Statistics and accuracy of magnetic null identification in multispacecraft data

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Eriksson, E., Vaivads, A., Khotyaintsev, Y V., Khotyayintsev, V M., Andre, M. (2015) Statistics and accuracy of magnetic null identification in multispacecraft data.

Geophysical Research Letters, 42(17): 6883-6889 http://dx.doi.org/10.1002/2015GL064959

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Statistics and accuracy of magnetic null identification in multispacecraft data

E. Eriksson

, A. Vaivads

, Yu. V. Khotyaintsev

, V. M. Khotyayintsev

, and M. André

1

Swedish Institute of Space Physics, Uppsala, Sweden,

Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden,

Department of Theoretical Physics, Taras Shevchenko National University of Kyiv, Kiev, Ukraine

Abstract Complex magnetic topologies are ubiquitous in astrophysical plasmas. Analyzing magnetic nulls, regions of vanishing magnetic field, is one way to characterize 3-D magnetic topologies. Magnetic nulls are believed to be important in 3-D reconnection and turbulence. In the vicinity of a null, plasma particles become unmagnetized and can be accelerated to high energies by electric fields. We present the first statistical study of the occurrence of magnetic nulls and their types in the Earth’s nightside magnetosphere. We are able to identify the nulls both in the tail and in the magnetopause current sheets.

On average, we find one null for every few current sheet crossings. We show that the type identification of magnetic nulls may be sensitive to local fluctuations in the magnetic field. We develop and demonstrate a method to estimate the reliability of the magnetic null type identification.

1. Introduction

Complex magnetic topologies are ubiquitous in astrophysical plasmas and have been observed, for example, in the solar corona [Freed et al., 2014] and in large-scale current sheets such as at the Earth’s magnetosphere [Dorelli et al., 2007]. One way to characterize complex three-dimensional (3-D) magnetic topologies is by ana- lyzing the distribution and the type of magnetic nulls present. Magnetic nulls are regions with vanishing magnetic field—topological singularities; field lines surrounding a null often belong to regions of differ- ent topology. Magnetic nulls are believed to be important in 3-D reconnection [Priest and Forbes, 2000] and turbulence [Wendel and Adrian, 2013; Olshevsky et al., 2015]. Due to the low magnitude of the magnetic field strength in the vicinity of a null, plasma particles become unmagnetized and can be accelerated to high ener- gies via interaction with electric fields. These factors highlight the importance of studying magnetic nulls in order to improve our understanding of astrophysical plasmas.

There have been a number of case studies of magnetic nulls using in situ observations in the near-Earth space [Xiao et al., 2006, 2007; He et al., 2008; Deng et al., 2009; Wendel and Adrian, 2013; Olshevsky et al., 2015].

However, there is a lack of in situ statistical studies, similar to statistical analysis of predicted magnetic nulls in the solar corona [Freed et al., 2014], so it is unclear how common magnetic nulls are in the near-Earth space.

Moreover, it is uncertain how reliable the identification of magnetic null type is using in situ data. In this letter we perform a statistical study of magnetic nulls in the Earth’s nightside magnetosphere. We demonstrate that the null type identification can be erroneous due to local disturbances and measurement error and present a method to estimate the reliability of the null type identification.

2. Method

Two different analytic methods can be used to find magnetic nulls using measurements from four points, which span a tetrahedron volume, e.g., in multispacecraft missions such as Cluster [Escoubet et al., 1997] and the Magnetospheric Multiscale (MMS). In the first method, introduced by Greene [1992], the Poincaré index (PI) is calculated. When PI = ±1 the tetrahedron encloses an odd number of magnetic nulls and an even number when PI = 0. It is usually assumed that for a sufficiently small tetrahedron PI = ±1 implies that the tetrahedron encloses a single magnetic null, and PI = 0 implies that no magnetic nulls are enclosed. The second method, introduced by Fu et al. [2015], is based on a Taylor expansion (TE) near a null:

B(r) = ∇B ⋅ (r − r

) , (1)

RESEARCH LETTER

10.1002/2015GL064959

Key Points:

• Method for estimating the reliability of the magnetic null type identification

• Statistical study of magnetic nulls in the Earth’s nightside magnetosphere

• Results important for multispacecraft missions such as Cluster and MMS

Correspondence to:

E. Eriksson, elin.eriksson@irfu.se

Citation:

Eriksson, E., A. Vaivads, Yu. V.

Khotyaintsev, V. M. Khotyayintsev, and M. André (2015), Statistics and accuracy of magnetic null iden- tification in multispacecraft data, Geophys. Res. Lett., 42, 6883–6889, doi:10.1002/2015GL064959.

