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 measuredmagnetic fields and plasma on both sides using Rankine-Hugoniot (R-H) relations. It isnot reliable to determine an intermediate/slow shock only by the shock properties andfitting procedure based on one spacecraft observation, though previous reportedintermediate/slow shocks are confirmed in such a way. We investigated two shock-likediscontinuities, which satisfy the R-H relations well. One meets the criterions of slowshocks and was reported as a slow shock, and another has all the characters of intermediateshock based on one spacecraft observation. However, both discontinuities also meet therequirements of tangential discontinuities and were confirmed as tangential discontinuitieson large-scale perspective by using multi-spacecraft observations. We suggest thatintermediate/slow shocks should be identified as carefully as possible and had better bedetermined 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 allowfour types of magnetic directional discontinuities (DDs):contact discontinuity (CD), tangential discontinuity (TD),rotational discontinuity (RD) and shocks. Shocks and TDsare 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 magneticcomponent normal to the discontinuity and conservation oftotal plasma (thermal and magnetic) pressure on both sidesis required, that is to say, flows simply move with thediscontinuity surface. A number of statistical studies of TDshave been carried out [e.g., Burlaga et al., 1977; Tsurutaniand Smith, 1979; Behannon et al., 1981]. These investiga-tions include the ratios of RD to TD and their macroscopicproperties such as thickness and magnetic field rotationangle. Lepping and Behannon [1986] gave a ratio of TDs toRDs greater than unity and Soding et al. [2001] found thatthe ratio of RDs to TDs varied by 5 to 10% depending onthe 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 earlierstatistical population of the RD category is simply a resultof inaccurate normal estimates.[4] The R-H relations have six shock solutions: the fast

and slow shocks and four intermediate shocks (ISs). Thefast shocks are observed frequently in the interplanetary

space. The reported slow shocks (SSs) are relatively rare,and only a small number of SSs have been observed ininterplanetary space [Chao and Olbert, 1970; Burlaga andChao, 1971; Richter et al., 1985; Whang et al., 1996, 1998;Ho et al., 1998; Zuo et al., 2006]. However, in thegeomagnetic 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 anddownstream regions and shock normal ns to be in the samecoplanar plane. So, we define an orthogonal shock frame ofreference as shown in Figure 1, let s denote the unit vectornormal to the coplanar plane(viz. s ? ns), then define: t =ns � s. Therefore the t � s plane is just the shock front, thusboth the up- and downstream magnetic fields are in the ns �t plane. On the other hand, the up- and downstreammagnetic fields of a TD also lie on the same plane, whichis defined as TD front (plane), because a TD has no normalmagnetic field. Therefore the t � ns plane defined above isjust the TD front (plane), and the TD normal (nTD) is in thedirection of s.[6] According to the R-H relations, a TD requires only

two conditions: (1) the velocities and magnetic fields are alltangential to the TD front (plane), and (2) total pressures onboth sides are balanced. For an IS or a SS, in the shockframe of reference, the up- and downstream plasma flowsalso lie on the t � ns plane. It also meets the firstrequirement of a TD. In addition, for an IS or a SS, themagnetic pressure may decrease and the plasma thermal

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1Purple Mountain Observatory, CAS, Nanjing, China.2Institute of Space Science, NCU, Chungli, Taiwan.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2007JA012311$09.00

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pressure increases across the shock front. The total pressurecould be close to balance for some shock conditions, whichmeets the second condition required by a TD. Thereforeusing only the R-H relations based on one spacecraftobservation to determine a TD and IS/SS likely causesambiguities; one can mis-interpret a TD as IS (or SS) andvice versa. So far, to the best of our knowledge, all thereported IS/SSs were identified in such a way and using onesatellite only.[7] The ambiguity can be fixed by using multi-spacecraft

observations. Suppose that two spacecraft observed thesame DD at a different location and at a different time,the time difference Dt can be expressed as [e.g., Russell etal., 1983]

Dt ¼ DR � n=Vdd: ð1Þ

The method can be extended to three or more spacecraftobservations [Schwartz, 1998]. Here DR is the vectordisplacement of the two spacecraft and Vdd is thepropagation 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 calculatedby an IS/SS model are very different from that calculated bya TD model. Therefore the estimated Dt should be verydifferent between these two models. However, one caneasily 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 aSS in a recent work by Zuo et al. [2006], and another meetsall the requirement for ISs. However, both DDs satisfycriterions of TD entirely. We identify the two events as TDs

on a large-scale perspective by use of multi-spacecraftobservations.

