K. Maezawa1, T. Hori1, T. Mukai2, Y. Saito2, T. Yamamoto2, S. Kokubun3, and A. Nishida2
1Division of Particle and Astrophysical Sciences, Nagoya
University, Nagoya 464-01, Japan,
2Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, 229 Japan
3Solar Terrestrial Environment Laboratory, Nagoya University, Honoharu, 442 Japan
Structure of the distant magnetotail (x<-150 Re) in the y-z plane is studied statistically on the basis of the plasma and magnetic field data obtained by the GEOTAIL spacecraft. In order to reduce the scatter due to time variations in the solar wind direction, a coordinate system is introduced whose x-axis is taken parallel to the hourly direction of the solar wind measured in the upstream region. Our analysis shows that the distant tail has almost the same dimension in y and z directions for average IMF conditions. ln other words, there is no indication of the tail flattening along the y axis for average magnitude of IMF By. The distant tail current sheet is found to be tilted by 20 degrees on average either clockwise or anticlockwise depending on the sign of the IMF By. However, the location of the core lobe, which is defined to be the region of maximum probability of observing the lobe, is not tilted from the z axis. This means that the IMF By effect is not an overall rotation of the tail structure but involves a skewing of the internal structure. The tilt angle of the current sheet is larger when the IMF is directed northward. The large tilt angle observed for the northward IMF implies that the northward magnetic field component in the tail does not necessarily mean the closure of field lines across the current sheet; this finding thus alleviates the difficulty regarding the net tailward transport of positive Bz during quiet times raised earlier.
Early observations of the earth's magnetotail have established that, at least within lunar distances, the tail has a cylindrical geometry, with circular or slightly elliptical cross section. The cross section of the tail comprises two different regimes, the northern and southern tail lobes and the plasma sheet which is located between the two lobes. The conventional view is that the tail lobes are low-beta, high field intensity regions threaded by polar cap field lines, and the plasma sheet is a high beta, higher plasma density region associated with the current sheet. This view has been modified to some extent by Hardy et al. (1975). They reported the existence of relatively dense plasma within the tail lobe. Hardy et al. presented evidence that the occurrence of the lobe plasma has a strong dependence on the y component (By) of the IMF; the dense plasma is found in the north-dawn and south-dusk quadrants of the tail when IMF By is positive, and in other two quadrants when IMF By is negative. The region of dense plasma within the tail lobe has been interpreted to be a tailward continuation of the high latitude boundary layer, or the mantle, found on the tailward side of the polar cusp (Paschmann et al, 1974, 1976; Hardy et al., 1975; Gosling et al., 1985; Siscoe, 1994).
It has been predicted theoretically and supported by many observations that, when the IMF By is nonzero, the torque exerted by the reconnected field lines pulls the convection path toward dawn or dusk depending on the sign of By. The sense of this asymmetry is consistent with the asymmetry of the lobe plasma density mentioned above. Cowley (1979,1981) discussed such asymmetry and suggested that, as a result of the presence of the magnetic field component normal to the tail magnetopause, a certain percentage of IMF By will penetrate into the tail. He also suggested a possibility that the tail current sheet will tilt because of the torque. Both the penetration of By and the tilt of the current sheet have been confirmed with the near-earth tail observations. However, in the near-earth tail region, these effects would be a relatively small modification of the overall tail structure, because relative contribution of By is small as compared to the main Bx component of the tail lobe.
As we go downstream along the tail axis, the relative contribution of By would become large as the tail lobe field magnitude diminishes. Thus the structure of the distant magnetotail is expected to be more strongly dependent on the IMF By. Observationally, the ISEE-3 mission to the distant tail region provided us basic knowledge of the structure of the distant magnetotail prior to the GEOTAIL mission. ISEE-3 observations have indicated that the tail at the distances of x=-200Re has the same two-lobe structure as observed nearer to the earth (Tsurutani et al., 1986 and the references therein). The IMF-By associated asymmetry of the lobe plasma density was confirmed to occur even in regions adjacent to the current sheet (Gosling et al., 1985). The tilt of the tail current sheet was observed as expected (Sibeck and Siscoe, 1985; Sibeck et al., 1986, Macwan, 1992; Owen, 1995). However, the estimates of the tilt angle varied among the studies of different authors. For example, Sibeck et al. (1986) derived the average tilt of 18o for a fixed polarity of IMF By. Macwan (1992) suggested much larger twist on the basis of the current sheet normals determined for selected events. Owen et al. (1995) obtained smaller values of 5-12o for the southward IMF cases and somewhat larger values of 13-24o for the northward IMF cases. Some of the discrepancies among the authors may be attributable to the different magnitudes of IMF By that occurred for individual examples, and to different methods of analysis utilized by different authors.
