2. Discontinuity Analyses

There are many discontinuities in space. These can be classified by where they are and what function they play, eg. the bow shock, the magnetopause, interplanetary shocks, solar wind discontinuities, the neutral sheet, and the heliospheric current sheet. They can also be classified by the physical nature of the boundary, fast shock, slow shock, rotational discontinuity or tangential discontinuity for example. The field and plasma properties usually change significantly across a discontinuity. In most theoretical studies a discontinuity is treated, for simplicity, as a one dimensional problem, namely, the physical quantities change only along the normal of the discontinuity. Observationally, while a spacecraft moves relative to a discontinuity, it measures the upstream and downstream conditions in time series. For a scalar quantity, the time series can be easily converted into a function of distance relative to the discontinuity if the motion of the discontinuity relative to the spacecraft is known assuming stationarity of the upstream conditions. For a vector quantity, to understand the physical behavior of a discontinuity and the processes near it, it will be convenient if the measurements are presented in a boundary normal coordinate system as has been discussed in section 1.2.2. Such a system can often result in variations only in two dimensions and allow easier visualization and understanding of the behavior of the plasma. To find such a coordinate system, the normal direction of the discontinuity has to be determined. Different methods of discontinuity analysis have been developed to allow this determination. In section 2.1, we briefly introduce the background and principles for discontinuity analysis. In sections 2.2 to 2.4, we describe and discuss the three most useful methods in discontinuity analysis. In section 2.5, we describe a method which has been proposed most recently.

2.1 Background.

2.1.1 Rankine-Hugoniot relations.

The plasma conditions on the two sides of a discontinuity are linked by the Magnetohydrodynamic (MHD) equations which describe the requirements for macroscopic continuity, pressure balance, and energy budget. If the discontinuity is planar and stationary, the MHD equations can be simplified to the form known as the Rankine-Hugoniot (R-H) relations. In isotropic plasmas, the R-H relations are

\begin{displaymath}[ \rho u_n ]
= 0 
\eqno (2.1)\end{displaymath}

\begin{displaymath}[ {\bf E}_T ]
= 0 
\eqno (2.2a)\end{displaymath}

\begin{displaymath}[ B_n ]
= 0
\eqno (2.3)\end{displaymath}

\begin{displaymath}[ \rho u_n^2 + P + B_T^2 / 2 \mu_0 ]
= 0
\eqno (2.4)\end{displaymath}

\begin{displaymath}[ \rho u_n {\bf u}_T - B_n {\bf B}_T / \mu_0 ]
= 0
\eqno (2.5)\end{displaymath}

\begin{displaymath}
\left[ ({\rho u^2 \over 2} + {P \over {\gamma - 1}} + P) u_n + S_n \right] = 0 
\eqno (2.6a)\end{displaymath}

where $\rho , {\bf u}, P, {\bf E}, \gamma, \mu_0,$ and ${\bf S} = {\bf E} \times
{\bf B} / \mu_0$ are the density, velocity, pressure, electric field, ratio of specific heats, magnetic permeability in vacuum and Poynting vector, the square brackets denote the changes across the discontinuity, and the subscripts n and T denote the normal and tangential components to the discontinuity. For most of the problems in space physics, the frozen-in condition is applicable, or ${\bf E} = -{\bf u} \times {\bf B}$. Thus, equations (2.2a) and (2.6a) can be written as

\begin{displaymath}[ ({\bf u} \times {\bf B})_T ]
= 0
\eqno (2.2b)\end{displaymath}

\begin{displaymath}
\left[ ({\rho u^2 \over 2} + {P \over {\gamma - 1}} + P) u_n...
 ...cdot ({\bf B}_T u_n - B_n 
{\bf u}_T)
\right] = 0 
\eqno (2.6b)\end{displaymath}

Equation (2.2b) holds upstream and downstream from a discontinuity even if the frozen-in condition is broken within the thin layer of the discontinuity. The R-H relations as written above hold in the shock frame rather than in the spacecraft frame, so that

\begin{displaymath}
{\bf u = V - V}_{disc}
\eqno (2.7)\end{displaymath}

where ${\bf V}_{disc}$ and V are the velocity of the spacecraft relative to the discontinuity and the velocity of the flow measured in the spacecraft frame.

The R-H relations contain 8 equations and 19 parameters including the velocity of the discontinuity ${\bf V}_{disc}$. It is important to point out that the goal of data analysis is often not to simply apply the R-H relations but to verify them, or to determine how well these relations hold in the situation being studied since several approximations have been made in applying the R-H relations. Ideally, one should substitute measured parameters in the left-hand-side of equations (2.1) to (2.6). The difference between the upstream value and downstream value of the quantity in each equation should be much smaller than either the upstream or the downstream value if the R-H relations are verified, or

\begin{displaymath}
{[Q] \over \vert Q\vert} \sim 0
\eqno (2.8)\end{displaymath}

where Q is the quantity in each of the R-H relations. The ratio on the left hand side of Equation (2.8) gives the uncertainty of the analysis and Equation (2.8) is used as the basis of the discussions of the uncertainty of each method in the following subsections. If the R-H relations are not verified, one or more of the approximations made may not be valid. There may be temporal variations and/or curvature of the discontinuity, the anisotropy of the plasma, and/or significant presence of heat flow. Perhaps the identification of the nature of the discontinuity is incorrect. In these cases, conclusions should be drawn carefully from the analysis. However, at the present time, even in the best situation, with two spacecraft and full three dimensional measurements, observations provide only 18 parameters, (16 plasma and field parameters, one timing difference and one distance measurement). Thus the R-H relations cannot be verified completely from observations and some additional assumptions must be made. Usually, one may assume a subset of the R-H relations as given, and then, use the remaining relations as confirmation. It is however not appropriate to assume all the R-H relations as given and then to determine remaining unmeasured parameters using, for example, optimization because different equations in the R-H relations have different uncertainties and because the results of such fittings are often found not to be a solution of the R-H relations [Chao, 1995]. A cluster of four closely spaced satellites will enable us to verify the R-H relations independently. We will discuss this issue later in section 4.2.

