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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A10104, doi:10.1029/2007JA012311, 2007
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From Rankine-Hugoniot relation fitting procedure: Tangential discontinuity or intermediate/slow shock? H. Q. Feng,1 C. C. Lin,2 J. K. Chao,2 D. J. Wu,1 L. H. Lyu,2 and L.-C. Lee2 Received 29 January 2007; revised 15 July 2007; accepted 2 August 2007; published 12 October 2007.
[1] To identify an observed intermediate/slow shock, it is important to fit the measured
magnetic fields and plasma on both sides using Rankine-Hugoniot (R-H) relations. It is not reliable to determine an intermediate/slow shock only by the shock properties and fitting procedure based on one spacecraft observation, though previous reported intermediate/slow shocks are confirmed in such a way. We investigated two shock-like discontinuities, which satisfy the R-H relations well. One meets the criterions of slow shocks and was reported as a slow shock, and another has all the characters of intermediate shock based on one spacecraft observation. However, both discontinuities also meet the requirements of tangential discontinuities and were confirmed as tangential discontinuities on large-scale perspective by using multi-spacecraft observations. We suggest that intermediate/slow shocks should be identified as carefully as possible and had better be determined by multi-spacecraft. Citation: Feng, H. Q., C. C. Lin, J. K. Chao, D. J. Wu, L. H. Lyu, and L.-C. Lee (2007), From Rankine-Hugoniot relation fitting procedure: Tangential discontinuity or intermediate/slow shock?, J. Geophys. Res., 112, A10104, doi:10.1029/2007JA012311.
1. Introduction [2] The MHD Rankine-Hugoniot (R-H) conditions allow four types of magnetic directional discontinuities (DDs): contact discontinuity (CD), tangential discontinuity (TD), rotational discontinuity (RD) and shocks. Shocks and TDs are commonly observed in interplanetary solar wind. [3] TDs can be considered to be boundaries between distinct flows of plasma. There is no mass flow or magnetic component normal to the discontinuity and conservation of total plasma (thermal and magnetic) pressure on both sides is required, that is to say, flows simply move with the discontinuity surface. A number of statistical studies of TDs have been carried out [e.g., Burlaga et al., 1977; Tsurutani and Smith, 1979; Behannon et al., 1981]. These investigations include the ratios of RD to TD and their macroscopic properties such as thickness and magnetic field rotation angle. Lepping and Behannon [1986] gave a ratio of TDs to RDs greater than unity and So¨ding et al. [2001] found that the ratio of RDs to TDs varied by 5 to 10% depending on the algorithm used to identify and select the discontinuities. Using a reliable triangulation method, Knetter et al. [2004] found that there is no clearly identified RD at all, and earlier statistical population of the RD category is simply a result of inaccurate normal estimates. [4] The R-H relations have six shock solutions: the fast and slow shocks and four intermediate shocks (ISs). The fast shocks are observed frequently in the interplanetary 1 2
Purple Mountain Observatory, CAS, Nanjing, China. Institute of Space Science, NCU, Chungli, Taiwan.
Copyright 2007 by the American Geophysical Union. 0148-0227/07/2007JA012311$09.00
space. The reported slow shocks (SSs) are relatively rare, and only a small number of SSs have been observed in interplanetary space [Chao and Olbert, 1970; Burlaga and Chao, 1971; Richter et al., 1985; Whang et al., 1996, 1998; Ho et al., 1998; Zuo et al., 2006]. However, in the geomagnetic tail, slow shocks are observed more often [e.g., Feldman et al., 1984, 1985, 1987; Smith et al., 1984; Cattell et al., 1992; Saito et al., 1995; Ho et al., 1994, 1996; Seon et al., 1995, 1996; Hoshino et al., 2000; Eriksson et al., 2004]. Observations of ISs are very rare; only one case has been reported by Chao et al. [1993]. [5] For an MHD shock, the coplanarity theorem requires the magnetic field vectors B1 and B2 in the upstream and downstream regions and shock normal ns to be in the same coplanar plane. So, we define an orthogonal shock frame of reference as shown in Figure 1, let s denote the unit vector normal to the coplanar plane(viz. s ? ns), then define: t = ns s. Therefore the t s plane is just the shock front, thus both the up- and downstream magnetic fields are in the ns t plane. On the other hand, the up- and downstream magnetic fields of a TD also lie on the same plane, which is defined as TD front (plane), because a TD has no normal magnetic field. Therefore the t ns plane defined above is just the TD front (plane), and the TD normal (nTD) is in the direction of s. [6] According to the R-H relations, a TD requires only two conditions: (1) the velocities and magnetic fields are all tangential to the TD front (plane), and (2) total pressures on both sides are balanced. For an IS or a SS, in the shock frame of reference, the up- and downstream plasma flows also lie on the t ns plane. It also meets the first requirement of a TD. In addition, for an IS or a SS, the magnetic pressure may decrease and the plasma thermal
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on a large-scale perspective by use of multi-spacecraft observations.
