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Originally published in: Adv. Space Res., 25(7/8), 1271-1484, 2000.
The location and shape of the magnetopause are among the important parameters in space physics because they specify the size of the magnetosphere and responds to the physcial processes occurring therein. In the past few years, several empirical models of the magnetopause shape have been developed using large in situ data sets of magnetopause crossings. These models were derived from best-fits to observed magnetopause locations; however, the data sets, the functional forms of the magnetopause, and the specific dependence of the shape on the upstream solar wind conditions used by these models are different, so are their ranges of validity. In this paper, we provide a comprehensive review of these models, compare the differences among them, and discuss their limitations. In addition, we also show the results of validation of these models for space weather forecasts using the January 1997 magnetic cloud event.
Chapman and Ferraro (1931) first introduced the concept of the magnetopause boundary that depends on solar wind dynamic pressure (Dp). Ferraro (1952) calculated the shape of the magnetosphere. Subsequently, Aubry et al. (1970) recognized that the orientation of the interplanetary magnetic field (IMF) also affects the distance from the Earth to the magnetopause boundary. More quantitative and empirical models of magnetopause locations were developed later using large in situ data sets of magnetopause crossings from various satellites (Fairfield, 1971; Howe and Binsack, 1972; Holzer and Slavin, 1978; Formisano et al., 1979; Sibeck et al., 1991; Petrinec et al., 1991; Petrinec and Russell, 1993b, 1996; Roelof and Sibeck, 1993; Shue et al., 1997, 1998; Kuznetsov and Suvorova, 1998; Kawano et al., 1999). Since these models were derived using data that were mainly from low-latitude satellites, the magnetopause shape calculated by these models refers to low-latitude magnetopause locations.
In Formisano et al.'s (1979) and Sibeck et al.'s (1991) data sets, not only low-latitude magnetopause locations but also high-latitude magnetopause locations are included. The high-latitude magnetopause was assumed to have the same radius as the low-latitude magnetopause. Zhou and Russell (1997) was the most recent paper to study locations of the high-latitude magnetopause. They found no significant indentation of the magnetopause near the cusp region, but that the cross section of the magnetosphere simply becomes greater suddenly from in front of to behind the cusp. Slavin et al. (1985), Sibeck et al. (1986), Fairfield (1992), and Nakamura et al. (1997) estimated the distant tail radius from observations of magnetopause crossings in the distant tail. Meng (1970), Holzer and Slavin (1978), and Petrinec and Russell (1993a) studied effects of ring current and auroral electrojet on the shape of the magnetopause However, these papers are beyond the scope of this review. We focus on models of ``low-latitude'' magnetopause locations that have an expression as a function of IMF Bz and Dp in the near-Earth region.
Until recently much of our understanding of the shape of the magnetopause is summarized by three review papers (Fairfield, 1991, 1995; Sibeck, 1995). Herein, we comprehensively review magnetopause locations and develop a model that is somewhat different from those in these three review papers.
From a mathematical point of view, low-latitude magnetopause location models were derived from best-fits to observed magnetopause locations. The primary differences among them are data set, solar wind data resolution, focus of curves, functional forms, range of validity, and dependence of the fitting parameters on solar wind conditions. Tables 1 and 2 summarize these quantities for various models, which will be useful to those who want to understand the magnetopause and choose a magnetopause model for their research. Besides these quantities shown in Tables 1 and 2, we also compare quantitatively the differences of magnetopause locations.
The data sets used by various models are summarized in Table 1. Sibeck et al. (1991) and later Roelof and Sibeck's (1993) compiled a large data set of magnetopause crossings by collecting the crossings from different investigators who might have used different identification criteria. Thus, their data set may not be homogeneous. Kuznetsov and Suvorova (1998) added more crossings from geosynchronous satellites to Roelof and Sibeck's (1993) data set. They attempted to extend the range of validity of the Roelof and Sibeck (1993) model. Herein, we caution that including geosynchronous crossings in a data set could add a bias to the data set because the position of geosynchronous crossings is always at 6.6 RE, independent of the ultimate radial distance of the magnetopause in response to a new level of solar wind pressure.
There are three important parameters in a data set: locations of magnetopause crossings, IMF Bz, and Dp. One should note that the locations of the magnetopause crossings could be different when using different identification criteria. For a given crossing, the values of the upstream conditions assigned to the crossing critically affect the results of best-fits.