If some assumptions must be made, choosing the right subset of R-H relations can minimize the uncertainty of the results. Among the R-H relations, the continuity of the normal magnetic field, equation (2.3), has the least uncertainty since it is not affected by time variations (see equation 1.2) and usually the magnetic field is the most accurately measured quantity with relatively high time resolution as discussed in section 1.3. Almost all the present methods of discontinuity analysis are in fact based on this assumption. However, as will be discussed later in this section, there are ambiguities in some instances. An alternative is to use the continuity of the tangential electric field, equation (1.3 or 2.2), if either the electric field or the plasma velocity can be measured accurately in three dimensions with a relatively high time resolution. We will discuss this method briefly in section 2.5. Here we emphasize that comparing equation (1.2) with (1.3), the required assumptions for the continuity of the tangential electric field are more than that for normal magnetic field conservation. Unless the velocity can be measured accurately (see discussion in section 1.3.3), equation (2.2b) should not be used since it involves the cross product of two vectors and the uncertainties in the measurements will be amplified in the calculations. In this case, the conservation of mass, equation (2.1), may provide a relatively smaller uncertainty than other relations except equation (2.3). However, as has been discussed in section 1.3.3, the calibration factors of plasma moments may change significantly across the bow shock or the magnetopause. One has to be extremely careful when using these moments. At the present time, it is suggested not to assume equations (2.4) to (2.6) as given, rather to use them as confirmation, since they involve more complicated calculations and the effects of the temperature anisotropy and the intercalibration between the magnetic field and plasma measurements may become important.

Ideally, application of the Rankine-Hugoniot relations would involve simultaneous measurements both upstream and downstream of a discontinuity using two independent measuring platforms. In practice such application is usually performed using a single observatory moving across the discontinuity under the assumption that the external conditions do not change. Often the discontinuity is encountered because there has been a temporal change in these conditions, so caution must always be exercised and the time stationarity assumption verified when using the R-H relations.

2.1.2 Types of discontinuities

There are several simplified types of discontinuities which have been commonly used to characterize and classify discontinuities (see the recent review by Lin and Lee [1994]). A discontinuity is called a tangential discontinuity (TD) if there is neither magnetic flux nor mass flux across it, or un = Bn = 0 in equations (2.1) and (2.3). A TD is a current sheet separating two different plasmas.

A discontinuity is called a rotational discontinuity (RD) if there is magnetic flux across it but the density and the field strength (in an isotropic plasma) are same on the two sides of it, or $B_n \not= 0, u_{1n} = u_{2n} \not= 0, [B] = 
[\rho] = 0$, where subscripts 1 and 2 denote the values upstream and downstream of the discontinuity. The field strength and density may change within an RD. An RD is a propagating, usually non-linear, Alfven wave front and satisfies equation (2.2) in the form of

\begin{displaymath}[{\bf u}_T ]
= {u_n \over B_n} [{\bf B}_T ]
\eqno (2.9)\end{displaymath}

and $u_n = B_n/\sqrt{\rho\mu_\circ}$. Equation (2.9) is the so-called Walen relation for isotropic plasmas and will be further discussed in section 2.7. As Bn, u1n and u2n go to zero, an RD may degenerate into a TD. However, this TD is different from a general TD because its velocity change has to be parallel to its field change but a general TD has not.

A discontinuity is called a shock if there are both magnetic flux and mass flux across it and if there is a change in the density, or $B_n \not= 0, u_{1n} \not= u_{2n} \not= 0$, and $\rho_1 < \rho_2$. A shock is associated with a dissipation process and usually with heating of the plasma. Across a shock, the flow velocity decreases from above to below a characteristic speed, such as the fast mode speed, intermediate mode speed or slow mode speed (for more discussion on the modes, see section 3.3.1), in the frame at rest to the discontinuity. Similar to a shock but with $\rho_1 \gt \rho_2$, a discontinuity is called a rarefaction wave. A rarefaction wave may occur in an expansion fan such as formed when flow moves across a ledge and expands into a vacuum. In theory, an expansion fan cannot steepen but in reality because of the rapid motion between an expansion fan and the spacecraft, it can appear sharp in the time series data. An observed discontinuity may be a superposition of these elementary discontinuities and also may not be in steady state, but in the regions far from where a discontinuity is generated, these elementary discontinuities are expected to separate because of their difference in speed.

2.2 The Minimum Variance Analysis.

This method is based on the Principal Axis Analysis (section 1.4) and the fact that the magnetic field is divergence-free, $\grad \cdot {\bf B} = 0$, the derivative form of equation (2.3), (see also the discussion of equation (1.2)). For an infinitesimally thin discontinuity, equation (2.3) should hold across it if it is planar. If a structure consists of many such discontinuities and they are parallel to each other, the sum of the fluctuations normal to the discontinuity should be zero. In reality, the fluctuations within the structure are equivalent to distortions of these thin discontinuities. Thus locally, the normal direction of a discontinuity may not be the same as of the overall structure, and thus there can be magnetic fluctuations along the direction of the average normal. The minimum variance method assumes that the distortions of these thin discontinuities from the overall structure are small compared to the changes in the magnetic field in the plane of the boundary. Thus the field fluctuations are smallest in the direction normal to the overall structure [Sonnerup and Cahill, 1967]. Therefore, to determine the normal direction of a discontinuity with internal structure is equivalent to finding the minimum eigenvector direction of the principal axis analysis, (see section 1.4).

