Estimates of magnetic cloud expansion at 1 AU

In this study we analyze 53 magnetic clouds (MCs) of standard profiles observed in WIND magnetic field and plasma data, in order to estimate the speed of MC expansion ( VE) at 1 AU, where the expansion is investigated only for the component perpendicular to the MCs’ axes. A high percentage, 83%, of the good and acceptable quality cases of MCs (N(good)=64) were actually expanding, where “good quality” as used here refers to those MCs that had relatively well determined axial attitudes. Two different estimation methods are employed. The “scalar” method (where the estimation is denoted VE,S) depends on the average speed of the MC from Sun-to-Earth ( ), the local MC’s radius (RO ), the duration of spacecraft passage through the MC (at average local speed ), and the assumption that =. The second method, the “vector determination” (denotedVE,V ), depends on the decreasing value of the absolute value of the Z-component (in MC coordinates) of plasma velocity ( |VZ|) across the MC, the closest approach distance ( YO ), and estimatedRO ; the Z-component is related to spacecraft motion through the MC. Another estimate considered here, V ′ E,V , is similar toVE,V in its formulation but depends on the decreasing |VZ| across part of the MC, that part between the maximum and minimum points of |VZ| which are usually close to (but not the same as) the boundaries points. The scalar means of estimating VE is almost independent of any MC parameter fitting model results, but the vector means slightly depends on quantities that are model dependent (e.g. |CA|≡|YO |/RO ). The most probable values of VE from all three means, based on the full set of N=53 cases, are shown to be around 30 km/s, but VE has larger average values of =49 km/s, =36 km/s, and=44 km/s, with standard deviations of 27 km/s, 38 km/s, and 38 km/s, respectively. The linear correlation coefficient for VE,S vs.V ′ E,V is 0.85 but is Correspondence to: R. P. Lepping (ronald.p.lepping@nasa.gov) lower (0.76) forVE,S vs. VE,V , as expected. The individual values ofVE from all three means are usually well below the local Alfvén velocities, which are on average (for the cases considered here) equal to 116 km/s around the inbound boundary, 137 km/s at closest approach, and 94 km/s around the outbound boundary. Hence, a shock upstream of a MC is not expected to be due to MC expansion. Estimates reveal that the errors on the “vector” method of estimating VE (typically about±7 km/s, but can get as large as ±25 km/s) are expected to be markedly smaller than those for the scalar method (which is usually in the range ±(15⇔20) km/s, depending on MC speed). This is true, despite the fact that |CA| (on which the vector method depends) is not always well determined by our MC parameter fitting model (Lepping et al., 1990), but the vector method only weakly depends on knowledge of|CA|.