In addition to using in situ crossings in deriving models, Petrinec and Russell (1996) and Kawano et al. (1999) used flare angle data obtained from the pressure balance between the solar wind and magnetosphere.
Upstream solar wind conditions are very important factors for a data set of magnetopause crossings. Two major issues are concerned with how to choose the upstream conditions for a given magnetopause crossing: time shift and time resolution. A common practice is to use the distance of a solar wind monitor to the magnetopause divided by the solar wind velocity as the time shift. This value may be either longer or shorter than the real convection time. An underlying assumption is that the statistics will wash out fluctuations. When the solar wind velocity changes drastically, the time shift should be handled more carefully. The propagation delay across the magnetosheath and the time delay caused by the effect of attitude of the normal to the solar wind structure should also be taken into account.
As solar wind conditions change frequently and suddenly, the resolution of data becomes very important. A long-period average may not correctly represent the real values of upstream conditions. Gaps in the solar wind conditions may worsen the situation: the value in a long-period average could be taken from a single measurement during the period. Roelof and Sibeck (1993) used 1-hour averages of solar wind data, while Shue et al. (1997) used 5-min resolution. Shue et al. (1997) also applied their approach to Roelof and Sibeck's (1993) data set. They found that the trend of standoff distances, r0, for the two models are different, as shown in Figure 1. For example, Shue et al.'s (1997) r0 has a change in slope at Bz = 0, but Roelof and Sibeck (1993) find a linear trend for r0 with a slight change in slope at Bz of 8 nT.
Figure 1. Comparison between the Roelof and Sibeck (1993) and Shue et al. (1997) data sets. This variation of r0 vs. Bz is for Dp = 1.915 nPa. Diamond symbols show the best-fit values of r0. Error bars represent probable errors of the best-fit values. A number indicated above or below each error bar shows the number of data points in each bin.
All models use a family of curves with a focus, but these focuses may or may not be at the same point. Some models fix the focuses at the center of the Earth and some models do not. For models that do not fix the focus at the center of the Earth, one more parameter is needed for the fitting process. The additional parameter increases the flexibility of family curves but reduces the statistical significance of results, in particular, in ranges that have few data points. Kawano et al. (1999) argued that, for the data set they were using, models with off-center focuses are better than models with a center focus, based on an objective criterion which is called the Akaike Information Criterion (AIC) in statistics. Readers who are interested in the AIC technique should refer to Kawano et al. (1999) for more details.
There are two issues related to choosing a functional form. The first issue is how well a functional form can mimic the actual physical behavior of the magnetopause and how flexibly the functional form can represent the magnetopause shape with a larger range of Bz and Dp. The second issue concerns the analytic simplicity of the functional form. The number of fitting parameters is also a concern. More parameters give more flexibility but less statistical significance. This interplay between apparent goodness of fit and its true statistical significance is the rationale for Kawano et al. (1999) use of the AIC criterion. In the Functional Forms column of Table 1, it can be seen that most of the models used the equation of an ellipse as the basic functional form. Since an ellipse must cross at some point on the nightside, it cannot represent a magnetopause with an open magnetotail (Fairfield, 1995). To improve the behavior of the magnetopause shape function on the nightside, Petrinec and Russell (1996) combined an elliptic model on the dayside (Petrinec et al., 1991) with their sinusoidal model on the nightside (Petrinec and Russell, 1993b) and with a smooth connection at the terminator. Later, Kawano et al. (1999) used a cylinder (R = sqrt(Y2+Z2) = constant) to represent the magnetopause shape where X is less than the X position of the maximum R of a family of elliptic curves.
This functional form has two parameters, r0 and alpha, representing the geocentric standoff distance at the subsolar point and the level of tail flaring, respectively. The value of r (= sqrt(X2+Y2+Z2)) is the radial distance at an angle, theta, between the Sun-Earth line and the direction of r. Unlike the equation of an ellipse, this functional form is flexible producing an open or closed tail magnetopause: see Figure 2 for a demonstration of the behavior of this function.
Figure 2. Behavior of Equation (1) as functions of r0 and alpha. The left figure shows fixed alpha (= 0.5) and variable r0. The right figure shows fixed r0 (= 10 RE) and variable alpha.
