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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A01101, doi:10.1029/2007JA012426, 2008

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A Shock Fitting Procedure Based on Monte Carlo Calculations: Application to Slow Shocks C. C. Lin,1 H. Q. Feng,1,2 J. K. Chao,1 L. C. Lee,1 L. H. Lyu,1 and D. J. Wu2 Received 28 March 2006; revised 1 January 2007; accepted 10 October 2007; published 3 January 2008.

[1] Recent derived procedure of fitting the Rankine-Hugoniot shock jump relations for

fast MHD shocks (Lin et al., 2006) is here applied to slow-mode shocks. The present paper is a continuation of Lin et al. (2006). We modify the method to be applicable to slow shocks. We have tested the new method via a simulated slow shock. The result shows that the method can find a best fit solution that is close to the ‘‘true answer’’ as long as the systematic errors in the synthetic data are not too large. We also apply the method to one interplanetary slow shock and two slow shocks observed in the magnetotail. The results have been compared to the previous studies. The new method of shock fitting is demonstrated successfully for identifying slow shocks. Citation: Lin, C. C., H. Q. Feng, J. K. Chao, L. C. Lee, L. H. Lyu, and D. J. Wu (2008), A Shock Fitting Procedure Based on Monte Carlo Calculations: Application to Slow Shocks, J. Geophys. Res., 113, A01101, doi:10.1029/2007JA012426.

1. Introduction [2] The MHD slow shocks in space have been studied for more than 3 decades through spacecraft observations. Chao and Olbert [1970] first reported two forward (anti-solar) propagating interplanetary slow shocks observed by Mariner 5 with upstream slow-mode Mach numbers (Msl) of 1.5 and 1.8 respectively. Here Msl
Vn/Vsl, where Vn is the incident plasma flow speed in the shock frame of reference, and Vsl is the upstream slow-mode wave speed. Burlaga and Chao [1971] reported the first reversed slow shock in interplanetary space (0.9 AU) via the Pioneer 6 observation. The Mach number of this shock is Msl  1.2. At the region close to the Sun (0.3AU), Helios 1 observed a forward slow shock [Richter et al., 1985]. The calculated slow-mode Mach number is as large as Msl  3. On the other hand, Ulysses observed a forward and a reversed slow-mode shock within a co-rotating interaction region at 5 AU [Ho et al., 1998]. The slow Mach number of the forward shock is reported to be 1.1, and that of the reversed shock is 2.2. Recently, the Wind spacecraft had also observed several interplanetary slow shocks [e.g., Whang et al., 1996, 1998; Zuo et al., 2006]. The reported slow Mach number is 1.3– 1.7. In addition, the slow shock reported by Zuo et al. [2006] is found to be just in front of a magnetic cloud. [3] Many slow shocks have been observed in the geomagnetic tail. Most shocks were found at the mid- and distant tail [Feldman et al., 1984a, 1984b, 1985; Smith et al., 1984; Ho et al., 1994, 1996; Seon et al., 1995, 1996; Saito et al., 1995, 1998; Hoshino et al., 2000]. Several shocks were reported to be observed at the near-Earth tail region [Feldman et al., 1987; Eriksson et al., 2004], 1 2

Institute of Space Science, National Central University, Taiwan. Purple Mountain Observatory, Academic Sinica, Nanjing, China.

Copyright 2008 by the American Geophysical Union. 0148-0227/08/2007JA012426$09.00

however, Cattell et al. [1992] suggested that the discontinuities observed in the near-Earth tail are usually not slow shocks. The shock normal vectors of these observed shocks are usually, nearly the Y-Z (GSE) directions, while most shocks are reported to be quasi-perpendicular. For these shocks the slow-mode Mach number is 2.5– 5.5 [Feldman et al., 1984a, 1984b; Ho et al., 1994; Seon et al., 1995, 1996; Saito et al., 1998]. [4] Many shocks observed at the lobe-plasma sheet boundary are found to be associated with the Petschek-type reconnection in the tail [Feldman et al., 1984a, 1984b, 1985, 1987; Smith et al., 1984; Ho et al., 1994; Saito et al., 1995, 1998; Seon et al., 1995; Hoshino et al., 2000; Eriksson et al., 2004]. Ho et al. [1994] reported a slowshock pair with full set of signatures of Petschek-type reconnection. In statistical survey, Saito et al. [1995] found 32 slow shocks from 303 plasma sheet-lobe boundary crossings of the Geotail spacecraft in the mid- and distant tail during September 1993 and February 1994, while Ho et al. [1996] found 86 slow shocks over 439 plasma sheet-lobe crossings of the ISEE 3 spacecraft in the distant tail. [5] Some slow shocks are also found at the magnetopause [Walthour et al., 1994; Scudder, 2005]. However, cases are few. The shock investigated by Walthour et al. [1994] is reported to be associated with magnetic reconnection of the dayside magnetopause. [6] The shock analysis based on one spacecraft observation is in general be tested by satisfying the RankineHugoniot (R-H) relations. The well-known method is the magnetic coplanarity theorem [Colburn and Sonett, 1966]. One can determine shock normal only with up- and downstream averaged magnetic fields. This theorem is very frequently used [e.g., Chao and Goldstein, 1972; Chao and Hsieh, 1984; Winterhalter et al., 1985; Wu et al., 2000]. A mixed data method called velocity-magnetic field coplanarity was proposed by Abraham-Shrauner [1972] and also Chao [1970]. This method requires up- and downstream magnetic fields and velocities to determine the shock