Received 15 JUN 2015 Accepted 11 AUG 2015

Accepted article online 14 AUG 2015 Published online 3 SEP 2015

©2015. American Geophysical Union.

All Rights Reserved.

Geophysical Research Letters 10.1002/2015GL064959

where r is the location in space, r

is the null position, B is the magnetic field at r, and ∇B is the gradient of the magnetic field derived using four spacecraft measurements, assuming the magnetic field changes linearly in space [Chanteur, 1998]. The gradient is thus assumed to be constant in the space surrounding the spacecraft.

In general, equation (1) will always yield a null position, but expansions with large (r − r

) are not valid. For simplicity, we only consider the magnetic null position reliable if it is located in a box defined by the spacecraft location: the edges of the box in each direction (x , y, z) are given by the maximum and minimum positions of the Cluster spacecraft. We disregard the nulls with positions outside of the box. We expect to find more nulls using the TE method because it looks for nulls in a volume which is ∼7 times larger than the tetrahedron used by the PI method.

In our analysis the nulls are also type identified. Depending on the surrounding structure of the magnetic field, magnetic nulls are classified as either A, B, As, or Bs [Cowley, 1973; Greene, 1988; Lau and Finn, 1990]. The classification is based on the eigenvalues, 𝜆

, 𝜆

, 𝜆

, of the tensor ∇B. The eigenvalues are either all real, or two of them are complex conjugates of each other with a real third value. Since ∇ ⋅ B = 0, the eigenvalues satisfy the condition 𝜆

+ 𝜆

+ 𝜆

= 0. The two eigenvectors, corresponding to the eigenvalues whose real parts have the same sign, span the fan plane of the null. If two of the eigenvalues are complex, the magnetic field lines will spiral about the null point in the fan plane. Such magnetic nulls are referred to as spiral nulls (As, Bs). The eigenvector corresponding to the third eigenvalue, opposite sign to the other two, defines a direction in space, called the spine. The sign of det(∇B) = 𝜆

⋅ 𝜆

⋅ 𝜆

determines the direction of the field along the spine. When det(∇B) > 0, magnetic field lines in the fan plane converge toward the null point and diverge away from the null point along the spine. These types of nulls are called A and As magnetic nulls, which we refer to as A-kind nulls. For the B and Bs nulls (B kind) the magnetic field direction about the null is reversed and det(∇B) < 0 [Lau and Finn, 1990].

3. Statistical Study

We perform a statistical study of magnetic nulls in the Earth’s nightside magnetosphere, where we expect magnetic nulls due to the thin tail current sheets present in the center of the magnetotail, as well as at the mag- netopause current sheet along the boundary of the magnetotail. We use in situ magnetic field data from the Fluxgate Magnetometer (FGM) on board the Cluster spacecraft [Balogh et al., 1997]. We consider the Cluster measurements between 1 July 2003 and 1 January 2004, as this was when the spacecraft separation was the smallest at the nightside magnetosphere. At this time the spacecraft separation was ∼200km, i.e., less than the ion inertial length. We also limit our study to the period when the spacecraft were close to the equa- torial part of the magnetotail, such that X < −4R

, and |Z|< 10R

in geocentric solar magnetospheric (GSM) coordinates, where R

is the Earth radius.

We apply the TE and PI methods to our data set in order to find magnetic nulls, and then we characterize their type. The estimate of ∇ ⋅ B based on the assumption of linear change in space of magnetic field measured by spacecraft [Chanteur, 1998] typically does not satisfy ∇ ⋅B = 0 due to nonlinearity of magnetic field, measure- ment errors, and finite number of measurement points. Therefore, we limit our data set to events satisfying the following constraint [Fu et al., 2015]:

|| || ∇ ⋅ B max(|𝜆

|) ||

|| ≪ 1 , (2)

which is a necessary but not sufficient condition for linearity of the magnetic field around the null. Evaluating the linearity of the field requires at least five measurement points, which are not available in the case of Cluster or MMS. For practical use we choose 0.4 as the limit in equation (2); we note that making this number smaller does not affect the result but significantly reduces the number of data points available for the study.

Figure 1 shows the location of all identified nulls. We find more magnetic nulls at the magnetopause than in the magnetotail current sheet. To estimate the relative probability of finding a null, we also plot the dwell time of the spacecraft in each of the spatial bins. In some regions nulls are not observed because the spacecraft spent little time there due to the orbit sampling. For example, there are no null observations inside the tail current sheet earthward from about X = −15R

when Cluster is close to the Sun-Earth meridional plane.

On average, in the magnetotail current sheet we observe one null per 25h, while in the magnetopause we

observe one null per 11h. In practice this corresponds to about one positive magnetic null identification per

few current sheet crossings. The magnetopause current sheet is more dynamic than the tail current sheet,

resulting in many crossings of the current sheet, which explains why we observe more nulls there. We found

Figure 1. The position of the magnetic nulls for both the Poincaré index (PI) and Taylor Expansion (TE) method. The gray background gives the dwell time of the spacecraft in each of the spatial bins.

