Originally published in: Adv. Space Res., 25(7/8), 1413-1424, 2000
The polar cusp is a region in which the magnetosheath plasma has direct access to the ionosphere. It exists whether the interplanetary magnetic field is northward or southward. In a non-reconnecting magnetosphere the location of the cusp depends on the shape of the magnetopause but when the magnetosphere reconnects with either southward or northward interplanetary magnetic field the location of the cusp is altered. Since the polar cusp was discovered simultaneously at both low and high altitudes in 1971, its exploration has been mainly carried out at low altitudes. The launch of the POLAR spacecraft has allowed similar exploration of the high altitude polar cusp in the northern hemisphere. Where the high altitude properties of the cusp have direct counterparts at low altitudes, we find consistency between the measurements in the two regions. The invariant latitude of the cusp depends on the tilt angle of the cusp producing a significant north-south asymmetry at the solstices. The cusp moves equatorward when the IMF turns southward but also moves somewhat equatorward for increasingly northward IMF. As the IMF By component becomes more negative, the cusp moves to earlier local times in the northern hemisphere. This is consistent with a motion of the reconnection site away from the noon meridian when the IMF is not due southward. When it is positive, the northern cusp moves to later local times. When the solar wind dynamic pressure increases the polar cusp becomes wider in both local time and latitude. Surprisingly simple models such as the vacuum model of Tsyganenko predict the cusp location better than the more recent Tsyganenko 96 empirical model. MHD models appear to be quite successful in predicting the location of the cusp, its magnetic configuration and plasma properties.
The polar cusp is an explicit feature of Chapman and Ferraro's (1931) model of the magnetosphere in which a planar superconducting boundary forms the magnetopause. Mathematically this magnetosphere is described by the superposition of two dipole magnetic fields, one upstream from the magnetopause a distance equal to that of the magnetopause from the center of the Earth. This magnetosphere is shown in the top left panel of Figure 1. The cusp is clearly seen as the point where the magnetic field lines diverge. This occurs at 77o invariant latitude, i.e. the latitude at which the magnetic field line intersects the surface of the Earth. The latitude of the cusp obviously depends on the shape of the magnetopause. For example if the magnetopause is a sphere and not a plane, symmetry tells us that the cusp will be directly over the dipole axis.
If the magnetopause has an intermediate shape then the cusp will be between 77o and 90o . For example the vacuum model of Tsyganenko (1989a) has a cusp at 81o invariant latitude as sketched in the lower left-hand panel. The cusp location also depends on the distribution of plasma in the magnetosphere and the rate and location of reconnection. The lower right panel shows the magnetic configuration of the 1989 empirical model of Tsyganenko (1989b). This model implicitly takes account of the real conditions in the magnetosphere by fitting the observed magnetic field. Here the cusp is at 78o invariant latitude. In short, the position of the cusp is an important property of the magnetosphere to determine as it gives us a means of testing magnetospheric models.
|Fig. 1. Four magnetospheric models demonstrating the dependence of the polar cusp on the shape of the boundary and the distribution of magnetospheric plasma. (Top left) Chapman-Ferraro image dipole model. (Top right) Spherical magnetopause. (Lower left) Tsyganenko (1989a) vacuum magnetosphere with dipole inside a superconducting magnetopause. (Lower right) Tsyganenko (1989b) empirical magnetosphere fit to observed magnetic field values.|
Because it was long a part of magnetospheric models, the discovery of the polar cusp was not a surprise. Perhaps the biggest surprise was that it took until 1971 for the first reports on the polar cusp to be made. These discovery papers were published nearly simultaneously: observations at low altitudes with the ISIS satellite by Heikkila and Winningham (1971); observations at high latitudes with IMP-5 by Frank (1971) and with OGO-5 by Russell et al. (1971). These observations showed that magnetosheath plasma was able to reach the ionosphere near the weak spot in the magnetic field at the magnetopause where the field diverged. In the ideal, inviscid, non-diffusing, non-reconnecting case the single field line that forms the cusp spreads across the magnetopause and covers the entire surface. In this case the cusp has zero thickness and no plasma reaches the ionosphere. The 1971 observations by IMP-5, ISIS-1 and OGO-5 showed that some process enabled magnetosheath plasma to become entrained on magnetospheric field lines. They did not show how this entrainment occurred, but whatever the process the cusp existed for both northward and southward IMF (Russell et al., 1971). Thus there was some opinion that multiple processes (e.g. diffusion and reconnection) could produce this entrainment.
