Three dimensional shape of the magnetopause: Global MHD results

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010ja016418, 2011 Three dimensional shape of the magnetopause: Global MHD results J. Y. Lu, 1 Z. Q. Liu, 2 K. Kabin, 3 M. X. Zhao, 1 D. D. Liu, 1 Q. Zhou, 4 and Y. Xiao 5 Received 30 December 2010; revised 26 May 2011; accepted 1 July 2011; published 30 September [1] The numerical results from a physics based global magnetohydrodynamic (MHD) model are used to examine the relationship between the shape and size of the magnetopause and the solar wind conditions. The magnetopause location is identified by tracing three dimensional streamlines through the simulation domain and is fitted by simple analytical functions. The resulting model is applicable for approximating magnetopause location for dipole tilt angle 0 and interplanetary magnetic field (IMF) B X and B Y = 0 nt at both low and high magnetospheric latitudes. In both regions the results are compared with available empirical models. It is shown that IMF B Z mainly affects the flaring angle (the magnetopause shape) and has smaller effects on the magnetopause size. In contrast, the solar wind D P mainly affects the magnetopause standoff distance (magnetopause size) and has little effect on the magnetopause shape. Both conclusions are consistent with empirical models. Citation: Lu, J. Y., Z. Q. Liu, K. Kabin, M. X. Zhao, D. D. Liu, Q. Zhou, and Y. Xiao (2011), Three dimensional shape of the magnetopause: Global MHD results, J. Geophys. Res., 116,, doi: /2010ja Introduction [2] The highly variable magnetopause size and shape reflects the upstream solar wind conditions controlling the geomagnetic activities. As the result of the interaction between supersonic solar wind and the Earth s magnetic field,the magnetopause has been observed by spacecraft since 1960 s and numerous empirical models have been developed from the spacecraft observations. Early theoretical models assumed that the standoff distance depended only on the solar wind dynamic pressure [e.g., Spreiter et al., 1966; Olson, 1969]. As more observation became available, it was found that the standoff distance is also affected by the IMF B Z [Fairfield, 1971], and so called static empirical models were established to fit spacecraft crossing positions of the magnetopause by using conic section or other functions [Fairfield, 1971; Howe and Binsack, 1972; Holzer and Slavin, 1978;Formisano et al., 1979]. As the number of spacecraft crossing increased, the empirical models were first parameterized as a function of solar wind pressure or IMF B Z [Sibeck et al., 1991; Petrinec and Russell, 1993] and, finally included contributions from both solar wind pressure and IMF B Z [e.g., Roelof and Sibeck, 1 National Center for Space Weather, China Meteorology Administration, Beijing, China. 2 Chinese Academy of Meteorological Science, Graduate University of Chinese Academy of Science, Beijing, China. 3 Department of Physics, Royal Military College of Canada, Kingston, Ontario, Canada. 4 Department of Geophysics and Geomatics, China University of Geoscience, Wuhan, China. 5 Department of Mathematics and Information Engineering, Puyang Vocational and Technical College, Puyang, HeNan, China. Copyright 2011 by the American Geophysical Union /11/2010JA ; Petrinec and Russell, 1996; Shue et al., 1997, 1998; Kuznetsov and Suvorova, 1998; Boardsen et al., 2000; Chao et al., 2002; Lin et al., 2010]. Besides solar wind dependence and IMF Bz dependence, the composite models of Boardsen et al. [2000] are also parameterized by the dipole tilt angle. Recently, Dušík et al. [2010] found a strong IMF cone angle dependence in the THEMIS magnetopause crossings. These models are also referred to as parameterized empirical models. [3] Most of the empirical models are restricted to lowlatitudes, that is, they only describe the shape and size of the magnetopause in the vicinity of the magnetospheric equatorial plane. They were developed typically based on an elliptic function [e.g., Roelof and Sibeck, 1993], the Shue model function [e.g., Shue et al., 1997], or a parabolic function [e.g., Kuznetsov and Suvorova, 1998]. The elliptic or parabolic functions are generally not appropriate for describing the distant nightside magnetopause, because ellipse is closed and parabola flares too much at the far nightside. In order to overcome the limitation of the elliptic function, Petrinec and Russell [1996] used the inverse trigonometric functions to describe the nightside magnetopause, and Kawano et al. [1999] used a cylinder to represent the tail magnetopause from the position where its transverse radius reaches the maximum. A good choice for the fitting function was introduced by Shue et al. [1997], which can produce an open or a closed magnetopause, and is generally considered appropriate for the distant night side magnetopause. [4] The parameterized empirical models are limited by a number of factors, such as the number of magnetopause crossings, and data resolution of upstream solar wind, the scope of application. Most spacecraft crossings of the magnetopause occur relatively close to the magnetospheric equatorial plane, thus severely restricting the datasets avail- 1of21