Introduction
In the past, observations of the speed profile of the solar wind within an interplanetary magnetic cloud (MC) at 1 AU were used in determining whether the cloud was expanding or not locally (e.g.Burlaga, 1990;Farrugia et al., 1992aFarrugia et al., , b, 1993)).For MC expansion the speed must show a marked, and approximately linear, decrease across the cloud or across most of it; see Fig. 1.Earlier studies have, in fact, shown that it was not uncommon for MCs at 1 AU to be expanding (e.g.Burlaga, 1995;Hidalgo, 2003Hidalgo, , 2005)), and it was determined from WIND data that a large percentage of MCs seen at 1 AU are expanding (Lepping et al., 2002).Briefly, a MC was defined empirically by L. Burlaga and coworkers as a (usually large) interplanetary structure having enhanced field R. P. Lepping et  (≡|V Z,CL |) gradient of velocity as the spacecraft passes through a MC that is expanding; the subscript CL refers to the MC coordinate system (see Sect. 2), where the X CL -axis is aligned with the estimated local axis of the MC.Also shown (in green) are the magnetic field magnitude, field latitude angle, and proton plasma beta.The duration of the MC passage is T and the interval from the MAX to MIN of |V Z,CL | is t .A upstream shock ramp is indicated for this MC, even though not all MCs possess upstream shocks.The red dashed curve for |V Z,CL | holds for a case where the MAX and/or MIN points for this quantity are markedly different from its values at the boundaries, and the black curve for |V Z,CL | holds for the case when its MAX and MIN values are at or very near to the boundaries.magnitude, a relatively smooth change in field direction as the observing spacecraft passes through the MC, and lower proton temperature (and proton beta) than the surrounding solar wind.MC properties were first discussed by Burlaga et al. (1981), Goldstein (1983), and Burlaga (1988Burlaga ( , 1995)).Many believe that eruptive prominences are the main source of MCs (e.g.Bothmer and Schwenn, 1994).Also it is believed that MCs are essentially the "core" of Interplanetary Coronal Mass Ejections (ICMEs); e.g.see Gopalswamy et al. (1998), and also see early reviews by Gosling (1990Gosling ( , 1997) ) that compare CMEs to large magnetic flux ropes in the solar wind which are usually the essence of MCs.
The main purpose of this study is to analyze quantitatively MC expansion using WIND magnetic field and solar wind velocity data.The ultimate goal of the study is to use our resulting understanding of MC expansion to modify, as accurately as possible, a static MC parameter fitting program (Lepping et al., 1990) to accommodate 3-D expanding MCs, as well as to accommodate other features (e.g.non-circular cross-section), in a systematic production mode; the original fitting program (using only a static model) was also capable of working in a production mode.For this study 100 WIND MCs, covering the 11-year period from early 1995 to about August 2006 have been identified and parameter-modeled, and after "editing" in two stages was reduced to N =53 cases.The two stages consist of, first, quality editing, described in detail below, and then a test for appropriateness (i.e.we ask: Was the MC actually expanding or not?).For the MC fitting itself (Lepping et al., 1990) only an average MC speed was required, in order to transform from the time domain to the space domain.Then the resulting estimated radius, called R O , from the static flux rope modeling is assumed useful for carrying over to the actually expanding cases by viewing R O as a weighted average of a continuum of radii during cloud passage.The view was (and is) that at first encounter the flux rope's radius was at a minimum and expanding to a maximum upon departure.It is also assumed that any noncircular cross-section effects are less important, even though it is very likely that cases of perfectly circular cross-sections at 1 AU are probably rare; see, e.g.Riley and Crooker (2004).There have been many hypotheses concerning the true nature of this radial expansion of MCs and how it is detected (e.g.Marubashi, 1986Marubashi, , 1997;;Farrugia et al., 1992a, b;Burlaga, 1995;Osherovich et al., 1993Osherovich et al., , 1995;;Mulligan and Russell, 2001;Berdichevsky et al., 2003).In such studies usually only one or two examples of MCs are examined in determining the nature of the velocity profile, but the results may not be characteristic of interplanetary MCs in general.Modern models of MC parameter fitting usually take into consideration MC expansion, as well as other important features, such as the good probability of a MC having a non-circular cross-section; see, for example, Hidalgo et al. (2002), Riley et al. (2004), andNieves-Chinchilla et al. (2005).
In this study we aim: (1) to estimate quantitatively the rate of expansion of a large number of MCs at 1 AU using two different methods and compare the results of the two methods, and (2) to ascertain the importance of expansion in MC parameter fitting models, for the practical purpose of modifying such a model, which assumes a MC is a simple static force free structure with a circular cross-section.And we examine a large number of cases.We deal here only with MC expansion that is perpendicular to the MC axis.It has been determined that MC expansion in actuality is also along the MC's axis as well, as it must be for full 3-D expansion, but this is much more difficult to determine empirically and accurately; see Berdichevsky et al. (2003), who describe such expansion and give examples of it.We argue that if the expansion is approximately isotropic in 3-D at 1 AU, then by determining the 2-D expansion, i.e. the expansion perpendicular to the MC axis, we are obtaining important information on the axial expansion as well.

Coordinate system utilized
It is necessary to carry out this analysis in what we refer to as Cloud (CL) coordinates, where the X CL -axis is aligned with the estimated local axis of the MC and positive in the direction of the magnetic field along that axis, Z CL is the positive projection of the trajectory of the passing spacecraft on the cross-section of the MC, and Y CL =Z CL ×X CL .[Note that since the MC moves directly outward from the Sun, the relative path of the spacecraft through the MC is positive inward toward the Sun, and therefore, is parallel to the X GSEaxis.For example, for the special case of a MC whose axis is parallel to either (+)Y GSE or (−)Y GSE , we see that the +Z CL axis is exactly parallel to the +X GSE axis, i.e. positive toward to the Sun.Then within the MC the plasma velocity exactly along the -X GSE axis, for this case, is along −Z CL and should be expressed as −V Z,CL .]The following Website shows how to develop the specific orthonormal matrix needed to transform any data from geocentric solar ecliptic (GSE) coordinates to CL coordinates for any particular MC: http://lepmfi.gsfc.nasa.gov/mfi/ecliptic.html.Such a matrix, as expected, depends on knowledge of the latitude (θ A ) and longitude (φ A ) of that MC's axis and on the polarity of the +Z CL axis with respect to the GSE system, as described above.