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normal. Therefore the shock normal, magnetic fields, and velocity difference (V2  V1) lie on the same plane, called a coplanar plane. These two methods can be reliable if the data are accurate. However, the velocity-magnetic field coplanarity method should more reliable than the magnetic coplanarity theorem if the data have more or less errors, since the previous one requires not only magnetic fields but also velocities to be on the same plane. [7] Shock normal can also be determined by least squares minimization. A well-known method is minimum variance method (MVA), which is based on the conservation of the normal component of the magnetic field across the shock layer [Sonnerup and Cahill, 1967]. This method, however, has been applied mainly to tangential and rotational discontinuities [e.g., Lepping and Behannon, 1980; Sonnerup et al., 2004]. Recently, Horbury et al. [1998] suggested that when there are surface waves on a discontinuity, the estimated normal by MVA could be inaccurate. Knetter et al. [2003, 2004] also suggested that the MVA is much less reliable than has been previously assumed. When applied to shock waves, it is well-known that the eigenvector degeneracy of the MVA causes that the method is not useful for finding the normal of the shock. In order to remedy this difficulty, Scudder [2005] proposed a novel scheme called CVA (Coplanarity Variance Analysis), which in the search for an accurate geometry exploits the eigenvalue degeneracy in MVA, at planar structures, to enforce coplanarity. It is found that CVA is much better than MVA at finding the shock normal. [8] Least squares minimization is also apply for a full set or subset of the R-H relations [e.g., Lepping and Argentiero, 1971; Lepping, 1972; Vin˜as and Scudder, 1986; Szabo, 1994; Kessel et al., 1994]. This method requires not only the up- and downstream magnetic fields but the plasma velocities and densities to satisfy the R-H relations. However, a set of non-linear algebra system with multiple variables derived from the partial derivative should be solved from this problem. Sometimes it is difficult to obtain a unique solution. If the procedure does not force the variables to be within the limitation of data, one could obtain a solution that is far from the observed values. [9] Lin et al. [2006] propose a method of shock fitting based on the full set of the R-H relations and coplanarity property, within which they use a Monte Carlo calculation together with least squares technique. The method was applied to interplanetary fast shocks. The method uses a straight forward calculation with the variable values based on observation. It finds the solution of the R-H relations without taking care of a complicate set of non-linear algebra equations as in the previous least-squares methods described above. The solution that is close to the observed value and satisfies the R-H relations is guaranteed. In addition, the method finds the shock tangential vectors as well as the corresponding de Hoffmann-Teller (HT) frame of reference simultaneously. The frame is important for determining a shock. [10] The method can easily be modified to apply to slow shocks, since both these two types of shock satisfy the R-H relations and have a property of coplanarity. A slow-mode shock is different from a fast-mode shock in two aspects: 1) for a slow shock, the ratio of the down- to upstream tangential magnetic fields is smaller than unity; and 2) for a

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slow shock, in the HT frame of reference, the ratio of downto upstream tangential velocities is smaller than unity. The modification is described in detail in section 2. [11] This paper is a continuation of Lin et al. [2006]. In this paper, we test the proposed method via a simulated slow shock, and we also apply the method to the slow shocks observed in interplanetary space and the magnetotail. In section 3 one synthetic shock is used to do the hypothetical test, while in section 4 we apply the method to one interplanetary shock and two shocks observed in the geomagnetic tail. Finally in section 5 a conclusion is given.

2. Model [12] The model is based on ‘‘Method A’’ of Lin et al. [2006], where the up- and downstream magnetic fields (B1 and B2), plasma densities (r1 and r2), plasma velocity difference (W = V2  V1), plasma betas (b1 and b2), and temperature anisotropy parameters (x 1 and x 2) are considered as variables in the fitting procedure. Here, x is defined as x = 1  (pk  p?)/(B2/m0), where pk and p? are parallel and perpendicular plasma thermal pressure defined in the magnetic field-aligned coordinates, and m0 is the magnetic permeability in a vacuum. In their model the information from two spacecraft observations are included. The time difference due to shock propagating between two spacecraft, Dt, are taken into account. Their method generates a massive set of self-consistent solutions that satisfy the R-H relations from the data and finds one that is close to the time-averaged value. The one is called the best fit solution. [13] A slow-mode shock is different from a fast-mode shock in two aspects: 1) for a slow shock, the downstream tangential magnetic field is smaller than the upstream one, that is, B2  ^t /B1  ^t < 1, where ^t
±(B2  B1)/jB2  B1j is the shock tangential vector; and 2) for a slow shock, in the HT frame of reference, V*2  ^t/V*1  ^t < 1, where V*1 and V*2 are the up- and downstream plasma velocities in the HT frame. The expression of velocity difference (W) for the fitting method of Lin et al. [2006] is rewritten as follows: W ¼ Wn ^ n þ Wt^t ¼ MAn VA ð1  yÞ cos qbn ^ n þ MAn VA ð yu  1Þ sin qbn^t;

ð1Þ

where MAn
Vn/VAcosqbn is the upstream normal Alfve´n Mach number, where VA = B1/(m0r1)1/2 is the upstream n/B1) is the angle Alfve´n wave speed and qbn
cos(B1  ^ between the upstream magnetic field and the shock normal. In addition, we here define y
r1/r2, u
B2  ^t /B1  ^t. For a slow shock, u < 1 and y < 1. Thus we have Wt < 0, which is different from a fast shock where Wt > 0. It should also be noted that the expression of u in equation (12) of Lin et al. [2006] should be written as u



2 1=2 Bt2 m  cos2 qbn ¼ Bt1 sin2 qbn

ð2Þ

in order to include the solution for intermediate shocks (u < 0). To avoid ambiguity between slow/fast and intermediate shocks, numerically we suggest to estimate u by u = B2  ^t /B1  ^t instead of the expression on the right side of equation (2).

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Table 1. The ‘‘True Answer’’ of the Synthetic Slow Shock Parameter (GSE)

Value (5.0, 6.0, 4.0), (4.5, 1.8, 3.6) 7.5, 10.0 1.0, 1.0 1.06, 3.70 (0.772, 0.151, 0.618) (0.118, 0.989, 0.094) 0.750 0.384 0.688 0.288 51.8, 26.0 69.9 0.93, 0.81, 1.28, 0.84 (388, 13, 10), (400, 21.4, 0.4) (12, 34.4, 9.6) 335.5 (100, 100, 0) 29.2

Magnetic field B1, B2 (nT) Plasma density Neq1, Neq2 (cm3)a Anisotropy parameter x 1, x2 Plasma beta b1, b 2 Shock normal vector ^n Shock tangential vector ^t Density ratio y Tangential magnetic field ratio u Magnetic field intensity ratio m Tangential velocity ratio z = yu Shock normal angles qbn, qbn2 (deg.) Upstream Alfve´n speed VA (km/s) Mach numbers MAn, MAn2, Msl, Msl2 Flow velocities V1, V2 (km/s) Velocity difference W (km/s) Shock speed Vs (km/s) Spacecraft displacement DR (RE) Time difference Dt (min.)