2 able to constrain the high latitude magnetopause models [Boardsen et al., 2000; Lin et al., 2010]. Boardsen et al. [2000] appears to be the first model to use high latitude observation data of the crossings from Hawkeye, and to include the influence of the Earth s dipole tilt angle in addition to solar wind dynamic pressure and IMF B Z. Unfortunately, this model is only valid near the subsolar region and at high latitudes in the near Earth region with X GSM 5 R E and is not appropriate for the low latitudes away from the subsolar region. [5] An additional cause of uncertainty inherent in the empirical magnetopause modeling, discussed, for example by Lin et al. [2010], is the treatment of the time delay between the solar wind measurements and their effect on the magnetopause. Some models did not consider the time shift from the upstream satellite to the magnetopause crossing and used the upstream solar wind parameters with 1 hour resolution [e.g., Roelof and Sibeck, 1993]. Other models assumed that the time delay was constant [e.g., Shue et al., 1997], while in reality the time delay depends on the highly variable solar wind conditions [King and Papitashvili, 2004]. Chao et al. [2002] used the uniform propagation method to calculate the solar wind delay, however this method has deviations for the calculated propagation time of the solar wind. [6] The empirical models are also significantly limited by the relatively narrow range of the solar wind conditions under which the most of the magnetopause crossings occurred. Thus, these models should be used with extreme caution for non typical solar wind conditions. Nevertheless, the models of Petrinec and Russell [1996], Shue et al. [1998], and Chao et al. [2002] have been used with some success to forecast the magnetopause crossing of the geosynchronous orbit under the extreme solar wind conditions [Yang et al., 2002]. A recent model developed by Lin et al. [2010] made efforts to overcome some of these disadvantages for previous empirical models. They used hourly solar wind parameters from OMNI data, and calculate the shift time of solar wind propagating from the upstream satellite to the magnetopause crossings by matching the clock angle of the IMF or the interplanetary plasma parameter variable profile with that of the magnetosheath. This approach can compensate for the propagation time errors inherent in the uniform propagation method, but is applicable only when the auxiliary satellite is sufficiently close to the satellite responsible for the magnetopause crossing identification. In summary, the empirical models introduce all sorts of hard to assess averaging effects which may or may not be compensated by the long history of observations. [7] An alternative to using satellite measurements to identify magnetopause location is to employ a physics based global circulation model. A physics based model can easily overcome some of the disadvantages discussed above, for example, (1) it free of any orbital bias that is present in all empirical models based on in situ satellite observations; (2) removing the wind shock effect, which becomes strong as one moves down the tail. The directional accuracy of in situ solar wind velocity measurements is highly overstated; (3) It removes the observational bias of the empirical models associated with the relative sparsity of the magnetopause crossings database for the far tail. This approach gives precise upstream solar wind conditions and their corresponding magnetopause, and also allows straightforward investigation of extreme events where the sub solar point is near or inside geosynchronous orbit. A study of such extreme events, however, would require an adequate ring current model which was not included in our simulations. Therefore, we limited our analysis to moderate values of solar wind dynamic pressure. Although new uncertainties associated with the accuracy of the global model itself may be introduced, over the last decade the global circulation modeling has become a mature tool in space science, and its validation has been achieved under variety of conditions (see section 2). In this paper we use the numerical results from a physics based, global MHD model to examine the global magnetopause location and investigate the relationship between the shape and size of the global magnetopause and the solar wind conditions. Moreover, one aspect of the current work is to achieve further validation of the global circulation modeling by comparing the magnetopause location in the model with well established empirical models in the low latitude region of the magnetosphere. We shall modify the Shue model function and show the new function to be more suitable for global magnetopause description at both low and high latitudes. [8] The paper is arranged as follows. In section 2, we describe the global circulation model we use to simulate the solar wind magnetosphere coupling, and describe the automated procedure for the magnetopause detection. In section 3, we extend Shue97 model to a fully global model of the magnetopause with azimuthal dependence and correlate the magnetopause configuration parameters with the solar wind conditions. In section 4, we present the results of the magnetopause shape and size from the new global model in the equatorial and meridional planes, respectively. We also compare the new model results based on MHD simulations with empirical low/high latitude models (Shue98 and Boardsen model) and discuss the differences in the magnetopause location in the equatorial and meridional planes. The paper concludes with a brief summary of results in section Global MHD Model and Identification of the Magnetopause [9] We use the Space Weather Modeling Framework (SWMF), developed by University of Michigan, to simulate the interaction between solar wind and magnetosphere [see Tóth et al., 2005]. The SWMF consists of several numerical modules, such as the ideal MHD solver (BATS R US) [Powell et al., 1999; De Zeeuw et al., 2004], Ionospheric electrodynamics (IE) model [Ridley et al., 2002], and Rice Convection Model (RCM) [Toffoletto et al., 2003]. The SWMF family of models has been used extensively to study various solar wind influences on the magnetosphere, for example, convection under northward IMF [Song et al.,1999],imfb Y [e.g., Kabin et al., 2004], Parker spiral [Gombosi et al., 2000] conditions and storm dynamics [Zhang et al., 2007;Tóth et al., 2007]. Welling and Ridley [2010] discussed the validation of SWMF magnetic field and plasma using satellite measurements. In another example of model validation work, Rae et al. [2010] compared the locations of the observed and modeled open/ closed field line boundaries and further demonstrated the effectiveness and reliableness of SWMF. [10] In a typical simulation used in this work, the computational domain is defined by 70 R E X 20 R E, 60 R E Y, Z 60 R E, with the grid size of 1.25 R E. Inside 40 R E X 2of21

