ArticlePDF Available

Spatial gradients from irregular, multiple-point spacecraft configurations

Authors:

Abstract and Figures

1] We present a generalized multipoint analysis of physical quantities, such as magnetic field and plasma flow, based on spatial gradient properties, where the multipoint data may be taken by irregular (distorted) configurations of any number of spacecraft. The methodology is modified from a previous, fully 3-D gradient analysis technique, designed to apply strictly to 4-point measurements and to be stable for regular spacecraft configurations. Here, we adapt the method to be tolerant against distorted configurations and to return a partial result when fewer spacecraft measurements are available. We apply the method to a variety of important physical quantities, such as the electric current density and the vorticity of plasma flows based on Cluster and THEMIS multiple-point measurements. The method may also have valuable applications on the coming Swarm mission. Citation: Shen, C., et al. (2012), Spatial gradients from irregular, multiple-point spacecraft configurations, J. Geophys. Res., 117, A11207, doi:10.1029/2012JA018075.
Content may be subject to copyright.
Spatial gradients from irregular, multiple-point spacecraft
configurations
C. Shen,
1
Z. J. Rong,
2,3
M. W. Dunlop,
4,5
Y. H. Ma,
1,6,7
X. Li,
8
G. Zeng,
1,7
G. Q. Yan,
1
W. X. Wan,
2,3
Z. X. Liu,
1
C. M. Carr,
5
and H. Rème
9,10
Received 29 June 2012; revised 28 September 2012; accepted 29 September 2012; published 14 November 2012.
[1]We present a generalized multipoint analysis of physical quantities, such as magnetic
field and plasma flow, based on spatial gradient properties, where the multipoint data
may be taken by irregular (distorted) configurations of any number of spacecraft.
The methodology is modified from a previous, fully 3-D gradient analysis technique,
designed to apply strictly to 4-point measurements and to be stable for regular spacecraft
configurations. Here, we adapt the method to be tolerant against distorted configurations and
to return a partial result when fewer spacecraft measurements are available. We apply the
method to a variety of important physical quantities, such as the electric current density and
the vorticity of plasma flows based on Cluster and THEMIS multiple-point measurements.
The method may also have valuable applications on the coming Swarm mission.
Citation: Shen, C., et al. (2012), Spatial gradients from irregular, multiple-point spacecraft configurations, J. Geophys. Res.,117,
A11207, doi:10.1029/2012JA018075.
1. Introduction
[2] The exploitation of the multipoint measurements from
the cluster spacecraft (S/C) have increasingly become the
manner in which the current exploration of space physics is
carried out, due to their ability to access spatial structure and
often separate the temporal and spatial variations of physical
quantities. The successful operation of multiple spacecraft
missions, as the earlier dual spacecraft ISEE1 and ISEE2
[Ogilvie et al., 1977; Russell and Elphic, 1978], AMPTE
missions [Bryant et al., 1985], Cluster (comprising four
identical S/C) [Escoubet et al., 1997, 2001], currently with
more than 10 year operation, and recently the THEMIS
mission (comprising five identical S/C) [Angelopoulos,
2009], has obtained many significant and original scientific
achievements. In the future, the Swarm mission (comprising
three S/C) [Friis-Christensen et al., 2006], is also due for
imminent launch. Nevertheless, it was the maintenance of the
close formation of the 4 S/C of Cluster which initially
allowed full 3-D spatial structure to be obtained regularly,
and the Magnetospheric Multiscale (MMS) mission [Curtis,
1999], due for launch in 2014, follows this close formation
flying, but at smaller spatial scales. Analysis tools which use
such close spacecraft configurations, however, need to be
selective in order to minimize errors as a consequence of
changing scales, evolution of the constellation shape and
complex or time dependent physical structure. Analysis of
the spacecraft data therefore benefits from robust methodol-
ogy which is tolerant to these factors as much as possible,
such as extremely distorted shapes of the S/C configuration,
and which is able to return particular results when fewer than
four point measurements are available.
[3] Previously developed analysis methods have demon-
strated furthermore that the radical separation of temporal-
spatial variation requires a cluster of at least four spacecraft.
As a paragon, the Cluster mission, which formed a small-
scale tetrahedron (hundreds km to thousands km) in the early
mission stages (20012004) can reveal well the local spatial
gradient of measured quantities. Taking the magnetic field B
as an example, the four-point measurements of Cluster tetra-
hedron, when accessed by a range of similarly developed
analysis methods [e.g., Harvey, 1998; Chanteur, 1998;
Chanteur and Harvey,1998;Shen et al., 2003, 2007; Shen and
Dunlop, 2008], allow the spatial gradients of B,including
magnetic field gradient rB, current density via m1
0rB,
and curvature of the magnetic field lines (MFLs) (B/Br)
(B/B), to be successfully derived in principle, although dif-
ferent methodology [Dunlop et al., 1988; Dunlop and
1
State Key Laboratory of Space Weather, National Space Science
Center and Center for Space Science and Applied Research, Chinese
Academy of Sciences, Beijing, China.
2
CAS Key Laboratory of Ionospheric Environment, Institute of
Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.
3
Beijing National Observatory of Space Environment, Institute of
Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.
4
Rutherford Appleton Laboratory, Didcot, UK.
5
Imperial College of Science, Technology and Medicine, London, UK.
6
Space Science Institute, Macau University of Science and Technology,
Macao, China.
7
College of Earth Science, University of Chinese Academy of Sciences,
Beijing, China.
8
Laboratory for Atmosphere and Space Physics, University of Colorado
Boulder, Boulder, Colorado, USA.
9
IRAP, UPS-OMP, University of Toulouse, Toulouse, France.
10
IRAP, CNRS, Toulouse, France.
Corresponding author: C. Shen, State Key Laboratory of Space
Weather, National Space Science Center and Center for Space Science
and Applied Research, Chinese Academy of Sciences, No.1 Nanertiao,
Zhongguancun, Haidian District, Beijing 100190, China. (sc@nssc.ac.cn)
©2012. American Geophysical Union. All Rights Reserved.
0148-0227/12/2012JA018075
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A11207, doi:10.1029/2012JA018075, 2012
A11207 1of19
Woodward, 1998; Robert et al., 1998a] and complementary
techniques [Dunlop et al., 2002; Haaland et al., 2004a,
2004b; Shi et al., 2005, 2006] have treated the control of
errors in different ways or provided a variety of contextual
information for the interpretation of gradients.
[4] All these methods have advanced enormously our
knowledge of the dynamic magnetosphere. In the determina-
tion of spatial gradients, the inverse of the volumetric tensor
Rjl ¼1
NP
N
a¼1rajral,i.e.,R
jl
1
,(wherer
a
is the position vector of
the spacecraft arelative to the mesocenter of cluster, and j,
lare Cartesian components) plays an essential role in these
analysis methods [Harvey, 1998; Shen et al., 2003, 2007;
Shen and Dunlop, 2008, and references therein]. For the
regular Cluster tetrahedron, the direct operation of R
jl
1
works
well, whereas when the tetrahedron becomes much distorted
or irregular, especially when all the S/C are almost in the
same plane, R
jl
becomes an ill-matrix and the direct calcula-
tion of R
jl
1
then yields significant error. Thus, no useful
information about the gradients can be directly obtained for
that case.
[5] Nevertheless, in addition to the much distorted Cluster
tetrahedron, the plane- or line-like configuration of the
spacecraft cluster is common; such as the three closed-S/C
configuration within the first pass of the THEMIS mission
[Angelopoulos, 2009], and the planned operations of the
forthcoming Swarm mission [Friis-Christensen et al., 2006].
Thus, faced with the various cluster of spacecraft missions
having arbitrary configuration, a serious question is pre-
sented. That is, whether we can develop a new universal
method which cannot only avoid calculating the R
jl
1
directly,
but also may derive the correct information about the gra-
dients. For the three S/C array situation, some authors have
used spatial interpolation or reciprocal vectors method to
deduce the gradients of the physical quantities within the S/C
plane [Li and Chen, 2008; Vogt et al., 2009]. Vogt et al.
[2009], in particular, have made a systematic treatment of
the methodological framework of the planar reciprocal vec-
tors. Nevertheless, in this study, we aim to develop a general
method for determining the gradients of physical quantities
based on data from multiple-S/C with arbitrary shapes and
any number (1, 2, 3, 4, or more); suitable for linear or planer
S/C constellation. The three S/C situation is only one appli-
cation of this new approach.
[6] In the following, the approach for the new method is
presented in Section 2. In Section 3, we apply the new
method to the cases of a much distorted Cluster tetrahedron,
and to the cases of a three-point S/C array among the THE-
MIS and Cluster spacecraft. A discussion and summary are
given in Section 4.
2. Approach
2.1. General Theory
[7] We may deduce the gradient of a certain physical quan-
tity F(e.g., the density of plasmas, the components of magnetic
field or bulk plasma velocity) from its measurements by N
identical spacecrafts. At a specific time, the number aspace-
craft at the position r
a
(a=1,2,,N) yields one measurement
of the physical quantity F,i.e.,F
a
(a=1,2,,N).
The barycenter coordinates [Harvey, 1998; Chanteur, 1998]
are used here, i.e., the mesocenter, c, is taken as the origin
point, thus the position vector of the mesocenter rc¼
1
NP
N
a¼1ra¼0.
[8] Generally, the physical quantity Fis spatially varying
in space around the Nspacecraft. So that we may expand F
around the mesocenter cas
Fa¼Fcþr
nFðÞ
cranþ1
2rnrkFðÞ
cranrakþð1Þ
where the subscripts n,kdenote Cartesian components. F
c
and rnFðÞ
care the value and the gradient of Fat the
mesocenter, and F
a
is the measured value at the position of
the spacecraft a.
[9] Averaging over the value of Fmeasured by the N
spacecraft yields
1
NX
a
Fa¼Fcþr
nFðÞ
c
1
NX
a
ranþ1
2rnrkFðÞ
c
1
NX
a
ranrakþ
Then, F
c
can be obtained as
Fc¼1
NX
a
Fa1
2rnrkFðÞ
c
1
NX
a
ranrakþ¼1
NX
a
Fa
L=DðÞ
2Oþð2Þ
where, Dis the characteristic size of the physical structure,
and Lis typical scale of the spacecraft cluster. Thus, from
equation (2), F
c
is the average of all the measured quantities
Fby the Nspacecraft with the error of order of (L/D)
2
.
[10] In order to get the gradient rnFðÞ
c, we multiply both
sides of equation (1) by the lcomponent of r
a
, i.e., r
al
, and
further make average over Nspacecraft to yield
1
NX
a
Faral¼Fc
1
NX
a
ralþr
nFðÞ
c
1
NX
a
ranralþ1
2rnrkFðÞ
c
1
NX
a
ranrakralþ;
or (since P
N
¼1
ra¼0)
1
NX
a
Faral¼r
nFðÞ
cRnl þ1
2NrnrkFðÞ
cX
a
ranrakralþ;ð3Þ
where Rnl ¼1
NX
a
ranralis the volume tensor [Harvey,
1998].
[11] Generally, the gradient of Fcan be obtained by mul-
tiplying both sides of equation (3) by R
ln
1
as
rnFðÞ
c¼1
NX
N
a¼1
FaralR1
ln þL=DðÞO:ð4Þ
[12] The procedure to estimate equation (4) is the old
method that was used in the previous studies [Harvey, 1998;
Shen et al., 2003, 2007]. If the polyhedron of the spacecraft
cluster is distorted considerably, the corresponding volume
tensor R
nl
would become abnormal, however, and the above
formula fails to yield the gradient of Faccurately. In order to
overcome this difficulty, we may process the equation in the
space of the eigenvectors of R
nl
.
[13] The volume tensor R
nl
is symmetric, and has three
nonnegative eigenvalues w
1
,w
2
and w
3
(w
1
w
2
w
3
0)
SHEN ET AL.: TECHNIQUE A11207A11207
2of19
with three corresponding eigenvectors ^
k1ðÞ
,^
k2ðÞ and ^
k3ðÞ
(where ^
klðÞ^
kmðÞ
¼dlm)[Harvey, 1998]. ^
k1ðÞ
,^
k2ðÞand ^
k3ðÞ
are the three characteristic directions of the spacecraft clus-
ter, which can constitute an orthogonal coordinates with
^
k1ðÞ¼^
k2ðÞ^
k3ðÞ and form the eigenvector space. ffiffiffiffiffi
w1
p,
ffiffiffiffiffi
w2
pand ffiffiffiffiffi
w3
pare the characteristic half widths of the
spacecraft cluster in these three characteristic directions,
respectively [Harvey, 1998]. It is noted here that the char-
acteristic size of the spacecraft cluster is defined as L¼
2ffiffiffiffiffi
w1
p, the elongation as E¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi
w2=w1
p, and the planarity
as P¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi
w3=w2
pin Robert et al. [1998b]. The volume
tensor R
ij
can be rewritten as
Rij ¼wlklðÞ
iklðÞ
j:ð5Þ
[14]IfFis scalar, e.g., the density, temperature, pressure
etc., then in the eigenvector space of R
ij
, equation (3) can be
expressed as
1
NX
a
Fa~
ral¼~
rjF