2. The 18 September 1997 Slow Shock-Like TD
Figure 1. Shock frame of reference (orthogonal coordinate system), where shock normal is in the ns direction, magnetic fields are in the ns t plane and the shock front in the s t plane. As a TD, the normal is the unit vector s and the ns t plane is the TD surface. pressure increases across the shock front. The total pressure could be close to balance for some shock conditions, which meets the second condition required by a TD. Therefore using only the R-H relations based on one spacecraft observation to determine a TD and IS/SS likely causes ambiguities; one can mis-interpret a TD as IS (or SS) and vice versa. So far, to the best of our knowledge, all the reported IS/SSs were identified in such a way and using one satellite only. [7] The ambiguity can be fixed by using multi-spacecraft observations. Suppose that two spacecraft observed the same DD at a different location and at a different time, the time difference Dt can be expressed as [e.g., Russell et al., 1983] Dt ¼ DR n=Vdd :
ð1Þ
The method can be extended to three or more spacecraft observations [Schwartz, 1998]. Here DR is the vector displacement of the two spacecraft and Vdd is the propagation speed of the DD in the rest frame of reference. n and Vdd are calculated from the local property of the DD. Their values depend on the model used in the calculation. As mentioned above, for the same DD, the values calculated by an IS/SS model are very different from that calculated by a TD model. Therefore the estimated Dt should be very different between these two models. However, one can easily have an observed Dt if two spacecraft data are used. [8] In this paper we demonstrate the ambiguities by use of two shock-like DDs. One is a SS-like DD reported as a SS in a recent work by Zuo et al. [2006], and another meets all the requirement for ISs. However, both DDs satisfy criterions of TD entirely. We identify the two events as TDs
2.1. Modeling as a Slow Shock [9] This shock-like discontinuity was observed at about 0255:15 on 18 September 1997 by Wind located at (83.51, 13.58, 1.45) RE in GSE coordinate system. Figure 2 shows the observed values of the parameters as functions of time for this event. The magnetic field data obtained from Magnetic Field Investigation (MFI) magnetometer and the proton data obtained from the 3-Dimension Plasma (3DP) analyzer are all shown in the GSE coordinate. The data have a time resolution of 3 s. As investigated by Zuo et al. [2006], this discontinuity has typical SS characters: (1) the density increases across the discontinuity, while the magnitude of magnetic field decreases; (2) the observed parameters all satisfy the R-H relations; (3) The upstream normal bulk velocity in the shock frame of reference is larger than local slow magnetoacoustic speed and smaller than local normal Alfve´n speed, and the downstream velocity is smaller than the local slow magnetoacoustic speed. They used a coplanarity method and a self-consistent method to determine the shock normal (ns) and the other two axes of the shock coordinate system (Listed in Table 1). The selfconsistent method utilizes the entire R-H relations and a minimization technique to determine the shock normal. The detailed descriptions can be found in the work of Zuo et al. [2006]. The angle between the estimated shock normals using the two methods is only 7°. In addition, the predicted R-H solutions based on the shock normal determined by a self-consistent method are in better agreement with the observations, so they considered the self-consistent method more accurate. Table 1 lists the up- and downstream magnetic fields, plasma velocities, and densities on both sides. The estimated shock speed Vsh is also given in Table 1. [10] In addition, this discontinuity is likely a 18– 20 s long solar wind reconnection exhaust transition according to the criteria first reported by Gosling et al. [2005a]. A number of recent reports have outlined this subject further [e.g., Gosling et al., 2005b; Davis et al., 2006; Huttunen et al., 2006; Gosling et al., 2006; Phan et al., 2006; Gosling et al., 2007]. The region between two dotted vertical lines (See Figure 2) has the characteristic features of a reconnection exhaust. Namely it has higher proton density, higher proton temperature, weaker magnetic field strength, and intermediate field orientation as compared with the surrounding solar wind. The changes in V and B are anticorrelated in the leading portion and correlated in the trailing portion of the event as expected. 2.2. Modeling as a Tangential Discontinuity [11] As mentioned above, from the MHD consideration, the observed discontinuity may also satisfy the criteria for a TD, since, in the frame of a discontinuity, there is no normal mass flow, and since the total plasma (thermal and magnetic) pressures are balanced on both sides. As in our analysis, the estimated TD normal nTD(=(B1 B2)/jB1 B2j) is (0.43, 0.88, 0.17), where the values of B1 and B2 are from Table 1. This normal is just in the s axis of the shock coordinates obtained by Zuo et al. [2006] (see Table 1). The dot product of nTD and the difference between the downstream and