a The plasma number density, Neq, is defined as Neq = r/mp, where mp = 1.6726  1027 kg is proton mass.

by another spacecraft at another time is used to test the fitting method discussed here. The assumed shock parameters, called ‘‘true answer’’, are listed in Table 1. We also provide the definitions of the parameters. GSE coordinates system is used. We arbitrarily assume that the two spacecraft are at two positions such that the vector displacement (DR
R2  R1, where R is position vector) at the GSE coordinates is (100, 100, 0) RE. We assume the shock is planar and the propagating speed is steady, thus, the two spacecraft observe the same shock at a time difference, Dt, of 29.2 min. This is calculated from the timing method that n is the Dt = DR  ^ n/Vs, where Vs = MAnVA + V1  ^ propagating speed seen in the spacecraft frame of reference. nj > 0, the shock is As can be seen from Table 1, Vs  jV1  ^ a forward-propagating shock. In addition, we assume that no inter-species slippage occurs, thus, the coplanarity is true for the shock. [17] The downstream normal Alfve´n Mach number MAn2 and the up- and downstream fast Mach numbers Msl and Msl2 can be expressed in terms of MAn, y, m, qbn, and the plasma betas. They are written as (see also Lin et al. [2006] for the derivation of fast-mode Mach numbers) pffiffiffi MAn2 ¼ MAn y;

[14] In a least squares approach, Lin et al. [2006] use a loss function defined by 16 variables, which is written as LðiÞ


 16  X Xk ðiÞ  hYk i 2 k¼1

sk

;

3. Hypothetical Test [16] A synthetic oblique slow shock assumed to be observed by one spacecraft at a specific time and, thereafter,

ð4Þ

2

ð3Þ

where Xk(i) represents the predicted values of the variables, and hYki represents the time-averaged values from observations. Here, notation i = 1, 2, , , N, indicates the calculated elements by Monte Carlo method. Moreover, sk is error obtained from observations. Here, the sampling and guessed systematic errors are included in sk. Therefore the best fit can be found by searching for the minimum value of L(i). The loss function in equation (3) is for two spacecraft observation. If only one spacecraft observation is available, the number of the variables reduces to 15. That is, in equation (3) only 15 terms are added. [15] This model can be used under an assumption only that slippage pressure contributed by the particle species other than protons is small in comparison to the thermal pressure. The modified R-H relations that involve the slippage pressure tensor will have additional terms in the conservation equation of tangential momentum flux. If the slippage is significant, the coplanarity can be no longer valid for the shock solution [Lin et al., 2006]. The coplanarity property was often used in the shock analysis [e.g., Lepping and Argentiero, 1971; Abraham-Shrauner, 1972; Chao et al., 1995; Wu et al., 2000; Berdichevsky et al., 2000; Scudder, 2005]. Lin et al. [2006] have investigated the pressure tensor due to the alpha particle slippage for 54 observed fast shocks in the ecliptic plane using the WIND spacecraft data. It is found that the slippage pressures are very small in comparison with the thermal pressures. The slippage effect, therefore, can be ignored.

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31=2 2 2MAn

2

cos qbn 6 7 Msl ¼ 4 h i1=2 5 2 ð1 þ gb1 =2Þ  ð1 þ gb1 =2Þ 2gb 1 cos2 qbn

; ð5Þ

2

31=2 2 2MAn2

2

2

cos qbn =m 6 7 Msl2 ¼ 4 h i1=2 5 2 2 2 ð1 þ gb2 =2Þ  ð1 þ gb2 =2Þ 2gb 2 cos qbn =m ð6Þ

where g = 5/3 is used. [18] We test the fitting procedure for slow shock as the way Lin et al. [2006] tested the procedure for fast shock. First, we generate a set of up- and downstream data which is simulated by giving some random errors upon the ‘‘true answer’’ values. The given errors represent the systematic and sampling errors obtained from observation. That is, data = true answer + systematic error + random error. In each side of the shock, only the random error is time-dependent, and the mean data value as well as the systematic error is embedded in the data fluctuations (random error). Here, the data have a systematic error, and the size of this error is 1/2 of the sample standard deviations (SD). The size is not the same for each variable, and it is randomly assigned. In addition, the up- and downstream data are connected using a hyperbolic tangent function, in which we simulate a shock transition layer. However, the data in the shock layer are not used in the fitting method. The simulated data are shown in Figure 1. Here, the equivalent proton number density, Neq, is defined as Neq
r/mp, where mp = 1.6726  1027 kg is the proton mass. For the up- and downstream regions two time intervals (indicated by the vertical lines) of 5 min are

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Figure 1. The simulated data for the synthetic slow shock. From up to bottom are the magnetic field, plasma density, anisotropy parameter, plasma beta, and velocity. It is assumed that GSE coordinate system is used. Neq = r/mp, is defined as the equivalent proton number density, where mp is the proton mass. The vertical lines indicate the data interval for the upstream (t1  t2) and downstream (t3  t4) sides. The shock layer is between t = 6 and t = 7 min.

selected for analysis. The time-averaged parameter values and the SDs are shown in Table 2. This selection is arbitrary. We simulate the data in a manner that in the up- and downstream sides the fluctuations are stable. If we select a different time interval, only slight difference in the calculated mean and standard deviation can be made. For example, we select from 3 to 6 min for the upstream and 7 to 10 min for the downstream. The calculated mean and standard deviation of the upstream magnetic field are (5.00, 5.91, 4.19) and (0.2, 1.0, 0.2), respectively. In addition, for the downstream magnetic field the values are (4.54, 1.77, 3.50) and (0.2, 0.5, 0.5), respectively. They have a slight difference from the values shown in Table 2. This slight difference is contributed by the random error (fluctuations) described above. In addition, the fitting method

Table 2. The Up- and Downstream Time-Averaged Values and SDs of the Synthetic Slow Shock Parameter (GSE)

Time-Averaged

SD

B1 (nT) B2 Neq1 (cm3) Neq2 x1 x2 b1 b2 V1 (km/s) V2 (km/s) Dt (min.)