735 data points of nulls with the TE method (green dots in Figure 1), and 84 of those data points correspond to all the magnetic nulls found with the PI method (red tri- angles). One null typically corresponds to several data points. As expected, due to the larger volume used, the TE method found more nulls. Figures 2 and 3 show two typical examples of the nulls found in the study using the TE method. In Event I (Figure 2: 6 August 2003), a magnetic null is observed for several data points crossing the spacecraft box in the −X, −Y direction.

Event II (Figure 3: 28 October 2003) shows a magnetic null crossing the spacecraft box in the +X, −Y direction with a few more data points than Event I; the null is moving slower than the null in Event I and there- fore stays within the tetrahedron longer.

As explained before, as soon as the posi- tion of a null for one data point is outside the spacecraft box it is no longer consid- ered valid, which is why none of the nulls in Figures 2 and 3 are outside the straight black lines.

We classify the observed nulls and find that 80% of them are spiral nulls (As and Bs).

This is consistent with the analysis performed in the turbulent magnetosheath plasma [Olshevsky et al., 2015].

The result of 80% is very close to the 76% of spiral nulls we find when we analyze a fully random magnetic field. Thus, the data do not show a tendency to have a different ratio of spiral versus nonspiral nulls than purely random magnetic field fluctuations. This may indicate that the physical processes responsible for the null formation do not favor formation of particular types of nulls.

To see how reliable type identification is, we have to understand the possible effects that can lead to errors in null type identification. For example, two nulls are observed for many time steps in Figures 2 and 3 consistently showing the same null type, except for the two time steps in the middle of Event I (Figures 2d–2f ) where the null is identified as an As type. This is most likely the same null, and we interpret the change as being due to local fluctuations in the magnetic field. Therefore, we want to investigate how fluctuations can affect the null type identification.

4. Accuracy of Type

Figures 2 and 3 are two typical examples showing the null type estimates from spacecraft measurements during the time a null moves through the spacecraft box. In the first event, Figure 2, the null type changes, while in the second event, Figure 3, the null type stays the same. One cause of the observed null type change could be local disturbances in the magnetic field introducing additional nonlinearity in the field. When using Cluster spacecraft data, the largest disturbances originate from local plasma processes (e.g., waves or localized structures on spatial scales smaller than the spacecraft separation) but can also be due to instrumental errors.

In order to investigate the effect of local disturbances, we take each data point and rotate its ∇B into the coordinate system of the null using the Parnell et al. [1996] method, which gives a new ∇B:

∇B = s 𝜇

⎛ ⎜

⎜ ⎜

⎝

1

2

(q − j

) 0

1

2

(q + j

) p 0

0 j

−(p + 1)

⎞ ⎟

⎟ ⎟

⎠

, (3)

Geophysical Research Letters 10.1002/2015GL064959

Figure 2. An example of magnetic null observations using TE method. (a–c) The magnetic field components in GSM for all four Cluster spacecraft. (d–f ) The distance to the magnetic null from the center of the four spacecraft, the mean value of all the spacecraft positions. The straight black lines indicate the edges of the spacecraft box volume.

Figure 3. Event II; see caption to Figure 2.

Figure 4. (a) Comparison of threshold current,

, with observed

values for all magnetic nulls; (b) a zoomed-in view of Figure 4a. Events I and II correspond to the examples shown in Figures 2 and 3.

where s is the scaling of matrix with units of nTkm

and p and q describe the potential (current-free) part of magnetic field [Parnell et al., 1996]. As mentioned earlier, B estimated from spacecraft data will not be diver- gence free and therefore the second and third diagonal terms will have slightly different values of p. The

∇B tensor allows us to estimate all the parameters that define the magnetic topology of the null: p, q, j

, j

, and j

, where j

and j

are the currents parallel and perpendicular to the spine of the magnetic null, while j

= √

(p − 1)

+ q

is a threshold current [Parnell et al., 1996]. The parameters j

and j

are derived by rotating the calculated current using four spacecraft method, j = ∇ × B∕𝜇 [Chanteur, 1998], into the nulls coordinate system in accordance to Parnell et al. [1996]. A magnetic null is of a spiral type (As/Bs) when j

> j

[Parnell et al., 1996]. Figure 4a shows the values of j

versus j

for all magnetic nulls found in the statistic study, while Figure 4b is a zoomed-in version. The majority of the magnetic nulls have 𝜇

j

s , 𝜇

j

s < 0.05 nTkm

. Many of the magnetic nulls are also close to the line 𝜇

j

s = 𝜇

j

s, making their type identification easily affected by local magnetic field disturbances.