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Figure 2. The interplanetary magnetic field and plasma data measured by the Wind spacecraft in GSE coordinate system on 18 September 1997. The region between two dotted vertical lines is likely a reconnection exhaust transition. upstream velocities (V2 V1) is only 0.66 km/s, which indicates that there is nearly no mass flow through the discontinuity. In addition, Table 2 shows the thermal, magnetic, and total pressures on both sides. The difference in the total pressures is only 3 percent of the total pressures in the upstream region. By including the systematic error and sampling errors, one can consider that the total plasma pressures are conserved across the discontinuity. Therefore from these two conditions the local parameters of this discontinuity satisfy the TD requirements. 2.3. Identify the Discontinuity as TD With ThreeSpacecraft Observations [12] This discontinuity was also observed at 0221:01 UT by ACE located at (193.31, 24.78, 20.78) RE. Figure 3 shows the corresponding magnetic field profiles measured by ACE and Wind. Here the dotted lines are for ACE, for which the time sequences were shifted by 34.2 min. As seen in Figure 3, the two sets of the profiles are very close to each other, while there are only a few differences seen in the detail structures. Therefore it can be confirmed that ACE observed the same discontinuity as Wind. [13] SS and TD models are used respectively to verify the time difference (Dt) between the two spacecraft. From the SS model, Dts = DR ns/Vsh, where ns and Vsh are from Table 1, and DR = (109.80, 11.23, 22.23)RE is the vector displacement between the Wind and ACE spacecraft.
The calculated time is 13.5 min, which is very different from the observed time difference (34.2 min). On the other hand, from the TD model, DtTD = DR nTD /VTD, where VTD is the TD propagation speed in the rest frame of reference. Since TD is a non-propagating discontinuity with respect to the solar wind, its speed (in the normal direction) in the rest frame can be estimated by VTD = nTD V1, where V1 is the upstream flow velocity in the rest frame. The estimated value of VTD is 160.22 km/s. The estimated DtTD is 35.6 min, which is very close to the observed time difference. Note that since there is no mass flux across the transition, using V1 and V2 to calculate VTD almost makes Table 1. The Observed Parameters of the 18 September 1997 SSLike Discontinuity, and Shock Speed Vsh, the Shock Normal ns and Other Two Axes of the Shock Coordinate System (From Zuo et al. [2006]) Parameter
Value
B1, nT B2 N1, N2, cm3 V1, km/s V2 ns s t Vsh, km/s
(3.1, 0.02, 7.9) (3.4, 2.3, 3.3) 22.1, 32.7 (341, 11, 23) (356, 5, 20) (0.44, 0.38, 0.81) (0.43, 0.88, 0.17) (0.79, 0.28, 0.55) 205
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Table 2. The Pressures on Both Sides of the 18 September 1997 Discontinuity Upstream Thermal pressure, Pa Magnetic pressure Total pressure
11
4.00 10 2.87 1011 6.87 1011
Downstream 5.98 1011 1.10 1011 7.08 1011
no difference here (nTD V2 = 160.88 km/s). With the value calculated from V2, the estimated DtTD is 35.5 min. According to the above results, we conclude that this discontinuity should be interpreted as a TD and not as a SS. This interpretation is based on a large-scale perspective due to the selection of up and downstream intervals. [14] Assuming that (1) a DD surface can be approximated by a plane thin sheet (1-D structure) and that (2) the speed of the DD is constant in time and space, Knetter et al. [2004] used four Cluster spacecraft to determine the discontinuity normal, namely via triangulation. Four spacecraft can give three independent equations associated with the propagation time and one equation that requires unity of the shock normal vector. They are Dti ¼ DR1i n=Vn ; i ¼ 2 to 4;
ð2Þ
jnj ¼ 1:
ð3Þ
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Here Vn is the propagation speed of the DD in the rest frame of reference, and DR1i represents the vector displacement between the Cluster 1 and Cluster i spacecraft. In addition, Dti represents the time difference between Cluster 1 and Cluster i spacecraft. With these four equations, n and Vn can be found. The method requires no magnetic field and plasma data and can be more accurate. [15] Note that there is a limitation in this method. If the four spacecraft are coplanar, the method of using timing cannot find the DD normal vector and its propagation velocity. As pointed out by Schwartz [1998] (pages 257 and 309), if the four spacecraft are coplanar, the determinant of the matrix of DR in Equation (2) is zero. The three linear equations reduce to two. Therefore we cannot find the full set of unknowns (n/Vn). For example, if four spacecraft are on the plane perpendicular to the Z axis, one can not get nz /Vn from the linear algebraic system in Equation (2). [16] Burlaga and Ness [1969] and Horbury et al. [2001] use the method similar to that of Knetter et al. [2004]. They apply only three spacecraft observations. With this method they have only two independent equations for propagation time. In order to have a close system, they assume that the DD moves with the plasma bulk velocity measured at one of the spacecraft. Under such assumption, their method is applicable only for TDs. Any DDs propagating with respect to the solar wind frame of reference cannot be studied by their method.