3 20 R E, 45 R E Y, Z 45 R E the grid size is R E and inside 25 R E X 12.5 R E, 30 R E Y, Z 30 R E, grid size is R E. The inner boundary is a sphere at 2.5 R E. [11] Our work requires automatic identification of the magnetopause surface in a significant number of numerical simulations corresponding to different steady state solar wind conditions. The traditional definition as the boundary between the volume of space controlled by the Earth magnetic field and that controlled by the IMF is, somewhat vague and can take to imply different things in different contexts. Sometimes it may be taken to refer to the openclosed field line boundary, but that would make magnetosheath to come down to all the way to the Earth in the cusps. Another possibility is to associate the magnetopause with the current layer, which the streamline method can identify but probably has difficulty in identifying the cusps or maps out the outer boundary of regions. However, those two definitions are different, and some streamlines starting in the solar wind actually cross all the way into the closed field line region (otherwise we would not have magnetospheric convection). Around the subsolar point all these definitions will be extremely close, but in the tail they may be different (and possibly affected by the numerical viscosity and resistivity). It is hard to say that any single one of these definitions is more physical than any other, but some of them might be more appropriate for different problems. For the magnetopause identification in global model results, Palmroth et al. [2003] presented an automated technique based on the streamlines, and more recently Němeček et al. [2011] developed a procedure by searching for the peaks of the current density, and maxima of density and velocity gradients. It has been shown that the density gradients, especially in the magnetotail region, are not sufficiently sharp for an automated recognition of the magnetopause surface. In this paper, we do not discuss the detail of the cusp region and will use the streamline method like Palmroth et al. [2003]. The magnetopause is identified in the simulation from the solar wind streamlines diverting around the magnetosphere. A set of streamlines is created at X GSM =20R E, outside the bow shock. The initial streamline grid is defined in the Y GSM Z GSM plane in a R E box with the X GSM axis at the center. The distance between neighboring initial streamlines is R E along each axis, giving in total 4,225 streamlines. [12] At the beginning of the magnetopause search, the set of 4,225 streamlines is mapped in steps of R E in the anti Sunward direction. For each R E step in the X GSM direction, the algorithm searches for a void of streamlines starting from the X GSM axis. Finding such a void of streamlines indicates that the streamlines have started to bend around the magnetosphere. When the void becomes larger than 1 R E, the algorithm finds an inner boundary, which defines the magnetopause in the Y GSM Z GSM plane. The search for the inner boundary starts by dividing the Y GSM Z GSM plane into 10 sectors. In each sector the streamlines are sorted by their distance from the X GSM axis. Three closest streamlines are excluded, and the magnetopause is defined to be the arithmetic mean of the radii of the four next closest streamlines. The three closest streamlines are excluded from the magnetopause determination because some streamlines may enter the magnetosphere and would hence produce erroneous results, which was clearly seen when the method was tested. However, varying the number of streamlines excluded from the evaluation and the number of streamlines used to compute the mean produced only small changes to the magnetopause location, which gives confidence in the selection method. The validation of our method are also carefully manually assessed. With this procedure, the resulting resolution of the magnetopause surface is R E in the X GSM direction and 10 in the Y GSM Z GSM plane. Above method was detailed by Palmroth et al. [2003]. [13] In our calculation, we consider the following combinations of IMF B Z and solar wind dynamic pressure (45 data set): B Z = 10, 7.5, 5, 2.5, 0, 2.5, 5, 7.5, 10nT, and D P =1,2,3,4,5nPa, IMF B X = B Y =0nT and 0 dipole tilt angle. Figure 1a illustrates one example of the global magnetopause surface identified from the SWMF simulation for zero IMF and solar wind dynamic pressure 1nPa strength (0nT, 1nPa). Figure 1b shows the density contour and the identified magnetopause (red line) in three coordinate planes. The magnetopause search is carried out in the Y GSM Z GSM plane until X GSM reaches 40 R E. Note that the magnetosphere itself does not need to be aligned with the X GSM axis, and the magnetospheric boundary in the Y GSM Z GSM plane is not necessarily circular. 3. Three Dimensional Model [14] We describe the coordinate position of the magnetopause with (r,, ), and the design formulas are: p r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 þ Y 2 þ Z 2 ; ð1þ ¼ arccosðx =rþ; ð2þ 8 p cos 1 Z= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < Y 2 þ Z 2 ; Y > 0 ¼ p cos 1 Z= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð3þ : Y 2 þ Z 2 ; Y 0 where r is the radial distance from the Earth, is the angle between the radial direction and the Earth Sun line, is the azimuth angle Construction of Surface Functions [15] In this section we shall extend Shue97 model to a fully global model of the magnetopause and show how the configuration parameters change with the azimuth angle. The magnetopause identified from solar wind streamline consists of 36 curves, each curve corresponds to one azimuth angle, we fit every curve with Shue97 model function (hereafter called as Shue97 function). We use Shue97 function because of its simplicity and advantage mentioned in Introduction. The Shue97 function can represent open or closed magnetopause shape by changing the level of the tail flaring a: 2 r ¼ r 0 ; ð4þ 1 þ cos where r 0 and a are the standoff distance and the flaring angle, and hereafter they are called as configuration parameters of the magnetopause. This model function produces a magnetopause which is closed when a < 0.5, asymptotes to a finite 3of21

4 Figure 1. (a) One example of the global magnetopause surface identified from the SWMF simulation by using the streamline tracing method for zero IMF and solar wind dynamic pressure 1nPa strength (0nT,1nPa). The magnetopause search is carried out in the YZ plane until X reaches 40 R E. (b) The density contour and the identified magnetopause (red line) in three coordinate planes (XY, XZ, andyz(x =0)). tail radius for a = 0.5, or expands with increasing distance when a > 0.5. In the series models based on Shue97 function, the configuration parameters (r 0, a) are functions of dynamic pressure D P and IMF B Z. Here we extend this model to describe the global magnetopause shape, so the parameters (r 0, a) will also depend on the azimuth angle. Because of the existence of the cusp, we fit every curve with dayside and nightside to obtain two groups of configuration parameters (r 0, a). Figure 2 gives the magnetopause in the equatorial plane and meridional plane under solar wind conditions (0nT, 1nPa). [16] Following Shue et al. [1997], we first rewrite equation (4) in a linear form: 2 lnðþ¼ln r ðr 0 Þþ ln ; ð5þ 1 þ cos Figure 2. The fitting results for the magnetopause in the (left) equatorial and (right) meridional planes under solar wind conditions (0nT, 1nPa). The squares stand for the magnetopause curve identified from the SWMF simulation, the lines stand for the fitting results. 4of21