Note that the minimum variance analysis assumes only that the variations are smallest along the normal but the normal component of the steady field is not necessarily smallest. As will be discussed next and in sections 2.3 and 2.4, in theory, the minimum variance method has a large uncertainty for tangential discontinuities and shocks (see Figure 2.1 [Lepping and Behanon, 1980]) since both theoretically consist of linearly polarized variations of the field and the normal may lie anywhere perpendicular to this direction of the maximum change, (see further discussion in section 2.7). In one situation minimum variance will give an accurate shock normal, when there is a standing whistler mode precursor propagating upstream along the shock normal. This is usually seen for subcritical shocks [Mellott and Greenstadt, 1984] and it assumes the upstream whistler wave propagates in the same direction as the shock wave does. The minimum variance method is most useful for rotational discontinuities and other more complicated situations.

Fig. 2.1. The errors of the minimum variance analysis from a numerical experiment [Lepping and Behannon, 1980]. The upper (lower) panel shows the errors for TDs (RDs). $\omega _T$ is the shear angle of the field across a discontinuity. The errors for TDs are much greater than for RDs.

In principle, one may also apply the minimum variance method to the mass flux, $\rho {\bf V}$, to determine the normal direction [Sonnerup et al., 1987], since in steady state we have $\grad \cdot \rho {\bf u} = 0.$However, in practice, due to temporal variations, the most important cause of which comes from the relative motion between the discontinuity and the spacecraft (see equation (1.1)), and relatively large uncertainties in the plasma measurements, for example, due to changes in composition and/or calibration factor (see section 1.3.3) across the discontinuity, this method has a much larger uncertainty than the magnetic field minimum variance in addition to the uncertainties discussed below.

What limits the accuracy of the minimum variance method? Following the steps of the description discussed above, we know that the minimum variance method can be limited by the data resolution and wave activity within the discontinuity. As the determination of the principal axes is equivalent to a three-free-parameter fit, a small number of data points will lead to a large uncertainty in the fit. In an extreme case, if there is no measurement within the discontinuity, this method should be used with caution since the minimum variance direction would then be determined by the wave activity upstream and downstream. Therefore, high resolution measurements and slow motion of the discontinuity relative to the spacecraft will minimize the uncertainty. On the other hand, as the resolution increases, one may be able to resolve the wave activity within the discontinuity. These waves may cause uncertainty in the determination of the normal direction as well. As discussed earlier, the assumption made in this method is that the infinitesimally thin surfaces within the overall structure have only small perturbations. Waves and small structures within the discontinuity may destroy the validity of this assumption. For example, if the magnetic perturbations within the structure due to structure and waves are mainly along the normal, the minimum variance direction will not be the normal direction for a discontinuity with a small field change across it. Filtering the data to pass only frequencies consistent with the thickness of the structure will help in reducing the uncertainty of the normal determination. The uncertainty of the minimum variance analysis was first discussed quantitatively by Sonnerup [1971] (in using equation (13) of Sonnerup [1971] note that there is a typographical error that <B2> should be <B>2 ) and then investigated comprehensively and numerically by Lepping and Behanon [1980]. Recently, Kawano and Higuchi [1995] used the bootstrap method to estimate the errors in the minimum variance analysis.

In principle, one may select many different time intervals for the minimum variance analysis and each of them provides a different normal. How to evaluate a result of the minimum variance analysis? Here are the several key issues to check.

1) Check the normal component of the field. After rotating the field into the minimum variance coordinates, the average fields in the minimum variance direction on the two sides of the discontinuity should be the same at least in the regions close to the discontinuity. Often a visual inspection can quickly determine whether the rotation is good. Since the minimum variance analysis provides the direction of smallest field fluctuations only within the selected time interval, if the interval does not include all the major field changes, one may find that the average fields in the minimum variance direction are different on the two sides of the discontinuity. In the case of the magnetopause, the field in the minimum variance direction may increase or decrease continuously on the magnetospheric side due to the curvature of the magnetospheric field.

2) Check the ratios of the eigenvalues. The square root of an eigenvalue is the standard deviation of the field along that direction. Ideally the minimum eigenvalue should be zero. In practice, if the minimum eigenvalue is much smaller than the other two eigenvalues, the minimum variance direction is well determined. Usually, a normal direction is considered as to be well determined if the minimum eigenvalue is one order smaller than the intermediate eigenvalue, or the amplitudes of the perturbations along the normal are less than one third of the smaller one of the two components perpendicular to the normal.

3) Check the minimum eigenvalue. Select a different time interval across a discontinuity to provide a set of normal directions. Since a smaller minimum eigenvalue indicates a better determination of the normal direction, one may choose the normal direction with a smaller minimum eigenvalue but also with a smaller ratio of the minimum and intermediate eigenvalues. However, usually, a shorter interval, or a smaller number of data points provides a smaller minimum eigenvalue. In the extreme case, if only three data points are selected, the minimum eigenvalue may go to zero since the ellipsoidal surface degenerates into a plane. In this case, the smaller minimum eigenvalue is obtained with some sacrifice in statistics. In practice one should include in the analysis only the variations associated with the discontinuity being analyzed. In principle, one should choose the normal direction with a smaller minimum eigenvalue and a longer time interval. The ratio of the minimum variation of the field and the strength of the average field should be very small, less than few percent, or $\sqrt{\lambda_3} / \vert B\vert \sim 0$, where $\lambda_3$ is the minimum eigenvalue.

