawn‐dusk symmetric for the cusp width in the X direction. Based on topology information, a new normalized statistical methodology is developed to organize the measurements of cusp crossings to obtain distributions of magnetic field and plasma parameters in the XZ plane. The statistical results show that (1) Bx is mostly negative and Bz is always negative; (2) proton velocity is found to be positive for Vx and Vz at low altitudes, while Vx and Vz are negative on the equator side and negative Vx and positive Vz on the pole side at high altitudes; (3) proton density is higher on the equator side than on the pole side. Results reported here will be useful in suggesting directions for future cusp research.


Introduction
first discussed the Earth's magnetospheric cusp, through which the solar wind plasma can directly enter the magnetosphere and the ionosphere without crossing geomagnetic field lines. Heikkila (1985) described vividly the shape of the magnetospheric cusp region as a narrow funnel. The magnetospheric cusp is a key region for studies of flux transfer events (Russell and Elphic, 1979), the signatures of magnetic reconnection at the magnetopause (e.g., Dungey, 1961;Lockwood and Smith, 1994;Cao D et al., 2017;Peng FZ et al., 2017), and heavy ion outflow from ionosphere (Yu YQ and Ridley, 2013). Heikkila and Winningham (1971) used observations of ISIS 1 spacecraft to observe the polar cusps for the first time. The plasma mantle (Rosenbauer et al., 1975) and the entry layer  are, respectively, the poleward and equatorward regions of the cusp. The plasma mantle, a loss channel for the plasma of the cusp, is formed by reconnected field lines (e.g., Sckopke et al., 1976;Sckopke and Paschmann, 1978): the entry layer is an important structure at which magnetosheath plasma can enter the closed magnetic field lines through diffusion processes (e.g., Haerendel and Paschmann, 1982), turbulence (e.g., Haerendel et al., 1978), and magnetic reconnection (e.g., Shi QQ et al., 2009a).
Because of the importance of the magnetospheric cusp, many studies have been extensively carried out, for example, on the interaction between the cusp region and interplanetary magnetic fields (IMF) (e.g., Chen JA and Fritz, 1998;Savin et al., 1998Savin et al., , 2002Savin et al., , 2004Le et al., 2001;Nykyri et al., 2004Nykyri et al., , 2006Nykyri et al., , 2011Zhang H et al., 2005;Shi QQ et al., 2009a;Liu WL et al., 2016;Xiao C et al., 2018), on the topology of the cusp (e.g., Zong Q-G et al., 2004Zong Q-G et al., , 2006Cao JB et al., 2005;Dunlop et al., 2005;Zhang H et al., 2006Zhang H et al., , 2007Shi QQ et al., 2009b;Shen C et al., 2011), on dynamic processes of the cusp (e.g., Chen JS et al., 2005;Duan SP et al., 2006;Yu YQ and Ridley, 2009;Wang TY et al., 2014) and on the trajectories of particles in the cusp region and their connections with the IMF and the equatorial magnetosphere (e.g., Sheldon et al., 1998;Delcourt and Sauvaud, 1999;Zhou XZ et al., 2006). Several studies have used data from different instrumentations aboard different spacecraft to determine the location, size, and properties of the magnetospheric cusp at different altitudes (e.g., Lockwood and Smith, 1992;Russell et al., 1998;Savin et al., 1998;Lavraud et al., 2002). For the location of the cusp, the dipole tilt has an important impact on the magnetic latitude of the cusp (e.g., Zhou XW et al., 1999;Palmroth et al., 2001) and the magnetic latitude of the cusp becomes more sensitive to dipole tilt as the altitude increases (Mĕrka et al., 2002). The magnetic latitude of the cusp is related to the southward IMF B z (Newell and Meng, 1987;Escoubet and Bosqued, 1989;Zhou XW et al., 2000;Palmroth et al., 2001;Pitout et al., 2006;Prölss, 2006;Niehof et al., 2010;Johnsen and Lorentzen, 2012). Previous studies have also showed that the size of the cusp is determined primarily by altitude, ranging from a few hundred kilometers at ~835 km (Newell and Meng, 1994) to several Earth radii at ~8R E (Chen JS et al., 2005). The solar wind is considered to be another important factor affecting the size of the cusp (Zhou XW et al., 2000;Pitout et al., 2006). For plasma and magnetic properties of the cusp, some statistical studies have been carried out, using data from Cluster. For example, Lavraud et al. (2004Lavraud et al. ( , 2005, using a sampling method first introduced by Grigoriev et al. (1999), have studied statistically the spatial distributions of the magnetic field and plasma parameters in the high-latitude cusp region. Walsh and Fritz (2011) used the same method to perform a seven-year statistical study of the energetic electron population in the high-altitude cusp and presented energetic electron populations consistent with local acceleration.
From the previously mentioned studies, we have learned that the cusp is a highly disturbed region in terms of position and size, varying with solar wind and IMF conditions, which leads to difficulty in the statistics studies when the determination of cusp boundary is necessary. Use of long-term observations from the Cluster mission allows us to propose a new normalized method for statistical description of cusp properties. We first study the topology of the cusp at mid-and high-altitudes (from ~2R E to ~9R E ), and, by organizing observations in a normalized grids of cusp region, subsequently use these findings to develop a new methodology for study of the statistical distributions of magnetic field and plasma parameters. The remainder of this paper is organized as follows. Section 2 describes typical cusp crossings and provides criteria for determining the cusp region. Section 3 presents statistical results on the location, size, and (the plasma and magnetic) properties of the cusp. Section 4 is devoted to discussion and conclusions.