cwldjl þ1
2N
~
rj~
rkF

cX
a
~
raj~
rak~
ralþ;
ð6Þ
where, the quantity with wavy superscript represents the
projection component in the eigenvector space. For example,
~
rlis the lth component of the vector rin the eigenvector
space.
[15] From equation (6), ~
rlF

ccan be expressed easily as
~
rlF

c¼1
Nwl1X
a
Fa~
ral1
2Nwl1~
rj~
rkF

cX
a
~
raj~
rak~
ralþ
¼~
GF
lþL=DðÞO;ð7Þ
where,
~
GF
l¼1
Nw
lX
a
Fa~
ral:ð8Þ
[16] Equations (7)(8) are the formulae for calculating the
gradient of a scalar physical quantity (at the mesocenter of the
cluster polyhedron) which demonstrates the calculation has a
truncation error of order L/D. The formula (7)(8) have a clear
meaning that, the gradient of a scalar physical quantity along
one eigenvector direction can be estimated by the weighted
average of the difference between the measurements at these
spacecraft over the corresponding characteristic length of
spacecraft cluster.
[17] For a vector physical quantity, e.g., the magnetic field
B, its gradient in the eigenvector space is
~
rl~
Bi

c¼1
Nwl1X
a
~
Bai~
ral1
2Nwl1~
rj~
rk~
Bi

cX
a
~
raj~
rak~
ralþ
¼~
GB
il þL=DðÞO;ð9Þ
Where
~
GB
il ¼1
Nw
lX
a
~
Bai~
ral:ð10Þ
[18] The formulae (9)(10) may also be used to calculate
the gradient of the bulk velocity of plasmas, for example. It is
implied distinctly by formulae (9)(10) that, the vector field
gradient of one component along one eigenvector direction
can be estimated by the weighted average of the difference
between the measurements at these spacecraft divided by the
corresponding characteristic length of spacecraft cluster.
[19] Since the formulae (7)(8) and (9)(10) calculate the
gradients of scalar or vector physical quantities in the eigen-
space of the volumetric matrix, they avoid the difficulty of an
abnormal volume matrix when the polyhedron of the space-
craft cluster is severely distorted. If one of the eigenvalues
equals zero, then the gradient of Falong the corresponding
eigenvector cannot be determined. For example, if the four
S/C of Cluster are in a plane, then we can only determine the
components of gradient in that plane. Nevertheless, even if the
spacecraftcluster is aligned or is in a plane, the above formulas
are still valid in some directions, and moreover the eigenvec-
tors, being the principle directions for the configuration, nat-
urally order the calculation.
[20] In Appendix A, the errors of this method have been
evaluated and discussed. The advantage of this method is
that, it is very natural and general, and represents the essence
of multipoint data analysis. This approach is valid for situa-
tions with any number S/C and arbitrary kind of shape of the
S/C array.
[21] The transformation of the full gradients in the eigen-
vector space to the Cartesian coordinates can be expressed as
GB
ij ¼klðÞ
i