The basic concept of our method to estimate the reliability of a magnetic null is to compare typical magnetic fluctuations seen in the data with theoretical minimum disturbances capable of altering the type of the null.

A magnetic null can be altered in two ways: it can shift either to/from a spiral type or between A kind and B kind. The theoretical minimum disturbance required to alter the null type from/to a spiral type is 𝛿B

= 𝜇

sL( j

− j

), where L is the characteristic separation between the spacecraft. Figure 5 shows estimates of 𝛿B

for all the observed nulls. We look more closely at Events I and II (the examples given in Figures 2 and 3).

For Event I, see Figure 5b (black triangles); the disturbance needed to change the null type from A to As is 𝛿B

=0.3nT, which means that magnetic field fluctuations ≥0.3nT can cause a change in magnetic null type.

The largest fluctuation amplitude seen in the data of one spacecraft is 0.4nT, which explains the observed

type alteration for two points in the middle of the interval shown in Figure 2. For Event II (green triangles in

Geophysical Research Letters 10.1002/2015GL064959

Figure 5. The minimum disturbance, at one spacecraft, necessary for changing the null type (see text for details).

For visualization purposes the fluctuation values of

have negative sign for the B kind nulls. (a) The minimum disturbances

and

for all the nulls found in data; (b) zoomed-in view of Figure 5a containing 30% of all nulls.

Nulls colored black (I) and green (II) correspond to the nulls in Figures 2 and 3, respectively.

Figure 5a) 𝛿B

=7nT is needed to shift from Bs to B. This is much larger than the observed fluctuations (2nT), consistent with the stable null type for Event II (Figure 3).

Type can also alter between A kind and B kind. The sign of det(∇B) determines whether the magnetic null is of A kind or B kind. Thus, in a linear approximation we can derive the theoretical minimum disturbance, 𝛿B

, at one of the spacecraft capable of causing such a type change as

𝛿B

= min ( |B

⋅ (B

× B

) |∕|(B

× B

) |) , (4) where B

= B

− B

and i , j, k, l are arbitrary permutations of the four spacecraft (1, 2, 3, 4).

We estimate 𝛿B

for all the observed nulls and once again look in more detail at the examples given in Figures 2 and 3, Events I and II. In Event I (Figure 5b, black triangles), the disturbance needed to change the null type from A to B is 𝛿B

=0.5nT, which is larger than the fluctuations seen in the data for the event. For Event II (Figure 5a, green triangles) 𝛿B

=2.2nT, which is still larger than the observed fluctuations for the event. Therefore, we expect no change between A kind and B kind for these two events. In summary, our estimates show that only a spiral null type shift is expected for Event I.

Figure 5 shows that more nulls tend to be closer to 𝛿B

= 0 line than the 𝛿B

= 0 line. Thus, alteration between A kind and B kind is more likely than from/to a spiral. A large fraction of the data points are close to the origin ( 𝛿B

= 𝛿B

= 0), and fluctuations of the order 2–3nT would alter the type of half of the data points. We rarely observe such strong fluctuations in our data set, and also, instrumental errors are typically below 1nT.

Therefore, we expect that at least 70% (see Figure 5 caption) of the magnetic nulls in our data set have reliable type identification.

5. Conclusions

We present the first statistical analysis of magnetic nulls in the Earth’s nightside magnetosphere using multi-

spacecraft Cluster data. We found nulls both in the tail current sheet and in the magnetopause current sheet.

The relative frequency of the null observations was higher in magnetopause current sheet. On average, we observed one null per each few current sheet crossings. We identified 80% of the observed nulls as spiral nulls (As and Bs). This ratio is close to the one obtained for magnetic nulls forming in a fully random magnetic field and therefore suggests that physical processes responsible for the null formation do not favor the formation of particular types of nulls.

We present a method to estimate the reliability of the null type identification by comparing the observed local fluctuations of the magnetic field for a particular event with the minimum theoretical disturbances required to alter the null type, 𝛿B

and 𝛿B

. We found that even relatively small disturbances in the magnetic field can alter the null type. The observed nulls are more likely to alter between A kind and B kind, rather than between spiral and nonspiral types. Our results also show that at least 70% of the magnetic nulls in our data set have a reliable type identification.

The results presented in this letter are important for analyzing magnetic topology with the recently launched NASA MMS mission, as well as for analyzing multiprobe measurements in laboratory plasmas.

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Acknowledgments

We thank the FGM team for the magnetic field data used in this study and the Cluster Science Archive [Laakso et al., 2010] for giving us access. This work was supported by the Swedish Research Council grant 2013-4309.

The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.