The formulation
We will develop two means of estimating a MC's expansion velocity, one (called the scalar means) which depends on the size (i.e.radius, R O , in this case), the duration (T ) of the MC passage, and the average local MC speed; we call this expansion velocity, V E,S .The other technique (called the vector means) depends on the gradient of the speed across the MC (strictly on the gradient of V Z,CL across most of the MC (more on that below) and on the relative closest approach distance |Y O |/R O (≡|CA|); we denote this expansion velocity as V E,V .Finally, we compare the values of V E,S and V E,V (and a modified form, called V E,V , explained below, will be considered) for our set of 53 MCs.Below we give the specifics of these two techniques.Some of our results will obviously depend on the technique employed (vector or scalar) and on the MC parameter fitting model, because that is how we obtain needed quantities: MC axis direction (especially for the coordinate transformation), Z CL -direction, Y O , and R O ; these are especially needed for the vector technique.For the scalar technique only R O is need from the MC fitting model, which is usually well determined (if there is not a gross deviation from a circular cross-section), so this technique is only weakly dependent on the model.
Since modern techniques of estimating MC fitting parameters and global models of MC tend to agree that most MCs do not have circular cross-sections, we do not demand that such be the case either; see e.g.Lepping et al. (1998), Vandas et al. (2002Vandas et al. ( , 2006)), Vandas and Romashets (2003), Riley et al. (2004), andNieves-Chinchilla (2005).However, we assume that the semi-minor axis (R MIN ) and semi-major axis (R MAX ) of the MC's cross-section are not vastly different from each other.(We will think of the cross-section as oval, but not necessarily an ellipse, centered at the MC's axis.)That is, we assume that R MAX /R MIN 2, or so.Then we think of R O , as estimated by the model of Lepping et al. (1990) as being the average of these two axes lengths, i.e.
where the brackets <> further represent a time average over t, which is the time delay from the first sighting (t EN , the time of the front boundary crossing) to the rear boundary (at t EX , the exit crossing), i.e. t=t EX −t EN , where the clock starts when the MC lifts off the Sun.(Notice then that t EN is just the Sun-to-Earth travel time for the MC.)So t in this case is identified as T , the duration of the spacecraft passage through the MC, and sometimes simply called "duration".We acknowledge that the approximation in Eq. ( 1) is usually a source of error in our estimates of V E , but not usually a significant one in the vector method.

The scalar derivation of V E
We start with the formulation of the "scalar derivation" of V E .Farrugia et al. (1992a, b) show that where all quantities are measured in a frame of reference where the MC's average velocity is zero, and where r O is the radius of the MC as initially observed at the MC's front boundary at time t EN (at 1 AU in our case).Simply put, this formula is derived from the fact that a relatively small structure, the MC at the Sun, must expand when going from the Sun to Earth, since its cross-section is observed to be a large fraction of an AU at 1 AU, and it was further assumed that it does so uniformly at constant speed over 1 AU.Next we assume that the average speed from the Sun to Earth (<V S−to−E >) is almost the same as the average speed <V C > of the plasma within the MC, as observed at 1 AU.Hence, (We assume that Eq. (3) holds, even though it has been established that there is some acceleration or deceleration of ejecta generally (see, e.g.Gopalswamy, 2000), since this apparently occurs mainly near the Sun, and therefore does not negate the good approximation of Eq. (3).)Then Hence, from Eqs. (2) and (4), we see that where we identify r O as approximately R O .As mentioned above and confirmed here, V E,S depends on MC duration, speed, and size, all scalars.We now check V E,S for reasonableness by using typical values, for a low speed MC case, on the right side of Eq. ( 5 , 26, 1919-1933, 2008 as R O =0.125 AU, <V C >=450 km/s, T =20 h; see e.g.Lepping et al. (2006) which provides these average values.This gives V E,S =46 km/s, which is within a typical range of values for the MC expansion speed at 1 AU for the slower MCs (see, e.g.Lepping et al., 2002).For MCs moving at, say, 650 km/s and keeping all other values in Eq. ( 5) the same, we obtain a V E,S of 62 km/s.Both of these are markedly lower (by a factor of about two) than the value of 114 km/s derived by Burlaga (1995, p. 100) for the expansion speed of a particular case (14/15 January 1988) under somewhat similar circumstances.