2. The 18 September 1997 Slow Shock-Like TD

2.1. Modeling as a Slow Shock

[9] This shock-like discontinuity was observed at about0255:15 on 18 September 1997 by Wind located at (83.51,�13.58, �1.45) RE in GSE coordinate system. Figure 2shows the observed values of the parameters as functions oftime for this event. The magnetic field data obtained fromMagnetic Field Investigation (MFI) magnetometer and theproton data obtained from the 3-Dimension Plasma (3DP)analyzer are all shown in the GSE coordinate. The data havea time resolution of 3 s. As investigated by Zuo et al.[2006], this discontinuity has typical SS characters: (1) thedensity increases across the discontinuity, while the magni-tude of magnetic field decreases; (2) the observed parame-ters all satisfy the R-H relations; (3) The upstream normalbulk velocity in the shock frame of reference is larger thanlocal slow magnetoacoustic speed and smaller than localnormal Alfven speed, and the downstream velocity issmaller than the local slow magnetoacoustic speed. Theyused a coplanarity method and a self-consistent method todetermine the shock normal (ns) and the other two axes ofthe shock coordinate system (Listed in Table 1). The self-consistent method utilizes the entire R-H relations and aminimization technique to determine the shock normal. Thedetailed descriptions can be found in the work of Zuo et al.[2006]. The angle between the estimated shock normalsusing the two methods is only 7�. In addition, the predictedR-H solutions based on the shock normal determined by aself-consistent method are in better agreement with theobservations, so they considered the self-consistent methodmore accurate. Table 1 lists the up- and downstreammagnetic fields, plasma velocities, and densities on bothsides. 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 thecriteria first reported by Gosling et al. [2005a]. A number ofrecent reports have outlined this subject further [e.g.,Goslinget 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 temper-ature, weaker magnetic field strength, and intermediate fieldorientation as compared with the surrounding solar wind. Thechanges in V and B are anticorrelated in the leading portionand 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 aTD, since, in the frame of a discontinuity, there is no normalmass flow, and since the total plasma (thermal and magnetic)pressures are balanced on both sides. As in our analysis, theestimated 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 coordinatesobtained by Zuo et al. [2006] (see Table 1). The dot productof nTD and the difference between the downstream and

Figure 1. Shock frame of reference (orthogonal coordi-nate system), where shock normal is in the ns direction,magnetic fields are in the ns � t plane and the shock front inthe s � t plane. As a TD, the normal is the unit vector s andthe ns � t plane is the TD surface.

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upstream velocities (V2 � V1) is only 0.66 km/s, whichindicates that there is nearly no mass flow through thediscontinuity. In addition, Table 2 shows the thermal, mag-netic, and total pressures on both sides. The difference in thetotal pressures is only 3 percent of the total pressures in theupstream region. By including the systematic error andsampling errors, one can consider that the total plasmapressures are conserved across the discontinuity. Thereforefrom these two conditions the local parameters of thisdiscontinuity satisfy the TD requirements.

2.3. Identify the Discontinuity as TD With Three-Spacecraft Observations

[12] This discontinuity was also observed at 0221:01 UTby ACE located at (193.31, �24.78, 20.78) RE. Figure 3shows the corresponding magnetic field profiles measuredby ACE and Wind. Here the dotted lines are for ACE, forwhich the time sequences were shifted by 34.2 min. As seenin Figure 3, the two sets of the profiles are very close toeach other, while there are only a few differences seen in thedetail structures. Therefore it can be confirmed that ACEobserved the same discontinuity as Wind.[13] SS and TD models are used respectively to verify the

time difference (Dt) between the two spacecraft. From theSS model, Dts = DR � ns/Vsh, where ns and Vsh are fromTable 1, and DR = (�109.80, 11.23, �22.23)RE is thevector displacement between the Wind and ACE spacecraft.