5 Figure 3. (left) The fitting results for the relationship (a) between r 0 and azimuth angle and (right) between a and azimuth angle under solar wind conditions (0nT, 1nPa). The up/down triangles stand for the relationship between configuration parameters and solar wind conditions of the dayside and nightside magnetopause respectively, the solid/dotted lines stand for the fitting results of the dayside and nightside magnetopause respectively. then we fit the terms ln(r) and ln(2/(1 + cos)). It should be noted that for the observation data fitting in empirical models, the subsection treatment has to be used because of the non one to one corresponding relationship between configuration parameters of the magnetopause and the observed solar wind data. This treatment will introduce non physical effects. Here the fitting does not need the subsection treatment for the data in a physics based model. [17] We assume that the magnetopause is symmetric about the X axis in the equatorial and meridional planes, so the configuration parameters (r 0, a) are periodic functions of, with the period equal p. To a good accuracy, we can approximate (r 0, a) by cosine functions: r 0 ¼ a 0 þ a 1 cos 2 ; ¼ b 0 þ b 1 cos 2 : Figure 3 gives the fitting results for one example corresponding to IMF B Z =0nT and solar wind dynamic pressure D P =1nPa. Figure 3 shows that the assumed expansions equations (6) and (7) can adequately describe the relationship between two groups (r 0, a) and. The coefficients are parameterized by the solar wind pressure and IMF B Z. The standoff distance of the nightside magnetopause is only used to get the nightside magnetopause shape. Under other solar wind conditions the relationship of (r 0, a) and is cosine functions too, similar to the case of (0nT, 1nPa). The existence of cusp requires a model to be able to describe the magnetopause shape at dayside and nightside, so we divide the magnetopause into two segments with cusp and construct two surface equations based on Shue97 function to fit the two segments separately. [18] At first, we fit magnetopause in the equatorial plane with Shue97 function to obtain configuration parameters (r 0, a): 2 r ¼ r 0 : ð8þ 1 þ cos ð6þ ð7þ [19] In Figure 3 the standoff distance and the flaring angle of the dayside magnetopause are cosine functions of, but the standoff distance of the dayside magnetopause should be constant when is changing in theory, so we assume: The Standoff distance is r 0 ; the flaring angle in the equatorial plane is a, the factor of trigonometric function of is b 1. We can fit the dayside magnetopause with following surface equation: þ1cos 2 ; ð9þ r ¼ r þ cos where b 1 is the rate of change in flaring angle with respect to, the flaring angle corresponding to is a + b 1 cos 2, the flaring angle of the magnetopause in the meridional plane is a + b 1. [20] For the nightside magnetopause the standoff distance and the flaring angle are also cosine functions of, similarly we assume: The standoff distance in the equatorial plane is r 0, the factor of cosine function of is r c ; the flaring angle in the equatorial plane is a, the factor of cosine function of is b 2. The nightside magnetopause can be fitted by using following surface equation: r ¼ r 0 þ r c cos 2 2 þ2cos 2 ; ð10þ 1 þ cos where r c is the rate of change in standoff distance with respect to, the standoff distance corresponding to is r 0 + r c cos 2, the standoff distance of the magnetopause in the meridional plane is r 0 + r c. b 2 is the rate of change in flaring angle with respect to, The flaring angle corresponding to is a + b 2 cos 2, the flaring angle of the magnetopause in the meridional plane is a + b 2. As mentioned above, the standoff distance of nightside magnetopause is only used to represent nightside magnetopause. [21] Now we have obtained a group of new configuration parameters (r 0, a, b 1, r c, b 2 ) corresponding to solar wind conditions. In this group, r 0 represents the size and a represents the shape of the whole magnetopause, b 1 represents the azimuthal asymmetry of the dayside magnetopause, r c and b 2 represents the azimuthal asymmetry of the nightside magnetopause. 5of21

6 Figure 4. (a) The three dimensional magnetopause from the new model for solar wind condition IMF Bz =0nT and Dp =1nPa. (b) The density contour and magnetopause projection (red line) on the equatorial, meridional and terminator plane Relationship Between the Magnetopause Configuration Parameters and Solar Wind Conditions [22] Now we identify the relationship of the parameters of our model (r 0, a, b 1, r c, b 2 ) and the upstream solar wind conditions. Schield [1969] suggested that r 0 and D P satisfy r 0 = kd P 1/a, where k is a constant, and a = 6.0 if the magnetospheric magnetic pressure is balanced by solar wind dynamic pressure. More recently, Lin et al. [2010] and Dušík et al. [2010] have shown that an exponent closer to 1/5 is more accurate. The fitting results for the relationship of (r 0, a) and the solar wind conditions by Shue et al. [1997] is: r 0 ¼ ða 1 þ a 2 B Z ÞD 1=a3 P ; ð11þ ¼ b 1 þ b 2 B Z þ b 3 D P þ b 4 B Z D P ; ð12þ where (a 1, a 2, a 3, b 1, b 2, b 3, b 4 ) are unknown coefficients. We can assume the relationship between(r 0, r c ) and (B Z, D P ) should be similar to equation (11), the relationship between (a, b 1, b 2 ) and (B Z, D P ) is similar to equation (12), that is: 1=a r 0 ¼ ða 1 þ a 2 B Z ÞD 3 P ; ð13þ ¼ b 1 þ b 2 B Z þ b 3 D P þ b 4 B Z D P ; ð14þ 1 ¼ c 1 þ c 2 B Z þ c 3 D P þ c 4 B Z D P ; ð15þ 1=d r c ¼ ðd 1 þ d 2 B Z ÞD 3 P ; ð16þ 2 ¼ e 1 þ e 2 B Z þ e 3 D P þ e 4 B Z D P : ð17þ [23] After we apply the multiple parameter fitting for our data set, we find that the best fit function is: 8 < ð11:494 þ 0:0371B Z ÞD 1=5:2 P ; B Z 0 r 0 ¼ ; ð18þ : ð11:494 þ 0:0983B Z ÞD 1=5:2 P ; B Z < 0 8 < 0:543 0:0225B Z þ 0:00528D P þ 0:00261B Z D P ; B Z 0 ¼ ; : 0:543 þ 0:0079B Z þ 0:00528D P 0:00019B Z D P ; B Z < 0 ð19þ 8 < 0:263 þ 0:0045B Z 0:00924D P 0:00059B Z D P ; B Z 0 1 ¼ ; : 0:263 0:0259B Z 0:00924D P þ 0:00256B Z D P ; B Z < 0 ð20þ 6of21

