The Solar Wind Control of the Magnetopause Shape:A Comparison of A Model Magnetopause and Empirical Models

The solar wind control of the magnetopause shape is studied with a model magnetopause that results from three sources of magnetic field. One source is the geomagnetic field produced by the Earth's dipole. Another results from an image dipole placed in front of the magnetopause to pro­ duce the effect of the magnetopause current which limits the spatial extent of the geomagnetic field. The third is a southward interplanetary magnetic field (IMF) in the Earth's vicinity. With an image dipole strength and the location to account for the effect of the solar wind dynamic pressure (D2), the shape of the model magnetopause can be regarded as controlled by DP and the southward IMF. As an image dipole strength is about fourteen times the Earth's dipole and the location is at thirty times the Earth's radius, the shape of the model magnetopause is consistent with observational results for a southward IMF. A comparison of the shape of the model magneto­ pause and an empirical model by Shue et al. (1997) shows that the image dipole strength at a fixed location correlates linearly with DP for a south­ ward IMF. In addition, they agree qualitatively for high DP and a south­ ward IMF. (


INTRODUCTION
The Earth's magnetosphere has generally been thought to be determined by the solar wind, the interplanetary magnetic field (IMF), and the geomagnetic field. Magnetospheric configuration is widely considered a useful means to describe the global picture of the Earth's magnetosphere. It is also conducive to understanding physical mechanisms of the formation of the Earth's magnetosphere (e.g. Nishida, 1978). In the past, there were two main models of magnetospheric configuration to describe the Earth's magnetosphere. From the balance be tween the dynamic pressure of the impinging solar wind and the magnetic pressure of the geomagnetic field, Chapman and Ferraro (1931) suggested that the magnetosphere might be closed. In order to study the cause of magnetospheric convection, Dungey (1961) proposed an open model of the magnetosphere in which the Earth's dipole is permeated by a southward IMF. Based on these two models of the magnetosphere, many studies have been conducted to explain various local magnetospheric phenomena.
In recent years, the spacecraft and satellites launched to probe the Earth's environment have collected valuable data from which some features of the magnetosphere are revealed.
High-latitude REOS 2 measurements near the dayside magnetopause showed that the mag netic field converges toward a cusp which presumably results from Chapman-Ferraro currents (Fairfield, 1977;Haerendel et al., 1978). On the night side, geomagnetic field lines are inter connected with the interplanetary space (Fairfield, 1987 and references therein). The observa tional evidence indicates that both the convergence of the magnetic field toward the cusp and the penetration of the IMF may take place jointly. In other words, the magnetosphere could be partially closed and partially open. Thus, it is useful to construct a theoretical model to exhibit some of these features of the magnetosphere.
The terrestrial magnetopause is generally identified as the surface of separation between the geomagnetic field and the IMF. It has long been known that the shape of the magnetopause is mainly controlled by the solar wind dynamic pressure DP and the IMF Bz component (see review by Fairfield, 1995). Recently, with the availability of large numbers of magnetopause crossings, the magnetopause shape has been fitted as a function of D P and the IMF Bz com ponent (Sibeck et al., 1991;Petrinec et al., 1991;Roelof and Sibeck, 1993;Petrinec and Russell, 1993;Shue et al., 1997). Due to the limit of the fu�ctional form or satellite orbits, some em pirical models are appropriate for the dayside region (e.g. Roelof and Sibeck, 1993) and some for the nightside region (e.g. Petrinec and Russell, 1993). Nevertheless, a common trend in these studies exists whereby increased pressure compresses the magnetopause earthward while IMF Bz remains steady. Moreover, the flank expands with increasing IMF Bz and the subso lar distance reduces conspicuously at low DP and less at high DP. Thus, comparison of a model magnetopause with empirical models helps in realizing how the magnetopause shape is controlled by the solar wind.
Recently, Yeh ( 1997) proposed a mathematical model to study magnetospheric structure with a southward IMF. In the model, the IMF is assumed to be due south and moderate as its magnitude is less than the geomagnetic field at the subsolar point. The model magnetosphere possesses features of both a closed model and an open model, as revealed by observations.
Since the realistic magnetopause shape is more affected by IMF Bz than other components such as Bx and B y , it seems appropriate that a model magnetopause for a southward IMF can be used to study how the magnetopause shape is controlled by the solar wind. Hence, the main goal of this study is to examine qualitatively the solar wind control of the magnetopause shape by comparing a model magnetopause based on Yeh ( 1997) with empirical models. This paper is organized as follows. Section 2 mathematically shows that a model magnetopause is con structed on three magnetic fields. Section 3 discusses the shape of the model magnetopause and compare it with empirical models. Finally, the last section discusses other factors affect ing the comparison of the model magnetopause and empirical models, and summarizes this study.