The calculated time is 13.5 min, which is very differentfrom the observed time difference (34.2 min). On the otherhand, from the TD model, DtTD = DR � nTD /VTD, whereVTD is the TD propagation speed in the rest frame ofreference. Since TD is a non-propagating discontinuity withrespect to the solar wind, its speed (in the normal direction)in the rest frame can be estimated by VTD = nTD � V1, whereV1 is the upstream flow velocity in the rest frame. Theestimated value of VTD is 160.22 km/s. The estimated DtTDis 35.6 min, which is very close to the observed timedifference. Note that since there is no mass flux across thetransition, using V1 and V2 to calculate VTD almost makes

Table 1. The Observed Parameters of the 18 September 1997 SS-

Like 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 (�3.1, 0.02, �7.9)B2 (3.4, 2.3, �3.3)N1, N2, cm

�3 22.1, 32.7V1, km/s (�341, 11, 23)V2 (�356, 5, 20)ns (�0.44, �0.38, 0.81)s (�0.43, 0.88, 0.17)t (�0.79, �0.28, �0.55)Vsh, km/s 205

Figure 2. The interplanetary magnetic field and plasma data measured by the Wind spacecraft in GSEcoordinate system on 18 September 1997. The region between two dotted vertical lines is likely areconnection exhaust transition.

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no difference here (nTD � V2 = 160.88 km/s). With the valuecalculated from V2, the estimated DtTD is 35.5 min.According to the above results, we conclude that thisdiscontinuity should be interpreted as a TD and not as aSS. This interpretation is based on a large-scale perspectivedue 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 speedof the DD is constant in time and space, Knetter et al.[2004] used four Cluster spacecraft to determine the dis-continuity normal, namely via triangulation. Four spacecraftcan give three independent equations associated with thepropagation time and one equation that requires unity of theshock normal vector. They are

Dti ¼ DR1i � n=Vn; i ¼ 2 to 4; ð2Þ

jnj ¼ 1: ð3Þ

Here Vn is the propagation speed of the DD in the rest frameof reference, and DR1i represents the vector displacementbetween the Cluster 1 and Cluster i spacecraft. In addition,Dti represents the time difference between Cluster 1 andCluster i spacecraft. With these four equations, n and Vn canbe found. The method requires no magnetic field andplasma 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 timingcannot find the DD normal vector and its propagationvelocity. As pointed out by Schwartz [1998] (pages 257and 309), if the four spacecraft are coplanar, the determinantof the matrix of DR in Equation (2) is zero. The three linearequations reduce to two. Therefore we cannot find the full setof unknowns (n /Vn). For example, if four spacecraft are onthe 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]. Theyapply only three spacecraft observations. With this methodthey have only two independent equations for propagationtime. In order to have a close system, they assume that theDD moves with the plasma bulk velocity measured at one ofthe spacecraft. Under such assumption, their method isapplicable only for TDs. Any DDs propagating with respectto the solar wind frame of reference cannot be studied bytheir method.

Table 2. The Pressures on Both Sides of the 18 September 1997

Discontinuity

Upstream Downstream

Thermal pressure, Pa 4.00 � 10�11 5.98 � 10�11

Magnetic pressure 2.87 � 10�11 1.10 � 10�11

Total pressure 6.87 � 10�11 7.08 � 10�11

Figure 3. The magnetic fields measured by the Wind and ACE in GSE coordinate system, where thedotted lines are for ACE, and the ACE time sequences were shifted by 34.2 m.

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[17] We examined all the solar wind data observed byother spacecraft, which are near the Earth, and found thatthe DD discussed in this section was also observed by theGeotail spacecraft at 0312:27 UT and at (24.85, �11.57,�1.52) RE (GSE). Figure 4 shows the correspondingmagnetic field profiles measured by Geotail and Wind.Here the dotted lines are for Geotail, for which the timesequences were shifted by 17.2 min. Therefore a total ofthree spacecraft are available. In general, the measuredmagnetic fields have smaller uncertainty than that of veloc-ities. 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 onefor any type of one dimensional DDs than the Equation (2).In addition, the systematic errors in magnetic field measure-ments are likely to be eliminated. This equation uses onlythe 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 (timedifference) between the Wind and ACE spacecraft. Here themeasured magnetic fields are from the data of Wind. Onecan obtain the solutions of n and Vn from these fourexpressions. 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), andwe obtain Vn = 164.99 km/s, which is also consistent withVTD (160.22 km/s). It, again, confirms that this event is aTD. On the other hand, this derived normal and propagatingspeed do not agree with the slow shock solution. For a detailcomparison of the results of the SS and TD models with theabove derived normal and speed, we summarize theparameters 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 linearalgebraic system as follows.