T~
GB
lmkmðÞ
j:ð11Þ
2.2. Linear Configuration Case
[22] When the spacecraft array is aligned, we may obtain
the gradient along the S/C line, as indicated by the formulas
(9)(10). We may check the simplest case when there is only
two S/C. Assuming the distance between them is L, the
position vectors of the two S/C in the mesocenter coordinates
are r1¼1
2L^
k1ðÞ
,r2¼1
2L^
k1ðÞ
, respectively. Then the vol-
ume tensor of the two S/C array is
R¼1
2X
2
a¼1
rara¼L
2

2
^
k1ðÞ
^
k1ðÞ
:ð12Þ
[23] The first eigen vector ^
k1ðÞis along the S/C line. The 3
eigenvalues of the volume tensor are w1¼L
2

2,w
2
=w
3
=0,
respectively.
[24] Only the gradient in the direction of the S/C line or
^
k1ðÞcan be obtained, according to the formula (9) and (10),
that is
^
k1ðÞr

B¼1
2w1X
2
a¼1
Ba~
ral¼B2B1
L;ð13Þ
where B
1
and B
2
are magnetic field measured by the two S/
C, respectively; obviously a reasonable result.
2.3. Planar Configuration Case
[25] If the spacecraft lie in a plane, the least eigenvalue of
R
jl
,w
3
equals zero, and the third eigenvector ^
k3ðÞ
is aligned to
SHEN ET AL.: TECHNIQUE A11207A11207
3of19
the normal ^
nof the spacecraft plane. Although it is impos-
sible to get the full gradient of physical quantities, we can still
obtain the gradient of physical quantities at the mesocenter in
the directions of the first eigenvector ^
k1ðÞand second eigen-
vector ^
k2ðÞfrom the equations (7)(8) and (9)(10), i.e.,
~
rlF1
Nwl1X
a
Fa~
ral;l¼1;2;ð14Þ
~
rl~
Bi1
Nwl1X
a
~
Bai~
ral;l¼1;2;ð15Þ
where the subscript cis omitted for clarity.
[26] When the spacecraft are in a plane, we are not able to
deduce the complete curl of magnetic field, but we can still
obtain the component of the curl of the magnetic field or the
current density along the normal of the spacecraft plane or
the third eigenvector ^
k3ðÞ
.
[27] The current density is the curl of the magnetic field, i.e.,
rBðÞ¼m0J:ð16Þ
The 3rd component of the current density is
m0~
J3¼^
k3ðÞm0J¼^
k1ðÞ^
k2ðÞ

m0J¼^
k2ðÞm0J

^
k1ðÞ
¼^
k2ðÞrB

^
k1ðÞ
;¼rBðÞ
^
k2ðÞ^
k2ðÞr

B
hi
^
k1ðÞ¼^
k1ðÞrBðÞ
^
k2ðÞ^
k2ðÞrBðÞ
^
k1ðÞ
:ð17Þ
The above equation is the formula for the normal component
of the current density, which may be further expressed as
m0~
J3¼~
r1~
B2~
r2~
B1¼1
Nw
1X
a
~
Ba2~
ra11
Nw
2X
a
~
Ba1~
ra2:
ð18Þ
Here, ~
ri~
Bj¼^
kiðÞrBðÞ
^
kjðÞ:The above formulas can
obviously be applied to deduce the normal component of the
current density from present THEMIS 5-point magnetic mea-
surements, where, in their string of pearls configuration during
the first operational pass, and later in the mission, often only 3
spacecraft formed a close array.
[28] For the case of the present Cluster and THEMIS
3-point plasma measurements, similar analysis may be done
with the gradient of the plasma velocity as is already carried
out for magnetic field. In the directions of the first eigen-
vector ^
k1ðÞand second eigenvector ^
k2ðÞ
, the gradients of the
plasma velocity are
~
rl~
Vi1
3wlX
a
~
Vai~
ral;l¼1;2:ð19Þ
Accordingly, the vorticity of the velocity along the normal ^
n
of the spacecraft plane or the third eigenvector ^
k3ðÞcan be
obtained as
~
W3¼~
r1~
V2~
r2~
V1¼1
Nw
1X
a
~
Va2~
ra11
Nw
2X
a
~
Va1~
ra2:
ð20Þ
The new approach developed here can yield as much useful
information on the gradient of the physical quantities for the
multiple point measurements as possible when the spacecraft
polyhedron becomes abnormal and the old method fails.
[29] In Appendix B, in order to check the correctness of the
new method, a test calculation has been made for obtaining
the normal current density from three S/C measurements in
the dipolar geomagnetic field, which has confirmed that the
new method is correct.
[30]Vogt et al. [2009] argued that, under certain con-
strains, for example, as the gradient becomes parallel or
perpendicular to a given vector, or when the stationarity
assumption is valid, the normal component of the gradient
can also be derived via the three-point analysis. Here, we
stress one common and important situation when the current
is field aligned or the force free condition is satisfied, which
has not been discussed by Vogt et al. [2009]. Under the force-
free condition, along with the solenoidal condition of the
magnetic field, our approach can also yield the normal
component of the magnetic gradient so that the full gradient
of the magnetic field is obtained. Assuming that the electric
current in the explored regions is field aligned, or the mag-
netic field is force-free; that is, the current density is J¼^
bJk,
we have
~
J3¼^
k3ðÞJ¼^
k3ðÞ^
b

Jk;Jk¼~
J3=^
k3ðÞ^
b

;ð21Þ
Then the current density is
J¼^
bJk¼^
b~
J3=^
k3ðÞ^
b

:ð22Þ
With equation (18), we get
J¼^
b1
m0^
k3ðÞ^
b

~
r1~
B2~
r2~
B1

:ð23Þ
So that the current density is completely determined when the
current is field-aligned.
[31] For 3-point S/C measurements, the gradient of the
magnetic field can be deduced in the S/C plane, as illustrated
in equation (15). Therefore, the following 6 components of
the gradient rBare already known:
~
r1~
B1;~
r1~
B2;~
r1~
B3

;and ~
r2~
B1;~
r2~
B2;~
r2~
B3

:
[32] Using the Ampere law and solenoidal condition of the
magnetic field (rB= 0), we may further determine the
other 3 component of the gradient rBalong the eigenvector
^
k3ðÞ
. That is
m0~
J1¼~
r2~
B3~
r3~
B2;ð24Þ
m0~
J2¼~
r3~
B1~
r1~
B3;ð25Þ
~
r1~
B1þ~
r2~
B2þ~
r3~
B3¼0:ð26Þ
SHEN ET AL.: TECHNIQUE A11207A11207
4of19
Then we further get
~
r3~
B1¼~
r1~
B3þm0~
J2¼~
r1~
B3þ^
k2ðÞ^
b
^
k3ðÞ^
b
~
r1~
B2~
r2~
B1