The range of validity among the models is different. In an ideal case, it should depend only on the data sets used in deriving the various models. However, some of the family curves derived from each model may well describe the observations. The other curves, which are constrained by a functional form, may have very large errors in representing data points under extreme conditions and in the tail region. When the errors become very large, the validity of a model breaks down. Therefore, the range of validity could be greater or smaller than the range represented in the observations, depending on how well a function form represents the intrinsic magnetopause. For example, for an elliptic function which cannot represent an open tail magnetopause, the quality of the description of the magnetopause rapidly deteriorates as the flaring angle increases with southward IMF.
In the 1970s, there were not many observations of magnetopause crossings with available solar wind conditions. The derived magnetopause shape was independent of solar wind conditions (Fairfield, 1971; Holzer and Slavin, 1978; Formisano et al., 1979), and then it represents the shape at the average value of Dp during each of the periods that was assumed to be dependent on the solar cycle. The positions of crossings were assumed to be independent of IMF Bz statistically, although in case studies, people have noted that ``erosion'' of the magnetopause occurred for southward IMF. Later, many more crossings were recorded from numerous missions. Roelof and Sibeck (1993) were the first to attempt to obtain the magnetopause shape as a bivariate function of Bz and Dp. However, this model was claimed to be valid in a very limited parameter range (1 nPa < Dp < 4 nPa and -3.5 nT < Bz < 3.5 nT).
Shue et al. (1997) used the range represented in their observations (0.5 nPa < Dp < 8.5 nPa and -18 nT < Bz < 15 nT) as their range of validity. Shue et al. (1998) extended the range of validity of Shue et al. (1997) to 0.5 nPa < Dp < 60 nPa and -20 nT < Bz < 20 nT by introducing nonlinear functions for r0 vs. Bz and alpha vs. Dp. Kawano et al. (1999) did not state the range of validity of Dp and X in their paper. They prefer to provide standard errors of the magnetopause shape for any given set of Dp, Bz, and X. Kawano et al. (1999) derived their magnetopause location model only for Bz > 0. Their results show that the magnetopause shape on the dayside is not affected by the magnitude of northward IMF.
The most recent empirical models (Roelof and Sibeck, 1993; Petrinec and Russell, 1996; Shue et al., 1997, 1998; Kuznetsov and Suvorova, 1998; and Kawano et al., 1999) provide mathematical expressions for coefficients of fitting parameters as a function of the north-south component of IMF and the solar wind dynamic pressure, as shown in Table 2. An important consequence emerging from these models is that locations of the magnetopause become predictable for a very wide range of upstream solar wind conditions. These expressions are useful for space weather operations. Contours of r0 as a function of Bz and Dp for various models are shown in Figure 3. These contours can be used, for example, to predict when satellites at geosynchronous orbit are in the magnetosheath. In Figure 3, we find that r0 predicted by both the Petrinec and Russell (1996) and Shue et al. (1997) models is within geosynchronous orbit for small Dp when the southward IMF is extremely strong. However, from both Shue et al. (1998) and Kuznetsov and Suvorova (1998) models, Dp needs to be at least a certain value in order for a geosynchronous satellite to have a magnetopause crossing, even when the IMF is strongly southward. This might be due to the limited capability for reconnection to erode the magnetopause.
Figure 3. Contours of subsolar distances, r0, as functions of IMF Bz and solar wind dynamic pressure, Dp, for various models. The shaded (unshaded) region shows the r0 is within (beyond) geosynchronous orbit.
The specific dependence of a model critically determines the success and the range of validity of the model, in particular, in extreme solar wind conditions. Some models have chosen complicated forms of the dependence, as shown in Table 2. Shue et al. (1997, 1998) not only chose the relatively simple function for the magnetopause shape, but also simple forms for the dependence of r0 and alpha on Bz and Dp. Further analyses have shown that these simple functional forms represent the intrinsic behavior of the magnetopause very well and hence provide a greater range of validity both in spatial parameter and in the domain of solar wind values. For example, from the relations of Shue et al. (1997), the dependence of r0 on IMF Bz changes at Bz = 0. A greater slope for southward IMF reflects the erosion of the magnetopause associated with dayside reconnection. More magnetic flux is removed from the dayside and added to the nightside (Aubry et al., 1970) when the IMF is southward, and alpha increases. For northward IMF, high-latitude reconnection may transfer magnetic flux back to the dayside magnetosphere (Gosling et al., 1991; Song and Russell, 1992) and slightly increase the standoff distance. Variations in dynamic pressure also change r0 and alpha. The parameters r0 and Dp are related with a power law index -1/6.6, which is slightly different from the index -1/6 for a pure dipole geomagnetic field.