DRWGx DRWGy DRWGz

DRWAx DRWAy DRWAz

DBx DBy DBz

0@

1A

mx

my

mz

0@

1A ¼

DtWG

DtWA

0

0@

1A ð8Þ

Figure 4. The magnetic fields measured by the Wind and Geotail in GSE coordinate system, where thedotted lines are for Geotail, and its time sequences were shifted by 17.2 m.

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where DB (B2 � B1) and m n/Vn. With the solutionof m, we can calculate the value of Vn as follows (viaEquation (7)).

Vn ¼ m2x þ m2

y þ m2z

� ��1=2ð9Þ

[19] For this system, if DB, DRWG, and DRWA are not onthe 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 thesame plane. As obtained from computation, the determinantof the matrix in Equation (8) is non zero. Here weemphasize that the calculation of Equations (4)–(6) isindependent of the type of DD. The normal and thepropagation speed derived in our calculation should corre-spond to the actual type it belongs to.[20] There is another way to understand the capability of

Equations (4) – (7). The magnetic field conservationEquation (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 SSthe 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 beused to determine which set, the SS or the TD, is the correctsolution. For the present case, we found that only the valuesfor the TD can satisfy Equations (4)–(7) well in comparisonwith the values for the SS.[21] For this case, we have checked the frozen in condi-

tion 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- anddownstream flow velocities in the spacecraft (rest) frame ofreference. The result shows that they are close to the prop-agation speed of the DD calculated from the Equations (4)–(7) (Vn = 165 km/s in the rest frame). This means that theDD is a non-propagating discontinuity with respect to thesolar wind plasma. In other words, V2-V1, B1 and B2 arealmost parallel to the DD plane. If we use the slow shockmodel, the calculated DD propagation speed is 205 km/s inthe rest frame which is very different from Vn = 165 km/s.In addition, we found that the total pressures across theplane are close to one another. Therefore the obtained DD ismore likely a TD.[22] On the other hand, we also compare our calculated

normal with the local derived normals using the R-Hrelations, 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 oneanother. 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 alsoconsider 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 ACEmagnetic field data is (�0.47, 0.87, 0.18). This normal is�2.3� off from the normal calculated from the Windmagnetic filed data. The normal calculated from the Geotailmagnetic field data is (�0.52, 0.83, 0.21). This normal is�6.7� off from the normal calculated from the Windmagnetic filed data. The normals are close to one another.This shows that the discontinuity is stable during the periodbetween ACE and Geotail. The using of the multiplespacecraft timing method in this paper should be appropriate.

3. The 8October 2001 Intermediate Shock-LikeTD

3.1. Modeling as an Intermediate Shock

[24] This shock-like discontinuity was observed by Windat RW = (37.20, �59.72, 4.91) RE in GSE coordinate systemat �0117:30 UT on 8 October 2001. Figure 6 shows themagnetic field and plasma data of this event. It is wellknown that shock fitting is very important for investigationof interplanetary shocks. One main problem related to shockfitting is to search for an accurate shock frame of reference.

Table 3. The Comparison of the Results From the SS and TD Models

Parameter SS Model TD Model Equations (4)– (7) Observation

Normal n (�0.58, 0.36, 0.73) (�0.43, 0.88, 0.17) (�0.42, 0.89, 0.15)VSS, km/s 205VTD, km/s 160.2 (162.9)a 165.0Dt(ACE to Wind) 13.5 m 35.6 m (35.5 m)a 34.2 m

aThe value in the parentheses is calculated from the downstream flow velocity.

Figure 5. The r � B = 0 allows only two normals - the SSnormal (ns) and the TD normal (nTD) for the system ofEquations (4)–(7). For the TD normal B1 � nTD = B2 � nTD =0, while for the SS the normal magnetic field to the DD is aconstant value. In this sketch the relationship of the normalsof the slow shock and the TD is demonstrated. The unitvectors nTD, ns, and t are orthogonal. Here, ns is determinedby the coplanarity theorem [Zuo et al., 2006], and nTD isobtained by nTD = (B1 � B2)/jB1 � B2j.