ISEE-3 observations also raised a question regarding the cross-sectional shape of the distant tail. Within the lunar distance, the tail has an almost circular cross section which may be slightly elongated in the GSM z direction. However, it has been suggested theoretically that the distant tail may be elongated in the y direction, since the contribution from the magnetosheath magnetic pressure in the pressure balance equation tends to compress the tail lobes from above and below for the nominal spiral direction of the IMF. For ISEE-3 observations, there have been contradicting analyses on this matter and a definite answer has not been obtained. Further, if the tail cross section is elongated in any direction, that direction may be tilted towards dawn or dusk, according to the sign of By. This point has not been addressed in previous studies with the exception of Sibeck et al. (1986).
The present analysis is motivated by the fact that the study of the distant tail structure must be made taking into account the possible excursions of the tail axis from its nominal position in response to the changing solar wind direction. Thus, although the satellite path is usually confined in low latitude regions of the GSM yz plane, the satellite does actually encounter various portions of the magnetotail cross section, as they are brought to the satellite location by the shift of the tail axis. To deal with this situation correctly, we introduce the coordinate system whose x axis is aligned with the solar wind direction as observed simultaneously by the IMP 8 satellite in the upstream region. When an off-ecliptic deviation of the solar wind flow dislocates the tail axis latitudinally, this coordinate system allows us to study the high latitude portions of the tail in a proper perspective, which cannot be done with the GSM coordinate system.
In this paper, in the framework of the above coordinate system, we are going to study the average tilt of the distant tail current sheet and the overall cross sectional shape of the distant tail. The location of the current sheet is obtained as the demarcation line between the regions of the northern and southern lobe field polarities, as statistically determined in the y-z plane. We will be discussing the overall tilt of the demarcation line between the northern and southern tail lobes, not to be influenced by local effects such as small scale deformations of the current sheet. We will show that at x<-150 Re, the average current sheet is tilted by 20o for the average IMF conditions of By = 3 nT and Bz = 0. When we restrict our analysis to the cases of northward IMF (Bz = + 3 nT on average), the tilt angle increases to 32o. We will also show that the distant tail cross section has almost the same dimension (50-55 Re) in both y and z directions, for average IMF conditions.
METHOD OF ANALYSIS
l. Selection of Data
The GEOTAIL data used in this analysis are the bulk plasma parameters (ion density N, ion temperature T and the three components of the bulk velocity Vx, Vy, and Vz) obtained from the LEP experiment, and the three magnetic field components (Bx, By, and Bz) obtained from the MGF experiment. One minute averages are used for time intervals when GEOTAIL was in the distant magnetosheath or magnetotail beyond x=-150Re. Details of the instrumental characteristics have been given elsewhere (Mukai et al., 1994).
As discussed in the Introduction, we utilize the coordinate system whose x axis is aligned with the solar wind flow direction with the origin taken at the earth. The sign of the x axis is positive in the direction of the sun. The definition of the y and z axis is exactly the same as that of the GSM coordinate system once the x axis is specified (that is, the y axis is parallel to the vector product of the earth's dipole direction and the x direction; z axis is chosen to complete the right handed Cartesian coordinates). In the following we call this coordinate system `Geocentric Solar-Wind Magnetospheric coordinates', abbreviated as GSW coordinates. As for the data of solar wind flow direction and IMF conditions, we utilize the hourly averages of the IMP 8 solar wind measurements provided from the NSSDC.