(5.03, 5.72, 4.16) (4.56, 1.69, 3.46) 7.7 9.8 1.0 1.0 0.99 3.68 (388.7, 13.8, 10.3) (400.7, 21.1, 0.6) 29.2

(0.2, 1.1, 0.3) (0.2, 0.5, 0.5) 0.6 0.5 0.01 0.01 0.2 0.2 (3, 5, 4) (3, 4, 5) 0.3

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Table 3. The Best fit Values for the two Spacecraft and Single Spacecraft Methods Parameter (GSE) B1 (nT), B2 (nT) Neq1, Neq2 (cm3) x 1, x 2 b 1, b 2 W (km/s) Dt (min) ^n ^t y u m z qbn, qbn2 (deg.) MAn, MAn2, Msl, Msl2 Vs (km/s)

Case 1: Two spacecraft Case 2: Single spacecraft method sethod (5.01, 5.89, 4.11), (4.51, 1.69, 3.67) 7.5, 10.0 1.0, 1.0 1.04, 3.63 (12.2, 34.4, 10.1) 29.3 (0.768, 0.156, 0.621) (0.117, 0.988, 0.102) 0.747 0.376 0.691 0.281 51.3, 25.1 0.93, 0.80, 1.29, 0.84 334.9

(5.01, 6.09, 4.06), (4.58, 1.77, 3.63) 7.4, 10.0 1.0, 1.0 1.04, 3.67 (11.9, 35.7, 10.1) N/A (0.780, 0.138, 0.611) (0.099, 0.990, 0.097) 0.746 0.369 0.688 0.275 51.3, 24.7 0.93, 0.81, 1.29, 0.84 340.2

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requires all the observed values to satisfy the R-H relations and can get a solution that is close to the ‘‘true answer’’. Therefore a slight difference due to sampling could not significantly change the result. [19] The magnitude of the SD of Dt is measured by eye from the data profile. As can be seen in Figure 1, the thickness of the shock layer is 3/4 min. If we identify that the shock time is at the center, then the uncertainty is ±(1/2)  (3/4) = ±0.375. Therefore we assume 0.3 min of the SD of Dt. In addition, the fitting method can not handle the error in the spacecraft position data, thus, no uncertainty about the spacecraft positions is given here. [20] We apply two spacecraft method (Case 1) and single spacecraft method (Case 2) to the shock data, respectively. In Case 2, one does not use the variable Dt. To calculate the loss function, the guessed systematic error in equation (3) was set to be 1/2 of the SD. The systematic error of data is usually unknown in the analysis of real cases.

Figure 2. The result of the two spacecraft method. The solid lines indicate the best fit values of the magnetic field, plasma density, anisotropy parameter, and plasma beta. In addition, the dotted lines indicate the upstream time-averaged values and downstream predicted values of the velocities. 5 of 14

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Figure 3. The result of the one spacecraft method. The solid lines indicate the best fit values of the magnetic field, plasma density, anisotropy parameter, and plasma beta. In addition, the dotted lines indicate the upstream time-averaged values and downstream predicted values of the velocities.

3.1. Case 1: Two Spacecraft Method [21] The result is shown in Table 3. It can be seen that the best fit solution is very close to the ‘‘true answer’’. The best fit shock normal has an angle of only 0.3° from the ‘‘true answer’’ value. The best fit values are also indicated in Figure 2. The solid lines on both the up- and downstream sides indicate the best fit values for the magnetic fields, plasma densities, anisotropy parameters, and plasma betas. For the velocities, the variables used for the fitting is W, thus, we do not have best fit values of V1 and V2, respectively. Here we set that the best fit values of V1 is the time-averaged values, then we calculate the predicted (best fit) values of V2 by V2 = V1 + W, where for W the best fit value is used. The time-averaged values of the upstream velocity are indicated by the dotted lines in Figure 2, and the predicted downstream velocity values

are indicated by the solid lines. It can be seen that the fit is good. 3.2. Case 2: Single Spacecraft Method [22] For this case the variable Dt is not used, and the loss function has only 15 terms. The result is shown in Table 3 as well as in Figure 3. As can be seen, the best fit solution for the one spacecraft method is very close to the ‘‘true answer’’. The best fit shock normal has an angle of 1.0° from the ‘‘true answer’’. [23] We have also tested many cases of a wide range of qbn (not shown here). The fits are good. However, for a shock with qbn2  0° (or u  0), the solution shows ambiguity due to errors. The best fit solution could be a slow shock with qbn2 very close to zero degree, a switch-off shock, or an intermediate shock with qbn2 very close to zero. Since the intrinsic properties for these three types of

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Figure 4. The value of the minimum loss function and the values of the estimated parameters with various shifts, s, which are applied to the simulation case: The s is normalized by the SD. The a represents the angle between the estimated and the ‘‘true answer’’ shock normal. shock are very similar to each other, this problem is difficult to resolve by the present fitting method. [24] The best fit solutions have been investigated given different sizes of systematic errors. First we give different size of shifts of the means for the synthetic simulation case. We define a shift parameter, s, s
(data mean  true answer)/s, where s is the data SD. The s is the discrepancy of the data mean from the ‘‘true answer’’ value of the fitting variables normalized by their SDs. It represents the size of systematic error. We look for the variations of the best fit solution and the loss function when a different s is applied. The result is shown in Figure 4. The Lmin is the minimum loss function value for each case, and a is the angle between the best fit and the ‘‘true answer’’ shock normal. As can be seen, the minimum loss function value increases with s, while the solution deviates more from the ‘‘true answer’’. Furthermore, the angle a increases with s as well. This result is similar to the analysis of fast shock in Lin et al. [2006]. Therefore we show the estimation of the fitting method is reliable as long as the difference between the observed time-averaged value and the ‘‘true answer’’ value, namely the systematic error in data, is not too large.

4. Application to Observed Slow Shocks [25] We apply the fitting procedure to one interplanetary slow shock, which have been investigated and identified by Whang et al. [1998]. We also apply the method to two slow shocks observed in the geomagnetic tail. These two shocks were investigated and identified by Saito et al. [1995, 1998].