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Such as the coplanarity theory and Minimum VarianceAnalysis (MVA) are frequently used to analyze interplane-tary shocks. Here we use a new shock fitting procedureproposed recently by Lin et al. [2006]. They use a whole setof the R-H relations and modified R-H relations. The mod-ified 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., Chaoand Goldstein, 1972; Davison and Krall, 1977; Yoon andLui, 2006]. Lin et al. [2006] separated their procedure undertwo conditions. One is called Method A, which utilizes theclassical R-H relations. Another one is called Method B andutilizes the modified R-H relations. With this, a best fitsolution that satisfies the R-H relations within the limitationof the data error is obtained. For more details of theprocedure please refer to Lin et al. [2006].[25] The second column of Table 4 lists the observed data

means and the corresponding parameters directly calculatedfrom the data means of the observed magnetic fields andplasma. The derived parameters are the shock normal vectorns, other two axes of the shock coordinate system t and s,the plasma beta (b), the normal Alfven-Mach number (MAN =Vn/VAn), the fast-mode Mach number (MF = Vn/Vf), theslow-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 upstreamto downstream plasma densities (y = N1/N2), the ratio ofdownstream to upstream tangential magnetic fields (u = Bt2/Bt1), the angle, qBN = cos�1(B1 � ns/B1), between the shocknormal and the upstream magnetic field (also called theshock normal angle). In the above expression, VAn is theAlfven speed based on magnetic field component normal toshock front (VAn = Bn/(m0r)

1/2), Vn is the component of thebulk velocity to the shock front and measured in the shockframe of reference, and Vf and Vsl are the speeds of the fast-and slow-mode magnetosonic waves in the direction of theshock normal, respectively. We applied both methods to thisdiscontinuity. The third and fourth columns of Table 4respectively list the fitting results form Method A and Bas well as corresponding parameters calculated from thesebest fit values. From Table 4 one can find that both methodsgive very similar results. Figure 6 also shows the best fitvalues (Method A) of upstream and downstream regions asdots. As can be seen in Table 4 and Figure 6, the best fitvalues are in very good agreement with the observed values.According to the MHD theory of shocks, an IS has thefollowing properties. (1) The normal Alfven-Mach number(MA) is greater than unity in the preshock state and less thanunity in the postshock state. (2) The tangential componentsof both the preshock and postshock magnetic fields on the

Figure 6. The interplanetary magnetic field and plasma data measured by the Wind spacecraft in GSEcoordinate system on October 8, 2001 and the best fitting valves of upstream and downstream regions(dotted lines).

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shock front have opposite signs. (3) The plasma numberdensity increases from the upstream region to the down-stream region. (4) Of the four types of intermediate shock,the 2 ! 4 type has a larger density jump across the shockfront than the 1 ! 3, the 1 ! 4, and the 2 ! 3 types [Chaoet al., 1993]. Figure 7 also shows the magnetic field data inthe shock coordinate system. From Figure 7, it can be seenthat the tangential magnetic field Bt changes sign across theshock front, and Bn approximately keeps constant, and Bs

component is approximately zero. Combining the shockparameters listed in Table 4, it is clear that this discontinuity

satisfies the criteria for an IS. In addition, the fast-modeMach number is less than unity, and both the slow-modeMach number in the upstream and downstream regions aregreater than unity. Thus the discontinuity has all theproperties of the 2 ! 3 type IS. So it is very likely thatone will interpret this DD as an IS on the basis of onespacecraft observation.

3.2. A TD From Two-Spacecraft Observations

[26] Table 5 lists the thermal, magnetic and total pressuresof up- and downstream sides of this discontinuity. As can be

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 (3.49, �3.38, �0.24) (3.49, �3.39, �0.24) (3.48, �3.37, �0.24)B2 (0.89, 0.38, �4.14) (0.89, 0.37, �4.14) (0.88, 0.36, �4.14)N1, N2, cm