7 Figure 5. The standoff distance r 0 as functions of solar wind conditions (B Z, D P ), The squares are the results from our model, the up triangles are the results from Shue98 model, and the down triangles are the results from Boardsen model. The standoff distance r 0 (left) as a function of IMF B Z, and (right) as a function of D P. 8 < ð 1:191 0:034B Z ÞD 1=5:2 P ; B Z 0 r c ¼ ; ð21þ : ð 1:191 0:189B Z ÞD 1=5:2 P ; B Z < 0 8 < 0:0924 þ 0:0121B Z 0:00115D P 0:00115B Z D P ; B Z 0 2 ¼ : : 0:0924 0:0069B Z 0:00115D P þ 0:00115B Z D P ; B Z < 0 ð22þ The correlation coefficient of every fitting reaches 95%, we can conclude our assumption is reasonable. Note that the 1/5.2 exponent for the pressure dependence of the magnetopause location is very consistent with the empirical model from observation data by Lin et al. [2010] and Dušík et al. [2010]. [24] Equations (9) and (10), and equations (18) (22) form the final three dimensional magnetopause model, which allow us to easily predict the global magnetopause shape for 7of21

8 Figure 6. The flaring angle a as functions of solar wind conditions (B Z, D P ), the squares are the results from our model, and the up triangles are the results from Shue98 model. (left) The flaring angle a as a function of IMF B Z, and (right) a as a function of D P. any given solar wind conditions. That is, from solar wind B Z and D P, we can calculate the (r 0, a, b 1, r c, b 2 ) from equations (18) (22), then the location parameters (r,, ) of the magnetopause can be obtained from equations (9) and (10) based on the values of the new configuration parameters. Figure 4a shows the three dimensional magnetopause from the above new model under solar wind condition IMF B Z =0nT and D P =1nPa. Figure 4b shows the magnetopause projection on the equatorial, meridional, and terminator planes respectively. [25] Figure 5 shows r 0 as functions of solar wind conditions (B Z, D P ), the squares are the MHD subsolar point from the streamline method, the lines indicate the results from our model, and the up/down triangles are the results from Shue98/Boardsen model, respectively. The left panel of Figure 5 is the standoff distance r 0 as the function of IMF B Z, and the right panel is r 0 as the function of D P. Figure 5 demonstrates that the standoff distance in our model agrees very well with the Shue98 model, and the difference is within 0.5 R E. r 0 from Boardsen model are larger, but its rate 8of21

9 Figure 7. b 1 as functions of solar wind conditions (B Z, D P ), (left) b 1 as a function of IMF B Z, and (right) b 1 as a function of D P. of change is similar to our model. The left panels show that the standoff distance decreases with increasing south IMF B Z. This is expected: active reconnection on the dayside for southward IMF erodes the magnetopause and decreases the standoff distance. Conversely, with increasing northward IMF B Z, the standoff distance increases only slightly and the effects of northward IMF B Z on the standoff location r 0 are far smaller than these at the southward IMF B Z. In theory, when IMF B Z is northward, cusp and nightside reconnection processes could occur and the open flux is destroyed, decreasing the polar cap area and increasing the r 0. With increasing D P, the change of r 0 with IMF B Z decreases as expected. Above results are consistent with the Shue98 model. The right panels of Figure 5 show the standoff location r 0 in both models decreases with increasing D P.As compared to Shue98 model, the change of r 0 with D P is a little larger because the power index is 1/6.6 in Shue98 model but is 1/5.2 in our model. We also find that the variance with D P decreases for southward IMF B Z. These results agree well with the Shue98 model, especially for small D P. 9of21

10 Figure 8. r c as functions of solar wind conditions (B Z, D P ), (left) r c as a function of IMF B Z, and (right) r c as a function of D P. [26] Figure 6 shows a as functions of solar wind conditions (B Z, D P ), the squares are the MHD results from the streamline method, the lines indicate the results from our model, and the up triangles are the results from Shue98 model. The left panel of Figure 6 is a as functions of IMF B Z, and the right panel is a as functions of D P. Let s look at a in the left panel, there are significant differences between the two models for a. In Shue98 model, the flaring angle a decreases with increasing northward IMF B Z, and increases with increasing southward IMF B Z. In our model, a also decreases with increasing northward IMF B Z, however its variance is larger than that from Shue98 model. Furthermore, as southward IMF B Z increases, a still decreases, which is contrary to Shue98 model result. We also find that with increasing D P, the slope of a with IMF B Z decreases. a in the right panel of Figure 6 again show differences between the numerical model result and Shue98 model result. Generally to say, the a values in Shue98 empirical model are larger than in our model. For the IMF B Z = 0, both models give comparable a. But when IMF B Z is northward or southward, the difference becomes large. On the other hand, the variance properties of a with D P in both models 10 of 21