4.1. Interplanetary Slow Shock on 23 May 1995 [26] The shock was observed at 17:20 UT by the WIND spacecraft and had been investigated by Whang et al. [1998]. In Whang et al. [1998] the WIND-Magnetic Fields Investigation instrument (MFI) data and WIND-3D Plasma and Energetic Particles instrument (3DP) data were used for the measured magnetic field and plasma moments. In addition, the Minimum Variance Method was used to estimate the shock normal. [27] Here we apply the present fitting method to this shock. The data used in the present analysis are obtained from Coordinated Data Analysis Web (CDAWeb). We use the WIND-MFI data for the measured magnetic field. The data have a time resolution of 3 s. The ion velocity, density, and temperature are from the WIND-3DP data set, which also have a time resolution of 3 s. The alpha particles in solar wind are included in the present analysis. The electron temperature is also from the 3DP data set, but the data have a time resolution of 90 s. Moreover, the anisotropy information is considered in the present model. Since the 3DP data set lacks of anisotropy data for proton, we use the WIND-Solar Wind Experiment instrument (SWE) data set. It has a time resolution of 90 s. In addition, we do not have the anisotropy data for the alpha particles, therefore, the anisotropy of the alpha particles is assumed to equal to that of the protons. For electron anisotropy, we also use the WINDSWE data set, which has a time resolution of 9 s. The total plasma moments are calculated from the moments of protons, alpha particles, and electrons (refer to equations (23) – (28) of Lin et al., 2006).

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Figure 5a. The data of the magnetic field, plasma density, and velocity for the shock on 23 May 1995. The GSE coordinate system is used. The time unit used is in minutes counted from the beginning of a day. Neq = r/mp, is the plasma number density, where mp is the proton mass. The vertical lines indicate the data interval for the up- and downstream sides. The shock layer is at 1040.8 min. The horizontal lines indicate the best fit values of the magnetic field and plasma density. In addition, the horizontal dotted lines indicate the time-averaged upstream velocity and the predicted downstream velocity. [28] The data of the magnetic field, plasma density, and plasma velocity are shown in Figure 5a, while the data of the plasma beta and anisotropy parameter (x) are shown in Figure 5b The plasma beta and anisotropy parameter were calculated from the electron temperature, anisotropy, and the proton anisotropy, which have a lower time resolution. Therefore the data shown in Figure 5b have a time resolution of 90 s. The time-averaged values of the up- and downstream parameters are shown in Table 4. The selected up- and downstream time intervals (15 s) for the magnetic fields, plasma densities, and plasma velocities are very close

to that of Whang et al. [1998]. However, for the plasma betas and anisotropies we select a larger time interval of 5 min, since the data are of low time resolution. [29] We do not have observation from any other spacecraft around WIND for this shock. Therefore we only can estimate it using the one spacecraft method. The best fit solution is shown is Table 5. We also show the best fit values in Figures 5a and 5b. The horizontal solid lines indicate the best fit values of the magnetic fields, plasma densities, plasma betas, and anisotropy parameters. Since only the difference of velocities (W) is used in the fitting,

Figure 5b. The data of the plasma beta and the anisotropy parameter for the shock on 23 May 1995. The vertical lines indicate the data interval for the up- and downstream sides. The horizontal lines indicate the best fit values. 8 of 14

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Table 4. Up- and Downstream Observed and Best Fit Parameters of the 23 May 1995 Shock Parameter (GSE)

Time-averaged

B1 (nT) B2 Neq1 (cm3) Neq2 x1 x2 b1 b2 V1 (km/s) V2 W

(11.17, 0.78, 9.98) (8.04, 1.57, 9.93) 16.2 19.1 1.03 1.13 0.76 1.56 (477.5, 25.4, 27.8) (497.8, 24.3, 34.8) (20.3, 1.1, 7.1)

SD

Best fit

(0.4, 0.7, 0.4) (11.11, 1.37, 9.89) (0.3, 0.2, 0.2) (8.29, 1.40, 10.04) 0.7 15.2 0.6 19.6 0.15 1.06 0.02 1.13 0.12 0.81 0.03 1.56 (1, 1, 2) (477.5, 25.4, 27.8)a (4, 2, 3) (496.9, 24.2, 36.5)a (4, 3, 4) (19.4, 1.2, 8.7)

a The downstream predicted velocity is calculated by V2 = V1 + W, where V1 is the observed upstream velocity and W is the best fit parameter.

we do not have best fit values for V1 and V2, respectively. We give V1 an observed time-averaged value and then calculate the predicted V2 by V2 = V1 + W, where the best fit value of W is used. The values for the time-averaged V1 and the predicted V2 are indicated by the horizontal dotted and solid lines, respectively, in Figure 5a. We also show the magnetic field and plasma flow velocity in the shock (n-t-s) coordinate system in Figure 6. Here, the notation n represents the normal direction, the notation t represents the tangential direction, and the notation s represents the direction perpendicular to the n-t plane. As can be seen in Figures 5a, 5b and 6, our fit is very good. The up- and downstream normal Alfve´n Mach numbers in Table 5 are both less then unity. The upstream slow-mode Mach number is larger than unity, while the downstream slow-mode Mach number is less than unity. The slow-mode shock criteria are fulfilled. [30] According to the best fit solution, the shock has properties described as follows. The shock front is almost lying on the ecliptic plane, and it propagates almost toward the negative Z (GSE) direction with a propagation speed (Vs) of 93 km/s in the spacecraft frame of reference. This shock seems not propagating directly from the Sun. Since Vs  j^n  V1j > 0, this shock is a forward propagating shock relative to the solar wind flow. The normal magnetic field is Bn  10.6 nT (in the n-t-s coordinate system). The upstream tangential magnetic field is Bt1  10.6 nT, while the downstream tangential magnetic field is Bt2  7.7 nT. We calculate the normal and tangential flow velocities in the HT frame of reference by using the upstream time-averaged velocity and the predicted downstream velocity. The up- and downstream normal plasma velocities are Vn1  43.7 km/s and Vn2  33.8 km/s, respectively. In addition, the up- and downstream tangential plasma velocities are Vt1  43.6 km/ s and Vt2  24.8 km/s, respectively. Therefore we can obtain that Vt1/Vn1  Bt1/Bn and Vt2/Vn2  Bt2/Bn, which satisfy the criterion that the electric fields on both sides of a shock vanish in the HT frame of reference. [31] The alpha particles contribute  10% in the number density for this case. Moreover, the velocity difference between the proton and alpha particle flows is  30 km/s. We have investigated the slippage pressure contributed by the alpha particles (see also Discussion by Lin et al. [2006]).