Figure 3. The magnetic fields measured by the Wind and ACE in GSE coordinate system, where the dotted lines are for ACE, and the ACE time sequences were shifted by 34.2 m. 4 of 12
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Figure 4. The magnetic fields measured by the Wind and Geotail in GSE coordinate system, where the dotted lines are for Geotail, and its time sequences were shifted by 17.2 m.
[17] We examined all the solar wind data observed by other spacecraft, which are near the Earth, and found that the DD discussed in this section was also observed by the Geotail spacecraft at 0312:27 UT and at (24.85, 11.57, 1.52) RE (GSE). Figure 4 shows the corresponding magnetic field profiles measured by Geotail and Wind. Here the dotted lines are for Geotail, for which the time sequences were shifted by 17.2 min. Therefore a total of three spacecraft are available. In general, the measured magnetic fields have smaller uncertainty than that of velocities. We use the equation: n B1 = n B2 (from r B = 0) to replace one of the Equation (2) used by Knetter et al. [2004]. The equation n (B2 B1) = 0 is a more reliable one for any type of one dimensional DDs than the Equation (2). In addition, the systematic errors in magnetic field measurements are likely to be eliminated. This equation uses only the measured magnetic fields on both sides of the DD. Therefore we have DtWG ¼ DRWG n=Vn ;
ð4Þ
DtWA ¼ DRWA n=Vn ;
ð5Þ
n B1 ¼ n B2
ð6Þ
jnj ¼ 1:
ð7Þ
where DRWG (DtWG) represents the vector displacement (time difference) between the Wind and Geotail spacecraft, DRWA (DtWA) represents the vector displacement (time difference) between the Wind and ACE spacecraft. Here the measured magnetic fields are from the data of Wind. One can obtain the solutions of n and Vn from these four expressions. As the result, we obtain that n = (0.42, 0.89, 0.15), which is very close to nTD (0.43, 0.88, 0.17), and we obtain Vn = 164.99 km/s, which is also consistent with VTD (160.22 km/s). It, again, confirms that this event is a TD. On the other hand, this derived normal and propagating speed do not agree with the slow shock solution. For a detail comparison of the results of the SS and TD models with the above derived normal and speed, we summarize the parameters in Table 3. [18] In Equations (4) – (7), the magnetic fields are included to replace one of the equations in four spacecraft method (Equation (2)). With Equations (4) – (6) we obtain a linear algebraic system as follows.
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0
DRWGx @ DRWAx DBx
DRWGy DRWAy DBy
10 1 0 1 DRWGz mx DtWG DRWAz A@ my A ¼ @ DtWA A DBz mz 0
ð8Þ
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Table 3. The Comparison of the Results From the SS and TD Models Parameter
SS Model
TD Model
Equations (4) – (7)
Normal n VSS, km/s VTD, km/s Dt(ACE to Wind)
(0.58, 0.36, 0.73) 205
(0.43, 0.88, 0.17)
(0.42, 0.89, 0.15)
160.2 (162.9)a 35.6 m (35.5 m)a
165.0
13.5 m
Observation
34.2 m
a
The value in the parentheses is calculated from the downstream flow velocity.