In summary, our data set consists of all the available one minute values for which the following conditions are met: (1) GEOTAIL was in the distant tail beyond x=-150 Re, (2) the IMP 8 solar wind data are available, and (3) both MGF and LEP data are available. GEOTAIL was in the distant tail orbits beyond x=-150 Re several times from 1992 to 1994. Altogether, 80863 data points are available for the analysis within the region x<-150 Re, |y|<60 Re, and |z|<60 Re in the GSW coordinates.
2. Region Identification
An important procedure for making statistical analysis of the magnetotail shape is to identify the tail region correctly. The distinction between the plasma sheet plasma and the magnetosheath plasma is relatively easy, since the ion temperature of the magnetosheath plasma rarely exceeds 2x106 K while the ion temperature in the plasma sheet exceeds 2x106 K, being usually higher than 3x106 K. On the other hand, distinction between the lobe plasma and the magnetosheath plasma is not so straightforward, because of the existence of the region called "mantle" where magnetosheath plasma has access to the magnetospheric field lines. We may even say that the distinction depends on how we define the `magnetospheric field lines', since, according to merging models, the reconnected field lines have one end on the earth and the other end threading the solar wind.
As the reconnected field lines are convected tailward, the field lines are stretched tailward and hence the field line direction begins to assimilate with that of other field lines in the lobe. At the same time, the density and speed of plasma on the reconnected field lines, which were initially comparable with those of the adjacent magnetosheath plasma, begin to decrease monotonically by the action of slow expansion wave traveling inward (Siscoe et al., 1994). Therefore, in principle, if we have any means to know the plasma density or speed in the adjacent magnetosheath, we can identify the tail lobe to be the region with a reduced plasma density and a reduced plasma speed as compared to those in the magnetosheath. Figure l (a) shows that this kind of definition really works for the plasma data obtained by GEOTAIL. For all the data points obtained at x <-150 Re, the azimuthal angles of the measured magnetic field are plotted against the tailward plasma speed Vx normalized by the IMP-8 solar wind speed. Data points with ion temperature T>3x106 k have been excluded from the figure as representing the plasma sheet. Otherwise, all the data points are plotted. A characteristic distribution with the shape of a reversed letter `E' emerges in the figure. The vertical bar sitting at Vx =1 is interpreted to represent the magnetosheath plasma population. Its speed is almost comparable to the upstream solar wind speed measured by IMP-8. The magnetic field direction for this magnetosheath population is rather uniformly distributed, showing the randomness of the magnetosheath field. On the other hand, the triple horizontal bars around the azimuthal angle values of 0o, 180o and 360o are interpreted to represent the plasma population in the northern and the southern lobes. These populations occur with magnetic fields pointing sunward (0o and 360o) or antisunward (180o) as expected for the tail lobe field lines.
Figure 1 (a) For the distances beyond x =-150 Re, the azimuthal angle of the measured magnetic field is plotted against the tailward component of the plasma speed normalized by the IMP-8 solar wind speed. Plasma sheet data with ion temperature T > 2 x 106 k are not included. (b) The number of occurrences for each bin of Vx as analyzed in Figure l(a). The occurrence frequency has a very sharp peak at Vx=1 reflecting the magnetosheath flow. A fairly flat distribution ranging from Vx=0 to 0.9 represents the lobe population.
Figure 1 (b) shows the number of data points for each bin of Vx. Each bin has the width of 0.05 in the normalized units of Vx. It is seen that the number of occurrence has a very sharp peak at unity (exactly speaking, at the bin Vx=0.975) representing the flow speed of the magnetosheath plasma. On the other hand, there is a fairly flat distribution occupying the range Vx=0 to 0.9, connecting at the upper end to the magnetosheath population. As stated earlier, this represents the lobe population, and faster portions of the lobe population may be identified as the mantle plasma. The flat shape of distribution suggests that the mantle plasma, which initially had a magnetosheath speed (Vx 1) upon entry to the tail, is continuously decelerated toward the null speed as the plasma drifts deeper into the tail lobe.
Comparing Figure 1 (a) and (b), we see that the field angle distribution makes a rather sharp transition from the lobe to the magnetosheath population at the normalized speed values of 0.8-0.85. In other words we can set the threshold value at Vx=0.8-0.85 to differentiate the two plasma populations. In the following analysis we apply the threshold value of Vx =0.8 to define the tail lobe region. We combine the lobe data points, as defined by the inequality Vx <0.8, and the plasma sheet points, as defined by T>3x106 k, to constitute what we define by the `magnetotail' in the following analysis.