;ð27Þ
~
r3~
B2¼~
r2~
B3m0~
J1¼~
r2~
B3^
k1ðÞ^
b
^
k3ðÞ^
b
~
r1~
B2~
r2~
B1

;ð28Þ
~
r3~
B3¼~
r1~
B1~
r2~
B2:ð29Þ
where, in equation (27) and (28), the first and second com-
ponents of Jare already known, i.e., ~
J1¼^
k1ðÞJand ~
J2¼
^
k2ðÞJ, and equation (23) has been used. Thus, the gradient
of the magnetic field along the eigenvector ^
k3ðÞor the nor-
mal of the S/C plane is readily derived.
[33] When the magnetic field is non-curl or the electric
current is ignorable in the measured region, equations (27)
and (28) reduce to
~
r3~
B1¼~
r1~
B3;ð30Þ
~
r3~
B2¼~
r2~
B3;ð31Þ
So far all the components of the gradient of magnetic field
(rB) are determined, as shown in equations (15), (27), (28),
and (29), under the assumption that the electric current is
field-aligned. For the special situations when there is no or
only very weak electric current (magnetic field is non-curl),
the above equations (15), (29), (30), and (31) can readily
yield the complete gradient of the magnetic field.
[34] In Appendix C, a test calculation of a three-point S/C
crossing force-free flux rope is made. The test results have
demonstrated that our method can recover well the full
gradient of the magnetic field in the model.
[35] It is worth noting that Vogt et al. [2009], have yielded
a different formula for determining the gradient from three
S/C measurements, using reciprocal vectors. They have put
forward new kinds of reciprocals, qa¼srbg
s
jj
2a¼1;2;3ðÞ,
where sis the normal vector defined as s=r
12
r
13
, and the
gradient in the S/C plane is G¼P
3
a¼1
qaFa. We note that their
reciprocals q
a
are different from those for deducing the full
gradient from the 4-point measurements [Chanteur, 1998].
In Appendix D, we have discussed the relationship of these
two approaches. It is shown that the new method can yield
general formulae of the reciprocals for various spacecraft
arrays with any number and any shape. The uniqueness of
the reciprocal vectors has been verified. Therefore, for the
four spacecraft array forming one tetrahedron, the 4 recip-
rocal vectors as got in the new method are equivalent to
those of Chanteur [1998]. For the three spacecraft array, the
two methods are equivalent on obtaining the reciprocals as
well as the gradient.
[36] In the following Section, we will apply the new
method to several cases and explore its usefulness.
3. Applications of the New Method
[37] In order to exhibit best the value of the new method,
in this section we concentrate on cases of much distorted
tetrahedron, or planar configurations. The geocentric solar
magnetospheric (GSM) base coordinates are used throughout
this study and the spherical coordinates for the vector direc-
tion (q,8) in the frame of GSM have also been used. The
polar angle q(0q180) is the angle between the positive
Z-axis and the vector direction; the azimuth angle 8(08
360) is the angle between the positive X-axis and the line
from the vector direction projected onto the XY-plane. For
example, the dawn direction or Y-direction is (90, 270),
while the dusk direction or +Y-direction is (90,90
).
3.1. Deducing the Current Density From
Cluster 4-Point Measurements
[38] In this section, with the multipoint magnetic field
measurements of Cluster [Balogh et al.,2001],wewillcal-
culate the current density in the tail lobe region by using the
new method developed above and also compare with that
given by the old method. As known from the last Section, if
the Cluster tetrahedron is planar, the old method will fail to
yield a reasonable current density; however, the new method
can still yield the gradient of the magnetic field in the space-
craft plane and thus obtain the component of the current den-
sity along the normal of the spacecraft plane. We compare
results from both methods below for the same observations.
[39] We consider the case during the period 11:0015:00 UT
on 05 September 2005, when the average location of Cluster
tetrahedron was at X = 17.5 R
E
,Y=2.1 R
E
,Z=6.8 R
E
.
As shown in Figure 1, the Cluster array is stably located in the
southern lobe region as B
x
25 nT (Figure 1a). The mag-
netic field strengths measured by the four S/C have little dif-
ferences between them (Figure 1b). In the interval investigated,
the configuration of Cluster tetrahedron is highly distorted
because the minimum eigenvalue (w
3
) of the volumetric tensor
is much small (Figure 1c). In particular, at 13:08, the highly
distorted tetrahedron becomes planar as indicated by w
3
being
nearly zero and the dramatic change appears in the estimates of
current density.
[40] As shown in Figure 1d, if the old method is applied
(using equation (4)), the derived current density (m1
0rB)is
considerably enhanced when the distorted tetrahedron
becomes planar around 13:08 UT, that is obviously nonphys-
ical and unreal. Therefore, the old method fails to calculate the
magnetic gradients during planar S/C configuration period and
no correct information about the current density can be
obtained around this time. The reason for this is that, if the
separation of the S/C in one characteristic direction is so small
that the difference of the real magnetic field at along this
dimension is less than the absolute error of magnetic detector,
the gradient of the magnetic field along this direction cannot be
calculated correctly. Therefore, the validity of old method
requires that each of the three eigenvalues of volume tensor is
sufficiently large, i.e.,
ffiffiffiffi
wi
pdB
rB
jj
;i¼1;2and3 ð32Þ
SHEN ET AL.: TECHNIQUE A11207A11207
5of19
where, |rB| is the measure of the gradient of the magnetic
field, and dBis the absolute error of the magnetic field
which is less than 0.1 nT [Dunlop et al., 1990]. It is noted
that, equation (32) is equivalent to (A11) in Appendix A.
However, if the critical condition (32) is violated, i.e.,
ffiffiffiffi
wi
pdB
rB
jj
,i= 1, 2 or 3, the old method will totally fail, and
the new method should be applied to draw useful information.
[41] Furthermore, we may re-examine this case with the
new method, which expresses the magnetic gradients in the
eigenspace of volume tensor as in equation (15), so that the
current density can be obtained in that eigenspace with
Figure 1. From top to bottom: (a) the magnetic field at the mesocenter of Cluster tetrahedron; (b) the
strength of magnetic field for Cluster four-S/C respectively; (c) the square roots of the three eigenvalues
of the volumetric tensor; (d) the current density calculated by the old method; (e) the component of current
density along ^
k1ðÞand the directional angles (q
1
,j
1
) of the first eigenvector ^
k1ðÞ
; (f) the component of cur-
rent density along ^
k2ðÞ
and the directional angles (q
2
,j
2
) of the second eigenvector ^
k2ðÞ
; (g) the component
of current density along ^
k3ðÞand the directional angle (q
3
,j
3
) of the third eigenvector ^
k3ðÞ
; (h) the com-
parison of the total current density between the old method and new method.
SHEN ET AL.: TECHNIQUE A11207A11207
6of19
equation (18) (Figures 1e1g). It is noted that the directions
of eigenvectors ^
k1ðÞ
,^
k2ðÞ
, and ^
k3ðÞare approximately along
the directions of Y, Z and +X axis, respectively, for the
whole interval. Around 13:08, although the derived J
1
(along ^
k1ðÞ
) and J
2
(along ^
k2ðÞ
) components are considerably
enhanced and obviously unreal, we still can reasonably
derive the J
3
(along ^
k3ðÞ
) component. It can be seen from
Figure 1g that the earthward component of current density
(J
3
) is very small and nearly zero, which is in consistence
with the traditional understanding on the current distribution
in the tail lobe region. On contrast, we cannot obtain any
component of the current density in this interval using the
old method (Figure 1d).
[42] However, when the above critical condition (32) is
satisfied, the old method and new method will yield the same
results, as shown in Figure 1h. Actually, throughout the whole
interval, the strength of total current density derived from the
old method and new method is the same (Figure 1h). Around
13:08, both methods yield unreal values of the strength of the
current density although the new method yields correct com-
ponent J
3
.
3.2. Deducing the Current Density From THEMIS
Three-Point Measurements
[43] THEMIS mission [Angelopoulos, 2009; Auster et al.,
2009], which are composed of five identical spacecraft and
have been launched into space on 17 Feb 2007, aims to
explore the global large scale evolution processes of the
magnetospheric substorms. During the early phase (in 2007)
of the mission, the spacecrafts were placed into a close string
of pearlsconfiguration. From Sep 2007 through Sep 2009,
the outmost two spacecraft are far distant from the other three
space-craft. In addition, since 29 Dec 2009, the outside two
spacecraft (P1 and P2) have been placed near the lunar orbit,
and the remaining three spacecraft (P3, P4 and P5) have
separations of about 5003000 km. Therefore, we may apply
the new method to three point (P3, P4 and P5) measurements
and deduce the component of the current density along the
normal of the S/C plane from the 3-point magnetic field
measurements of THEMIS from equation (18).
[44] For illustration, we may make analysis on one case on
29 May 2010 when the THEMIS array is inbound from the
near-Earth tail (radial distance 11 R
E
) to the inner mag-
netosphere and at the same time a moderate geomagnetic
storm (the minimum Dst index is 85 nT) occurs. The
orbits of the three THEMIS spacecraft (P3, P4 and P5)
during the period 00:0014:00 UT on 29 May 2010 are