Example of a Cusp Crossing Event
A typical cusp crossing event observed by Cluster on 8 Sep 2002 is shown in this section, helping us to understand the signatures of a cusp crossing. Figures 1a and 1b illustrate the Cluster-3 (C3-Samba) spacecraft position in Geocentric Solar Magnetospheric (GSM) coordinates; the GSM X-axis is from the Earth to the Sun; its positive Z-axis is chosen to be in the same sense as the Earth's northern magnetic pole; its Y-axis is defined to be perpendicular to the Earth's magnetic dipole, so that the XZ plane contains the dipole axis. We use the magnetic field lines calculated by the Tsyganenko and Stern (1996) model. The satellite moves from dayside to nightside magnetosphere with a possible cusp crossing. Overview of magnetic field and plasma parameters recorded by C3 from 05:00 UT to 07:00 UT are shown in Figures 1c-1g. From the three components of the magnetic field shown in Figure 1c, in GSM coordinates, there is no obvious magnetic field change associated with cusp crossing in this event. However, the cusp region can be distinguished clearly in ion measurements from the Hot Ion Analyzer (HIA) of the Cluster Ion Spectrometry (CIS) instrument (Rème et al., 2001), as shown, respectively, in Figure 1d for the spectra of ion omni-directional energy fluxes, in Figure 1e for ion density, in Figure 1f for ion bulk flow velocity in GSM coordinates, and in Figure 1g for ion temperature. From Figure 1 we can distinguish that, before 06:05 UT, the spacecraft was in the dayside magnetosphere, which is characterized by strong magnetic field, low ion density and velocity, and high ion temperature. At about 06:05 UT, the satellite entered into the cusp region: the ion energy spectrogram shows a broadband signature, ion number density is increasing rapidly from ~0.4 cm −3 to ~20 cm −3 , and ion temperature is decreasing from 30-35 MK to 3-10 MK. The Cluster satellite remained in the cusp region until it entered the dayside magnetosphere at around 06:36 UT: the ion density begins to fall from ~20 cm −3 to the low densities (~0.2 cm −3 ) and the ion energy spectrogram becomes a narrowband signature, the ion temperature descending to 1−3 MK. The signatures of the cusp region recorded in the ion energy spectrogram, ion number density, and ion temperature, as revealed in this event, can be used to identify a cusp crossing and to study the characteristic of the cusp region, as discussed below in the following sections.

Statistical Study
In this study, we analyze Cluster data collected from 2001 to 2009 in the region of −1 < X < 10R E and |Y| < 8R E in the Solar Magnetic coordinate system (SM): the positive SM Z-axis is chosen parallel to the Earth's north magnetic pole; its Y-axis is perpendicular to the Earth-Sun line towards dusk. We manually selected 470 cusp crossing events observed from the Cluster-3 spacecraft in the northern hemisphere, based on the crossing signatures described in the previous section. Histograms of the distribution of the number of events are plotted for year and month in Figures 2a and 2b, respectively. Due to the precession of the Cluster trajectory, most of the crossings were observed in the spring and fall seasons.