where DB (B2 B1) and m n/Vn. With the solution of m, we can calculate the value of Vn as follows (via Equation (7)). 1=2 Vn ¼ m2x þ m2y þ m2z
ð9Þ
[19] For this system, if DB, DRWG, and DRWA are not on the same plane, one can obtain all three components of m. Therefore although the three spacecraft are always coplanar, in the present case DB, DRWG, and DRWA are not on the same plane. As obtained from computation, the determinant of the matrix in Equation (8) is non zero. Here we emphasize that the calculation of Equations (4) – (6) is independent of the type of DD. The normal and the propagation speed derived in our calculation should correspond to the actual type it belongs to. [20] There is another way to understand the capability of Equations (4) – (7). The magnetic field conservation Equation (6) allows only two normals - the SS normal (ns) and the TD normal (nTD) as demonstrated in Figure 5. For the TD normal B1 nTD = B2 nTD = 0, while for the SS the normal magnetic field to the DD is a constant value. With these two normals and one of the Equations (4) and (5), one can find their corresponding propagating speeds, Vns and VnTD. The other one of Equations (4) or (5) can be used to determine which set, the SS or the TD, is the correct solution. For the present case, we found that only the values for the TD can satisfy Equations (4) – (7) well in comparison with the values for the SS. [21] For this case, we have checked the frozen in condition of the DD by the quantities of V1 nTD (= 160.2 km/s), V2 nTD (= 160.9 km/s), where V1 and V2 are the up- and downstream flow velocities in the spacecraft (rest) frame of reference. The result shows that they are close to the propagation speed of the DD calculated from the Equations (4)– (7) (Vn = 165 km/s in the rest frame). This means that the DD is a non-propagating discontinuity with respect to the solar wind plasma. In other words, V2-V1, B1 and B2 are almost parallel to the DD plane. If we use the slow shock model, the calculated DD propagation speed is 205 km/s in the rest frame which is very different from Vn = 165 km/s. In addition, we found that the total pressures across the plane are close to one another. Therefore the obtained DD is more likely a TD. [22] On the other hand, we also compare our calculated normal with the local derived normals using the R-H relations, which correspond to either a shock or a TD. The normal obtained from Equations (4) – (7) is (0.42, 0.89, 0.15), while the normal obtained from (B1 B2)/ jB1 B2j is (0.43, 0.88, 0.17). They are close to one another. However, the normal obtained from the slow shock
solution of Zuo et al. [2006] is (0.79, 0.28, 0.55), which is perpendicular to the normal obtained from (B1 B2)/jB1 B2j. From the above result, one should also consider this DD as a TD than a SS. [23] We have also checked the TD normals (B1 B2)/ jB1 B2j using the data from the other two spacecraft (ACE and Geotail). The normal calculated from the ACE magnetic field data is (0.47, 0.87, 0.18). This normal is 2.3° off from the normal calculated from the Wind magnetic filed data. The normal calculated from the Geotail magnetic field data is (0.52, 0.83, 0.21). This normal is 6.7° off from the normal calculated from the Wind magnetic filed data. The normals are close to one another. This shows that the discontinuity is stable during the period between ACE and Geotail. The using of the multiple spacecraft timing method in this paper should be appropriate.
3. The 8 October 2001 Intermediate Shock-Like TD 3.1. Modeling as an Intermediate Shock [24] This shock-like discontinuity was observed by Wind at RW = (37.20, 59.72, 4.91) RE in GSE coordinate system at 0117:30 UT on 8 October 2001. Figure 6 shows the magnetic field and plasma data of this event. It is well known that shock fitting is very important for investigation of interplanetary shocks. One main problem related to shock fitting is to search for an accurate shock frame of reference.
Figure 5. The r B = 0 allows only two normals - the SS normal (ns) and the TD normal (nTD) for the system of Equations (4) – (7). For the TD normal B1 nTD = B2 nTD = 0, while for the SS the normal magnetic field to the DD is a constant value. In this sketch the relationship of the normals of the slow shock and the TD is demonstrated. The unit vectors nTD, ns, and t are orthogonal. Here, ns is determined by the coplanarity theorem [Zuo et al., 2006], and nTD is obtained by nTD = (B1 B2)/jB1 B2j.