The ISIS, IMP-5 and OGO-5 data revealed the gross properties of the polar cusp, although arguments continued for many years about its exact geometry. Measurements by Aureole, DE-1 and the DMSP satellites at low altitudes (Cambou and Galperin, 1974; Sauvaud et al., 1980; Burch et al., 1982; Meng, 1984; Newell and Meng, 1988) and by HEOS 2 and Hawkeye at high altitudes added greatly to our understanding of the cusp (Paschmann et al., 1976; Farrell and Van Allen, 1990). Because of the great amount of data accumulated by the DMSP mission over several decades with multiple satellites the low altitude cusp became well characterized. However, the low data rates and small payloads of HEOS-2 and Hawkeye did not provide as comprehensive a study of the high altitude polar cusp. To rectify this, two missions of the ISTP (or IASTP) program were launched into the region of the polar cusp. The first, Interball 1, passed through the cusp and its interface with the solar wind on a regular basis. The second, POLAR, had an apogee of 9 RE over the north pole. It provided more regular cusp passages but remained within the magnetosphere most of the time. Some of the contributions of the Interball mission to the study of the cusp are given in papers by Savin et al. (1999) and by Fedorov et al. (1999) in this volume. In this paper we examine the statistical behavior of the cusp as seen by the Polar spacecraft and compare these results with those obtained at low altitude by the DMSP spacecraft. We use Polar observations obtained over the period March 1996 to December 1997 together with solar wind and interplanetary magnetic field observations from the WIND spacecraft. The time lag of the solar wind has been corrected by dividing the X displacement of the WIND spacecraft from the magnetopause by the average solar wind velocity. Five minute averages are used for the solar wind quantities in this study.
POLAR MEASUREMENTS OF THE CUSP
Figure 2 shows observations obtained on cusp crossings when the IMF is northward and southward on the left and right respectively. The top panel shows the magnitude of the magnetic field with the Tsyganenko 1996 model field removed (Tsyganenko, 1995; 1996). The broad depression around the cusp is characteristic of difference between the model and the Polar observations in this region (Zhou et al., 1998). The additional irregular depression from 0457 to 0532 UT on May 7 and from 1743 to 1750 on April 21 is due to the presence of plasma as indicated by the electron measurements from Hydra and the proton measurements from Timas shown in the four lower panels (Scudder et al., 1995; Shelley et al., 1995). We note that these data are preliminary measurements and are not intercalibrated. The proton data moreover cover only part of the energy range and hence is only a partial density. These two figures illustrate how we identify the cusp crossings. First, we identify the irregular depression in the magnetic field. If it is greater than 1 nT in depth below the smooth background depression observed on the orbit then we confirm the cusp identification with the plasma observations. The plasma in this region must be magnetosheath-like.