demonstrated in Figure 2 where the separations of the three
S/C are amplified by a factor of five for the resolution.
[45] The measured magnetic field vector at the mesocenter
of the three S/C array, i.e., Bc¼1
3P
3
a¼1Ba, and the magnetic
strength measured by each S/C are shown in Figures 3a and
3b, respectively. For the interval 00:0012:30, as indicated
by the B
z
component (Figure 3a, in blue) being rather small,
Figure 2. THEMISorbit (blue line for P5) during 03:0014:00 UT on 29 May 2010, (a) projected in the
X-Y plane, and (b) projected in the Y-Z plane in GSM coordinates. In both panels, the separated size
between P3 (red dots), P4 (green dots) and P5 (blue dots) are amplified by a fact of 5. The nominal mag-
netopause with standoff distance r
0
= 10 RE and tail flaring level a= 0.5 provided by Shues model [Shue
et al., 1997] is shown as dashed lines.
SHEN ET AL.: TECHNIQUE A11207A11207
7of19
Figure 3. From top to bottom: (a) the magnetic field at the mesocenter of the spacecrafts plane; (b) the
strengths of magnetic field at tha(P5), thd(P3) and the(P4), respectively; (c) the position of the mesocenter
in GSM spherical coordinates; (d) the square roots of the three eigenvalues of the volumetric tensor of P5,
P3 and P4; (e) the directional angle of the normal to the spacecraft (P5, P3 and P4) plane, i.e., the direction
of ^
k3ðÞ
; (f) the component of the current density along the normal of S/C plane; (g) the index of AE and
SYM_H.
SHEN ET AL.: TECHNIQUE A11207A11207
8of19
the three S/C are in the stretched magnetotail. Particularly
for the short interval 10:5012:30, the total field reaches the
minimum and B
x
component reverses the sign, the three S/C
are crossing the inner plasma sheet. Then, for the interval
12:3014:00, the three S/C transit from the tail region to the
dipolar field region as indicated by the gradually growing B
z
component and the decreasing radial distance (Figure 3d).
[46] As the three THEMIS S/C are coplanar, the volume
tensor is abnormal and one eigenvalue is zero, so that the old
method (equation (4)) is not fitful for analysis as indicated by
the violation of the criteria (32). Nevertheless, the new method
is still applicable for obtaining the component of the current
density along the normal of the S/C plane (equation (18)).
[47] During the whole interval of 00:0014:00 UT, as
indicated by the eigenvalues of volume tensor in Figure 3d,
the separation of the three S/C is generally less than 3000 km.
By using the new method (equation (18)), the component of
current density along the normal of the three-S/C plane
(Figure 3e), i.e., J
3
, is calculated (Figure 3f). As it happens, in
this case, a moderate geomagnetic storm occurs as indicated
by the SYM_H index (Figure 3g) and the minimum SYM_H
index is about 75 nT. The investigated period spans the
whole main phase of the magnetic storm, and there is per-
sistent substorm activities with the maximum AE index being
about 1800 nT. In the initial stage during 00:0006:00 UT of
the main phase, the three THEMIS S/C are operating around
their apogee in the near-Earth tail lobe region (r 11 R
E
,B
x
50 nT), and the eigenvector ^
k3ðÞ
, that is also the normal to
the S/C plane, is about along the direction of positive Yor
duskward (Figure 3e). The calculated J
3
, which is about the
duskward component of the current density, is constantly
around 3 nA/m
2
with small fluctuations (Figure 3f). In the
period of 08:0014:00 UT, which is the most active stage of
the main phase of geomagnetic storm, the three S/C move
into the inner magnetosphere while their radial distance is
decreasing from 10 R
E
to 5R
E
(Figure 3c). During this
stage, the eigenvector ^
k3ðÞis roughly along the direction of
negative Yor dawnward (Figure 3e), and the corresponding
J
3
derived was duskward or westward with great enhance-
ment. It is noted that, the enhanced J
3
reaches the maximum
value 25 nA/m
2
, for the short interval of 10:5012:30
when THEMIS are in the center part of inner plasma sheet.
[48] One may argue that, the enhanced J
3
is possible induced
by the spatial effect, i.e., S/C moves from one region to the other
region, instead of the result induced by the magnetic storm. To
check this argument, the THEMIS orbits before or after the case
for several days are also surveyed, but similar trend of enhanced
J
3
is not observed for those orbits. Therefore, the enhancement
of the westward J
3
for this case is directly driven by the mag-
netic storm. So that it is yielded that, during the main phase of
this geomagnetic storm, the electric current density enhances
greatly in the inner plasma sheet. It is unclear about the prop-
erties of the inner plasma sheet with strong current density, and
the roles it would play in the evolution of magnetic storms.
Theseissuesdeservetobeexploredinthefuture.
[49] Anyway, the enhancement of the westward J
3
obtained from THEMIS 3-point measurements during geo-
magnetic storm is consistent with the traditional picture of
well-known partial ring current, and also consistent with the
statistical observation of ring current as obtained from
Cluster 4-point analysis [Zhang et al., 2011].
[50] In this section, it has been confirmed that the new
method developed here may be used to derive the current
distribution in the near earth tail region, as well as in the ring
current region, based on THEMIS 3-point magnetic field
measurements.
3.3. Deducing the Vorticity of Plasma Flows From
THEMIS Three-Point Measurements
[51] We may also apply this method to obtain the gradient
of the velocity of plasma flows. Particularly for the three-
point observations of plasma flow, with the formula (19) and
(20) the gradients of flow in the plane of S/C cluster can be
calculated, and further the component of flow vorticity along
the normal of S/C plane can be derived. This is beyond the
ability of the old methods.
[52] One case analysis on THEMIS observations will be
made to show the usefulness of the method developed above.
During the period 10:2210:32 UT on 17 March 2008, the
five S/C of THEMIS are located in the near-Earth magneto-
tail with geocentric distances of about 8 R
E
to 14 R
E
. With
THEMIS simultaneous measurements of plasma flows for
this case, Panov et al. [2010] have claimed that the earthward
or tailward flow bursts can lead to the formation of flow
vortices. To check this argument of Panov et al. [2010] and
also to exhibit the ability of the new method developed
here, we may re-examine quantitatively this case with the
three-S/C measurements of THEMIS mission.
[53] During this interval, the locations of the three-S/C of
THEMIS in GSM are as follows: P1 at (12.7, 3.3, 0.2) R
E
,
P2 at (11.1, 2.7, 1.2) R
E
, and P4 at (10.2, 3.3, 1.6) R
E
.
The data from ESA (the reduced mode with 3-s resolution)
[McFadden et al., 2009] have been used to obtain the plasma
velocity in the frame of GSM coordinates. The plasma velocity
for P1, P2 and P4 are shown in Figures 4a4c, respectively. In
these panels, the V
x
,V
y
components and the total speed are
presented by the blue line, green line and black line, respec-
tively. As indicated by the eigenvalues of the volume tensor,
the separation size of these S/C exceeds 1 R
E
(Figure 4d). The
normal direction of the S/C plane, i.e., ^
k3ðÞ
(150,174
), points
roughly toward the southward (Figure 4e). With equation (20),
the flow vorticity along the direction of ^
k3ðÞ
,i.e.,W
3
,isesti-
mated and shown in Figure 4f. It can be seen from Figure 4f
that, the derived W
3
is about always negative, implying the
vorticity ofthe flow velocity has a significant component at the
northern direction.
Figure 4. From top to bottom: (a) the velocity of plasma flow of P1; (b) the velocity of plasma flow of P2; (c) the velocity of
plasma flow of P4; (d) the square roots of the three eigenvalues of the volumetric tensor; (e) the directional angles of the nor-
mal to the spacecraft plane, i.e., the direction of third eigenvector ^
k3ðÞ
; (f ) the component of the flow vorticity along the normal
of S/C plane (^
k3ðÞ
). The four vertical black lines mark the times when the calculated vorticity W
3
reach the maximum values.
For the lower part, the orientations of plasma flows at the locations of the three S/C are showed in the XY plane corresponding
to the marked times, where, P1, P2 and P4 are labeled as red, green and blue dots, respectively.
SHEN ET AL.: TECHNIQUE A11207A11207
9of19
Figure 4
SHEN ET AL.: TECHNIQUE A11207A11207
10 of 19
Figure 5. Evolution of the K-H waves observed by Cluster S/C (C1, C3 and C4) during the period
20:2620:42 UT on 20 Nov. 2001. From top to bottom: (a) the magnetic field at the mesocenter of triangle
plane of C1, C3 and C4; (b) the B
z
component of magnetic field measured by the three S/C; (c) the V
x
com-
ponent of CODIF H
+
flows measured by the three S/C; (d) the V
y
component of CODIF H
+
flows mea-
sured by the three S/C; (e) the V
z
component of CODIF H
+
flows measured by the three S/C; (f ) the
square roots of the three eigenvalues of the volumetric tensor; (g) the directional angles of the normal
to the spacecraft plane, i.e., ^
k3ðÞ
; (h) the component of current density along ^
k3ðÞ
; (i) the component of flow
vorticity along ^
k3ðÞ
. Five vertical black lines mark the time when the calculated J
3
reaches the extremes.
SHEN ET AL.: TECHNIQUE A11207A11207
11 of 19
[54] The four vertical black lines in Figure 4 mark the times
when W
3
reaches its extreme. To check whether the derived
W
3