The tail magnetopause derived from the Kuznetsov and Suvorova (1988) model does not flare when the southward IMF becomes larger. All other models predict that the dayside magnetopause moves earthward and the tail flaring increases when the IMF is southward. For various values of Dp, the magnetopause shape evolves smoothly when the IMF is northward in all these models. The Petrinec and Russell (1996), Shue et al. (1997, 1998), Kuznetsov and Suvorova (1998), and Kawano et al. (1999) models show little or no change in magnetopause locations for northward IMF. However, the Roelof and Sibeck (1993) model found significant dependence of the magnetopause shape on the magnitude of the northward IMF. Shue et al. (1997) predicts larger flared tail magnetopause for southward IMF than for northward IMF when Dp is large. For southward IMF, the Petrinec and Russell (1996) model has a smaller (larger) flared tail magnetopause than the Shue et al. (1997) model when Dp is large (small).
Results of the Kawano et al. (1999) model are consistent with those of the Petrinec and Russell (1996) model in general. Kawano et al. (1999) suspected that the differences between the Roelof and Sibeck (1993), Petrinec and Russell (1996), and Kawano et al. (1999) models is due to the assumed high-order dependence of parameters of the Roelof and Sibeck (1993) model on Dp. Large uncertainty is included in the calculation when Dp is either extremely small or large, due to the high-order terms.
Most models predict similar dayside magnetopause locations under normal solar wind conditions. Very few magnetopause crossings are available during periods when the solar wind is under extreme conditions. Statistical accuracy becomes low under these circumstances although these conditions are important for space weather forecasts. Models usually produce strikingly different predictions in such extreme conditions. A way to validate the applicability of models is to compare their predictions for these unusual events.
A magnetic cloud arrived at Earth on January 10-11, 1997. The WIND satellite observed strong southward IMF and slowly rotated back to Bz ~ 0 over a period of 12 hours on January 10. A sudden enhancement in the solar wind dynamic pressure occurred around 0200 UT on January 11, while the IMF was extremely strong and northward (~18 nT). This event has attracted a large amount of attention both in the scientific community and the media. For more details of various aspects of this event, readers can refer to Fox et al. (1998). The magnetopause was pushed inside the geosynchronous orbit during the solar wind dynamic pressure enhancement. The LANL 1994-084 and GMS 4 geosynchronous satellites crossed the magnetopause and moved into the magnetosheath. Also, the Geotail satellite was in the magnetosheath while the Interball 1 satellite observed several magnetopause crossings: see Figure 4 for the locations of the satellites. This event provides an excellent opportunity to test the prediction capabilities of the existing magnetopause location models for space weather forecasting.
Figure 4. Locations of the magnetopause and satellites at 0100 UT on January 11, 1997. A solid (dashed) curve is the prediction by Shue et al. (1997) (Petrinec and Russell (1996)). Satellites are designated as follows: L-90, LANL 1990-095; L-91, LANL 1991-080; L-94, LANL 1994-084; GO-8, GOES 8; GO-9, GOES 9; T-401, Telstar 401; GE, Geotail; I-1, Interball 1; GM-4, GMS 4.
Shue et al. (1998) compared the predictions of the Petrinec and Russell (1996) and Shue et al. (1997) models with the in situ observations for the January 1997 event. They did not compare with the Roelof and Sibeck (1993) model because the test conditions were outside the stated range of validity of that model. As stated above, the results of the Kawano et al. (1999) model are sufficiently consistent with those of the Petrinec and Russell (1996) model that comparing with the Petrinec and Russell (1996) model is equivalent to comparing with the Kawano et al. (1999) model.
Shue et al. (1998) calculated the distance between a satellite and a model prediction along the normal to the boundary as a function of time for the Petrinec and Russell (1996) and Shue et al. (1997) models. They found that the two models correctly predict the magnetopause crossings on the dayside, as shown in Figures 5 and 6.