4) Check the ratio of the minimum variation of the field to the average field in the minimum variance direction, or $\sqrt {\lambda_3 / (j - 2)} / B_{min}$ where Bmin is the average field in the minimum variance direction and j is the number of data points. A large value of this ratio indicates a large uncertainty in the analysis as will be discussed in the section of tangential discontinuity analysis. A large number of data points will reduce the uncertainty. When Bmin is extremely small, the discontinuity could be a tangential discontinuity which needs to take additional caution when using the minimum variance analysis.

The suggested procedures are as follows.

1) In the time interval selection, try to minimize the number of the data points on the two sides of the discontinuity but try to maximize the number of the data points within the discontinuity. Too many data points on the two sides of the discontinuity will put too much weight on the fields on the two sides. More data points within the discontinuity in general will increase the statistical significance of the determination.

2) Examine the fluctuations in the field during the crossing. If there are strong waves present that are not part of the discontinuity structure being analyzed, low-pass filter the data before a minimum variance analysis.

3) Perform the minimum variance analysis for several different selected time intervals and compare the results according to the discussion above. Experience indicates that if the results are essentially the same for several neighboring ``nested" data segments, they are perhaps believable (B.U.Ö. Sonnerup, private communication, 1992)

4) Compare with the normal directions predicted by geometric models if there are any. Occasionally, one may find the normal direction determined by the minimum variance is orthogonal to the model prediction. Most likely, this is caused by the 90$^{\circ}$ ambiguity to be discussed in section 2.7.

5) Since the difference among the normals determined from different time intervals provides a measure of the uncertainty of the analysis, never draw qualitative conclusions which may not be true given the uncertainty.

2.3 Tangential Discontinuity Analysis.

In theory tangential discontinuities are those with Bn and un zero. There is no magnetic field or mass flux across a tangential discontinuity. As discussed in section 2.1, equation (2.8), a good determination of the normal is indicated by a small ratio between the difference and the average of the quantity in the R-H relation across a discontinuity. Noting that the ratio of the standard deviation and the probable error of the mean is j-1/2, the ratio $j^{-1/2} \Delta B_n / <B_n\gt$,where $\Delta B_n$ and <Bn> are the minimum variation and the average of the field in the minimum variation direction, provides a measure of the uncertainty of the minimum variance method in verifying equation (2.3). For a TD, since <Bn> is small, the ratio becomes very large and hence the minimum variance has a large uncertainty. The effect of such a large uncertainty can be seen when one selects different intervals and finds different normal directions but <Bn> remains similar.

Since the magnetic field is tangential to a TD surface, the normal direction of the discontinuity is perpendicular to the fields upstream and downstream, or

\begin{displaymath}
{\bf n} \vert\vert {\bf B}_1 \times {\bf B}_2
\eqno (2.10)\end{displaymath}

where ${\bf B}_1$ and ${\bf B}_2$ are determined by selecting a relatively stable interval on each side of the discontinuity. Ideally, this is done using simultaneous data from two spacecraft. When one spacecraft is used care must be exercised to ensure that the changes observed are solely due to the spatial gradients across the discontinuity.

The uncertainty in the determination of the normal for a tangential discontinuity arises from the uncertainties in measurements of the two fields due to the fluctuations near the discontinuity. The uncertainties for the measurements of the two fields are the probable errors of mean, not the standard deviation. A large number of data points in measuring each of the fields may reduce the uncertainty. In reality, however, there may be temporal changes in the magnetic field near the discontinuity of interest. Some are oscillations and others may be either gradual or sudden changes. The effects of oscillations can be removed by averaging over many wave cycles. Again if there are temporal changes in conditions as the spatial discontinuity is crossed, the calculated normal will be affected. Finally, the uncertainty of the TD method becomes large when the fields on the two sides are nearly parallel to each other.

In summary, the tangential discontinuity analysis has a relatively small uncertainty in determining the normal of a tangential discontinuity if the fields on the two sides of the discontinuity are not parallel to each other (with a change only in magnitude). However, one has to verify carefully that a discontinuity is a tangential discontinuity before using the method. To minimize the uncertainty in the normal direction of the discontinuity, one should try to select time intervals as long as possible on the two sides of the discontinuity to minimize the effect of the wave but without major changes in the field on the two sides to minimize the effects of changing external conditions.

2.4 Coplanarity Analysis.

A discontinuity is called a shock if there are magnetic flux and mass flux through the discontinuity and the velocity changes from supersonic relative to the discontinuity to subsonic. This velocity change causes a change in the density across the discontinuity. A shock is called a fast (slow) shock if the density changes in (out of) phase with the magnetic field strength across the shock. Here we have ignored the intermediate shocks in which the density and the field strength may vary either in phase or out of phase but the rotation in the field tangential to the discontinuity must be exactly 180$^{\circ}$. From the R-H relations, one can show that the magnetic fields on the two sides of the shock and the normal of the shock are coplanar, and that the normal is also perpendicular to the vector of $({\bf B}_1 - {\bf B}_2)$ [Colburn and Sonett, 1966]. The normal direction, thus, is

\begin{displaymath}
{\bf n} \vert\vert ({\bf B}_1 - {\bf B}_2) \times ({\bf B}_1 \times {\bf B}_2)
\eqno (2.11)\end{displaymath}

The fields are coplanar only in the region in which there is no electric field along the normal. In boundary normal coordinates, the noncoplanar component is almost zero on the two sides of the shock. The normal component remains constant but with a finite value through the shock. Consistent with this theorem, non-coplanar magnetic fields are frequently observed within the quasi-perpendicular subcritical shock associated with the dissipation, see Fig. 2.2 for example. Under this circumstance the minimum variance analysis should be applicable as it is when there is a standing wave upstream of the shock along the normal. When there is a sizable non-coplanar component its magnitude can be used to derive the shock velocity from a single spacecraft [Newbury et al., 1997]. Since under typical conditions the variations in both the normal and noncoplanar components are small, the minimum variance analysis generally has a large uncertainty at the shock.