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Figure 6. The interplanetary magnetic field and plasma data measured by the Wind spacecraft in GSE coordinate system on October 8, 2001 and the best fitting valves of upstream and downstream regions (dotted lines). Such as the coplanarity theory and Minimum Variance Analysis (MVA) are frequently used to analyze interplanetary shocks. Here we use a new shock fitting procedure proposed recently by Lin et al. [2006]. They use a whole set of the R-H relations and modified R-H relations. The modified R-H relations include terms for equivalent ‘‘heat flow’’ and ‘‘momentum flux’’ possibly due to waves/turbulences, energetic particles, and/or other unknown causes [e.g., Chao and Goldstein, 1972; Davison and Krall, 1977; Yoon and Lui, 2006]. Lin et al. [2006] separated their procedure under two conditions. One is called Method A, which utilizes the classical R-H relations. Another one is called Method B and utilizes the modified R-H relations. With this, a best fit solution that satisfies the R-H relations within the limitation of the data error is obtained. For more details of the procedure please refer to Lin et al. [2006]. [25] The second column of Table 4 lists the observed data means and the corresponding parameters directly calculated from the data means of the observed magnetic fields and plasma. The derived parameters are the shock normal vector ns, other two axes of the shock coordinate system t and s, the plasma beta (b), the normal Alfve´n-Mach number (MAN = Vn/VAn), the fast-mode Mach number (MF = Vn/Vf), the slow-mode Mach number (MSL = Vn/Vsl) in the upstream/ downstream region, the ratio of downstream to upstream
magnetic field intensities (m = B2/B1), the ratio of upstream to downstream plasma densities (y = N1/N2), the ratio of downstream to upstream tangential magnetic fields (u = Bt2/ Bt1), the angle, qBN = cos1(B1 ns/B1), between the shock normal and the upstream magnetic field (also called the shock normal angle). In the above expression, VAn is the Alfve´n speed based on magnetic field component normal to shock front (VAn = Bn/(m0r)1/2), Vn is the component of the bulk velocity to the shock front and measured in the shock frame of reference, and Vf and Vsl are the speeds of the fastand slow-mode magnetosonic waves in the direction of the shock normal, respectively. We applied both methods to this discontinuity. The third and fourth columns of Table 4 respectively list the fitting results form Method A and B as well as corresponding parameters calculated from these best fit values. From Table 4 one can find that both methods give very similar results. Figure 6 also shows the best fit values (Method A) of upstream and downstream regions as dots. As can be seen in Table 4 and Figure 6, the best fit values are in very good agreement with the observed values. According to the MHD theory of shocks, an IS has the following properties. (1) The normal Alfve´n-Mach number (MA) is greater than unity in the preshock state and less than unity in the postshock state. (2) The tangential components of both the preshock and postshock magnetic fields on the
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Table 4. The Observed and Best Fitting Parameters of 8 October 2001 IS-like Discontinuity Parameter
Observed Valuesa
Best Fit Values (Method A)
Best Fit Values (Method B)
B1, nT B2 N1, N2, cm3 W(V2 V1)(km/s) b 1, b 2 ns S T MAN1, MAN2 MF1, MF2 MSL1, MSL2 y m u qBN
(3.49, 3.38, 0.24) (0.89, 0.38, 4.14) 14.16, 15.64 (14.9, 17.5, 16.5) 2.46, 3.35 (0.583, 0.355, 0.731) (0.687, 0.695, 0.212) (0.433, 0.626, 0.649) 1.021, 0.971 0.510, 0.490 1.262, 1.079 0.905 0.874 0.732 45.50°
(3.49, 3.39, 0.24) (0.89, 0.37, 4.14) 14.46, 15.49 (15.4, 21.7, 21.1) 2.44, 3.59 (0.582, 0.357, 0.731) (0.689, 0.694, 0.210) (0.432, 0.625, 0.650) 1.013, 0.979 0.455, 0.442 1.178, 1.056 0.933 0.873 0.730 45.48°
(3.48, 3.37, 0.24) (0.88, 0.36, 4.14) 14.19, 15.68 (15.8, 21.9, 20.7) 2.45, 3.36 (0.582, 0.358, 0.731) (0.688, 0.695, 0.208) (0.434, 0.624, 0.650) 1.021, 0.971 0.514, 0.494 1.266, 1.081 0.905 0.875 0.734 45.38°
a
The SD of B1 is (0.17, 0.19, 0.17), the SD of B2 is (0.12, 0.20, 0.10), the SD of N1 and N2 are 0.34 and 0.36, the SD of W is (1.51, 1.70, 2.47), where SD is the sample standard deviation.