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The results, however, shows that the slippage pressures for both the up- and downstream sides are all less than 3% of the thermal pressures. The low ratio of slippage pressure is due to a large thermal speed of the plasma. Therefore the coplanarity, and hence the present model is still applicable to the present case. 4.2. Slow Shock in the Geomagnetic Tail [32] We fit two slow-mode shocks observed by the Geotail spacecraft in the geomagnetic tail. The shocks have been investigated by Saito et al. [1995, 1998] and were considered to be associated with the Petschek-type magnetic reconnection occurring in the middle magnetotail. The shocks were identified to be at the lobe-plasma sheet boundary layer. For one of the two shock events, the Geotail spacecraft flew from the north lobe into the central plasma sheet, while for the other one, the Geotail spacecraft flew from the south lobe into the central plasma sheet. [33] The data used in the present analysis are obtained from the CDAWeb and kindly provided by Dr. Yoshifumi Saito. The magnetic field data are from the Geotail-Magnetic Field instrument (MGF). The data have a time resolution of 3 s. The ion data are from the Geotail-Low-Energy Particles instrument (LEP), which has a time resolution of 12 s. Electron data are also obtained from Geotail-LEP and are provided by Dr. Yoshifumi Saito. Since the data of ion temperature Txx (GSE) sometimes have wrong value when the ion bulk velocity Vx is high (Dr. Yoshifumi Saito, private communication), only Tyy and Tzz are used here. From only two-dimension temperature we can not have enough information to derive the anisotropy. Therefore we here assume that the plasmas are nearly isotropic (x1  x 2  1). This assumption is the same as adopted by Saito et al. [1998]. 4.3. Case 1: Shock on 12 January 1994 [34] The shock was observed at  15:41 UT at (93, 13, 4) RE in the GSE coordinate system. The observed magnetic fields indicate that the Geotail spacecraft flies from the north lobe (upstream) into the central plasma sheet (downstream). The magnetic field and plasma moment data are shown in Figure 7. We also show the up- and downstream time-averaged values and errors in Table 6. As can be seen, the magnetic field changes direction across the shock. The magnetic field magnitude deceases, while the density increases across the shock. This shows a situation of

Table 5. The Parameters Calculated by the Best fit Values in Comparison With the Parameters Calculated by Whang et al. [1998] Parameter (GSE)

This method

Whang et al. [1998]

^n ^t y u m z qbn, qbn2 (deg.) MAn, MAn2, Msl Msl2 Vs (km/s)

(0.053, 0.140, 0.989) (0.999, 0.010, 0.053) 0.775 0.732 0.876 0.567 45.0, 36.2 0.74, 0.65, 1.08, 0.77 93

(0.276, 0.755, 0.602)a N/A 0.714 N/A 0.82 N/A 47, 34 0.81, 0.68, 1.33, 0.79 182

We calculate the values of n from cos1 nx = 74°, cos1 ny = 41°, and cos nz = 53° given by Whang et al. [1998].

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a

1

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Figure 6. The data of the magnetic field and plasma velocity in the shock (n-t-s) coordinate system. The vertical lines indicate the data interval for the up- and downstream sides. The horizontal lines indicate the best fit values.

Figure 7. The data of the magnetic field, plasma density, plasma beta, and velocity for the shock observed at the magnetotail on 12 January 1994. The GSE coordinate system is used. The upstream is on the left-hand side, and the downstream is on the right-hand side. The time unit used is in minutes counted from the beginning of a day. Neq = r/mp, is the plasma number density, where mp is the proton mass. The vertical lines indicate the data interval for the up- and downstream sides. The shock layer is at 942.5 min. The horizontal lines indicate the best fit values of the magnetic field and plasma density. In addition, the horizontal dotted lines indicate the time-averaged upstream velocity and the predicted downstream velocity. 10 of 14

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Table 6. The Data and Best fit Values for the 12 January 1994 Shock Parameter (GSE)

Time-averaged

SD

Best fit

B1 (nT) B2 Neq1 (cm3) Neq2 b1 b2 V1 (km/s) V2 W

(10.21, 1.27, 1.04) (1.91, 0.90, 0.58) 0.017 0.045 0.013 16.7 (54, 20, 35) (925, 73, 102) (979, 53, 137)

(0.12, 0.16, 0.21) (1.53, 0.79, 0.59) 0.004 0.011 0.003 10.3 (49, 29, 68) (146, 55, 30) (160, 63, 74)

(10.16, 1.20, 1.03) (2.51, 1.29, 0.43) 0.021 0.060 0.013 12.3 (54, 20, 35)a (1176, 117, 82)a (1230, 97, 116)

a The downstream predicted velocity is calculated by V2 = V1 plus; W, where V1 is from the observed value and W is from the best fit value.

slow shock. In addition, there is a very strong downstream plasma velocity in the X direction. If the fast flow is associated with the magnetic reconnection, the reconnection site must be on the tail-ward side of this shock [Saito et al., 1995]. [35] The best fit values of the magnetic fields, plasma densities, and plasma betas are shown in Figure 7 with the horizontal solid lines. We give V1 an observed timeaveraged value and then calculate the predicted V2 by V2 = V1 + W, where the best fit value of W is given. The values for V1 and V2 are indicated by the horizontal dotted lines in Figure 7. We also list the best fit values in Table 6. As can be seen, the fit is good. The data satisfy the whole set of the R-H relations for slow shocks. We calculate the shock parameters from the best fit values. They are shown in Table 7. The up- and downstream normal Alfve´n Mach numbers are both less then unity. The upstream slow-mode Mach number is larger than unity, while the downstream slow-mode Mach number is less than unity. Therefore the discontinuity is identified as slow-mode shock. As can also be seen, the shock is quasi-perpendicular. The shock normal is (0.002, 0.984, 0.178), which points nearly to the negative Y (GSE) direction. In addition, the upstream slow-mode Mach number (= 8.7) indicates this shock has a very strong incoming normal flow (in the shock/HT frame of reference) in comparison to the local slow-mode wave speed. In Saito et al. [1998] they have obtained the values of the upstream shock normal angle (qbn = 80.7°) and plasma beta (= 0.01). Our results are very close to theirs. [36] From the best fit solution, we obtain that the normal magnetic field is Bn  1.34 nT in the shock (n-t-s) coordinate system. The upstream tangential magnetic field is Bt1  10.2 nT, while the downstream tangential magnetic field is Bt2  2.5 nT. In the HT frame of reference, the up- and downstream normal plasma velocities are Vn1  178 km/s and Vn2  64 km/s, respectively. In addition, the up- and downstream tangential plasma velocities are Vt1  1354 km/s and Vt2  120 km/s, respectively. Therefore we obtain Vt1/Vn1  Bt1/Bn and Vt2/Vn2  Bt2/Bn, which satisfy the criteria that the electric fields vanish on both sides of a shock in the HT frame of reference. 4.4. Case 2: Shock on 13 February 1994 [37] The shock was observed at  19:36 UT at (63, 7, 4) RE in the GSE coordinates. The observed magnetic