Fig. 2.2. An example of bow shock crossings. The normal direction is determined using the coplanar analysis. Note that Bm, the non-coplanar component, is near zero on both sides of the shock. If one used the tangential discontinuity analysis, the normal direction would be the m direction. In this case, because of a significant non-coplanar field within the shock, the minimum variance analysis will provide correct normal direction. Without such a noncoplanar component, the minimum variance analysis will have a very large uncertainty.

Similar to the tangential discontinuity analysis, ideally the calculation is made with simultaneous measurements on two sides of the discontinuity. If as usual the calculation is made from the measurements on a single spacecraft, uncertainty in the coplanar analysis is mainly caused by temporal variations. Since usually there are strong fluctuations near a shock, especially for a high Mach number shock, the two fields should be measured in the regions which are relatively quiet. The trailing wavetrains downstream from shocks are usually not coplanar with the normal. One should avoid selecting these regions as the downstream condition. For a weak shock, the uncertainty for this method may become large since the fields on two sides of the shock may be similar and the two vectors in the brackets in equation (2.11) are both close to zero [Russell et al., 1983].

For a quasi-parallel shock, the normal of which is nearly parallel to the upstream magnetic field, the average field change across the shock is small. The uncertainty in the normal determination is expected to be large. Furthermore and more importantly, there are usually large amplitude fluctuations present near such a shock, and the shock front is often not clearly defined. This makes the shock normal analysis extremely difficult. Computer simulations have shown that the shock front undergoes a continuous reformation process [Krauss-Varban and Omidi, 1993]. Therefore the stationarity approximation based on which the R-H relations are derived may not be valid. How to analyze quasi-parallel shocks has not been systematically investigated.

2.5 The Maximum Variance Analysis.

The determination of the normal of a discontinuity from the minimum variance analysis of the magnetic field has a very large uncertainty when the intermediate and minimum eigenvalues are close to each other. This is the usual situation when the field shear across the discontinuity is small, (e.g., when the major field change is in its strength) or when the observations cannot resolve the interior of the discontinuity either due to low time resolution of the measurements or fast motion of the discontinuity relative to the spacecraft. In these circumstances, the maximum variance analysis of the electric field may offer a better determination of the normal. The maximum variance analysis of the electric field is based on the fact that the electric field is curl free in steady state, (see also discussions on equation 1.3) or

\begin{displaymath}
\grad \times {\bf E} = - {\partial {\bf B} \over \partial t} = 0 
\eqno (2.12a)\end{displaymath}

Thus,

\begin{displaymath}
{\bf n} \times \Delta {\bf E} = 0 
\eqno (2.12b)\end{displaymath}

namely, the normal is along the electric field change. Analogous to the minimum variance analysis of the magnetic field, the normal direction of the discontinuity is along the maximum variance direction of the electric field.

The electric field data for the analysis can be from either direct measurements of the electric field or the convective electric field derived from the magnetic field and plasma velocity measurements according to the frozen-in condition, or ${\bf E} = -{\bf v} \times {\bf B}$. If the electric field measurements are two dimensional, the third component of the electric field can be obtained from the frozen-in condition, or ${\bf E} \cdot {\bf B} = 0$.

The major uncertainty in this method comes from the temporal variation terms in equation (1.3). Both unsteadiness of the discontinuity itself and changes of its motion relative to the observer will affect the results. In particular in the case of the magnetopause, the boundary usually oscillates instead of being in constant motion. Another important uncertainty comes from the relative motion between the spacecraft and the boundary, even if the motion is steady. From equation (2.2), one obtains a tangential electric field change in the spacecraft frame, $\Delta {\bf E}_T = {\bf V}_{disc} \times ({\bf B}_2 - {\bf B}_1)$.To remove this effect, one has to transfer the electric field into a frame which is at rest in the discontinuity. However, since the motion of the discontinuity is in general unknown before the normal direction of the discontinuity is determined, it is difficult to completely remove this effect. Sonnerup et al. [1987] developed a method and gave a comprehensive discussions on how to minimize this effect. One way to reduce this effect is as follows.

1) Find the maximum variance direction of the electric field, ${\bf
n}$.

2) Measure the average velocity and magnetic field along ${\bf n}, v_n$and Bn, within the discontinuity.

3) If the discontinuity is not a shock, calculate

\begin{displaymath}
u_n = \pm B_n / \sqrt {\mu_0 \rho}
\eqno (2.13)\end{displaymath}

where un is similar to the flow velocity across the discontinuity, and we have assumed that the discontinuity is a rotational discontinuity.

4) The relative velocity of the discontinuity to the spacecraft is approximately

\begin{displaymath}
V_{disc} = \nu_n - u_n
\eqno (2.14)\end{displaymath}

5) Subtract the electric field due to the relative motion

\begin{displaymath}
{\bf E}^{\prime} = {\bf E} - {\bf V}_{disc} { \times \bf B}
\eqno (2.15)\end{displaymath}

6) Using $E^{\prime}$as corrected electric field, repeat the procedures above, until ${\bf
n}$does not change.