�3 14.16, 15.64 14.46, 15.49 14.19, 15.68W(V2 � V1)(km/s) (�14.9, 17.5, �16.5) (�15.4, 21.7, �21.1) (�15.8, 21.9, �20.7)b1, b2 2.46, 3.35 2.44, 3.59 2.45, 3.36ns (�0.583, 0.355, 0.731) (�0.582, 0.357, 0.731) (�0.582, 0.358, 0.731)S (�0.687, �0.695, �0.212) (�0.689, �0.694, �0.210) (�0.688, �0.695, �0.208)T (0.433, �0.626, 0.649) (0.432, �0.625, 0.650) (0.434, �0.624, 0.650)MAN1, MAN2 1.021, 0.971 1.013, 0.979 1.021, 0.971MF1, MF2 0.510, 0.490 0.455, 0.442 0.514, 0.494MSL1, MSL2 1.262, 1.079 1.178, 1.056 1.266, 1.081y 0.905 0.933 0.905m 0.874 0.873 0.875u �0.732 �0.730 �0.734qBN 45.50� 45.48� 45.38�

aThe 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 ofW is (1.51, 1.70, 2.47), where SDis the sample standard deviation.

Figure 7. The observed Wind magnetic fields on October 8, 2001 in the shock coordinate system.

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seen, the total pressures on the two sides are almostequivalent. In addition, the scalar product of the TD normalnTD (nTD = (B1 � B2)/jB1 � B2j = (�0.689, �0.694,�0.210)) and W (observed velocity difference) is verysmall (�1.58 km/s). Therefore V2 � V1, B1 and B2 arealmost parallel to the plane of the discontinuity. One mayconsider that the pressures are in balance and there is noflow cross the discontinuity. Thus this discontinuity satisfiesthe 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 thisdiscontinuity. Figure 8 gives the overlapping magnetic fielddata observed by Wind and Geotail. Here the dotted linesare for the Geotail and its time sequences were shifted by14.6 min. It can be seen in Figure 8 that the two sets ofcurves are consistent. They are the same discontinuitystructure.

[28] We now calculate the time differences on the basis ofIS and TD models and compare them to the real timedifference (14.6 min) between Wind and Geotail observa-tions. We first apply an IS model. Figure 9 shows the sketchof positions of the two spacecraft (asterisk), the shocknormal vector (ns), and the vector displacement (DR =RG � RW). It can be seen that the dot product of the shocknormal ns(�0.58, 0.36, 0.73) and the vector displacementDR is positive. This means that Wind should have observedthe ‘TD’ earlier than Geotail. However, the Geotail space-craft observed this ‘TD’ earlier than Wind. On the otherhand, we apply the TDmodel to this event, then nTD �DR < 0.The causality is reasonable. In addition, the estimated timedifference, Dt, is 13.9 min, which agrees with the observedtime difference (14.6 min) well. Here,Dt = (DR � nTD)/VTD,where VTD is calculated fromV1 � nTD orV2 � nTD. Accordingto the above results, we consider that this DD should be a TDthan an IS.[29] In the present case, we do not identify the DD using

two-spacecraft timing method alone, but we check whetherthe derived solution from the R-H relations is consistentwith the arrival time from spacecraft Wind to Geotail. Underthe IS assumption, we obtain a normal ns from the localparameters 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 DDbefore Geotail. However, in fact Geotail observes the event

Table 5. The Pressures on Both Sides of the 8 October 2001

Discontinuity

Upstream Downstream

Thermal pressure, Pa 2.32 � 10�11 2.41 � 10�11

Magnetic pressure 9.42 � 10�12 7.19 � 10�12

Total pressure 3.26 � 10�11 3.13 � 10�11

Figure 8. The magnetic fields measured by the Wind and Geotail in GSE coordinate system, where thedotted lines are for Geotail, and the Geotail time sequences were shifted by 14.6 m.

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first. Therefore the causality is wrong. Under the TDassumption, the normal nTD is also derived from the localparameters (nTD = (B1 � B2)/jB1 � B2j) which is perpen-dicular to the IS normal (ns). When taking nTD, we obtain(V2 � V1) � nTD = 1.58 km/s, which is very small. It isshown that with the TD model the DD does not propagatewith respect to the solar wind. On the other hand, with theTD normal (nTD), we found both the causality and the timedelay are consistent with reality. With these evidences, weconclude that the DD should be a TD rather than an IS.Therefore the method used in the present case is first toderive the normals and propagating speeds from the localparameters and then to check the timing from the twospacecraft observations.