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fields showed that the Geotail spacecraft flies from the south lobe (upstream) into the central plasma sheet (downstream). The kinetic structure and the characters of this shock had been discussed by Saito et al. [1998] in detail. [38] The magnetic field and plasma moment data are shown in Figure 8. For this case, we select the up- and downstream time intervals similar to that by Saito et al. [1998]. For the upstream, we select the interval in which the fluctuations are stable. The interval is slightly different from that selected by Saito et al. [1998]. For the downstream region, there are strong fluctuations. It is difficult to define where the downstream is. We thus select the interval that is the same as that used by Saito et al. [1998]. They have checked the satisfaction of the R-H relations and analyzed the kinetic properties of this shock. [39] We show the up- and downstream time-averaged values and errors in Table 9. There is also a very strong downstream plasma velocity in the X (GSE) direction, which indicates that the reconnection site should be on the tail-ward side of this shock. The plasma data used here are provided by Dr. Yoshifumi Saito. They used the same set of plasma data in their paper ([Saito et al. [1998]). The best fit values of our estimation are shown in Figure 8 with the horizontal solid and dotted lines. We also show the best fit values in Table 8. As can be seen, the fit is good for this case. We calculate the shock parameters from the best fit values. They are shown in Table 9. We also list the values of Saito et al. [1998] in Table 9 for comparison. [40] As can be seen, the estimated shock normal is close to that obtained by Saito et al. [1998]. It is 13.7° off from the shock normal obtained by Saito et al. [1998]. The normal is more in Y (GSE) direction. The shock tangential vector (^t) estimated here is more in -X (GSE) direction. Since in the work of Saito et al. [1998] no shock tangential vector is shown, and there is no information that can help to derive the vector, it is thus impossible to compare the out results in the tangential direction with the results of Saito et al. [1998]. [41] From the best fit solution, we obtain that the normal magnetic field is Bn  3.46 nT (in the n-t-s coordinates). The upstream tangential magnetic field is Bt1  10.33 nT, while the downstream tangential magnetic field is Bt2  7.26 nT. In the HT frame of reference, the up- and downstream normal plasma velocities are Vn1  351 km/s and Vn2  120 km/s. In addition, the up- and downstream tangential plasma velocities are Vt1  1050 km/s and Vt2  251 km/s. The magnetic fields and flow velocities satisfy

Table 7. Parameters Calculated by the Best Fit Values for the 12 January 1994 Shock Parameter (GSE)

Value

^n ^t y u m z qbn, qbn2 (deg.) MAn, MAn2, Msl Msl2 Vs (km/s) b1, b 2

(0.002, 0.984, 0.178) (0.997, 0.012, 0.078) 0.360 0.248 0.278 0.089 82.5, 62 0.90, 0.54, 8.72, 0.56 190 0.013, 12.3

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Figure 8. The data of the magnetic field, plasma density, plasma beta, and velocity for the shock observed at the magnetotail on 13 February 1994: The shock layer is at 1176 min. The upstream is on the left-hand side, and the downstream is on the right-hand side. the criteria that the electric fields vanish on both sides of a shock in the HT frame of reference.

5. Conclusion [42] In this paper, a method of fitting the R-H shock jump relations for fast MHD shocks derived by Lin et al. [2006] is applied to slow-mode shocks. The method can be used under the situations for both one and two spacecraft observations. Using synthetic slow shocks, we have tested the method. The result shows that the method can success-

Table 8. The Data and Best fit Values for the 13 Feb. 1994 Shock Parameter (GSE)

Time-averaged

SD

Best fit

B1 (nT) B2 Neq1 (cm3) Neq2 b1 b2 V1 (km/s) V2 W

(10.93, 0.81, 0.97) (6.15, 0.66, 0.84) 0.026 0.046 0.039 1.140 (70, 35, 55) (528, 148, 12) (598, 113, 67)

(0.32, 0.36, 0.43) (1.67, 1.85, 1.30) 0.009 0.010 0.025 0.656 (56, 33, 50) (194, 40, 76) (203, 52, 88)

(10.82, 1.11, 0.73) (8.01, 0.15, 0.66) 0.018 0.054 0.039 1.010 (70, 35, 55)a (751, 153, 3)a (821, 118, 52)

a The downstream predicted velocity is calculated by V2 = V1 + W, where V1 is from the observed value and W is from the best fit value.

fully find the ‘‘true answer’’ from the noised data. We apply the method to one slow shock in interplanetary space and two slow shocks observed in the geomagnetic tail. The result shows that the method can find an R-H solution in agreement with the observations, which also re-identify the reported slow shocks. [43] For the two slow shocks in the geomagnetic tail, our results show that, in one spacecraft observation, the magnetic fields and plasma moments of the two reported shocks satisfy the R-H relations for slow-mode shock from the best fit procedure. However, our results are somewhat different from the results of the previous analysis proposed by Saito et al. [1998]. The R-H solution obtained by Saito et al. [1998] is not a best fit solution. Their solution for the

Table 9. The Parameters Calculated by the Best fit Values for the 13 Feb. 1994 Shock in Comparison With the Values Calculated by Saito et al. [1998] Parameter (GSE)

Value

Saito et al. [1998]