is reasonable, we have plotted the orientations of plasma
flows at the locations of the three S/C in XY plane at the four
marked times in the lower part of Figure 4. It can be clearly
seen from all the four plots that, the orientations of plasma
flows at the locations of the three S/C consistently satisfy the
expected swirl of vortex (denoted as the black arrowhead
circles) which has a northward flow vorticity. This means
that the derived W
3
by the new method is reasonable and can
be the estimation of the component of flow vorticity along
the normal of S/C plane.
3.4. Deducing the Vorticity of Plasma Flows From
CLUSTER Three-Point Measurements
[55] In contrast to the four-point measurements of mag-
netic field, Cluster mission can only obtain 3-point observa-
tions of the plasma measurement due to the failure of the CIS
onboard C2 [Rème et al., 2001]. It is impossible, therefore,
to derive the plasma-related gradients by the previous methods.
With the method developed in this study, we can obtain the
plasma-related gradients in the plane of three-S/C cluster as
having been carried out in section 3.3. In this section, to
further show such unique ability of the new method, we apply
it to the analysis of Cluster plasma data.
[56] Due to the strong flow shear at the flank magneto-
pause, the Kelvin-Helmhotz (K-H) instability can be driven
by the down-streamed magnetosheath flow as manifested by
the rolled-up K-H waves. One well-known case of K-H
waves observed by Cluster is the event occurred during the
period of 20:2620:42 UT on 20 Nov. 2001, which has been
previously investigated by Hasegawa et al. [2004]. Here, we
may apply the new method to re-examine this case, so that
some quantitative features of K-H waves can be revealed.
[57] In this case, Cluster is at the dusk flank of the mag-
netosphere, and the average location of Cluster tetrahedron is
at (3.6, 18.7, 2.7) R
E
in GSM. As shown in Figure 5, the
B
z
component of the magnetic field is enhanced periodically
while the magnetic strength has only small variations
(Figures 5a and 5b). The plasmas are flowing mainly anti-
sunward (Figures 5c5e). It is well known that, within the
magnetopause, the plasma temperature is higher and the
density is smaller, while in the magnetosheath the opposite is
true with down-streaming sheath flows. So repetitive varia-
tions of magnetospheric-like plasmas to magnetosheath-like
plasmas [see Hasegawa et al., 2004, Figure 2], as well as the
anti-sunward plasma flow, demonstrates that the Cluster
spacecraft are located in the low latitude boundary layer
region and crossing the flank magnetopause repetitively. The
Figure 6. Variations of the plasma properties of the K-H waves observed by Cluster S/C (C1, C3 and
C4) during the period 20:2620:42 UT on 20 Nov. 2001. From top to bottom, the B
z
component of mag-
netic field, the density of CODIF H
+
, and the plasma temperature, respectively. Five vertical black lines
mark the time when the calculated J
3
reaches the extremes.
SHEN ET AL.: TECHNIQUE A11207A11207
12 of 19
separation size between C1, C3, and C4 is about 1400 km
(Figure 5f), much less than the width of the K-H vortex and
the thickness of the boundary layer [Hasegawa et al. 2004].
Therefore, such three-point measurements can be well used
to determine the local gradients of the magnetic field and
plasma moments with the new method. The direction of ^
k3ðÞ
,
i.e., the normal of the three S/C plane, is (48.2, 6.0),
pointing about northward (Figure 5g). By using equation (18)
and equation (20), the component J
3
of the current density
and the component W
3
of the plasma flow (CODIF H
+
) vor-
ticity along ^
k3ðÞ are derived, which are illustrated in
Figures 5h and 5i, respectively.
[58] As shown from Figure 5 by the marked vertical black
line, both the derived J
3
and W
3
are periodically pulse-
enhanced simultaneously along with the jumps of B
z
. Based
on a detailed check of the plasma density and temperature as
shown in Figure 6, it is found that, the pulse-enhanced J
3
and W
3
occurred when S/C transits from magnetopause to
magnetosheath (see Table 1). Such periodical pulse-
enhanced flow vorticity can be regarded as a direct indictor
of the rolled-up K-H waves. In addition, the wave period
implied by Table 1 is about 34 min, and the detected
sheath flow is about 220 km/s, so the wavelength can
roughly be estimated as 68R
E
.
4. Summary and Discussions
[59] Multiple spacecraft measurements have increasingly
become the mainstream manner of space exploration. The 4
and 5 Cluster and THEMIS spacecraft have been successfully
launched into orbits, operating satisfactorily for more than a
decade in the case of Cluster. Fruitful results have been
achieved from the multipoint observation data, but the
methodology applied depends on the close arrays of space-
craft achieved. In the near future, Swarm (3 S/C) and MMS
(4 S/C) will also been launched into the magnetospheric
space. The satellite configurations of these missions are
diverse. The existent methods for multiple-S/C data analysis
are primarily defined for determining the full gradients from
the four-point measurements of the phased Cluster array in a
regular tetrahedron configuration.
[60] In this research, we have deduced the gradients in the
coordinates of the eigenvector space of the volume tensor of
satellite cluster. This new approach cannot only be applied to
the analysis of data from a S/C cluster with the regular tet-
rahedron configuration, as in the previous methods, but also
can be used successfully to draw the gradients from the data
observed by S/C cluster with the irregular (distorted) con-
figurations, e.g., in a planar or linear configuration. If the S/C
cluster is planar, or there is only 3 S/C, the gradients of the
physical quantities in the S/C plane can still be obtained,
although the gradient along the normal of the S/C plane is not
available; and furthermore, the component of the curl of
magnetic field along the normal of the spacecraft plane can
be deduced. If the S/C cluster is aligned, only the gradient
along the S/C line can be determined. It is also shown that,
under the force free assumption along with the divergence
free condition of the magnetic field, the full magnetic gradi-
ent can be obtained based on three-spacecraft magnetic
measurements. In Appendix C, a test calculation of three-
point S/C crossing force-free flux rope is made, which has
confirmed the new approach provides high accuracy.
[61] To demonstrate the abilities of the new approach
developed in this research, four case analyses have been
carried out. First we have studied the tail current density
calculation based on the 4 point magnetic field data of Cluster
with abnormal tetrahedron. Even if the Cluster tetrahedron
becomes planar, the component of the current density along
the normal of the S/C plane can be deduced and here its value
is almost zero in the tail lobe. In the second case, with the
3-point THEMIS data, we try to deduce the component of the
current density along the normal of the THEMIS 3-S/C
plane, and the enhanced near-earth duskward or westward
current density are readily yielded in the main phase of a
geomagnetic storm. In the last two cases, we have investi-
gated the flow vorticity determinations with 3-point obser-
vations. In the third case, with THEMIS 3-point plasma
measurements, the component of the flow vorticity along the
normal of the S/C plane has been estimated and the derived
results are reasonable. Last, in the fourth case, by using the
new method, we have quantitatively investigated the varie-
ties of the vortices created by the severe K-H instability at the
dusk flank of the magnetopause. Therefore, one component
of the current density has been obtained based on 3 S/C
magnetic field measurements for the first time; and also, one
component of the flow vorticity has been first obtained based
on 3-satellite plasma measurements.
[62] The new method can find more applications on data
analysis for Cluster, THEMIS and Swarm missions. For
Cluster, we may calculate the gradient of the magnetic field in
the inner magnetosphere as the spacecraft tetrahedron has
abnormal configurations. And furthermore, with the Cluster
C1, C2 and C4 CIS measurements, the gradient of the flow
velocity in the 3 S/C plane and the component of the vorticity
along the normal to the S/C plane can readily be deduced in
the regions with plasma flows. For THEMIS, the new method
may find similar usages to those for Cluster, e.g., it may be
applied to deduce the ring current and field aligned current
distributions in the inner magnetosphere. As for Swarm
mission (planned to be launched on July 2012), only if the
three S/C are sufficiently near to each other, the gradient of
the magnetic field in the 3 S/C plane can be determined, and
further one component of the current density can also be
obtained, as well as the field aligned current density.
Appendix A: The Error of the Method
[63] Considering equations (7)(8) for calculating the
gradient of a scalar physical quantity, the relative error of the
gradient of Fin direction of the l
th
eigenvector is
d~
@lF
~
@lF
dPaFa~
ral
PaFa~
ral
þdwl
wl
:ðA1Þ
Table 1. The Data of Pulse-Enhanced J
3
and W
3
Time J
3
(nAm
2
)W
3
(s
1
)
20:28:11 4.9 0.041
20:30:59 7.5 0.085
20:34:59 7.3 0.076
20:38:27 8.2 0.048
20:40:55 2.9 0.088
SHEN ET AL.: TECHNIQUE A11207A11207
13 of 19
The error of PaFa~
ralis
dX
a
Fa~
ral
jj
¼X
a
Fa
jj
d~
ral
jj
þ~
ral
jj
dFa
jj
ðÞ
¼X
a
Fa~
ral
jj
d~
ral
jj
~
ral
jj
þdFa
jj
Fa
jj