The vector derivation of V E
We now provide the "vector derivation" of the expansion velocity, called V E,V , which will depend on the gradient of the speed across the MC (i.e.strictly on the component V Z,CL across the part of the MC where the gradient is smoothest and steepest), so it depends strictly on local and relevant measured plasma velocities after coordinate transformation and to some extent on MC modeling results, but to a lesser degree.
Figure 1 shows the portions of the |V Z,CL | profile (in black and red dashed lines) that may be used for finding the "gradient" of |V Z,CL | across the MC; actually only a vector difference will be used, not the gradient itself.The MC expansion is assumed to be perpendicular to the MC axis, i.e. 2-D, and further it is assumed to be isotropic.The three panels below |V Z,CL | (in which green curves are shown) in Fig. 1 give profiles of the magnetic field (magnitude and latitude angle, in a GSE system, for example) and proton plasma beta that are commonly seen in interplanetary MCs at 1 AU, in order to put the associated change in |V Z,CL | in context.As Fig. 1 is meant to indicate, and we stress here, there are two distinct types of |V Z,CL |-profiles, where the maximum (MAX) and minimum (MIN) values occur at the boundaries (the black curve) or somewhere within those boundaries (the red dashed curve).We will treat each type separately below, but first we give a few examples of expanding MCs.
Figure 2 shows six examples of speed (V =|V |) profiles, emphasizing the gradient of plasma velocity, as the WIND spacecraft passes through a MC that is expanding.Also shown are the magnetic field magnitude (B), and field latitude angle (θ, in GSE coordinates).Black solid vertical lines indicate the identified start and end times of the MC, as given by Lepping et al. (2006)  We note that the average speeds for these MCs range from 323 km/s (a) to 880 km/s (f), and this average is the same, or almost the same, regardless of whether the average was taken over the full MC or over only t (giving <V > ; only case (f) shows any noticeable difference.However, the V s (and the |V Z |s, discussed later) can differ significantly between the black and red (taken over t ) type of averages; note especially case (f) where V is 149 km/s (black) and the other (red) is 229 km/s.In almost all cases the red cases of V are larger than the black, and for the one exception (case e) the two quantities are close in value.Notice also that these six examples cover almost all major "types" of MCs as described by Lepping and Wu (2007), where two (cases b and d) have nearly a full interval of southward field, two other cases (c and f) are nearly all northward, and remaining two (cases a and e) are about half northward and half southward.Most important is the fact that the velocity gradients usually come close to covering the full MCs.In fact, in case (e) all three types of gradient end-points are in very close agreement.Only in case (f) is there dramatic disagreement in the position of the vertical lines, in the front region; even for this case the end of the gradient shows remarkable agreement for the three estimates.Front vs. rear disagreement is evenly divided among these six examples.In three cases we see that the speed reaches a minimum several hours before the estimated rear boundary of the MCs; these are cases (b), (c), and (d).This is apparently due to the increased speed of the external plasma ramming into the MCs.This phenomena was first pointed out by Lepping et al. (2003b) where the superposition of many MCs were used to find this peculiar feature that occurs for many, but not all, MCs.
As the cases in Fig. 2 exemplify, the V-profile within a MC is not always simple or well behaved, and since |V Z | (now understood to be in CL coordinates) is directly related to V (as discussed in Sect.2), we will translate this assessment directly to the component |V Z |.For example, |V Z | is not always smoothly decreasing from spacecraft entrance to exit and, even when |V Z | does smoothly change in time, the MAX and MIN of |V Z | are not always at the entrance and exit points, respectively, as was briefly discussed for Fig. 1.Hence, we found it necessary to filter the |V Z | values by use of a running average of 2-h length, slipped every minute, to find the maximum value of , on the basis of initially one minute "sample rate" data.This was done in order to obtain the low frequency variation of |V Z | for analysis.From these filtered-|V Z | values, we find the MAX and MIN values, and from 15-min averages (from the smoothed 1-min averages) centered on the MAX and MIN positions.This approach will be utilized below in one way of finding V E,V .In another approach, we use the closest 15-min averages of |V Z | to the boundaries.In both approaches we decrease any possible errors due to peculiar noise-fluctuations in |V Z | (that is unrelated to the actual measure of the gradient) either at the boundaries or at MAX and/or MIN.In this way, any damage due to noise-fluctuations is at least minimized.where the spacecraft passes at a closest approach distance Y O from the axis, where t=t EN is the entrance time and t=t EX is the exit time; these times are separated by t=T , the "duration" of time that the spacecraft spends inside the MC.V E is the expansion velocity perpendicular to the MC's axis and shown for the entrance time, t EN .The magnitude of V E ideally holds for all γ angles.
Figure 3, which shows the cross-section of the ideal MC (circular for convenience) giving the relationship of expansion speed V E (moving out radially from the axis) and the velocity of the internal plasma relative to the motion of the center of the MC. Figure 3 indicates the passage of the spacecraft from the initial contact point, where the velocity is V EN , to the exit point where the velocity is V EX occurring over time t=T .As pointed out above, the relevant velocity-gradient of interest is that of the Z-component as rendered in CL coordinates (as in Fig. 1).The MC's center-velocity can be thought of as the average across the MC, called <V C >, taken along the spacecraft's path; this is hopefully a good approximation, since the spacecraft does not usually go exactly through the MC's center.This average is transformed to CL coordinates to give <V C > CL , and the Z CL -component obtained, called <V Z > CL .We then form V Z,EN,CL =(V Z,EN −<V Z >) CL (for inbound compared to average) and V Z,EX,CL =(V Z,EX −<V Z >) CL (for outbound compared to average), where |V Z,EN,CL | is the Z CL -component of velocity of the MC's plasma usually near |V Z,MAX,CL |, and |V Z,EX,CL | is the Z CL -component of velocity usually near |V Z,MIN,CL |, both in an inertial frame of reference.Hence, V Z,EN,CL and V Z,EX,CL are the V Z,CL -components of the MC's boundary velocities, essentially "inbound" and "outbound," respectively, in the MC's moving reference frame (with the average speed), in CL coordinates.We then form the difference between these two differences which is the Z CL -component of the velocity difference the across the MC.We choose ( V Z ) CL to be positive in all cases (as well as both | V Z,EN,CL | and | V Z,EX,CL | individually), where there is actual expansion.And since Z CL •1 X,GSE is always negative, we must use absolute values in Eq. ( 6).
From Fig. 3 it is ascertained that where the factor of 2 arises from the fact that ( V Z ) CL itself is twice the horizontal projection of the expansion speed, since values were taken at the two boundaries, rather than one at the center and the other at one of the boundaries. Since , as also seen in Fig. 3, then where , respectively), are the proper places to estimate the values of velocity to use in our vector formulation, since a gradient that is calculated based on velocities at the times of the actual MC boundaries (and using real duration) is often much smaller than a realistic one, as a broad review of many MCs reveals.We believe that the proper gradient then is V Z,CL / t , where t is determined by using the difference between the times of |V Z,MAX,CL | and |V Z,MIN,CL |, i.e. t =t MIN −t MAX ; see Fig. 1 which gives the pictorial representation of t .This is the portion of the MC where expansion is actually occurring and apparently would be so throughout the MC, if it were not for front and rear interaction with the solar wind.Hence, with this consideration Eq. ( 8) becomes where ( V Z ) CL is now understood to be based on i.e. based on |V Z,MAX,CL |, |V Z,MIN,CL |, separated by t .We assume that Eq. (9) will usually be the proper means of estimating vector velocity expansion, and it will be used for that estimate.But for comparison, we will also estimate expansion based on Eq. ( 8), understanding that it is almost always going to give a lower bound to the estimate.And, of course, our assumption will be tested.
Finally, we should stress that it is clear that the positions where |V Z,MAX,CL | and |V Z,MIN,CL | occur should not be considered to be indicators of the MC boundaries, in any case, because many other physical indicators are much better at determining MC boundaries, e.g.changes in |B|, proton temperature, proton plasma beta, direction of B, and indications from model fitting, etc.; see Lepping and Wu (2007).And, as expected from what we have said above, those other (reliable) quantities often disagree, even if only slightly, with using velocity as a means of determining the boundary.Even when velocity does appear to agree with these other means, its change is usually not sharp enough, at the start or end of the gradient, to pin down very well the time of occurrence of the boundary.As we see, Eqs. ( 8) and ( 9) depend on the Z CL component of a velocity change, and on the relative closest approach (which depends on Y O , the magnitude of the closest approach vector).
We now check V E,V for reasonableness by using typical values on the right side of Eq. ( 9), such as ( V Z ) CL =60 km/s, R O =0.125 AU, and Y O =0.05 AU.This gives V E,V =33 km/s, which, like V E,S (test), is within a typical range of values for the MC expansion speed at 1 AU, especially for the slower moving MCs.But it appears that using V E,V is generally going to give lower estimates of V E than using V E,S , and using V E,V (as in Eq. 8), on average, is expected to give even slightly lower values than V E,V .