shock front have opposite signs. (3) The plasma number density increases from the upstream region to the downstream region. (4) Of the four types of intermediate shock, the 2 ! 4 type has a larger density jump across the shock front than the 1 ! 3, the 1 ! 4, and the 2 ! 3 types [Chao et al., 1993]. Figure 7 also shows the magnetic field data in the shock coordinate system. From Figure 7, it can be seen that the tangential magnetic field Bt changes sign across the shock front, and Bn approximately keeps constant, and Bs component is approximately zero. Combining the shock parameters listed in Table 4, it is clear that this discontinuity
satisfies the criteria for an IS. In addition, the fast-mode Mach number is less than unity, and both the slow-mode Mach number in the upstream and downstream regions are greater than unity. Thus the discontinuity has all the properties of the 2 ! 3 type IS. So it is very likely that one will interpret this DD as an IS on the basis of one spacecraft observation. 3.2. A TD From Two-Spacecraft Observations [26] Table 5 lists the thermal, magnetic and total pressures of up- and downstream sides of this discontinuity. As can be
Figure 7. The observed Wind magnetic fields on October 8, 2001 in the shock coordinate system. 8 of 12
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Table 5. The Pressures on Both Sides of the 8 October 2001 Discontinuity Upstream Thermal pressure, Pa Magnetic pressure Total pressure
11
2.32 10 9.42 1012 3.26 1011
Downstream 2.41 1011 7.19 1012 3.13 1011
seen, the total pressures on the two sides are almost equivalent. In addition, the scalar product of the TD normal nTD (nTD = (B1 B2)/jB1 B2j = (0.689, 0.694, 0.210)) and W (observed velocity difference) is very small ( 1.58 km/s). Therefore V2 V1, B1 and B2 are almost parallel to the plane of the discontinuity. One may consider that the pressures are in balance and there is no flow cross the discontinuity. Thus this discontinuity satisfies the TD requirements entirely. [27 ] This discontinuity was also observed at about 0102:50 UT by Geotail located at RG = (29.91, 7.11, 4.13)RE, and only the two spacecraft are available for this discontinuity. Figure 8 gives the overlapping magnetic field data observed by Wind and Geotail. Here the dotted lines are for the Geotail and its time sequences were shifted by 14.6 min. It can be seen in Figure 8 that the two sets of curves are consistent. They are the same discontinuity structure.
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[28] We now calculate the time differences on the basis of IS and TD models and compare them to the real time difference (14.6 min) between Wind and Geotail observations. We first apply an IS model. Figure 9 shows the sketch of positions of the two spacecraft (asterisk), the shock normal vector (ns), and the vector displacement (DR = RG RW). It can be seen that the dot product of the shock normal ns(0.58, 0.36, 0.73) and the vector displacement DR is positive. This means that Wind should have observed the ‘TD’ earlier than Geotail. However, the Geotail spacecraft observed this ‘TD’ earlier than Wind. On the other hand, we apply the TD model to this event, then nTD DR < 0. The causality is reasonable. In addition, the estimated time difference, Dt, is 13.9 min, which agrees with the observed time difference (14.6 min) well. Here, Dt = (DR nTD)/VTD, where VTD is calculated from V1 nTD or V2 nTD. According to the above results, we consider that this DD should be a TD than an IS. [29] In the present case, we do not identify the DD using two-spacecraft timing method alone, but we check whether the derived solution from the R-H relations is consistent with the arrival time from spacecraft Wind to Geotail. Under the IS assumption, we obtain a normal ns from the local parameters using the R-H fitting method of Lin et al. [2006]. With this IS normal (ns), we found that (RG RW) ns > 0, which demonstrates that Wind should observe the DD before Geotail. However, in fact Geotail observes the event
Figure 8. The magnetic fields measured by the Wind and Geotail in GSE coordinate system, where the dotted lines are for Geotail, and the Geotail time sequences were shifted by 14.6 m. 9 of 12
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Figure 9. The sketch of shock normal and spacecraft locations: (a) take this discontinuity as an intermediate shock; (b) take this discontinuity as a tangential discontinuity. first. Therefore the causality is wrong. Under the TD assumption, the normal nTD is also derived from the local parameters (nTD = (B1 B2)/jB1 B2j) which is perpendicular to the IS normal (ns). When taking nTD, we obtain (V2 V1) nTD = 1.58 km/s, which is very small. It is shown that with the TD model the DD does not propagate with respect to the solar wind. On the other hand, with the TD normal (nTD), we found both the causality and the time delay are consistent with reality. With these evidences, we conclude that the DD should be a TD rather than an IS. Therefore the method used in the present case is first to derive the normals and propagating speeds from the local parameters and then to check the timing from the two spacecraft observations.