11 Figure 9. b 2 as the function of solar wind conditions (B Z, D P ), (left) b 2 as a function of IMF B Z, and (right) b 2 as a function of D P. are very similar, increasing with increasing D P, and having comparable variance ratio. When the solar wind pressure is increased, the compressed magnetosphere brings about smaller standoff distance and lowering magnetopause flank, but the magnetopause flank lowers slower than the decrease of standoff distance, resulting in the increase of the magnetopause flaring angle. We also find that the slope of a with D P is a little larger when IMF B Z is northward. [27] Figure 7 shows b 1 as functions of solar wind conditions (B Z, D P ), the squares are the MHD results from the streamline method, the lines indicate the results from our model. The left panel is the b 1 as functions of IMF B Z, and the right panel is b 1 as functions of D P. Figure 7 shows b 1 < 0, that is to say, the flaring angle is largest in the equatorial plane and smallest in the meridional plane. The left panels show the absolute value of b 1 decreases with increasing southward IMF B Z, with increasing northward IMF B Z, the absolute value of b 1 decreases and the effect of northward IMF B Z on b 1 are smaller than that under the southward IMF B Z. The right panels show the absolute value of b 1 increases slightly with increasing D P. 11 of 21

12 Figure 10. Magnetopause comparison in the equatorial plane among our model and Shue98 model. The solid line is from our model, the dashed line is from the Shue98 model, the arrowhead line is streamline, the squares is the largest current density. The color represents the density contour from SWMF simulation and the external boundary of dark blue area clearly shows the magnetopause. [28] Figure 8 shows r c as functions of solar wind conditions (B Z, D P ), the squares are the MHD results from the streamline method, the lines indicate the results from our model. The left panel of Figure 8 is the r c as a function of IMF B Z, and the right panel is r c as a function of D P. Figure 8 shows r c > 0 when B Z 7.5nT, and r c < 0 under other conditions, that is to say, the standoff distance of magnetopause is smallest in the equatorial plane and largest in the meridional plane when B Z 7.5nT, under other conditions is largest in the equatorial plane and smallest in the meridional plane. The left panels show r c increases with increasing southward IMF B Z. With increasing northward IMF B Z, r c decreases and the effects of northward IMF B Z on r c are smaller than these at the southward IMF B Z. The right panels show r c slightly increases with increasing D P. [29] Figure 9 shows b 2 as functions of solar wind conditions (B Z, D P ), the squares are the MHD results from the streamline method, the lines indicate the results from our model. The left panel of Figure 9 is the b 2 as a function of IMF B Z, and the right panel is b 2 as a function of D P. Figure 9 shows b 2 > 0, that is to say, flaring angle is smallest in the equatorial plane and largest in the meridional plane. The left panels show b 2 increases with increasing southward IMF B Z. With increasing northward IMF B Z, b 2 increases and the effects of northward IMF B Z on b 2 are larger than these at the southward IMF B Z. The right panels show b 2 slightly decreases with increasing D P. 4. Discussion 4.1. Magnetopause in the Equatorial and Meridional Plane [30] Based on the three dimensional magnetopause model, we can easily obtain the equation of the magnetopause in the equatorial plane. In the equatorial plane the azimuth angle is = p/2, we have: 2 r ¼ r 0 : ð23þ 1 þ cos [31] Figure 10 shows a comparison of the magnetopause in the equatorial plane among our model and Shue98 model. The solid line is from our model, the dashed line is from the Shue98 model, the arrowhead line is the streamline, and the squares are the largest current density. The color in Figure 10 represents the density contour and the external boundary of dark blue area clearly shows the magnetopause. It can be seen that our model agrees with the streamline very well, two model results are very close to each other especially when the IMF is northward, the magnetopause locations from the largest current 12 of 21

13 Figure 11. Magnetopause comparison in the meridional plane among our model and Boardsen model. The solid line is from our model, the dashed line is from the Boardsen model, the arrowhead line is streamline, the squares is the largest current density. The color represents the density contour from SWMF simulation and the external boundary of dark blue area clearly shows the magnetopause. density are in the middle between the two models results and consistent with our model. The standoff distance from our model is a little smaller (hard to see in the figure) and the difference is less than 0.5 R E, for increasing D P the difference for the two models becomes smaller. The flaring angle from our model is a little smaller than from Shue98 model, and this difference is larger when IMF B Z <0,buttheresultfromthe largest current density is closer to our model. [32] Similarly, we also can easily obtain the equation of the magnetopause in the meridional plane. In the meridional plane the azimuth angle is = 0, we have: 2 r ¼ r 0 1 þ cos r ¼ ðr 0 þ r c Þ ð þ1 Þ 2 1 þ cos ð þ2 Þ ; ð24þ : ð25þ [33] Figure 11 shows a comparison of the magnetopause in the meridional plane among our model (solid) and Boardsen model (dashed), the arrowhead line is the streamline, and the squares are the largest current density. The color in Figure 11 represents the density contour and the external boundary of dark blue area clearly shows the magnetopause. It can be seen that our model agrees very well with streamline especially when the IMF is northward, the two models give reasonably consistent magnetopause locations, and the magnetopause locations from the largest current density are in the middle between the two models and consistent with our model. The subsolar distance in the Boardsen is a slightly larger than that in our model. Though there are some differences between these two models, the magnetopause locations from the largest current density are closer to our model. The comparison result from these two figures shows our model s result is reasonable, this also demonstrates the SWMF model can provide a reliable tool for the magnetopause. [34] The comparison of the magnetopause curves between the equatorial plane and the meridional plane is shown in Figure 12, where the solid lines represent the magnetopause in the equatorial plane and dashed lines for the magnetopause in the meridional plane. Within the solar wind conditions we considered, the nose side flaring angle in the meridional plane is smaller than in the equatorial plane, but the tail side flaring angle in the meridional plane is larger than in the equatorial plane. This difference is significant for both northward and southward IMF B Z. On the other hand, the solar wind dynamic pressure produces little asymmetry between the equatorial and meridional planes, which is 13 of 21