Statistical Study of Cusp Boundaries
The locations of the two cusp boundaries are investigated statistically for poleward boundary (PoB) and equatorward boundary (EqB). The SM coordinate system is used to eliminate the impact of dipole tilt angle on cusp boundary locations Zhou XW et al., 1999;Palmroth et al., 2001;Mĕrka et al., 2002). Figures 2c-2e show projections of the positions of the two boundaries: onto the XZ plane (Figure 2c), onto the YZ plane (Figure 2d), and onto the XY plane ( Figure 2e); the red/blue dots represent PoB/EqB, respectively. Empirical relations between X and Z are fitted from the observations as X = 0.525Z − 1.129 (cc ≈ 0.72) for PoB and X = 0.88Z − 1.408 (cc ≈ 0.87) for EqB, as shown, respectively, by the red and blue lines in Figure 2c. Subsequently, the center of the cusp in the XZ plane can be described as X = 0.731Z − 1.43 (cc ≈ 0.84), as shown by the black line in Figure 2c. It should be noted that PoB or EqB is controlled by solar wind (Zhou XW et al., 2000;Pitout et al., 2006) and IMF conditions (Escoubet and Bosqued, 1989;Zhou XW et al., 2000;Palmroth et al., 2001;Pitout et al., 2006;Niehof et al., 2010;Johnsen and Lorentzen, 2012); however in this paper we calculate only the average locations of the two boundaries while leaving their solar wind and IMF dependence for future study. Figures  2d-2e show that the scale of the cusp in the Y direction increases nearly linearly with increasing X (Z), making the cusp region look intuitively like a funnel. Figures 2f and 2g display the corresponding MLT (Magnetic Local Time) distribution of PoB and EqB, respectively, showing that both boundaries are located mainly between 08:00 and 16:00 MLT.
The widths of all individual cusp crossings are investigated. Previous studies have shown that the width of the cusp increases with increasing X and Z, resulting in a funnel-like shape: the width of the cusp may be only a few hundred kilometers in the ionosphere (Newell and Meng, 1994) and increases to several Earth radii at high-altitudes (Chen JS et al., 2005). In this study, we define the where (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ).
Scatter plots of cusp widths are shown in Figure 3 for the 470 cusp crossing events. Figure 3a shows the scatter plot of W x as a function of Z. It is clear that W x increases with increasing Z. This relationship can be fitted by a quadratic function as W x = 0.43 − 0.17Z + 0.064Z 2 , with correlation coefficient of cc1 = 0.729 calculated for the data points and cc2 = 0.992 for bin averaged values. The scatter plot of W x and R (the radial distance to the center of the Earth) is shown in Figure 3d, in which a similar relationship can be observed.
Next we study W x along the dawn-dusk direction. Figure 3c shows W x as a function of Y for the events with geocentric distance (R) between 4 and 6R E . It can be seen that W x decreases with increasing |Y| and is almost dawn-dusk symmetric. Similar conclusions can be drawn from the relationship between the cusp W x and  (Tsyganenko and Stern, 1996). Panel (b) presents the orbit of Cluster C3 in the GSM XY plane through the cusp region. Panel (c) shows the FGM magnetic field measurements. Panels (d), (e), (f), and (g) show the ion energy spectrogram, number density, velocity, and temperature from C3, respectively.
MLT, as shown in Figure 3d. It is also clear from this scatter plot that W x decreases with increasing |MLT-12| value and is dawndusk symmetric.

Statistical Study of Plasma Properties
Based on topology information obtained in previous sections, spatial distributions of the cusp's physical parameters are studied, again using data from these cusp crossing events. The spacecraft position during a cusp crossing is normalized as 0 to 1 from PoB to EqB in X direction, dividing the cusp region into 21 by 21 grids (X: from 0 to 1; Z: from 2 to 8R E ). Magnetic field and plasma measurements for cusp crossings are subsequently averaged in each grid with the identified cusp crossing events to obtain the global statistical properties of the cusp. Figure 4a shows the number of observations in the normalized grids. From the previous section, the cusp width in the X direction is given by W x = 0.43 − 0.17Z + 0.064Z 2 and the X axis is given by X = 0.731Z − 1.43; thus the normalized grid can be converted to the XY plane in the SM coordinate system, as shown in Figure 4b. Consistent with previous studies (Heikkila, 1985), the cusp region appears to be funnel-shaped. The background magnetic field lines (the dashed lines) in Figure 4b are computed by the T96 magnetic field model (Tsyganenko and Stern, 1996).
By using this new methodology, the spatial distribution of the physical parameters of the cusp are obtained in SM coordinates ( Figure 5). Figures 5a and 5b show the spatial distribution of the X and Z components of the magnetic field B x and B z , respectively. It is obvious that the amplitude of B x (variation range: -1500 to 1000 nT) and B z (variation range: -3700 to 0nT) gradually increase with decreasing altitude. Figure 5c shows that the magnetic field vectors in the XZ plane are very consistent with values predicted by the T96 model. Combining Figures 5a, 5b, and 5c, we find that the B x is mostly negative and the B z is always negative. This result implies that the cusp region is bent primarily to the equator side in the SM coordinate system. Because B z is dominant in the cusp region, as shown in Figure 5b, the magnetic field amplitude on the pole side is larger than that on the equator side at the same altitude.
Similarly, Figures 5e, 5f and 5g, respectively, present the spatial distributions of the X and Z components of the proton velocity (V x and V z ) and the proton velocity vectors in the XZ plane. The statistical results show that V x (variation range: -30 to 30 km/s) is positive on the pole (equator) side below about 6R E (5R E ) and turns negative over this altitude. V z (variation range: -25 to 60 km/s) is positive in most areas of the cusp region except at high altitudes (greater than 6R E ) and on the equator side. For proton velocity (positive V x and V z ) at low altitudes, the cause may be upstream particle flow and the tilt of the cusp region. For proton velocity (negative V x and V z on the equator side, negative V x and positive V z on the pole side) at high altitudes, the cause may be dragging of the solar wind, the magnetic reconnection at the sub-solar magnetopause (e.g., Cargill et al., 2005;Lavraud et al., 2005) and the tilt of the cusp region. Figure 5d shows that the spatial distribution of the proton density (variation range: 0-9 cm −3 ) is higher on the equator side than on the pole side, and this may be caused by a higher magnetic pressure on the pole side. The proton temperature (0-12 MK), which is very high at EqB as shown in Figure  5h, may be caused by the high dynamic pressure near the day- Month