We first study the average size and shape of the cross section of the distant magnetotail. To do this we calculate the probability of observing the magnetotail, for each 4 x 4 Re2 bin in the GSW y-z plane, by dividing the number of data points identified as "magnetotail" for each bin by the total number of data points for each bin. (As stated earlier, one data point has a time span of one minite.) However, if we simply sum up all the observations, the possible dawn-dusk asymmetry due to the presence of IMF By will be smeared out because the asymmetry reverses for opposite signs of By. We will thus adopt the following method: Assuming that the tail configuration for the negative IMF By is simply a mirror image (about z axis) of that for the positive IMF By, we invert the GSW y position of the satellite for all the data points that occurred with IMF By <0. Figure 2 (a) shows the contour map, as seen toward the earth, of the occurrence probability of the tail thus obtained. This figure represents the case for the positive IMF By, though actually both the positive and negative IMF By cases contributed to this figure, as explained above.
Figure 2: a) False color contour map of the probability of observing the tail for each 4 x 4 Re2 bin in the GSW y-z plane, as seen toward the earth. Color coding, is such that reddish tones represent the occurrence probabilities greater than 50 per cent while greenish tones represent smaller probabilities. The white thick line represents the 50 per cent contour line which may be taken as the median location of the magnetopause. This figure is drawn to represent the case for the positive IMP By and all IMF Bz. (b) False color map of the difference Pn - Ps between the probabilities of observing the northern tail lobe field polarity and the southern tail lobe field polarity. The thick white line represents the zero contour line, which may be taken as the average location of the neutral line.
Figure 3: a) Same as Figure 2(a) but for the case IMF Bz <0, (b) Same Figure 2(b) but for the case IMF Bz <0.
Color coding in Figure 2 (a) is such that red to blue tones represent the occurrence probabilities greater than 50 per cent while greenish tones represent smaller probabilities. A thick white line represents the 50 per cent contour line. We take this 50 per cent contour line as representing the average boundary location of the tail at x<-150 Re.
Several things are clear from Figure 2(a). First, regarding the size of the tail, the 50 per cent contour line shows that the y dimension of the tail is about 55 Re, measured where it is maximum, while the z dimension of the tail is also about 55 Re. The z dimension of the tail may actually be slightly larger because a part of the southern high latitude contour lines is missing due to the lack of observation points. Thus, there is no indication of a major flattening of the tail in the y direction as proposed by some authors (Sibeck et al., 1986). Second, the shape of the 80 per cent contour line (highest contour level in the figure) is elongated in the z direction showing that the boundary location is more variable at lower latitudes. Third, there seems to be no tilt in the overall silhouette of the tail. This is also suggested from the fact that the 80 per cent contour line is elongated roughly in the z direction.
We now address ourselves to the possible tilt of the current sheet. As stated earlier, we define the current sheet as the demarcation line between the regions of northern and southern lobe field polarities. We are going to statistically derive the position of the current sheet in the following way. Let Pn be the probability of observing the tail region with positive Bx polarity. Similarly, let Ps be the probability of observing the tail region with negative Bx polarity. (We note that the sum P=Pn+Ps represents the total probability of observing the tail, plotted in Figure 2 (a). Then statistically the difference D=Pn - Ps should vanish at the position of the current sheet. In Figure 2(b), we show a contour map of the difference D= Pn - Ps in the GSW y-z plane. The thick white line shows the contour line D=0, which is expected to be the average position of the current sheet. It is clear that the current sheet is tilted toward dawn, as is consistent with theoretical expectations. (Remember that the figure is drawn for positive IMF By.) By making least squares line fitting to the portion of the D=0 contour inside the 50 percentage probability contour given in Figure 2(a), the tilt angle is obtained to be 20.0o.
It is to be noted that Figure 2 provides us information about the structure within the lobe as well. The region where D is close to 1.0 is the region where the north tail lobe is most stably observed. By the same token, the region where D is close to -1.0 is the region where the south lobe is most stably observed. (Of course, the absolute value of D is small near the current sheet.) We arbitrarily call the region where the absolute value of D exceeds 0.8 `core lobe', since it is interpreted to be the most stable part of the tail as long as the Bx polarity is concerned.