^n ^t qbn, qbn2 (deg.) MAn, MAn2, Msl, Msl2 Vs (km/s) b1, b 2

(0.402, 0.903, 0.146) (0.912, 0.410, 0.023) 71.5, 64.5 0.63, 0.37, 3.54, 0.53 283 0.039, 1.010

(0.25, 0.91, 0.33) N/A 77.3, 72.8 0.72, 0.59, 3.68, 0.88 144 0.048, 1.6

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nately, there is a large discrepancy in the calculated shock normals, but, for the scalar shock parameters the results are close. In the analysis we use the same data set as by Whang et al. [1998], but we have included the anisotropy effect and the contribution of the alpha particles in our analysis. However, even if the two considerations are not included, the large discrepancy between these two normals still remains. The discrepancy should be mainly due to the methods which are used. MVA requires the magnetic field to be conserved in the ‘normal’ direction. However, this method does not guarantee B1, B2, and the ‘normal’ to be coplanar. We have demonstrated that the result of Whang et al. [1998] have suffered such a discrepancy. The shock normal (0.276, 0.755, 0.602) of Whang et al. [1998] and B1 and B2 is actually not in the same plane.

Appendix A Figure 9. The sketch of the shock geometry in the shock coplanar (n-t) plane. The notation ^n represents the normal direction, and the notation ^t represents the tangential direction. The flow velocities shown in this figure are seen in the normal incidence frame of reference. downstream parameters has a discrepancy of up to 20% from the observed values. [44] An observation of very fast downstream flow in the X (GSE) direction for the two shock cases indicates that possible magnetic reconnection may occur on the tail-ward side (|XGSE|  100 RE) of the shocks. Such a problem was discussed by many authors [e.g., Feldman et al., 1984a, 1984b, 1985, 1987; Smith et al., 1984; Ho et al., 1994; Saito et al., 1995, 1998; Seon et al., 1995; Hoshino et al., 2000; Eriksson et al., 2004]. We found an interesting shock geometry. The observation of the fast downstream flow in the X (GSE) direction implies that in the HT frame of reference the upstream flow should have a very large angle with respective to the shock normal. As can be seen in our results, the shock tangential direction (^t) is nearly in the negative X (GSE) direction. Then, in the normal incidence frame of reference the downstream flow has a large component in the negative ^t direction. Figure 9 shows the sketch. If qv2 is defined as the angle between the downstream flow and the shock normal, we can derive that (see Appendix A) tan qbn

y ¼ tan qv2 : 1  yu

ð7Þ

For a slow shock, yu < 1. Thus 1  yu > 0. Therefore qbn is large when qv2 is large. This indicates that if there is a large ^t component of downstream flow, a slow shock should have a large shock normal angle qbn. In a statistical survey of Saito et al. [1998] with 10 slow shocks, all the slow shocks are quasi-perpendicular, while 6 cases have qbn > 80°. The survey of Ho et al. [1996] shows that for the 86 identified slow shocks, qbn mainly distributes between 65° and 85°. [45] For the interplanetary slow shock, we compare our result with that provided by Whang et al. [1998]. Unfortu-

[46] Figure 9 shows the sketch of the shock geometry in the shock coplanar (n-t) plane. Here, the coordinate ^ n is in the normal direction, and the coordinate ^t is in the tangential direction. The flow velocities shown in this figure are seen in the normal incidence frame of reference. As can be seen, 0 ) the normal component of the upstream flow velocity (V1n 0 (in the normal incidence frame) is negative, and V2n < 0, too. In addition, the tangential component of the upstream flow velocity (V1t0 ) is equal to zero, while V2t0 < 0. If we define V*1t and V*2t as the up- and downstream tangential flow velocities in the HT frame of reference, the relations between them and V1t0 and V2t0 are V1t* ¼ V1t0 þ VHT ;

ðA1Þ

V2t* ¼ V2t0 þ VHT ;

ðA2Þ

where VHT is the HT frame velocity with respect to the normal incidence frame of reference. With the above two equations, we have V1t* ¼ V1t0 þ V2t*  V2t0 :

ðA3Þ

Dividing the terms on both sides by the V*1n (the upstream normal flow velocities in the HT frame of reference), we obtain V1t* V0 V2t* V2t0 ¼ 1t þ  V1n* V1n* V1n* V1n*

ðA4Þ

Here, V*1n < 0, thus, V*1t/V*1n = tanqbn. V1t0 equals to zero, then, the first term on the right-hand side also equals to zero. In addition, the second term on the right-hand side can be written as V2t* V2t* V1t* ¼ ¼ z  ð tan qbn Þ: V1n* V1t* V1n*

ðA5Þ

V2t0 V 0 V2n* V2t0 V2n* ¼ 2t ¼ 0 ¼ tan qv2  y: V1n* V2n* V1n* V2n V1n*

ðA6Þ

Finally,

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0 Note that here V2n = V*2n; the normal velocity is the same for the shock and the HT frames. Over all, equation (A4) can be rewritten as

tan qbn ¼

y tan qv2 : 1  yu

ðA7Þ

[47] Acknowledgments. This work was supported by National Science Council (NSC) (Taiwan) under grants NSC 95-2111-M-008-035, NSC 95-2111-M-008-037, NSC 95-2811-M-008-034, and NSC 96-2111M-008-019 to National Central University, and it is also supported by National Nature Science Foundation of China (NSFC) under grant number 10425312, 10373026, 10603014, and 40574065 and by National Key Basic Research Special Funds (NKBRSF) under grant 2006CB806302. The authors thank R. Lin at UC Berkeley, R. Lepping at NASA/GSFC, K. Ogilvie at NASA/GSFC, S. Kokubun at Nagoya University (Japan), T. Mukai at ISAS (Japan), and CDAWeb for the use of the key parameters from the WIND and Geotail spacecraft. The authors especially thank Yoshifumi Saito for providing the Geotail data and his constructive suggestions. [48] Zuyin Pu thanks the reviewers for their assistance in evaluating this paper.

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J. K. Chao, H. Q. Feng, L. C. Lee, C. C. Lin, and L. H. Lyu, Institute of Space Science, National Central University, Taiwan 32001. (dannylin@ jupiter.ss.ncu.edu.tw) D. J. Wu, Purple Mountain Observatory, Academic Sinica, Nanjing 210008, China.

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