14 Figure 12. The comparisons of the magnetopause curves between the equatorial plane and the meridional plane, the solid lines represent the magnetopause in the equatorial plane and dashed lines for the magnetopause in the meridional plane. expected because D P effects are symmetric while B Z effects are clearly anisotropic. [35] Figure 13a shows the magnetopause curve family in the equatorial plane from our model for given solar wind conditions: left panel for (D P =1nPa; B Z =5,0, 5nT) and right panel for (B Z =0nT; D P =1,3,5nPa). For increasing southward IMF B Z, the standoff location r 0 slightly decreases, and the flaring angle decreases; while for increasing northward IMF B Z, the standoff location r 0 slightly increases, and the flaring angle decreases. From the right panel of Figure 13a, we see that for increasing D P, the standoff location obviously decreases but the flaring angle almost does not change. Above results tell us that IMF B Z mainly affects the flaring angle (the magnetopause shape) and has little effects on the magnetopause size; while the solar wind D P mainly affects the magnetopause standoff distance (the magnetopause size) and almost does not affect the magnetopause shape. [36] Figure 13b shows the magnetopause curve family in the meridional plane for given solar wind conditions: left panel for (D P =1nPa; B Z =5,0, 5nT) and right panel for (B Z =0nT; D P =1,3,5nPa). When southward IMF B Z increases, the standoff location r 0 has a slightly decrease, but the flaring angle increases; while for increasing northward IMF B Z, the standoff location slightly increases, and the flaring angle decreases. From the right panel of Figure 13b, we see that for increasing D P, the standoff location r 0 has an obvious decrease but the flaring angle hardly changes at all. Therefore, once again, IMF B Z mainly affects the flaring angle (the magnetopause shape) and has little effect on the magnetopause size; in contrast the solar wind D P mainly affects the magnetopause standoff distance (the magnetopause size) but little the magnetopause shape. Moreover, the magnetopause shapes in these two planes for the different IMF B Z show little differences, which implies that the solar wind mainly controls the size of magnetopause but hardly affects the magnetopause shape. This conclusion is consistent with Shue98 model. [37] Figure 14 shows the magnetopause curves in the = p/3 plane, and Figure 15 shows the magnetopause curves in the = p/6 plane, the squares are the MHD results from the streamline method, the lines indicate the results from our model. It can be seen that our model agrees with the streamline method very well, with the largest discrepancies occurring in the = p/3 plane; however, even those are not significant Azimuthal Asymmetry of the Magnetopause [38] Figure 16 shows the magnetopause curves in the YZ plane, the solid lines represent the curves corresponding to = p/4 and dotted lines represents the curves corresponding to =3p/4. The curves corresponding to = p/4 is an oval 14 of 21

15 Figure 13. (a) The magnetopause curve family in the equatorial plane from our model for given solar wind conditions: for (left) (D P =1nPa; B Z =5,0, 5nT) and(right)(b Z =0nT; D P =1,3,5nPa). (b) The magnetopause curve family in the meridional plane from our model for given solar wind conditions: for (left) (D P =1nPa; B Z =5,0, 5nT) and (right) (B Z =0nT; D P =1,3,5nPa). with long axis at Y axis, b 1 < 0 cause that magnetopause in the meridional plane is more closed than in the equatorial plane. The curves corresponding to =3p/4 is an oval with long axis at Z axis, the whole effect of r c < 0 and b 2 >0is that magnetopause in the meridional plane is more open than in the equatorial plane. [39] Figure 17a shows the magnetopause curve family in the YZ plane corresponding to = p/4 for given solar wind conditions: The left panel for (D P =1nPa; B Z =5,0, 5nT) and the right panel for (B Z =0nT; D P =1,3,5nPa). The left panel shows that B Z slightly affects the asymmetry of the dayside magnetopause, the right panel shows that D P never affect asymmetry, this result is expected because D P effects are symmetric as mentioned above. [40] Figure 17b shows the magnetopause curve family in the YZ plane corresponding to =3p/4 for given solar wind conditions: The left panel for (D P =1nPa; B Z =5,0, 5nT) and the right panel for (B Z =0nT; D P =1,3,5nPa). The left panel shows when northward or southward B Z is increasing the asymmetry increases obviously. From the right panel we can see D P does not affect the asymmetry of the nightside magnetopause, either. The azimuthal asymmetry of the nightside magnetopause is the total composite effect of r c and b 2. For example: When B Z = 5nT, r c is much larger than that when B Z =5nT, on the other hand, when B Z = 5nT, b 2 is little smaller than that when B Z =5nT, In this situation the effect of r c dominates, so we see the asymmetry when B Z = 5nT is larger. 5. Conclusions [41] In this paper, we analyze the numerical results from a physics based, global MHD model (SWMF) to examine solar wind and IMF control of the magnetopause location and shape. We use streamline technique to identify the magnetopause location, then use the multi parameter fitting method to expand the 2D Shue97 function to a new function which can describe the global magnetopause size and shape at both low and high latitudes. Finally we compare our results with the low latitude model (Shue98) and high latitude model (Boardsen), developed from the original satellite data. This comparison provides good foundation for future validation studies using individual events. Our models shows a significant dependence of the magnetopause standoff distance and other model parameters on IMF B z. Many of these were not taken into account in the earlier empirical models. Furthermore, recently Verigin et al. [2009] put into question the dependence on Bz regardless of the polarity. The main results from our numerical model are presented below: [42] 1. The Shue model function is extended from a curve equation to two surface equations with azimuth angle and introduce new configuration parameters (b 1, r c, b 2 ) in addi- 15 of 21