Another source of uncertainty in the maximum variance method is due to the assumption of the frozen-in condition to calculate the electric field if it is not measured directly in three dimensions. Under the frozen-in approximation the effects due to the Hall term, the resistivity term, electron pressure gradient term and electron inertial term in Ohm's law have been ignored. These effects may be important in sharp changes within a discontinuity.

The principles to evaluate a result of the maximum variance analysis is similar to some of that for the minimum variance analysis discussed in section 2.2. The success of the method requires a much larger maximum eigenvalue than the other two eigenvalues. The continuity of the normal component of the magnetic field and tangential electric field can be used as a check on the results. The maximum variance analysis does not require many data points within a discontinuity, a significant advantage over the minimum variance analysis. However, since the plasma velocity measurements have usually a lower time resolution than the magnetic field measurements, fewer data points are obtained within and near a discontinuity. If the fluctuations near the discontinuity are not small, the result may be very sensitive to the number of the data points used in the analysis.

In summary, the maximum variance analysis of the electric field can be used as an alternative in cases when the minimum variance analysis has a large uncertainty. It should be used with caution.

2.6 DeHoffmann-Teller Frame and Walen Relation Test.

The deHoffmann-Teller (HT) frame is one of the shock frames, ie. a frame at rest in the discontinuity. Here we recall that frames at rest in the discontinuity can have different tangential velocities. The HT frame moves along the shock front with a velocity such that the magnetic field and velocity are parallel, and hence the tangential electric field vanishes (see a brief review by Sonnerup et al. [1995]). The HT frame moves at a speed

\begin{displaymath}
{\bf V}_{HT} = {\bf V}_{i} \pm {\bf B}_{i} {{u_{in}} \over {B_{n}}} \ \ (i = 1,2)
\eqno (2.16)\end{displaymath}

relative to to the observer. The plus and minus signs correspond to the normal component of the velocity and magnetic field to be antiparallel and parallel, respectively. In the normal incidence case, because ${\bf V}_{1T}=0$,the downstream tangential velocity ${\bf V}_{2T}=({\bf B}_{2T} u_2 - {\bf B}_{1T} u_1)/B_n$is in general nonzero. The flow is accelerated tangentially in crossing the shock due to the kink force of the field. Since the electric field in the HT frame is zero, the electric field in the spacecraft frame is

\begin{displaymath}
{\bf E}_i = -{\bf V}_{HT} \times {\bf B}_i
\eqno (2.17)\end{displaymath}

The proportionality between the electric and the magnetic field variations can be used to determine the velocity of the HT frame. Practice indicates that a well-determined HT frame can often be found in the magnetopause [Sonnerup et al., 1990; Walthour et al., 1993]. Since the HT frame is a shock frame, (VHTn=Vdisc), it can be used to solve the difficulty in determination of a shock frame discussed in section 2.5. However, in general, since the normal direction is unknown, it needs iteration before a satisfactory result is reached. In its most developed form, using the information from the maximum variance analysis of the magnetic and electric fields and the minimum variance analysis of ${\bf V}_{HT}$, through iteration, one can derive not only the normal direction, but also the normal velocity and its acceleration, see Fig. 2.3 for example. The effect of the acceleration is important as seen in the last term of Equation (1.3).

Fig. 2.3. In the most advanced development [Sonnerup et al., 1987], the acceleration of the H-T frame can be introduced to improve the fit and hence the temporal variations of the H-T frame can be partially resolved.

Equation (2.17) can also be written in the form of finite field perturbations. If the magnetic field change is in the $\hat L$ direction, and when the discontinuity has no motion in the normal direction, the electric field change is in the $\hat N$ direction, the HT frame moves mainly along the $\hat M$direction. To derive ${\bf V}_{HT}$ in equation (2.17) is equivalent to a three-parameter fit or a minimum variance problem. Sonnerup et al.[1987] provided the expression for the covariance matrix. Here we should note that unless ${\bf B}_i$ and ${\bf V}_i$ outside the shock ramp are not coplanar, ${\bf V}_{HT}$ and the magnetic fields are all in a plane orthogonal to the shock surface. The derived electric field change is along the shock surface and not along the normal direction.

For RDs, combination of equations (2.5, 2.9 and 2.16) yields the Walen relation in the spacecraft frame,

\begin{displaymath}
{\bf V } = {\bf V}_{HT} \pm {\bf C}_A
\eqno (2.18)\end{displaymath}

where ${\bf C}_A = {\bf B} / \sqrt{\mu_{\circ}\rho}$ for an isotropic plasma and ${\bf C}_A = \sqrt{\xi_{\circ}}{\bf B} / \sqrt{\mu_{\circ}\rho}$ for an anisotropic plasma, $\xi_{\circ} = \mu_{\circ}(P_{\perp} - P_{\parallel})/B^2$ is the anisotropy factor [Chao, 1970], and $P_\perp$ and $P_\parallel$ are the pressures perpendicular and parallel to the magnetic field, respectively. The plus and minus signs denote the RDs propagate antiparallel and parallel to the magnetic field, respectively. The subscript i has been neglected assuming that the relationship applies to every measurement.

The Walen relation is a more restrictive test for a discontinuity. It requires not only a well determined HT frame but also that the discontinuity propagates with the Alfven speed relative to the flow. A positive result of the test verifies the discontinuity to be an RD. Here we should emphasize that a linear relationship between the plasma velocity and magnetic field (or the Alfven velocity) variations does not necessarily imply the discontinuity to be an RD unless the offset between the two, ${\bf V}_{HT}$ (see equation 2.16) equals the proportional factor between the electric and magnetic field variations (see equation 2.17).