|Fig. 2.Polar measurements in the cusp. The top panel shows the magnetic field magnitude observed by Polar less the magnitude of the model field of Tsyganenko 1996. The next panel shows the preliminary electron density measured by Hydra. The next panel shows the Hydra electron temperature. The next panel shows the proton density measured by Timas in the energy range 15 to 370 eV and the bottom panel the proton temperature in this same energy range. Left panel shows the cusp for a pass during northward IMF and right panel during southward IMF.|
TILT ANGLE DEPENDENCE OF THE INVARIANT LATITUDE OF THE CUSP
Southward interplanetary magnetic fields are known to erode flux from the dayside magnetopause and transfer it to the magnetotail (Aubry et al., 1970; Russell and McPherrron, 1973). This process certainly affects the location of the cusp (Russell et al., 1971). Thus we will restrict our discussion of tilt angle effects to the situation of northward IMF. Figure 3a, b, and c show the locations of the cusp detected by POLAR, according to the criteria described above, for three tilt angle ranges 0-10o , 10-20o and =20o respectively. The magnetic field lines drawn are those from the Tsyganenko 1989 vacuum magnetosphere (Tsyganenko 1989a) for tilt angles of 5o , 15o and 25o . Only crossings within 2 RE of the noon-midnight meridian are shown. The straight line segments are medians over radial distances of 1 RE . The locations shown have been scaled by the sixth root of dynamic pressure to account for the varying size of the magnetosophere in response to pressure changes. Some of variability is due to the fact that the location of the cusp is responsive to the history of the IMF over some interval of time and does not respond instantaneously to the IMF. We see from these three figures that the invariant latitude of the cusp varies with the tilt angle of the dipole both in the data and in the model. The amount of shift with tilt observed is consistent with that expected from the model. We note that we have chosen to compare the observations with Tsyganenko's (1989) vacuum model and not the recent 1996 model because there is a 2o invariant latitude shift in the 1996 model to lower latitudes that is not consistent with the POLAR data.
|Fig. 3.The location of the cusp as observed by Polar for tilt angle ranges of (a) 0-10o, (b) 10-20o, and (c) >20o. The cusp locations have been scaled by the sixth root of the dynamic pressure to account for the varying size of the magnetosphere. Only cusp crossings with 2 RE of the noon-midnight meridian have been used. Positive angles indicate a tilt of the northern magnetic pole toward the sun.|
Figure 4 shows the location of the equatorward boundary of the cusp as a function of the dipole tilt angle as observed by the DMSP spacecraft (Newell and Meng, 1989). The cusp varies by 4 degrees over a tilt change of 60o , quite consistent with the 2o shift seen over 20o as observed at POLAR. However, the DMSP positions have not been restricted to only northward IMF. This dependence implies that the cusp is at a significantly different invariant latitude in the northern and southern hemispheres at the solstices.
|Fig. 4.The location of the cusp as observed by the DMSP spacecraft within one hour of local noon as a function of the tilt of the dipole (Newell and Meng, 1989). Here an angle of 90o is perpendicular to the solar direction and less than 90o indicates the northern magnetic pole is tilted toward the sun.|
Southward IMF erodes flux from the dayside magnetopause and adds it to the tail. Thus we would expect the cusp location to move equatorward as the southward component of the IMF increased. This behavior is confirmed in Figure 5 that shows the invariant latitude of the center of the cusp as seen by POLAR as a function of IMF Bz in solar magnetospheric coordinates. The invariant latitude of the cusp center is 81.3 + 0.98 Bz for southward IMF when Bz is measured nT and is negative for southward fields. The latitude of the center of the cusp is 80.7 û 0.027 Bz for northward IMF. The correlation coefficient of these east squares fits to the cusp location are 98% and 53% respectively. The values have all been adjusted for the pressure of the solar wind and the crossings restricted to within 2 RE of the noon meridian. When the IMF is northward, there is a slight tendency to move equatorward with increasingly northward IMF. We can understand such a behavior in terms of where plasma is added to the magnetosphere for northward IMF. The plasma is added equatorward of the bifurcating field line for northward IMF. Thus if the bifurcation point remains fixed as it would if the size of the tail were constant, then a more rapid addition of plasma to field lines connected to the Earth will cause an equatorward motion of the cusp. However, this effect is small.
|Fig. 5.The location of the center of the cusp as a function of the IMF Bz. The locations have been adjusted for the solar wind pressure but not the dipole tilt angle.|
Figures 6a and b show the equivalent result from the DMSP data (Newell et al., 1989). Here the equatorward boundary of the cusp is at 73o for a 5 nT southward field and at 77o for 0 nT. For northward IMF there is little dependence and the equatorward boundary is near 77.5o . In comparing these data with POLAR observations we note that neither data set has been corrected for differing tilt angles and the POLAR data correspond to the center of the cusp not the equatorward boundary.