The data and results
Most of the 100 MCs initially considered in this study (i.e.82 of them) are parameter fitted and discussed by Lepping et al. (2006), based on data from WIND/MFI (Lepping et al., 1995) and WIND/SWE (Ogilvie et al., 1995).The start/end times of the full 100 MCs, along with their various derived and estimated characteristics are provided on the WIND/MFI Website with URL of http://lepmfi.gsfc.nasa.gov/mfi/mag cloud S1.html and referred to as Table 2 on that site.Of these only MCs of relatively good quality were used, meaning the MCs that possess quality indices of Q O =1,2 (where Q O =3 is poor), where Q O depends on the following MC parameters: the value of the chi-squared of the parameter fit, a comparison of two independent means of estimating the MC's radius, where only one means depends on t (or on duration, T ), the value of closest approach (CA) distance, reasonableness of the estimated diameter (2 R O ), reasonableness of profile-symmetry, comparison of the MC axis alignment to what an axis in the MC's flanks (viewed globally) would be, and a check of the sign/strength of the axial-field component in the CL coordinate system.(See Appendix A of Lepping et al., 2006, for a rigorous definition of Q O ).After this quality editing the set of N=100 MCs is reduced to N =64 cases.The reason for restricting our analyses to those of quality Q O =1,2 is because, as we saw, certain model quantities and abilities are required in our estimation of V E , such as R O , Y O , (and indirectly T ), and being able to accurately transform into the CL coordinate system which requires obtaining accurate estimates of the latitude and longitude of the MC's axis.The N =64 cases were individually inspected to see if there was a gradient across each MC, or across a major part of it, indicating that the MC is, indeed, expanding at the time of the observations.Another 11 cases were dropped because they did not have such a gradient, i.e. they were not good cases of expanding MCs where both ( V Z ) CL and ( V Z ) CL were positive.Hence, we arrive at N=53 good cases for analysis.So a high percentage, 83%, of the eligible 64 cases were actually expanding.
Table 1 shows, for the full 53 MCs, the start time, duration (T ), t , R O , |CA|, and various relevant speeds and velocity components, needed for use in Eqs. ( 5), ( 8) and ( 9), including <V C >, the difference quantities, ( V Z ) CL for both conditions (MAX/MIN) and for the boundaries, and the last three columns provide the estimates of expansion speed: V E,S , V E,V , and V E,V , in that order; all quantities are defined in the footnotes.At the bottom, in red, are the averages and standard deviations (σ ) for each quantity.It is clear that V E,S (≡V n in Table 1) is on average (as well as for most individual cases) closer to V E,V (≡V p ) than to V E,V (≡V o ).Also there is a relatively small spread of V E,S values (with a σ =27 km/s) compared to its average (49 km/s), i.e. a ratio (≡avg/σ ) of 0.55.This is especially so with regard to that ratio for V E,V , which is 1.06, or for V E,V , which is 0.86.In the four cases where V E,S was unusually large, say over 85 km/s, V E,V and V E,V were also very large.This is very noticeable in the case where V E,S is largest (i.e.case 2001, 11, 24); there we see that V E,S =151 km/s, V E,V =262 km/s, and V E,V =213 km/s, but in this case the latter two (although clearly being very large) are not very believable.In Fig. 4 we show a scatter diagram of V E,S vs. V E,V , based on the values in Table 1, with a least-squares fitted straight line; the c.c. for this correlation is 0.76.So as V E,V increases so also does V E,S , and the correlation is more-or-less linear.It is interesting that the majority of the values for V E,V lie between 5 and 70 km/s, and for V E,S they are mainly within 10 and 80 km/s, as Table 1 reveals.We now investigate the distributions of the V E,S and V E,V values.
Figure 5 shows histograms of the values derived for V E,S (black solid line) and V E,V (red dashed line) based on Eqs. ( 5) and ( 8), respectively.The peaks for both are at 30 km/s (with bucket widths of 20 km/s), and the averages and standard deviations (σ ) are shown for the two sets.Note that both distributions are skewed, so that the average value for V E,V is higher than its most probably value (30 km/s), and for V E,S the average is quite a bit higher than its most probable value (also 30 km/s).In fact, as pointed out above, there is one value of V E,V as high as 263 km/s (not shown in the histogram).There is obviously larger uncertainty on this large estimate, calling in doubt the fact that it is actually so high, but it is likely that there are some actual expansion velocities much higher than 30 km/s; see Burlaga (1995, p. 100).