2.7 Suggested Procedures for Discontinuity Analysis.

As discussed in sections 2.2 to 2.4, the three methods commonly used for discontinuity analyses are for different purposes. None is intrinsically better than others. However, different methods may provide different normal directions. For example, the noncoplanar direction in the coplanarity analysis is very close to the normal direction in the tangential discontinuity analysis for the same discontinuity (see Fig. 2.2 for an example). The minimum variance direction could be the noncoplanar direction of a shock. Thus, the question is how to analyze a discontinuity without a presumption about its type. The following is a suggested approach.

1) Collect as much information as possible for the interesting discontinuity in addition to the magnetic field, such as plasma measurements and the measurements from other close spacecraft if there are any. These measurements may help to constrain the results.

2) Make an overview of the discontinuity to decide where are the upstream and downstream regions and which major change is most interesting. There may be more than one choice. Keep in mind that a single discontinuity, when its internal structure can be resolved, may appear to consist of more than one sharp changes, that a discontinuity can oscillate back and forth, and that two distinct discontinuities could be very close to each other in time series records.

3) Begin the analysis by using the minimum variance analysis with caution as discussed in section 2.2. For low time resolution measurements or when the discontinuity being studied moves fast relative to the spacecraft, there may be no measurement within the discontinuity. In this case, the minimum variance method can provide only the direction of maximum variance and the normal of the discontinuity cannot be determined with only magnetic field measurements.

This step only provides hints to the nature of the discontinuity. One may only guess the nature of the discontinuity from the results of the minimum variance analysis. For example, if the field in the minimum variance direction is close to zero, the discontinuity may be a tangential discontinuity. If the field in the minimum variance direction is not small and the field strength is similar on the two sides of the discontinuity, the discontinuity may be a rotational discontinuity. If the major field change is in only one component and the field strength changes, the discontinuity may be a shock.

Further determinations of the properties of the discontinuity need plasma measurements or assumptions.

4) Make assumptions of the nature of the discontinuity if the processes associated with the discontinuity are known. In many of discontinuity studies, the nature of the discontinuity has been carefully studied previously with particle measurements and hence known, but the plasma moments are not available. In these cases, assumptions of the nature of a discontinuity can be made, but keep in mind that the results are conditional depending on the accuracy of the assumptions. For most of bow shock studies, coplanarity is a good assumption. In fact, most of these studies skip step 3 and use the coplanarity analysis directly. In most circumstances, the magnetopause can be considered as either a tangential discontinuity or a rotational discontinuity. Thus one may try the tangential discontinuity analysis if the minimum variance component is close to zero. The neutral sheet can be considered as a rotational discontinuity. However, to analyze the neutral sheet using the minimum variance method is difficult because the variance in one of the tangential components $(\hat
y)$ can be very small. The field aligned current sheets in the low beta magnetosphere can often be treated as tangential discontinuities. For most interplanetary discontinuities, since their natures are unknown without plasma measurements, an analysis using only magnetic field measurements cannot be conclusive except if the field strength remains nearly constant across the discontinuity. In this latter case, the discontinuity can be either a tangential discontinuity or a rotational discontinuity and can be treated similarly to the magnetopause situation just discussed above. However, because often an interplanetary discontinuity moves with a speed similar to that of the solar wind, the flow velocity relative to the discontinuity is not easy to resolve. Caution should be taken when interpreting it as either a TD or an RD. The main difficulty in analyzing an interplanetary discontinuity is the lower time resolution of the data caused by the fast passage of the discontinuity.

With an assumption of the type of a discontinuity, one can determine the normal direction of the discontinuity using the methods discussed in previous subsections.

5) Use plasma measurements. With plasma measurements, if the normal direction of the discontinuity has been determined in the last step, the normal velocity of the discontinuity can be determined, according to equations (2.1) and (2.7),

\begin{displaymath}
V_{disc} = {\rho_1 V_{1n} - \rho_2 V_{2n} \over \rho_1 - \rho_2}
\eqno (2.19)\end{displaymath}

If the plasma velocity measurements are not three dimensional, an additional assumption has to be made, hence more uncertainties are introduced, in the calculations. Although this relationship is widely used, we would like to point out that when Vdisc is much less than both V1n and V2n, it could be in the range of the uncertainty of the velocity measurements. In this case, equation (2.19) is not useful. The uncertainty in this calculation comes from these major sources. The first is in the direction of the normal. This error is discussed above. The second is the fluctuations upstream and downstream. In some cases, this source may not be important. The third is due to the change in the calibration factor of the instrument associated with the change of plasma state as discussed in section 1.3.3. While velocity measurements, at least of the solar wind ions are usually quite accurate, the plasma density upon which equation (2.19) depends are not so accurate. If the temperature and speed of the plasma remain similar across the discontinuity, this second source may not be important. Unfortunately under these circumstances, the density change is small, leading to a very large uncertainty in the denominator of equation (2.19). At shocks, both the temperature and speed vary across the discontinuity in such a way that the density calibration in some instruments can be drastically affected [Petrinec and Russell, 1993]. An example of where this calibration change may have seriously affected measurements across the bow shock can be found in Lepidi et al. [1996].