|Fig. 6.The dependence of the equatorward boundary of the polar cusp from the DMSP observations (a) southward IMF (b) northward IMF (Newell et al., 1989).|
The location of the polar cusp is also dependent on the Y-component of the IMF. Figure 7 shows the local time at the center of the cusp crossing as a function of the By GSM as observed by Polar in the northern hemisphere.
|Fig. 7.The dependence of the local time of the cusp as observed by POLAR as a function of the IMF By component when Bz was southward. The vertical line segments show the median value of the cusp location in steps of 2 nT. All cusp crossings used in this figure were obtained above the northern hemisphere.|
For small By there is no significant effect but when By is greater than 6 nT the cusp is displaced about 1 hour in local time to the afternoon side and when By is less than -6 nT the cusp is displaced about 2 hours toward morning. A similar effect is seen at low altitudes. Figure 8 shows the probability of detecting the cusp as a function of local time for Bz northward (upper two panels) and Bz southward (lower two panels) for By greater than 3 nT (left two panels) and By less than -3 nT (right two panels). There is a shift in the position of cusp of approximately the same magnitude and in the same direction as at low altitudes. The shift is somewhat smaller for northward IMF than for southward IMF. This effect may also be present in the northward IMF POLAR observations but the fewer cusp observations with POLAR do not allow the motion to be quantified. We can understand this effect in terms of the motion of the reconnection site away from noon for positive and negative By according to the calculations of Luhmann et al. (1984) following the conjecture of Crooker (1979). This should be more strongly observed at low altitudes where the reconnection process does not much alter the field direction. At high altitudes the field lines containing the cusp plasma are pulled across the noon meridian resulting in a lesser displacement there. We note that in the anti-parallel merging scenario that there are two reconnection sites on either side of noon and on either side of the equator both of which connect to the northern and southern hemisphere. However, the ionosphere that is closest to the reconnection site is the one that is affected most by it [Fenrich et al., unpublished manuscript, 1998].
|Fig. 8.The probability of observing the cusp by the DMSP spacecraft as a function local time for two ranges of IMF By and Bz. The top row shows the probability for positive Bz and the bottom row for negative Bz. The left column shows the probability for By greater than 3 nT and the right column for By less than -3 nT (Newell et al., 1989). All observations have been converted to northern hemisphere equivalents. The reverse local time shifts are observed in the south.|
DYNAMIC PRESSURE EFFECTS
While no dependence of the cusp has been reported at low altitudes we might expect some effects as the pressure of the solar wind increases. The cusp locations in Figure 3 were scaled for the effect of dynamic pressure on the size of the magnetosphere. Other than that obvious effect, there is no significant effect of dynamic pressure on the invariant latitude of the cusp. Figure 9shows, however, that the width of the cusp is affected (Zhou et al., 1999). As the pressure of the solar wind increases the cusp at POLAR is thicker. This effect may be due to the gradual widening of the cusp as the magnetopause is approached since the observations at high dynamic pressure are of necessity obtained on average closer to the magnetopause.
|Fig. 9.The width of the polar cusp in invariant latitude seen by the POLAR spacecraft as function of the solar wind dynamic pressure.|
The cusp also becomes wider in local time with increasing dynamic pressure. Figure 10 shows the local time of the start and end of cusp crossings and the solar wind dynamic pressure during the crossing of the cusp (Zhou et al., 1999). The horizontal line segments show the median dynamic pressure at each hour of local time. From 0900 to 1400 LT the dynamic pressure of the cusp crossings is that typically seen in the solar wind. However, to earlier and later local times the median pressure increases. This indicates that the cusp only reaches these regions during higher than normal dynamic pressures.