Fig. 4.
A scatter diagram of V E,S vs. V E,V showing a linear correlation coefficient (c.c.) of 0.76.(Note that there is one value of V E,V of 263 km/s that occurs off-scale and therefore is not shown.This value was included in the least-squares fit and in the calculation of c.c., however.) Figure 6 is a scatter plot of V E,S vs. V E,V , which is similar to that of Fig. 4, except V E,V of Fig. 4 is replaced by V E,V in Fig. 6; notice, however, that the scales are also different.As we see, there are similarities in the values in the two figures, but the c.c.'s are significantly different, i.e. c.c.=0.85 and 0.76 for Figs.6 and 4, respectively.This improvement in correlation is not unexpected, since we postulated that the MAX/MIN means was expected to give a more faithful representation of the gradient of velocity across the MC (and therefore better represent expansion) than the boundary value means.We now investigate the distributions of the values V E,V and again show V E,S for comparison.
Figure 7 shows histograms of the values derived for V E,S (black solid line) and V E,V (red dashed line) based on Eqs. ( 5) and ( 9), respectively; V E,S is again shown for comparison.The peaks for both are at 30 km/s (with  5) and ( 8), respectively, i.e.where the latter is based on the actual boundary values of V Z of the MCs.The peaks for both are at 30 km/s for bucket widths of 20 km/s and the averages and standard deviations (σ ) are shown for the two sets.(Note that there is one value of V E,V of 263 km/s that occurs off-scale and therefore is not shown.)bucket widths of 20 km/s) and the averages and standard deviations (σ ) are shown for V E,V : <V E,V >=44 km/s and σ (V E,V )=38 km/s (the comparable values for <V E,S > are given in Fig. 5).Note that both distributions are skewed, so that the averaged value for V E,V is higher than its most probably value (30 km/s), and again for V E,S (<V E,S >=49 km/s) the average is quite a bit higher than its most probable value (30 km/s), as discussed in connection with Fig. 5.There is obviously larger uncertainty on the very high values, e.g.those above 100 km/s, again calling into doubt that they are really so high.

Estimates of errors
For the scalar determination of V E , which depends on R O , <V C >, and T , the net error on V E results from the combined errors from each of these three quantities.The error on R O is greatest, since <V C > and T are generally quite well  =<V C > is the average (over T ) speed of the MC locally V f =<|V Z,CL |> is the average of absolute value ("ABS") of the Z-component of the velocity across the full MC V g =<|V Z,CL |> is the average of ABS of Z-component of the velocity across t V h =|V Z,MAX,CL | is ABS of the Z-component of the velocity at the maximum-point in CL coords.determined.The sources of error in R O are: (1) it is modeldependent with all of the model's sources of errors and (2) a simple value for R O (≈<(R MIN +R MAX )/2>) may not be adequate for a MC with an oval cross-section, as briefly discussed in the beginning of Sect. 3.But as we will see the structure of Eq. ( 5) is such that it propagates errors most seriously for large errors in T , not so much for R O .Here we attempt to estimate the typical impact of these errors.First, we will assume that: (1) <V C > is well known and essentially error free (or very small compared to the errors of the other two quantities), (2) T is known to an uncertainty of 10%, (3) the assumption that the average speed across the MC, <V C >, is approximately equal to <V S−to−E > (Eq. 3) is a very good one, and (4) R O is known to an uncertainty of about 30%.Hence, using the example at the end of Sect.3.1, where T is 20 h and R O is 0.125 AU, and where two values of <V C > were used, 450 km/s and 650 km/s, we see that these uncertainties yield these specific ranges for the three relevant quantities: <V C >=450 km/s, R O : (0.0875-0.163)AU, and T : (18-22) h.Then from Eq. ( 5) and for <V C >=450 km/s we obtain for the MAX value V E,S =61.3 km/s and a MIN value of 31.8 km/s, or V E,S is 47±15 km/s.But for a MC moving on average at <V C >=650 km/s we obtain V E,S to be 63±20 km/s.As we see, the error is dominated by the error on T which is usually fairly small.However, the assumption that <V C >≈<V S−to−E > may not be good in all cases.For example, if there were a marked deceleration of a MC (which is believed to occur occasionally near the Sun), then this assumption may not be very good, and therefore, be another source of error, one not easily estimated.Therefore, any estimated error on V E,S , as done above, must be considered a minimum estimate.