It is apparent from Figure 2 (b) that the centers of north and south core lobes are not tilted from the z axis. Therefore, the current sheet tilt induced by IMF By does not lead to an overall rotation of the tail structure. (As a matter of fact we will see later that for the northward IMF, the major axis of the tail tilts in a opposite way to the current sheet tilt.) It is because the major magnetic stress exerted by the reconnected field lines works at the mantle, which we find to be situated at relatively low latitudes. On the other hand, the core regions of the tail lobe are located at higher latitudes and may be free from the magnetic stress. This point will be further discussed later in this section.
Although the average tilt angle is 20o as deduced above, there is an indication that this angle increases with northward IMF Bz. Figure 3 (a) and (b) show the same statistics as given in Figure 2 (b) and (a) but made for the cases IMF Bz <0 (southward IMF). On the other hand, Figure 4 (a) and (b) show the same statistics made for the cases IMF Bz <0. First of all, the current sheet tilt seen in Figure 4 (a) is larger than that in Figure 2 (b). The least squares fit shows that the tilt angle is 32o for the northward IMF. On the other hand, the tilt angle seen in Figure 3(a) is much smaller and the least square fit gives 15o for the southward IMF case. Therefore, the tilt angle for the northward IMF (with the average Bz= +2nT) is more than twice as large as that for the southward IMF (average Bz= -2nT).
There is also an interesting difference between the northward and southward IMF cases regarding the overall shape of the tail cross section. Figure 3(b) shows that the major axis (vertical axis) of the tail cross section is tilted toward dawn when the IMF is directed southward. This sense of the tilt is the same as that for the current sheet. However, Figure 4(b) shows that, when the IMF is directed northward, the tilt of the major axis, as defined by the overall silhouette of the tail, opposes the current sheet tilt. A careful examination of Figure 4(a) and (b) shows that this is because the locations of the core lobes are not tilted and instead the portion of the tail near the current sheet is stretched in a slant way toward southdawn and north-dusk in the northern and southern hemispheres, respectively, in the expected direction of the magnetic tension exerted by IMF.
These observations indicate that the effect of IMF By is not a simple rotation but involves a skewing of the internal structure. This is reasonable because the major magnetic stress exerted by the reconnected field lines works not on the entire tail but at the mantle, which is localized and whose location varies with the IMF direction.
In order to see where within the tail the IMF By effect is the strongest, a regression analysis is made between the y component of the 1MF and that of the magnetic field measured by GEOTAIL for three regions in the tail, i.e. core lobe, mantle, and plasma sheet. Plasma sheet is identified against other two regions by a higher temperature T>3x106 K. Core lobe and mantle are defined by T<2x106 K, with the former distinguished by normalized plasma density less than 0.03. The results, shown in Table 1 show that for the northward IMF, the effect of By is strongest for the mantle region, second strongest in the plasma sheet, and the weakest in the core lobe region. For the southward IMF, the mantle and plasma sheet show nearly the same strength of the IMF By effect. The location of the mantle is shown in Figure 5.
|IMF Bz>0||IMF Bz<0|
|regression coef.||c.c||regression coef.||c.c|
SUMMARY AND DISCUSSION
Our analysis presented in this paper is summarized as follows:
(1) The distant tail current sheet (x<-150 Re) is tilted by 20o on the average (for IMF |By|=3nT), either clockwise or anti-clockwise depending on the sign of IMF By.
(2) The tilt angle is larger for the northward IMF. The average tilt angle for IMF Bz>0 is 32o, while the average for IMF Bz<0 is 15o.
(3) The core lobe region at higher latitudes is not tilted on the average.
(4) No significant tail flattening is observed for average IMF angles.
(5) The By effect is the strongest in the mantle region at the flanks, and the weakest in the core lobe regions at high latitudes.