|Fig. 10.The solar wind dynamic pressure during each passage through the polar cusp by POLAR. The local time at start and end of the crossing are shown. The horizontal line segments give the median dynamic pressure for each hour of local time.|
MHD MODEL COMPARISON
Above we compared the Polar observations with both theoretical/analytic approximate models of the magnetic field and empirical models. There is another class of models that is highly useful in the study of the magnetosphere, the global MHD simulation. These models now have sufficient resolution to resolve current systems and they can follow the time-varying changes in the magnetosphere in response to changing solar wind boundary conditions. The MHD equations properly take account of the inertia in the plasma that controls the time-scales over which processes occur. In particular the MHD simulations contain reconnecting fields, and can reproduce many processes that are attributed to reconnection, such as the erosion of the magnetosphere and substorms. The MHD simulations provide the dissipation required for reconnection through numerical resistivity and this resistivity does not necessarily match that of the process in the magnetosphere. Thus we must be careful to test these numerical models and validate them with actual observations. This has been done for several cases (e.g. Fedder et al., 1997). We examine one of these tests (Russell et al., 1998) below. First, we examine the simulation of these steady-state conditions at three canonical orientations of the magnetic field.
|Fig. 11.Magnetohydrodynamic simulations of the solar wind interaction with the magnetosphere under these steady-state conditions with (a) the IMF southward (b) eastward and (c) northward. The colors and contours show the density in the noon-midnight meridian. The projections of selected magnetic field lines are shown. See Fedder et al. (1997) for further details.|
Figure 11a, b and c show the noon-midnight meridian view of the magnetosphere for southward, eastward and northward IMF respectively. The color-coding shows the plasma density and selected field lines are traced. For northward and southward IMF the field lines in the noon-midnight meridian remain in that meridian but for the eastward IMF the field lines extend out of the meridian. Figure 11 shows these field lines projected on the noon-midnight plane.
There are many interesting features in these plots. First, we note the island of high density seen in the throat of the cusp for both southward and eastward fields. This feature appears to arise because of the indentation in the magnetopause at the cusp. Time sequences show that this island is time variable. The second important feature is the entrainment of magnetosheath plasma on magnetic field lines that have either one or two feet connected to the Earth. This entrainment is important because the solar wind has momentum. In the case of southward IMF the entrained plasma flows tailward in the mantle and adds magnetic flux and plasma to the tail. The solar wind is decelerated by the JxB force and Poynting flux flows into the tail. The energy content of the lobe then increases. For nearly due southward IMF the mantle is restricted to a narrow layer. For nearly eastward (or westward) IMF the mantle is spread in a thinner layer over a broader region of the tail magnetopause as inferred by the field geometry of Figure 11b.
For northward IMF, reconnection is still important for the formation of the cusp, but here the reconnection takes place behind the cusp. In this case the plasma is entrained on the field lines that are equatorward of the field line that bifurcates as seen in Figure 11c. Again the solar wind captured by the magnetosphere in this way has momentum and the field lines connected to the Earth on both ends can be stretched far behind the Earth. Thus reconnection leads to a longer tail than would occur in an inviscid situation.
These three simulations have an important implication for our comparison with the Tsyganenko model above. In Figure 3 we compared northward IMF cusp crossings with the Tsyganenko (1989a) model, using cusp crossings identified by their plasma signature (including the depression in the magnetic field in the cusp that is due to the presence of plasma) with the cusp as identified by its bifurcation. For northward IMF, however, the plasma should be found below the bifurcation point. Thus the DMSP and POLAR cusp locations should be equatorward of that predicted by the divergence of the field lines in the Tsyganenko model.