Start time
For the vector determination of V E , which depends on R O and Y O , (giving |CA|≡|Y O |/R O ), and on the differencevelocity ( V Z ) CL (where we recall that the prime refers to obtaining the difference from the MAX and MIN components of velocity).This difference-velocity, obtained straightforwardly from measurements, should be quite well determined, but not completely error-free.Both |Y O | and R O are sources of error, especially Y O , which is, unfortunately, one of the most poorly estimated quantities in the Lepping et al. (1990) fitting program; see Lepping et al. (2003a).However, the structure of Eq. ( 8) is such that the net error in V E will not depend strongly on the error in |CA|, as we will see.
Here we attempt to estimate the typical impact of these two errors (in |CA| and ( V Z ) CL ).The error in ( V Z ) CL is about 5% of its value, due only to the fact that the gradient is not always ideal (as in Fig. 1) nor measured exactly (e.g.choices of what intervals to use in obtaining the needed averages, etc. require judgement).We will also consider the typical uncertainty on |CA| to be 60%, which is large, but the resulting uncertainty on V E,V (for primed or un-primed) is not strongly dependent on |CA|.From Table 1 we see that the average ( V Z ) CL is 71 km/s, so for a 5% error we will have a range on this quantity of: (67.5-74.6)km/s.And a range on |CA| is: 0-0.6.Hence, from Eq. ( 9) we obtain for the MAX value V E,V =46.6 km/s and a MIN value of 33.7 km/s, or V E is 40±7 km/s.It is evident that for small percent errors in ( V Z ) CL , as we have here, there will be small errors on V E,V , distinctly smaller than for V E,S in general, which were typically in the range ±(15-20) km/s or larger, if <V C >≈<V S−to−E > is a poor assumption for any given case.Let us consider what the error would be for an unusually large ( V Z ) CL of say 255 km/s, our largest value (see col.V m of Table 1), at the same 5% level.This yields 144±23 km/s, i.e. with an error comparable to or slightly larger than those for the scalar V E 's error, but, of course, this is a highly unusual case.

Comparisons of V E to local Alfvén speeds
We now compare the MC expansion speed to various relevant local Alfvén speeds (V A s).In particular, we examine V A for three points within the ideal MC: the entrance-point (see t=t EN of Fig. 3), the closest approach-point, (at t=t CA ), and the exit-point (see t=t EX of Fig. 3).Table 2 shows the Alfvén speeds calculated for these three positions and compares them to the value of V E,V .In almost all cases the V A s are larger than V E,V (and recall that the V E,V estimate is usually comparable to the V E,S estimate).There are a few exceptions, however, and for those the V A is usually close to the value of V E,V .These local Alfvén velocities are on average (for the 53 MCs considered here) equal to V A,EN =116 km/s around the inbound boundary, V A,CA =137 km/s at closest approach, and V A,EX =94 km/s around the outbound boundary; see average values at the bottom of Table 2. Recall that the average values of scalar V E,S (<V E,S >=49 km/s), and vector V E,V (<V E,V >=44 km/s), are well below these average Alfvén speeds, and, in fact, it is rare that any individual Alfvén speed at these positions is smaller than the associated V E .The peaks for both are at 30 km/s for bucket widths of 20 km/s and the averages and standard deviations (σ ) are shown for the two sets.(Note that there is one value of V E,V of 213 km/s that occurs offscale and therefore is not shown.)

Summary and discussion
Here we have investigated expansion speed of a set (N =53) of well chosen WIND magnetic clouds that occurred over the period from early 1995 to April 2006 by using two separate means of estimation, scalar (V E,S ) and vector (V E,V ) methods.Only expansion with respect to the MCs local axis was considered.The "scalar" method uses a well established means of estimation that depends on the average speed of the MC from Sun-to-Earth (<V S−to−E >), the local MC's radius (R O ), the duration (T ) of spacecraft passage through the MC (at average local speed <V C >), and the assumption that <V S−to−E >≈<V C >.We actually formulated two vector means of estimating V E by: (1) using the decrease in |V Z | (in MC coordinates, where the Z-component is related to spacecraft motion through the MC, as described in Sect.2) over the full duration (T ) and (2) depending only on the decrease in |V Z | between the V Z,MAX and V Z,MIN values, occurring over t (usually a shorter interval than T - Where: a V E,V is the vector estimate of expansion velocity based on set, is smaller than the associated V E .Hence, we should not generally expect a shock to be driven by the relatively rapid expansion of any MC at 1 AU.This is consistent with the remarks of Burlaga (1995, Sect. 6.5.1) who studied this effect for earlier cases of MCs at 1 AU.However, upstream shock waves at MCs are observed, of course, and these obviously are due to the larger bulk speed of those MCs compared to their upstream fast mode speeds.