4. Discussion and Summary

[30] In the past more than thirty years, there are a largenumber of investigations for intermediate/slow shocks.Previously, it was argued that on the evolutionary condi-tions ISs could not exist. [e.g., Kantrowitz and Petschek,1966]. Numerical simulations [e.g., Wu, 1987, 1988; Wuand Hada, 1991] showed that ISs are admissible. Inaddition, Wu and Kennel [1992a, 1992b] have shown thatan MHD system that is almost hyperbolic but not strictlyhyperbolic in nature may lead to the formation of ISs. Chaoet al. [1993] reported an IS observed in 1980 when Voyager1 was approximately at 9 AU from the Sun. To ourknowledge, this event is the one and only case identifiedas an IS. For SSs, there are relatively more reported cases ininterplanetary space and in the geomagnetic tail. However,all the reported intermediate and slow shocks were identi-fied using one satellite only. As mentioned in Section 1,both TDs and intermediate/slow shocks satisfy the R-Hrelations. It is difficult to distinguish intermediate/slowshocks with TDs using only one satellite due to their weakshock strength.[31] In order to demonstrate the ambiguity, we analyzed

two shock-like DDs. One is on 18 September 1997, whichmeets all the requirement for SSs including the R-Hrelations. Zuo et al. [2006] et al. reported this event as a

SS. Another one is on 8 October 2001. The measured solarwind magnetic fields and plasma on both sides of the DDsatisfy the R-H relations. This DD also meets all thecriterions of the 2 ! 3 types IS. On the other hand, boththese two DDs meet the requirements of TDs. Because morethan one satellite are available for investigating the twoDDs, we estimated the time different between thecorresponding spacecraft using Dts = DR � n/Vdd. Ifthe DD on 18 September 1997 is considered as a SS, theestimated time difference (13.5 min) between Wind andACE deviated greatly from the observed value (34.2 min).In the same way, if the DD on 8 October 2001 is consideredas an IS, the estimated time difference is negative. So, it isunreasonable to consider them as shocks. On the contrary,considering the two DDs as TDs, one can find that theestimated time differences agree with observed time differ-ences well. In addition, the DD on 18 September 1997 wasobserved by three spacecraft. We use a novel method, whichis independent of the type of the DD, to determine the DD’snormal vector. With the observed time differences, vectordisplacements between corresponding spacecraft and mag-netic fields measured by Wind, the determined DD normalvector is n (�0.42, 0.89, 0.15). The normal vector isconsistent with nTD (�0.43, 0.88, 0.17). In addition, wealso checked the normals using magnetic data from the threespacecraft and found that the calculated normals are allconsistent. So the discontinuity is stable during its propa-gation form one spacecraft to another, and the multiplespacecraft timing method should be appropriate. Thus bothdiscontinuities should be interpreted as TDs rather thanshocks on large-scale perspective. Based on the above-mentioned two events, we speculate that some of thereported intermediate/slow shocks may possibly be TDs.Therefore we suggest that intermediate/slow shocks shouldbe identified as carefully as possible and had better bedetermined by multi-spacecraft observation, whenever pos-sible. However, if one determines the normal of a DD usingfour-spacecraft method, it should be noted that the methodfails 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.

Figure 9. The sketch of shock normal and spacecraft locations: (a) take this discontinuity as anintermediate shock; (b) take this discontinuity as a tangential discontinuity.

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In general, one tries to select relatively stable and long timeintervals on the two sides of the discontinuity to minimizethe effect of the waves but without major changes in thefield on the two sides. Since all four type of DDs need tosatisfy the R-H relations, here we select the intervals by trialand error to get the best average observed parameters, whichfit 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 obser-vations at the magnetopause often reveal rotational discon-tinuities (RDs) and reconnection events that open up theTD. In the same way, there may be another possibility forthe two DDs. Namely the TDs are large-scale equilibriumplane configurations, and a local IS/SS is formed some-where within the plane; the local IS/SS is a sub-structureexisting in the bigger TD structure. It is an interestingproblem for further study and is not the subject of thispaper. We propose to discuss this problem in another paper.

[34] Acknowledgments. This work was supported by NationalNature Science Foundation of China (NSFC) under grant Nos. 10425312,10373026, 10603014, and 40574065 and by National Key Basic ResearchSpecial Funds (NKBRSF) under grant 2006CB806302 and KJCX2-YW-T04, 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, andNSC 95-2811-M-008-034 to National Central University. The authors thankNASA/GSFC for the use of the key parameters from WIND, Geotail, andACE 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. (fenghq9921@163.com)

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