X
a
Fa~
ral
jj
dr
ffiffiffiffi
wl
pþdF
Fjj

dr
ffiffiffiffi
wl
pþdF
F
jj

NF ffiffiffiffi
wl
p;ðA2Þ
Where, Fis the typical value of the scalar physical quantity,
d
r
is the error of the position of the S/C, and Nis the number
of S/C. The relative error
dPaFa~
ral
PaFa~
ral
1
Nwl~
@lF
NF ffiffiffiffi
wl
pdr
ffiffiffiffi
wl
pþdF
F

D
ffiffiffiffi
wl
pdr
ffiffiffiffi
wl
pþdF
F

;ðA3Þ
Where, Dis the characteristic size of the structure, and dFis
the error of the measurement of the physical quantity.
Because the eigenvalues may be written as
wl¼1
NX
N
a
~
ral
ðÞ
2;ðA4Þ
So the errors of the eigenvalues are
dwl¼1
NX
N
a
2~
ral
jjd~
ral1
NX
N
a
2ffiffiffiffi
wl
pdr¼2ffiffiffiffi
wl
pdr:ðA5Þ
Then the relative errors the eigenvalues are
dwl
wl2dr
ffiffiffiffi
wl
p:ðA6Þ
Therefore, the total relative error is
d~
@lF
~
@lF
dPaFa~
ral
PaFa~
ral
þdwl
wlD
ffiffiffiffi
wl
pdr
ffiffiffiffi
wl
pþdF
F

þ2dr
ffiffiffiffi
wl
p
¼2þD
ffiffiffiffi
wl
p

dr
ffiffiffiffi
wl
pþD
ffiffiffiffi
wl
p
dF
F:ðA7Þ
The above formula will pose restraints on the measurement
errors of the S/C positions and the physical quantities. In
order to yield accurate results, i.e., d~
@lF
jj
~
@lF
jj
1, it requires that
2þD
ffiffiffiffi
wl
p

dr
ffiffiffiffi
wl
p1;ðA8Þ
and
D
ffiffiffiffi
wl
pdF
F1:ðA9Þ
Furthermore, the restraints on drand dFare
dr
ffiffiffiffi
wl
pffiffiffiffi
wl
p
2ffiffiffiffi
wl
pþDffiffiffiffi
wl
p
D;ðA10Þ
dF
Fffiffiffiffi
wl
p
D:ðA11Þ
Here, it has been assumed that ffiffiffiffiffi
wl
pD.
[64] On the other hand, if
dr
ffiffiffiffi
wl
pffiffiffiffi
wl
p
D;ðA12Þ
or
dF
Fffiffiffiffi
wl
p
D;ðA13Þ
the errors are too large that we cannot effectively deduce the
gradient of the physical quantity F. E. g., when the space-
craft array is growing planar, the third eigenvalue w
3
is so
small that
dF
Fffiffiffiffiffi
w3
p
D;ðA14Þ
then the gradient at the direction of the third eigenvector ^
k3ðÞ
cannot be obtained with reasonable accuracy. This can be
illustrated in section 3.1.
[65] Furthermore, if the spacecraft array is becoming lin-
ear, the second eigenvalue w
2
is so small that
dF
Fffiffiffiffiffi
w2
p
D;ðA15Þ
the gradient along the directions ^
k2ðÞ and ^
k3ðÞ cannot be
deduced, and only the gradient along the first eigenvector
^
k1ðÞcan be obtained by the new method.
[66] Therefore, the equations (A14) and (A15) define
when the spacecraft array has an abnormal configuration, or
when the spacecraft array has to be regarded as planar and
linear, respectively.
Appendix B: Test on the New MethodThree S/C
Observations on the Dipolar Magnetic Field
[67] We assume the three S/C array are measuring the
geomagnetic field as illustrated in Figure B1. The geomag-
netic field is approximated as dipolar one. The orbit of the
three S/C are regular circles with the same geocentric dis-
tance R. The three S/C constitute one regular triangle.
[68] The volume tensor is
R¼1
3X
3
a¼1
rara
¼1
3"d2^
q^
qþffiffi
3
pd
2^
fþd
2^
q