Figure 5. Distances between the prediction by the Shue et al. (1997) model to GMS 4. A negative distance with a solid shading indicates that the satellite is in the magnetosheath. A positive distance shows that the satellite is in the magnetosphere. The uncertainty of the prediction estimated by Shue et al. (1998) is shaded in gray. Thick horizontal bars below the curves indicate periods when the satellite was in the magnetosheath. Figure 6. The distance from LANL 1994-084 to the predicted magnetopause in the same format as Figure 5.
However, there are some differences in the predictions on the nightside: see Figures 7 and 8. The Shue et al. (1997) model correctly predicts the Geotail crossings and partially predicts the Interball 1 crossings. The Petrinec and Russell (1996) model correctly predicts the Interball 1 crossings and partially predicts the Geotail crossings. Shue et al. (1998) improved the Shue et al. (1997) model by introducing new functional forms to better represent the solar wind pressure effect on the magnetopause flaring and the IMF Bz effect on the subsolar standoff distance. They fit r0 to a hyperbolic tangent function for Bz and fit alpha to a natural logarithmic function for Dp, as shown in Figure 9.
Figure 7. The distance from Geotail to the predicted magnetopause in the same format as Figure 5. Figure 8. The distance from Interball 1 to the predicted magnetopause in the same format as Figure 5.
The new functions provide a better implicit description of the underlying physical conservation laws. For example, the r0 tends to be saturated when the dynamic pressure becomes large. If the flaring of the magnetopause increases without approaching a saturated level, the flux conservation would be broken. Also, the r0 may not approach zero when the IMF is extremely strong and southward. The new functions lead to a better agreement with the Interball 1 observations for the January 1997 event. The results of the Shue et al. (1998) model for the January 1997 event are shown in Figure 10.
Figure 9. Value r0 as a function of Bz and value alpha as a function of Dp. Figure 10. Distances between the prediction by the Shue et al. (1998) model to the Interball 1, Geotail, GMS 4, and LANL 1994-084 in the same format as Figure 5.
Homogeneous data set without orbital biases.
Flexible functional form of the size and shape which allows modeling both open and closed tail magnetopause.
Functional forms of the solar wind dependence that represents the intrinsic behavior of the magnetopause location as functions of the upstream conditions and hence with a large range of validity of the upstream conditions.
Consistency with observations in event studies.
Low-latitude magnetopause location models assume axisymmetry around the aberrated Sun-Earth line and have been derived with data sets that are for normal solar wind conditions. Petrinec and Russell (1995) showed that simple ellipsoidal approximation may not be appropriate at higher magnetic latitudes. Formisano et al. (1979) and Sibeck et al. (1991) indicated that the dimension of the equatorial magnetopause is slightly greater than that of the polar magnetopause. The axisymmetry assumption should be relaxed in future developments using more high-latitude data from future missions. Recently, Kartalev et al. (1996), Sotirelis (1996), and Elsen and Winglee (1997) made some progress in numerical simulations and theoretical models of the magnetopause, which are useful to extend the scope of empirical models. However, these models are limited in small ranges of Bz and Dp. Moreover, we cannot yet be sure that these models correctly mimic the behavior of the Earth's magnetosphere. We are still in the stage of validating numerical models.
Models treat the magnetopause as a rigid body which responds to the upstream changes instantaneously. However, the real magnetopause is not rigid and will respond to the upstream changes dynamically, especially during extreme solar wind conditions. The magnetopause will oscillate around its equilibrium state. Therefore, dynamic responses of the magnetopause to external and internal factors will be another topic in future development. At present, the magnetopause becomes predictable for a wide range of upstream conditions. A magnetopause location model could be used to remove the distance variation controlled by the solar wind. After this, one may be able to study the internal responses of the magnetopause associated with storms and substorms (quantified, for example, by the Dst, Kp, and AE indices).
This research was supported by the Center of Excellence (COE) program of Ministry of Education, Science, Culture, and Sports of Japan. The work at UCLA was supported by NASA under grant NAGW-3948 and by NSF under grant ATM 94-13081. The work at the University of Michigan was supported by NSF/ONR under grant ATM-9713492. We thank D. H. Fairfield of NASA/GSFC, H. Kawano of Kyushu University and G. Zastenker of IKI for valuable comments on the tables.
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