For interplanetary discontinuities, plasma measurements can help to determine the type of a discontinuity. As discussed in step 4, its type can be determined when there is no change in the field strength across the discontinuity. If there is a change in the field strength, in the simplest case, the discontinuity is either a tangential discontinuity or a shock. (If the plasma is anisotropic, even an RD can be accompanied by a change in field strength.) As discussed before, the minimum variance method has a large uncertainty in these two situations, and the tangential discontinuity analysis and the coplanarity analysis provide two orthogonal normal directions. Which of these two normals is correct? For a tangential discontinuity, since u1n = u2n = 0, or V1n = V2n = Vdisc, the velocity difference ${\bf V}_1 -{\bf V}_2$ is tangential to the discontinuity. Thus, comparing the direction of the velocity difference with the two normals may eliminate one of the two. There is a possibility that the three vectors are orthogonal to each other. Another way to distinguish a tangential discontinuity from a shock is to check the total pressure, the sum of the thermal pressure and the magnetic pressure $B^2 / 2 \mu_o$. For a tangential discontinuity, since un = Bn = 0, from equation (2.4), the total pressure remains constant across the discontinuity. However, this method relies on a good intercalibration between the magnetic field measurements and the plasma measurements. For slow shocks, since the change in the total pressure is expected to be small, it is difficult to eliminate the possibility of slow shocks solely based on the total pressure balance.

6) Use the timing difference from two spacecraft. Measurements from two spatially separated spacecraft may be able to determine the velocity of a discontinuity. If the separation of the two spacecraft along the discontinuity is smaller than the curvature of the discontinuity and the time scale of the variations of the discontinuity is much larger than the time scale of the time delay between the encounters of the discontinuity by the two spacecraft, (the determination of the time delay will be discussed in section 4.1), the discontinuity can be considered as a stationary planar wave front. The curvature of the discontinuity can be determined by comparing the two normals from the two spacecraft if only a single crossing is observed by each of them. The velocity of the discontinuity in the plasma frame can be obtained by

\begin{displaymath}
V_{disc} = {{\bf L} \cdot {\bf n} \over \Delta t} - V_{(1, 2)} 
\cos \eta_{(1, 2)} 
\eqno (2.20)\end{displaymath}

where ${\bf L}, \Delta t$ and $\eta$ are the separation vector between the two spacecraft, the time delay between the observations of the discontinuity from the two spacecraft and the angle between the flow velocity, either upstream or downstream, and the normal of the discontinuity. Note that equation (2.20) requires both 3-D plasma measurements and the normal direction. Therefore, this step is not independent of the last two steps. It provides additional information and can be used in combination with the last two steps or as a verification of the results from the last two steps.

2.8 R-H Relations Test.

In the previous steps, we have used two of the R-H relations, the mass conservation and the continuity of the normal field. These two relations have the least uncertainty in practice. However, in steps 4 and 5, to determine the normal of a discontinuity, we have approximated it by a simplified discontinuity. In fact, a discontinuity may not be any of these simple discontinuities assumed rather a combination of them. Also, an analysis may provide many different parameter sets as the result of multiple interval selections. One should use these parameter sets to test other R-H relations. With this step, one may find the best parameter set for the study and estimate the uncertainty of the analysis. The uncertainty is the ratio of the difference to the average of the quantity, equation (2.8), in each of the R-H relations. The conclusions of any analysis should be drawn consistent with this uncertainty.

In the procedure described above, the determination of the shock speed carries enormous weight in determining the nature of a discontinuity. As discussed in section 1.3.3, the calibration factor of plasma measurements may change across a shock. To accurately determine the shock velocity is extremely difficult. To solve this problem, Chao et al.[1995] recently developed a method to derive the properties of a shock without predetermining the shock velocity and then derive the shock velocity afterward. In their method, the R-H relations are combined and written in normalized parameters, such as ratios of upstream and downstream values, angles, plasma betas and Mach numbers. It is realized that to determine an isotropic shock requires only any three independent such parameters. Therefore, one can choose three observationally best determined parameters to substitute into the normalized R-H relations and then derive those remaining. We suggest using the ratio of the ratio of the magnetic field strengths, B2/B1, the upstream shock angle and downstream plasma beta. All these three parameters are independent of frame reference. The ratio of the magnetic field strengths has little uncertainty. The shock angle depends only on the accuracy of the shock normal determination. As shown in Fig. 1.2, among three plasma moments, the uncertainty in the pressure measurements is least and is even smaller for higher temperature plasmas. Thus the downstream plasma beta is the most accurately measured quantity although its absolute calibration needs to be verified.

An interesting finding by Chao et al.[1995] from a parametrical study is that the ratio of the upstream and downstream plasma densities is less sensitive than the ratio of plasma velocities. In other words, a small difference in the density ratio corresponds to a large range of shock parameters. A small uncertainty in the density measurements will hence cause drastically different conclusions about the shock properties. Such temperature and velocity dependence of the density calibration of several plasma instruments as found by Petrinec and Russell [1993] would cause such uncertainties across almost any shock crossing. Therefore, it is not recommended to use the density ratio as a primary parameter in shock normal determination unless the density calibration on both sides of the shock is well known.

Chao [1995] shows that in many cases, the parameters derived using the best fit to all R-H relations are actually far from possible solutions of the R-H relations. This result indicates that because of the complicated relationship among different variables, the values of the unmeasured quantities determined from a best fit actually do not represent well the real values of these quantities for each set of the parameters at a given point. It is not recommended to use a best fit of all R-H relations.

In summary, using only magnetic field measurements one can determine the normal direction of a discontinuity in simple cases. The minimum variance method applies only if there are measurements within a discontinuity. It has a large uncertainty for either a tangential discontinuity or a shock. The tangential discontinuity and coplanarity analyses impose strong assumptions on the type of a discontinuity. They should be used carefully in combination with the minimum variance analysis and plasma measurements.





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