A MODEL MAGNETOPAUSE FOR THE SOUTHWARD IMF
Referring to Yeh ( 1997), a mathematical model of the magnetopause shape results from three sources of magnetic field. One source is the geomagnetic field produced by the Earth's dipole. The second results from an image dipole placed in front of the magnetopause to have the effect of the magnetopause current which limits the spatial extent of the geomagnetic field. The third is a southward IMF in the Earth's vicinity. To simplify the mathematical analysis, there is no plasma in the model. Moreover, the model does not include magnetic fields induced by the field-aligned current and the magnetotail current which are important elements of the magnetosphere.
We describe the geomagnetic field by B -B 3-3xzx-3 y zy+(x 2 +y 2 -2z 2 )z G -E a (x 2 + y 2 + z 2 ) 5/2 (1) in terms of a southward magnetic dipole. Its magnetic moment is equal to the equatorial field strength BE times the cube of the Earth's radius a. The magnetic field resulting from an image dipole is described by in terms of a southward dipole placed at x = rM on the x-axis. This image dipole in front of the magnetopause has a magnetic moment BMa3 greater than the geomagnetic moment.
Namely, BM> BE and xM >a. The IMF is described by in terms of the strength of the IMF B1, and its direction points southward for B1 < 0 and northward for B1 > 0. Here we use the Cartesian coordinates with the origin located at the Earth's center, the x-axis pointing toward the Sun, the y-axis toward the dusk, and the z-axis toward geomagnetic north. The total magnetic field is thus given by x -rM + yz B z = E a + M a + I ( x 2 + y 2 + z 2)5 / 2 ((x-rM )2 + y 2 + z 2 )5/ 2 Since regular field lines are smooth curves, these can provide a vivid visualization of the magnetopause shape. Regular field lines emanate from and terminate at neutral points at which the magnetic field strength becomes zero. The model magnetopause is a flux surface in which regular field lines emanate from and terminate at cusps corresponding to neutral points. Hence, the location of cusp neutral points plays a predominant role in the magnetopause shape. Before proceeding to draw magnetic field lines, it is essential to find out the location of cusp neutral points in the model.
In the model, neutral points occur where B G + Bp is canceled out by B1. For the y = 0 plane on which a cusp neutral point is located, by combining (5) and (7), the location of the cusp neutral point is determined by Since the model is assumed to be permeated by a southward IMF, the magnetopause shape appears symmetrical in both the y = 0 and z = 0 planes. Accordingly, the location of north and south cusp neutral points are symmetrical in the z = 0 plane. In this study, the distance from the Earth is normalized by the Earth's radius a and an image dipole strength is normal ized by the magnetic field strength B E at the the Earth's equator. However, B1 is converted to SI units by multiplying B1 /BE by BE = 3.12X104 nT. Instead of directly solving (8) and (9) for the location of cusp neutral points, we superimpose the contours of various BM/ BE and B1 values with arbitrary rM in the xz plane and find out the location of north and south cusp neutral points using the intersection of both contours. In situ measurements (e.g. Roelof and Sibeck, 1993) have shown that on average, the subsolar distance is 1 Oa. The subsolar distance is the standoff distance of the subsolar point where the solar wind dynamic pressure is balanced by the Earth's magnetic pressure. Hence, it is appropriate to choose a normalized rM roughly twice the subsolar distance. For example, rM is given as 20 (see Figure 1) and contours of various values of BM/ BE and B1 are plotted over 4 :::; x :::; 9 and 4:::; z :::; 9 in the meridian plane. Note that dotted lines denote BM/ BE and solid lines B1. With , rM =20, BM/ BE =8.6 and B1 = -5nT, Figure 1 shows that the north cusp neutral point is located at ( x=5.7, z=6.2). This cusp location is consistent with satellite observations (cf. Frank, 1971). As a result, the location of cusp neutral points in the model is determined using rM, BM/ BE and B1.
For given values of rM, BM/ BE and B1, the flux surface of the magnetopause is con structed on the spatial extent of coplanar field lines leaving/entering north/south cusp neutral points. The integration of field lines is described by a differential equation as follows

THE MODEL MAGNETOPAUSE AND EMPIRICAL MODELS
As magnetohydrodynamic interaction of the terrestrial magnetosphere and flowing solar wind is in a state of equilibrium, the magnetopause current will be induced to shield the im pinging solar wind. To balance the solar wind dynamic pressure D P , the magnetic flux in duced by the magnetopause current will enhance magnetic pressure at the dayside magneto pause. In the steady state, the magnetic pressure at the subsolar point is equal to D P " In other words, the magnetic flux induced by the magnetopause current is proportional to D P . With an image dipole and the location to produce the magnetic flux induced by the magnetopause current, the model magnetopause is an approximation of the day side magnetopause. Hence, the shape of the model magnetopause can be regarded as being controlled by D P and a south ward IMF. In order to investigate how the magnetopause shape is controlled by the solar wind, we attempt to compare the shape of the model magnetopause with empirical models.