4. Discussion and Summary [30] In the past more than thirty years, there are a large number of investigations for intermediate/slow shocks. Previously, it was argued that on the evolutionary conditions ISs could not exist. [e.g., Kantrowitz and Petschek, 1966]. Numerical simulations [e.g., Wu, 1987, 1988; Wu and Hada, 1991] showed that ISs are admissible. In addition, Wu and Kennel [1992a, 1992b] have shown that an MHD system that is almost hyperbolic but not strictly hyperbolic in nature may lead to the formation of ISs. Chao et al. [1993] reported an IS observed in 1980 when Voyager 1 was approximately at 9 AU from the Sun. To our knowledge, this event is the one and only case identified as an IS. For SSs, there are relatively more reported cases in interplanetary space and in the geomagnetic tail. However, all the reported intermediate and slow shocks were identified using one satellite only. As mentioned in Section 1, both TDs and intermediate/slow shocks satisfy the R-H relations. It is difficult to distinguish intermediate/slow shocks with TDs using only one satellite due to their weak shock strength. [31] In order to demonstrate the ambiguity, we analyzed two shock-like DDs. One is on 18 September 1997, which meets all the requirement for SSs including the R-H relations. Zuo et al. [2006] et al. reported this event as a
SS. Another one is on 8 October 2001. The measured solar wind magnetic fields and plasma on both sides of the DD satisfy the R-H relations. This DD also meets all the criterions of the 2 ! 3 types IS. On the other hand, both these two DDs meet the requirements of TDs. Because more than one satellite are available for investigating the two DDs, we estimated the time different between the corresponding spacecraft using Dts = DR n/Vdd. If the DD on 18 September 1997 is considered as a SS, the estimated time difference (13.5 min) between Wind and ACE deviated greatly from the observed value (34.2 min). In the same way, if the DD on 8 October 2001 is considered as an IS, the estimated time difference is negative. So, it is unreasonable to consider them as shocks. On the contrary, considering the two DDs as TDs, one can find that the estimated time differences agree with observed time differences well. In addition, the DD on 18 September 1997 was observed by three spacecraft. We use a novel method, which is independent of the type of the DD, to determine the DD’s normal vector. With the observed time differences, vector displacements between corresponding spacecraft and magnetic fields measured by Wind, the determined DD normal vector is n (0.42, 0.89, 0.15). The normal vector is consistent with nTD (0.43, 0.88, 0.17). In addition, we also checked the normals using magnetic data from the three spacecraft and found that the calculated normals are all consistent. So the discontinuity is stable during its propagation form one spacecraft to another, and the multiple spacecraft timing method should be appropriate. Thus both discontinuities should be interpreted as TDs rather than shocks on large-scale perspective. Based on the abovementioned two events, we speculate that some of the reported intermediate/slow shocks may possibly be TDs. Therefore we suggest that intermediate/slow shocks should be identified as carefully as possible and had better be determined by multi-spacecraft observation, whenever possible. However, if one determines the normal of a DD using four-spacecraft method, it should be noted that the method fails if the spacecraft are nearly coplanar [Schwartz, 1998]. [32] In addition, the selection criteria and duration for the upstream and downstream periods are related to DD types.
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In general, one tries to select relatively stable and long time intervals on the two sides of the discontinuity to minimize the effect of the waves but without major changes in the field on the two sides. Since all four type of DDs need to satisfy the R-H relations, here we select the intervals by trial and error to get the best average observed parameters, which fit the R-H relations well. [33] It is a well-known fact that the Earth’s magnetopause is most often considered as a TD, while small-scale observations at the magnetopause often reveal rotational discontinuities (RDs) and reconnection events that open up the TD. In the same way, there may be another possibility for the two DDs. Namely the TDs are large-scale equilibrium plane configurations, and a local IS/SS is formed somewhere within the plane; the local IS/SS is a sub-structure existing in the bigger TD structure. It is an interesting problem for further study and is not the subject of this paper. We propose to discuss this problem in another paper. [34] Acknowledgments. This work was supported by National Nature Science Foundation of China (NSFC) under grant Nos. 10425312, 10373026, 10603014, and 40574065 and by National Key Basic Research Special Funds (NKBRSF) under grant 2006CB806302 and KJCX2-YWT04, and it is also supported by National Science Council (NSC) (Taiwan) under grants NSC 95-2111-M-008-035, NSC 95-2111-M-008-037, and NSC 95-2811-M-008-034 to National Central University. The authors thank NASA/GSFC for the use of the key parameters from WIND, Geotail, and ACE obtained via the CDA Web page. [35] Zuyin Pu thanks You-Qiu Hu and another reviewer for their assistance in evaluating this paper.
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J. K. Chao, L.-C. Lee, C. C. Lin, and L. H. Lyu, Institute of Space Science, NCU, Chungli, 32001, Taiwan. H. Q. Feng and D. J. Wu, Purple Mountain Observatory, CAS, Nanjing 210008, China. (
[email protected])
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