These results confirmed that the magnetic field tension exerted by the reconnected IMF field lines plays an important role in determining the shape and structure of the distant magnetotail. Particularly, our results clearly show that the Y component of IMF causes a twist of the tail current sheet. This is in reasonable agreement with some of the theoretical expectations and earlier observations. We stress that in our analysis the current sheet position has been obtained globally as a demarcation line between the regions of north and south tail field polarities. Therefore, our results are free from short time-scale and/or local variations in the current sheet orientation which might have affected other methods.
Figure 4. (a) Same as Figure 2(a) but for the case IMF Bz >0, (b) Same as Figure 2(b) but for the case IMF Bz>0.
Figure 5. False color map in the GSW y-z plane of the occurrence frequency of the tail plasma having a normalized density greater than 0.03 for the Southward IMF cases left panel), and for the northward IMF cases (right panel).
As for the overall average tilt angle (for a fixed polarity of IMF By) the value of 20o is in close agreement with the analysis of Sibeck et al. (1986). They showed from ISEE-3 data that occurrences of the northern (southern) lobe south (north) of the GSM equatorial plane is explainable if the current sheet is tilted by 18o on the average either clockwise or anti-clockwise depending on the sign of IMF By. An important finding presented in this paper is that the magnitude of the tilt angle is a function of IMF Bz. Thus, for the northward IMF cases, the average magnitude of the tilt angle is found to be 32o, which is more than 50 per cent larger than the overall average. It is noted that Owen et al. (l995) recently reported a similar tendency, but gave much smaller estimates for the tilt angle. It may be that their method of determining the tilt angle, based on the directional anisotropy of energetic ions, may be more sensitive to local effects such as surface waves, which are not controlled by IMF By.
The large tilt angle found for the northward IMF shows that the IMF strongly interacts with the earth's magnetosphere even when it is directed northward. Further, the By dependent torque suggests that the lobe field lines are under direct control of the IMF. They would not have their both ends in the ionosphere as predicted by some merging models for the northward IMF. Our results are basically consistent with the open lobe reconnection model first suggested by Russell (1972), which has been discussed and applied to ground observations by Maezawa (1976) and later supported by low-altitude satellite observations (e.g. Burke et al., 1979; for related theoretical models see Crooker, 1979, Reiff and Burch, 1985). It is noted that the closed field line reconnection models for the northward IMF such as the one first introduced by Dungey (1961) require almost due northward IMF. A due northward IMF may occur for individual cases, but statistically the probability of having a due northward IMF is much lower than that for having a less strictly northward IMF. Therefore, it is expected that the effect of Dungey type (closed field line) reconnection processes would not show up in our statistics of northward IMF even though it may exist.
It is not easily understood why the tilt angle of the current sheet should be larger for the northward IMF than for the southward IMF. A possible reasoning would be that the tilt angle is determined by the total torque exerted by the reconnected IMF field lines, and this torque should be integrated on the tail surface. Now, the integrated torque is a function of at least two factors, i.e. the total amount of reconnected field lines, and the force an individual reconnected field line would exert on the tail surface in the direction tangent to the tail boundary. The former quantity, which is proportional to the normal component of the magnetic field, Bn, integrated over the tail surface, is believed to be larger for the southward IMF than for the northward IMF. However, the latter quantity (tangential force) may depend on the clock angle of the IMF. (Take the case of positive IMF By. For the north dawn and south dusk quadrants where a large portion of reconnected field lines flow, the draped IMF has a larger tangential component Bt at the magnetopause for the northward IMF than for the southward IMF. Therefore, when Bn is fixed, the tangential stress (BnBt/4) would be larger when the IMF is directed northward.) Thus the two effects (Bn and Bt) compete each other, and the observation suggests that the latter effect actually dominates.
Figure 6. Schematic model of the tail cross section at x<-150 Re for the southward and the northward IMF.
Figure 6 shows our empirical model for the structure of the distant magnetotail as seen toward the earth. The model is drawn separately for southward (right panel) and northward IMF (left panel) cases, both with a positive IMF By. (For a negative IMF By, a mirror image about the z axis applies.) The results obtained in this paper are reflected in the model as follows.
(1) The current sheet inclines towards the dawn side both for northward and southward IMF cases. This is consistent with the fact that the torque is determined basically by the sign of IMF By, that is, by the dawn-dusk component of the magnetic tension exerted by the reconnected field lines.