With an image dipole and the location to account for the effect of D P , we first compare the shape of the model magnetopause with an empirical model by Roelof and Sibeck (1993). An ellipse of evolution was used by Roelof and Sibeck (1993) to fit the magnetopause shape, and they studied it by dividing magnetopause crossings into each bin of dynamic pressure D P according to IMF Bz and IMF Bz according to D P , respectively. Figure 9 of their cor-rected version (Roelof and Sibeck, 1994) shows that the subsolar distance is about 8a -12 a and the flank distance is about I Sa -25a for various values of DP and IMF Bz. In Figure 9 of Roelof and Sibeck (1994) there is a trend whereby for a constant IMF Bz> the magneto pause moves earthward in a similar shape with increasing pressure. In addition, their results show that the magnetopause around the subsolar point moves earthward and the flank expands with increasing IMF Bz for a constant dynamic pressure.

·
In the second section, we mentioned that the shape of the model magnetopause is deter mined by rM, BM/ BE and B1. This means that the subsolar distance and the flank distance in the model are determined by rM, BM/ BE and B1. For convenience of comparison with em pirical results, we have to choose adequate values of rM, BM/ BE and B1 to calculate the shape of tpe model � agnetopause. In the model, the subsolar distance is less than xM =rMf(l+(BM/BE) 114 ). Hence, the subsolar distance is determined by rM and BM/BE. In Figure 2, it is shown that xM varies as rM and BM/BE change. In the five · · · · ··· · · · · ·· ·· ·· · · · ···· ·-· · · ··· · · · ·· · ··· · . .  curves in Figure 2, the solid curve is for rM =20, the dotted curve for rM =30 , the dashed curve for rM =40 , the dashed-dotted curve for rM =50 and the dashed-dotted-dotted curve for rM =60. Figure 2 shows that x M is close to 10 when rM has values of 30 , 40 , 50 , 60 which correspond to BM/ BE =14, 72, 190 , 440 , respectively. However, in Figure 3, it is shown that the flank distance shrinks as rM and BM/ BE increase. In contrast, the flank distance for rM :::: 30 and BM/ BE =14 is consistent with the empirical fitting by Roelof and Sibeck (1994). By using this procedure , as shown in Figure   is rewritten as Ez. In the model, the magnetic pressure at the subsolar point is expressed as where Bzs is the magnetic field at the subsolar point and µ0 is the permeability in space. Thus, BM/BE=14correspondsto M p =3 . 02nPaand BM/BE=14 . 5for Mp =3. I7nPa. In Figure 4, the solid curve is for M p= 3 . 02nPa and the dashed curve for Mp = 3. 1 ?nPa.
Note that the shape of the model magnetopause moves earthward in a similar way with in creasing Mp for a constant Bz. It is evident from Figure 4 that there is a qualitative trend of dependence of the magnetopause shape upon the solar wind dynamic pressure for a constant Bz. The two curves in Figure 5 are for, rM =30, Mp = 3.3nPa (corresponding to BM/ BE =15 ) and for Bz= -2.5 (dashed) and -5.0nT (solid). In Figure 5, there is a trend of the flank expanding with increasing southward Bz for a constant D P . However, the subsolar distance in Figure 5 seems constant for varying southward Bz. This is due to the subsolar distance in the model being limited by XM = rM 7(1 +(BM / B E )114) which is independent of the south ward Bz. Roelof and Sibeck(l993) also studied dependence of the subsolar distance upon DP and southward Bz. They found that the subsolar distance varies less sensitively at high D P southward IMF. However, owing to a lack of variation of the subsolar distance in the model magnetopause, it is appropriate for high D P only by comparison. Nevertheless, it implies that the magnetopause shape can be fitted as a function of the southward Bz, the image dipole strength and the location.