16 Figure 14. The magnetopause curves in the = p/3 plane, the squares are the MHD results from the streamline method, the lines indicate the results from our model. tion to (r 0, a). The new model is applicable for approximating the three dimensional magnetopause as functions of IMF B Z and solar wind D P. [43] 2. The standoff distance slightly decreases and a decreases with increasing southward IMF B Z (the latter contradicts to the present understanding, see following discussion). While increasing the northward IMF B Z, standoff distance slightly increases and a decreases. As increasing D P, the standoff location obviously decreases but a almost does not change (just slightly increases). [44] 3.b 1 increases with increasing southward or northward IMF B Z ;whend P increases, b 1 decreases slightly. r c and b 2 increases with increasing southward IMF B Z, while increasing the northward IMF B Z, r c decreases and b 2 increases; as increasing D P, r c increases slightly and b 2 decreases slightly. [45] 4. We find good comparison between the new model and older low latitude empirical model (Shue98) and high latitude empirical model (Boardsen), which is somewhat better for northward than southward IMF. Although we presently do not attempt a direct comparison of our model to satellite measurements, this result provides a foundation for the validation of the new model, since the empirical magnetopause models are derived from statistical analysis of satellite data. [46] 5. IMF B Z mainly affects the flaring angle (the magnetopause shape) and has smaller effects on the magnetopause size; in contrast, the solar wind D P mainly affects the standoff distance (the magnetopause size) and has little effect on the magnetopause shape. This conclusion is consistent with Shue98 empirical model. [47] 6. The magnetopause is not axisymmetric with respect to the X axis. The dayside magnetopause location in the YZ plane is an oval a little stretched in the Y direction. B Z slightly affects the asymmetry of the dayside magnetopause and D P does not affect the asymmetry. The nightside magnetopause location is an oval stretched in the Z direction. When the northward or southward B Z increases the asymmetry increases obviously and D P does not affect the asymmetry of the nightside magnetopause, either. [48] The result of flaring angle in the equatorial plane decreasing with the southward IMF B Z seems to contradict to present understanding. The increasing southward IMF B Z could transport a part of the dayside magnetic flux to the nightside. As a result, the nightside magnetopause moves outward and the flaring angle should increase. Possibly this is partially caused by the method we used for the magnetopause determination. As we know, one shortcoming of streamlines method is that a part of the streamlines can penetrate into the magnetosphere and the penetration would strongly depend on IMF Bz: For northward IMF Bz, the streamlines almost never penetrate into the closed field line region; For increasing southward IMF Bz, more streamlines could possibly penetrate into the magnetosphere, the iden- 16 of 21

17 Figure 15. The magnetopause curves in the = p/6 plane, the squares are the MHD results from the streamline method, the lines indicate the results from our model. 17 of 21

18 Figure 16. The magnetopause curves in the YZ plane, the solid lines represent the curves corresponding to = p/4 and dotted lines represents the curves corresponding to =3p/4. 18 of 21

19 Figure 17. (a) The magnetopause curve family in the YZ plane corresponding to = p/4 for given solar wind conditions: for (left) (D P =1nPa; B Z =5,0, 5nT) and (right) (B Z =0nT; D P =1,3,5nPa). (b) The magnetopause curve family in the YZ plane corresponding to =3p/4 for given solar wind conditions: for (left) (D P =1nPa; B Z =5,0, 5nT) and (right) (B Z =0nT; D P =1,3,5nPa). 19 of 21

20 tification of the magnetopause becomes less clear cut. However, based on Figures 10 and 11 we can see our model agrees with the largest current density result better than the empirical models, while the flaring angle in the meridional plane is increasing with the southward IMF B Z (refer to Figure 13b. However we admit, currently we do not know the exact reason why the flaring angle in the equatorial plane decreases with the southward IMF in global MHD results. It maybe even related to the rationality of the MHD model, and we would like to leave these as open questions and do the discussion in the future. [49] It should be noted that the proposed model does not include the effects of IMF B Y and dipole tilt, and is, therefore, symmetric with respect to XY and XZ planes. It has been found that positive tilts can lead to the tail magnetopause displacement as large as 4 R E [Šafránková et al., 2005, and references therein]. Also, our treatment of the cusp is not adequate to model the cusp indentation. In future work, we intend to improve the identification of the cusp indentation and develop a three dimensional asymmetric magnetopause model taking into account the effects of dipole tilt and all IMF components. [50] Acknowledgments. The authors would like to thank H. L. Liu for helpful discussions. This work was partially supported by the National Natural Science Foundation of China (grants and ), by the China Meteorology Administration (the CMA grant GYHY ), and by the China Public Science and Technology Research Funds Projects of Ocean ( ). Simulation results were obtained using BATS R US, developed by the Center for Space Environment Modeling, at the University of Michigan with funding support from NASA ESS, NASA ESTO CT, NSF KDI, and DoD MURI. This research has been enabled partially by the use of computing resources provided by Westgrid and Compute/Calcul Canada. 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