(2) The tilt of the current sheet is larger for the northward IMF than for the southward IMF. As a result, the current sheet is nearly parallel to the incident IMF direction when IMF is directed northward.
(3) The mantle, as characterized by a relatively cold, dense plasma of the solar wind origin, occurs in the region adjacent to the magnetopause where the reconnected field lines penetrate the boundary. The mantle is located at lower latitudes for the northward IMF than for the southward IMF.
(4) The major axis of the tail is inclined in opposite ways for the northward and southward IMF, with a fixed polarity of IMF By.
(As stated earlier, if the sign of IMF By is reversed, the total configuration becomes the mirror image of what is shown in this figure.)
The tilt of the current sheet has an important implication for the discussion of the average Bz in the distant tail. It has been thought that the Bz component (in GSM coordinates) in the tail is the critical component that determines the amount of closed field lines across the current sheet. Accordingly, the topology of field lines in the tail has been discussed mainly on the basis of the statistics of the Bz component. However, when the current sheet is tilted, a simple coordinate transformation yields
B = Bz cos - By sin
where B is the magnetic field component perpendicular to the current sheet, By and Bz are components in the GSM coordinates, and represents the tilt angle of the current sheet (positive if IMF By>0). The first term on the RHS shows that the magnitude of the contribution from Bz is reduced (by the cos term) when the current sheet is tilted. Further, the second term is always negative as long as is non zero. (Remember the sign of By in the tail is controled by that of IMF By, and hence tends to be the same as that of .) Over all, the effect of the current sheet tilt would be such that the real closing component of the magnetic field is smaller than what has been estimated from Bz alone.
Therefore, the previous estimates of the magnetic field component perpendicular to the current sheet should be reconsidered taking into account the effect of a tilted current sheet. There are two distinct topics which may need reconsideration. First, the average location of the distant neutral line, which has been inferred from the position where the sign of Bz reverses as a function of distance from the earth, may need reconsideration. As stated above, the real B would be smaller than the simple average of Bz; this means that the location where B changes sign may be nearer to the earth than estimated previously. Accordingly, the distant neutral line would be nearer to the earth than estimated before. The second topic is concerned with the difficulty arising from the positive values of Bz in the distant tail during quiet times. Both ISEE-3 and GEOTAIL observations have shown that during quiet times, the plasma flow is tailward on the average, and Bz is positive on the average. Combination of a positive Bz and the tailward flow suggested a large loss of closed field lines from the near-earth magnetosphere during quiet times, and this has led to a difficulty. However, equation (1) indicates that this difficulty may go away if we estimate B correctly taking into account the tilt of the current sheet. Even a moderate estimate for By will reduce the loss of the close field lines considerably. This possibility has already been discussed by Nishida et al. (1996) (see their figure 8(b)).
Finally, we note that theoretically, the anisotropic magnetic pressure of the magnetosheath would compress the distant tail from above and below (ie. in the direction perpendicular to the average IMF direction which lies in the ecliptic plane). This effect was suggested by Sibeck et al. (1986) to be quite large, resulting in a flattening of the tail. At least for average conditions of IMF, we failed to confirm their prediction from our GEOTAIL data set. The reason why the tail cross section is not elongated in the y direction as predicted is not clear now. However, we should take into account at least three effects. First, there is a possibility that within lunar distances the tail cross section is elongated in the z direction. Therefore, the predicted effect of the anisotropic magnetic pressure of the IMF to flatten the tail in the distant region may be compensated for by the larger dimension in the z direction which already existed in the near-earth region. Second, the average magnitude of the IMF By is observationally not much different from the average magnitude of IMF Bz. Thus the direction of the compression by the IMF field lines is not necessarily perpendicular to the ecliptic plane but varies as a function of the IMF angle in the y-z plane.
Finally, we have shown that the effect of the magnetic tension exerted by the IMF plays an important role in determining the structure of the distant tail. It is reasonable to assume that magnetic tension also plays an important role in determining the shape of the distant tail. So far, the effect of magnetic tension has not been included in the theoretical estimates of the tail flattening. Probably, a reasonable comparison between theory and observation will become possible if such effects are included in the theoretical estimates.
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