In order to verify whether an image dipole strength and the location can account for the effect of D P , we investigate their relation with a comparison of the magnetopause shape fitted by a functional form proposed by Shue et al. (1991). According to Shue et al.(1991), the mag netopause shape can be fitted as where r is the radial distance from the Earth's center to the magnetopause, r0 is the distance from the Earth's center to the subsolar point and ex is the exponential factor related to magne topause flaring. For a southward IMF, the subsolar distance r0 is derived as (13) and the flaring factor ex is expressed as From (13), D P can be determined for given r0 and a southward IMF Bz component. The subsolar distance in the model is the location of one of the coplanar field lines emanating from the south cusp neutral point intercepted by the x-axis. Hence, the subsolar distance r0 can be acquired and substituted into (12) with a given Bz to get the corresponding value of D P ' In this study, the subsolar distance ro in the model is calculated with BM / BE varying from 14 to 16 for rM =30 and the southward IMF Bz= -2.5 and -5.0nT. Figure 6 shows that Mp increases with increasing BM/ BE. In Figure 6, the dotted line is for Bz = -2. SnT and the solid line for Bz = -5.0nT. The relation between D and BM /BE is shown in Figure 7, with a dotted line for Bz = -2. SnT and a solid line for Bz = -5 . 0nT. It is obvious from Figure 7 that D P increases as BM/ BE increases. Comparison of Figures 6 and 7 shows that for the southward IMF, the effect exerted by the solar wind dynamic pressure upon the dayside magnetopause can be approximated with an image dipole strength and the location.
To justify whether Figures 4 and 5 are for high D P and the southward IMF, we recon struct the magnetopause shape with the empirical model for the same parameters. By using (11), (12) and (13), the magnetopause shape based on the Shue et al.(1991) model is shown in Figures 8 and 9. Figure 8 shows the magnetopause shape for Bz = -2.5nT with D P = 3.02nPa (solid) and DP= 3.l 7nPa (dashed). The magnetopause shape in Figure 9 is for D P = 3.3nPawith the southward IMF values of Bz = -2.5nT(dashed) and Bz = -5.0nT

DICUSSION AND SUMMARY
In this study, the shape of the model magnetopause is obtained in the equatorial plane.
Since the dayside magnetospause is axis-symmetric, the shape of the model magnetopause can be regarded as an approximation of the magnetopause shape observed by satellites in low latitude orbits. For the same subsolar distance and the same flank distance, M p in Figures 4 and 5 seems larger than D P  al., 1986). In this study, the model magnetosphere is based on a mathematical model by Yeh (1997) which is characterized by a pair of cusp neutral points in the noon meridian and a segmental neutral line on the equatorial plane. In a steady state, magnetic reconnection in this model magnetosphere occurs at the flank with the exception of at the subsolar point. Hence, this model magnetosphere may not explain the flux erosion in the subsolar region. In Figure 5, there is also a trend whereby increased Bz enhances tail flaring as D P corresponding to M p remains constant. This result is consistent with the prediction by Petrinec and Russell (1993) that tail flaring is associated with a southward IMF Bz. It is worth noting that the flank dis tance in Figures 4 and 5 are inconsistent with Figures 8 and 9. This discrepancy shows that the dipole's strength BM/ BE in the theoretical model may not account for the total effect of the solar wind. Nevertheless, the model provides a preliminary theoretical explanation on how the magnetopause shape is controlled by the solar wind.
Instead of solving the location of a cusp neutral point analytically, we have presented an alternative approach to finding the location by using the intersection of both contours of BM/ BE and B1 for a specific rM in the meridian plane. By using the integration of coplanar field lines emanating from a south cusp neutral point, we have constructed the shape of the model magne topause which consists of the footpoints of coplanar field lines in the equatorial plane. With an image dipole strength and the location to account for the effect of the solar wind dynamic pressure, the shape of the model magnetopause is com p ared to the empirical model by Roelof and Sibeck (1993) for a southward IMF. With BMfBE�14 and rM=30 , the shape of the model magnetopause is consistent with the empirical fitting by Roelof and Sibeck (1994) for a southward IMF. The comparison shows that in both models the dayside magnetopause moves earthward with increased dynamic pressure for a constant southward IMF. Moreover, the flank in both models expands as the southward IMF increases for a constant D P . The relation be tween the image dipole strength at a fixed location and D P is examined by using a functional form by Shue et al. (1997), and it shows that they are linearly correlated. Hence, it is verified · thatthe effect exerted by the solar wind dynamic pressure can be approximated by an image dipole strength and the location. With the exception of the flank distance, for high D P and southward IMF, the magnetopause shape based on Shue et al. (1997) is consistent with our results.
In summary, this study provides a qualitative description on the solar wind control of the magnetopause shape and it shows that the magnetopause shape can be fitted as a function of the southward IMF, the image dipole strength and the location. In addition, the shape of the model magnetopause is consistent with empirical models for high DP with a southward IMF.