To illustrate that the plasma entrainment equatorward of the bifurcation in the field is observed, we show in Figure 12 a simulation of the magnetic field along the POLAR trajectory when the IMF was nearly northward and POLAR was near the magnetopause but inside it (Russell et al., 1998). We see that the simulation quite accurately predicts the magnetic field profile throughout this pass. This suggests that the amount of reconnection in the code is similar to that of the real magnetosphere.
|Fig. 12.Comparison of the magnetic field observed by POLAR and predicted along the POLAR trajectory for the observed solar wind conditions on May 29, 1996 when the IMF was strongly northward and Polar near the magnetopause. The residuals of the magnetic from the Earth main field (that is a dipole in the case of the simulation) are shown in SM coordinates (Russell et al., 1998).|
Finally, we show in Figure 13 the locations of the polar cusp in the MHD model shown in Figure 11b for eastward IMF. The cusp location is shown both as color contours of the plasma density and line contours of the magnetic field depression from the Tsyganenko 96 model [Fenrich et al., unpublished manuscript, 1998]. Because reconnection sites would be expected in the afternoon in the northern hemisphere and the morning in the southern hemisphere and since they should be connected to both northern and southern hemispheres, we might expect to see two cusps in the northern hemisphere. This is partially true but one is very very weak (see e.g. the 5 RE shell). Thus the apparent effect of a strong eastward IMF is to move the cusp to the afternoon in the north and to the morning side in the south. This explains the observations of DMSP at low altitude. This shift is not present at high altitudes in agreement with POLAR observations. We note that the Tsyganenko model is not accurate at highest altitudes in the cusp. Thus, the density contours and the field depression contours diverge at 9 RE. At this distance the plasma density gives the best indicator of cusp location.
|Fig. 13.The polar cusp at 6 different radial distances in the MHD model for eastward IMF. Colors show the plasma density. Contour lines show the difference between the magnetic field calculated in the MHD model and that of the Tsyganenko 96 model. The Tsyganenko model is less accurate near 9 RE.|
SUMMARY AND CONCLUSIONS
The POLAR spacecraft has enabled the regular probing of the cusp at high altitudes near its source. The observed response of the cusp to the tilt of the dipole, the interplanetary magnetic field and the solar wind dynamic pressure are consistent with earlier low altitude measurements. As the dipole axis tilts closer to the solar wind flow direction, the cusp moves toward the pole. When the IMF becomes more southward, the cusp moves further equatorward. This is consistent with the transfer of magnetic flux from the dayside magnetosphere to the tail. As the IMF By component becomes more negative the cusp moves to earlier local times. This is consistent with the motion of the expected site of reconnection away from the noon meridian as postulated by Crooker (1979) and calculated by Luhmann et al. (1984). At POLAR altitudes (around 9 RE ) the width increases in local time and latitude when the solar wind pressure increases. The latitude widening may in part be due to the approach of POLAR to the magnetopause under high solar wind pressure conditions. The Tsyganenko 96 model does not seem to predict the location of the cusp as well as the Tsyganenko 89 model. However, part of this difference is due to the fact that the cusp is on closed field lines equatorward of the bifurcation line when the IMF is northward.
MHD models appear to be very accurate and predict the behavior of the plasma and magnetic field in the vicinity of the cusp quite well. These models can be used to extrapolate our observations in key regions of the magnetosphere to a much larger volume and place better our observations in the context of global magnetospheric processes. Most importantly these models show us how the region around the polar cusp behaves under a variety of solar wind conditions. In particular they demonstrate how the plasma of the solar wind becomes entrained in the magnetosphere and leads to the energization of the magnetosphere. When the interplanetary field is southward, magnetic flux is eroded from the front side of the magnetosphere and the magnetic energy content of the tail increases. When the IMF is northward, the magnetosheath plasma is entrained on closed dayside field lines that convect to the nightside and forms an extended closed tail. Nevertheless, despite their extreme value in providing insight into global magnetospheric behavior, MHD models do not at present replicate the actual physical processes that enable the reconnection to occur. Thus until we understand better the processes that control the rates of reconnection, we must view these models as only qualitative predictors. Predicted quantities such as the length of the closed tail for northward IMF or the location of the X-point for southward IMF must yet be validated.
Much of the work reported herein is due to the able assistance and advice of X-W. Zhou and J. A. Fedder in our study of the polar cusp. We thank K. W. Ogilvie and R. P. Lepping for permission to use solar wind observations from the WIND spacecraft in the studies. This work was supported by the National Aeronautics and Space Administration under research grant NAG5-3171.
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