Summary and Discussion
In this paper, we have investigated the position, size, and global characteristics of the mid-and high-altitude cusp using data from a nine year (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009) statistical survey by the Cluster spacecraft that includes a total of 470 crossing events. A new orbitsampling methodology has been introduced for studying the global characteristics of the cusp region.
Previous studies (e.g., Zhou XW et al., 1999;Palmroth et al., 2001) have revealed that the dipole tilt angle has a great impact on the locations of the cusp boundaries. In order to eliminate this impact, the SM coordinate system has been used in this paper. The EqB and PoB for the altitudes from 2 to 9R E are identified from Cluster measurements. In the SM coordinate system, there is a clear separation between PoB and EqB, and the relations between X and Z are nearly linear for the poleward, equatorward boundaries and the center of the cusp. Especially, the relation for the center of the cusp can be described as X = 0.731Z − 1.43 (cc ≈ 0.84). We found that the scale of the cusp region in the Y direction increases nearly linearly with increasing X or Z, and that both of the boundaries are located primarily between 08:00 and 16:00 MLT.
Subsequently we studied W x . We found that (1) the relationship between W x and Z is consistent with previous studies and can be expressed by W x = 0.43 − 0.17Z + 0.064Z 2 ; (2) the relationship between W x and Y (MLT) suggests that the cusp region is nearly dawn-dusk symmetric.
A number of statistical studies of the cusp region have been previously presented (Zhou XW et al., 1999, 2000Lavraud et al., 2004Lavraud et al., , 2005Walsh and Fritz, 2011). However, except that Lavraud et al. (2004Lavraud et al. ( , 2005 and Walsh and Fritz (2011) used an elaborate orbitsampling methodology which uses the Tsyganenko and Stern (1996) magnetic field model and takes into account the geomagnetic activity and actual solar wind conditions, and most of these studies have ordered crossing data spatially by using simple coordinates systems (such as SM, GSE ans GSM). In this paper, to study the spatial distributions of the cusp's global properties we have employed a new orbit-sampling methodology that uses normalized coordinates to eliminate distortions of cusp topology. The statistical results of this method can be displayed intuitively. We report a spatial distribution of the magnetic field in which B x is mostly negative, B z is always negative, and the magnetic field vectors in the XZ plane are consistent with T96 model calculations. At low altitudes, the proton velocity is found to be positive for V x and V z , while at high altitudes, we report negative V x and V z on the equator side and negative V x and positive V z on the pole side. The proton density is higher on the equator side than on the pole side. The high dynamic pressure near the dayside magnetosphere may lead to the very high proton temperature at EqB. The methodology and statistical results shown in this paper suggest directions for future cusp studies.

Acknowledgments
This work was supported by the National Natural Science Foundation of China, grants 41821003, 41974194 and 41574154. The authors are thankful to the Cluster team and the Cluster Science Archive for providing Cluster data, which are available at https://csa.esac.esa.int/csa-web/#home. We also acknowledge use of the OMNIWeb service of NASA/GSFC's Space Physics Data Facility. The OMNI data are available at http://cdaweb.gsfc.nasa.gov/.