Fig. 1 .
Fig. 1.A cartoon stressing the profile of the observed |V Z | (≡|V Z,CL |) gradient of velocity as the spacecraft passes through a MC that is expanding; the subscript CL refers to the MC coordinate system (see Sect. 2), where the X CL -axis is aligned with the estimated local axis of the MC.Also shown (in green) are the magnetic field magnitude, field latitude angle, and proton plasma beta.The duration of the MC passage is T and the interval from the MAX to MIN of |V Z,CL | is t .A upstream shock ramp is indicated for this MC, even though not all MCs possess upstream shocks.The red dashed curve for |V Z,CL | holds for a case where the MAX and/or MIN points for this quantity are markedly different from its values at the boundaries, and the black curve for |V Z,CL | holds for the case when its MAX and MIN values are at or very near to the boundaries.
and depend only on magnetic field quantities; dotted vertical (blue) lines indicate identifications, made through visual inspection, of the positions of MAX and MIN in the speed profile; and dashed (red) vertical lines are points of MAX and MIN chosen automatically, as described in Fig. 2's caption.(a) is the MC with start day of 4 February 1998, (b) is for 8 November 1998, (c) is for 21 February 2000, (d) is for 22 April 2001, (e) is for 29 April 2001, and (f) is for 15 May 2005.

Fig. 2 .
Fig. 2. Six examples of the profiles of plasma speed (V =|V|), magnetic field magnitude (B), and field latitude angle (θ) as the WIND spacecraft passes through a MC.All data presented in 10-min average form.Black solid vertical lines indicate the identified start and end times of the MC; dotted vertical (blue) lines indicate choices, through visual inspection, of the points of MAX and MIN in the speed profile; and dashed (red) vertical lines are points of MAX and MIN chosen automatically, via computer searching, after V is smoothed via a running average of 2 h length.Average speed is given in the V-panel, as <V > for an average over the full MC and as <V > (in red) for the t region, both in units of km/s.Panels (a) through (f) are ordered according to date: (a) is the MC with start day of 4 February 1998, (b) is for 8 November 1998, (c) is for 21 February 2000, (d) is for 22 April 2001, (e) is for 29 April 2001, and (f) is for 15 May 2005.In each θ -panel the regions where the magnetic field goes southward are in yellow and when northward they are in blue.

Fig. 3 .
Fig.3.The cross-section of an ideal MC (circular for convenience) where the spacecraft passes at a closest approach distance Y O from the axis, where t=t EN is the entrance time and t=t EX is the exit time; these times are separated by t=T , the "duration" of time that the spacecraft spends inside the MC.V E is the expansion velocity perpendicular to the MC's axis and shown for the entrance time, t EN .The magnitude of V E ideally holds for all γ angles.

Fig. 5 .
Fig. 5. Histograms of the values derived for V E,S (black) and V E,V (in red) based on Eqs.(5) and (8), respectively, i.e.where the latter is based on the actual boundary values of V Z of the MCs.The peaks for both are at 30 km/s for bucket widths of 20 km/s and the averages and standard deviations (σ ) are shown for the two sets.(Note that there is one value of V E,V of 263 km/s that occurs off-scale and therefore is not shown.) Fig. 6.A scatter diagram of V E,S vs. V E,V showing a c.c. of 0.85.

Fig. 7 .
Fig. 7. Histograms of the values derived for V E,S (black) and V E,V (in red) based on Eqs.(5) and (9), respectively, and where the latter is based on [|V Z,CL (MAX)|, |V Z,CL (MIN)|], separated by t ].The peaks for both are at 30 km/s for bucket widths of 20 km/s and the averages and standard deviations (σ ) are shown for the two sets.(Note that there is one value of V E,V of 213 km/s that occurs offscale and therefore is not shown.) |V Z,MAX,CL |, |V Z,MIN,CL | and t b EN refers to the entrance point in Fig. 3 c C refers to the CA point in Fig. 3 d EX refers to the exit point in Fig. 3.
is the relative closest approach parameter.It is important to point out that the gradient of velocity within most MCs tends to be greatest in the central regions of the MCs, i.e. the points of |V Z,MAX,CL | and |V Z,MIN,CL | generally are not exactly at the boundaries of the MC.It appears that the times of |V Z,MAX,CL | and |V Z,MIN,CL | (i.e.t MAX and t MIN

Table 1 .
Magnetic cloud parameter values.

Table 1 .
Continued.T is the duration in hours b t is the interval between the points at V MAX and V MIN in hours c R O is the estimated radius of the MC, which is≈<(R MIN +R MAX )/2> |CA| is the relative closest approach distance = |Y O |/R O (in %) V e a d

Table 2 .
Alfvén speeds (V A ) compared to the V E,V expansion speed.