ffiffi
3
pd
2^
fþd
2^
q

þffiffiffi
3
pd
2^
fþd
2^
q

ffiffi
3
pd
2^
fþd
2^
q

#
¼d2
2^
q^
qþ^
f^
f

;ðB1Þ
SHEN ET AL.: TECHNIQUE A11207A11207
14 of 19
So that, the eigenvectors: ^
k1ðÞ¼^
q,^
k2ðÞ¼^
f,^
k3ðÞ¼^
r, the
eigenvalues: w1¼w2¼d
2
2,w
3
= 0. With the formula (14),
the components of the magnetic field gradient
~
r1~
B2¼1
3w1X
3
a¼1
~
Ba2~
ra1¼1
3w1
~
Baf~
raqþ~
Bbf~
rbqþ~
Bcf~
rcq

;
ðB2Þ
~
r2~
B1¼1
3w2X
3
a¼1
~
Ba1~
ra2¼1
3w2
~
Baq~
rafþ~
Bbq~
rbfþ~
Bcq~
rcf

:
ðB3Þ
[69] It is noted that
~
raf¼ffiffi
3
p
2d;~
rbf¼ ffiffi
3
p
2d;~
rcf¼0;
~
Baf¼~
Bbf¼~
Bcf¼0;
~
Baq¼MR3sin qþd
2R

;~
Bbq¼MR3sin qþd
2R

;
then
~
r1~
B2¼~
r2~
B1¼0:ðB4Þ
Therefore, from equation (18), the normal component of the
current density is
m0~
J30:ðB5Þ
So that the new method yields correct normal current
density.
Appendix C: Test on the New MethodThree S/C
Observations on the Force-Free Flux Rope
[70] As shown in Figure C1, we assume that a three
spacecraft array is crossing a stationary, force-free flux rope.
The spacecraft array is taken to be a regular triangle shape
with its side length being 0.1 R
E
(amplified by a factor of 5
in Figure C1). The flux rope can be described by the well-
known Lundquist-Lepping (L-L) model [Lundquist, 1950],
which is a particular solution of the force-free field condition
rB=aBwith the assumption of cylindrical symmetry.
The magnetic field in the force-free flux rope of the L-L
model can be expressed as
Bz¼B0J0arðÞ;Bj¼B0J1arðÞ:ðC1Þ
Figure B1. One ideal situation for three S/C measurements on the dipolar geomagnetic field. (left) The
orbits of the three S/C and (right) the configuration and orientation of the three spacecraft. The spherical
coordinates system is used. The center of the Earth is the origin, and ^
r,^
qand ^
fare the unit radial, polar
and azimuthal vectors, respectively.
SHEN ET AL.: TECHNIQUE A11207A11207
15 of 19
For the test, we arbitrarily adopt the parameters B
0
= 20 nT,
a¼1RE1. This can also be expressed in Cartesian coor-
dinates as
Bz¼B0J0arðÞ;Bx¼B0J1arðÞsinj;By¼B0J1arðÞcosjðC2Þ
Where, jis the azimuthal angle and ris the radial distance.
[71] The three-S/C array crosses the LL flux rope in a
straight line with a constant velocity. The mesocenter,
labeled by the asteroid marker, has a parallel motion from
the point (x = 0.55, y = 1.9711, z = 0) R
E
to the point (x =
0.55, y = 1.5289, z = 1) R
E
with a time interval 500 s. The
resolution of the magnetic field measurement is taken to be
1-s.
[72] The test results are shown in Figure C2, wherein the
thicker red lines and the thinner blues line in panels b to j are
the results based on equations (15), (27), (28), and (29) and
the exact analytic results at the mesocenter, respectively.
Obviously, the full derived magnetic field gradient compo-
nents are well consistent with the exact analytic results. As
shown in Figure C2, the absolute error of the components of
the calculated magnetic gradient from the exact ones is less
than about 0.2 nT/R
E
. In this case, the characteristic size of
the spacecraft array is L¼2ffiffiffiffiffi
w1
p0:08RE, the characteris-
tic spatial scale of the flux rope is D1/a=1R
E
, the typical
magnetic gradient is about aB
0
= 20 nT/R
E
. Thus the trun-
cation error is at the order of (L/D)aB
0
1.6 nT/R
E
.
Therefore, the errors in the calculated components of the
magnetic gradient are well within the truncation errors. The
approach put forward in this paper therefore has the ability
to recover the full gradient components of magnetic field
with good accuracy based on three-point measurements and
if the current is field-aligned.
Appendix D: Discussion on the Relationship
Between the New Approach and the Reciprocal
Vector Method
[73] We consider the situation when only 3 satellites make
observations. Vogt et al. [2009] yielded the reciprocal vec-
tors as
qa¼srbg
sjj
2;ðD1Þ
where, r
bg
=r
g
r
b
,s=r
12
r
13
. Note that s¼s^
k3ðÞ
.
The value of sis twice the area of the triangle of the three
satellites. If the three spacecraft are aligned, sis zero, and
Vogt et al. [2009] formula (D1) fails.
[74] For the three satellite situation, we may consider only
the two dimensions in the spacecraft plane. The spatial
scales in the two independent directions ^
k1ðÞ and ^
k2ðÞ are
ffiffiffiffiffi
w1
pand ffiffiffiffiffi
w2
p, respectively. Thus, the area s/2 of the triangle
of the three satellites is proportional to ffiffiffiffiffiffiffiffiffiffi
w1w2
p. Similarly to
the deduction in the paper of Harvey [1998], it is readily
obtained that
s¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
27w1w2
p:ðD2Þ
The coefficient in the above formula (D2) is easy verified as
shown in Appendix B in the situation of the regular triangle
spacecraft array.
[75] From the newly proposed approach in this paper, it is
straightforward to get the formulae of the reciprocal vectors
for various situations when the multiple spacecraft arrays
have shapes of either tetrahedrons, planes or lines.
[76] From section 2.1, we may get the gradient of the
quantity Ffrom the observations of N spacecraft as
rF¼~
rlF

^
klðÞ 1
NwlX
N
a¼1
Fa~
ral^
klðÞ
;ðD3Þ
Considering the gradient rF¼PN
a¼1Faqa, we may get the
formula of the N reciprocal vectors accordingly as
qa¼X
3
l¼1
1
Nwl
~
ral^
klðÞ
;a¼1;2;;N:ðD4Þ
The reciprocal vectors q
a
(a=1,2,,N) are determined by
the structure and orientation of the spacecraft array.
[77] There is one problem if there are other sets of recip-
rocal vectors for deducing the gradient rF. Here we may
verify that the uniqueness of the reciprocal vectors for a
spacecraft array. Assume that there are two different sets of
reciprocal vectors, q
a
(1)
(a=1,2,,N) and q
a
(2)
(a=1,2,,N),
for the same spacecraft array. Then
rF¼X
N
a¼1
Faq1ðÞ
a;
rF¼X
N
a¼1
Faq2ðÞ
a:
Figure C1. The three-S/C array is crossing the LL flux
rope in straight trajectory (thin black line). For this LL flux
rope, the characteristic magnetic field is B
0
= 2 nT and the
force free factor a¼1RE1. The three-S/C array, labeled
as the colored squares, constitutes one regular triangle with
side length 0. One R
E
(amplified by a factor of 5 in the plot),
The mesocenter is denoted by asteroids, which moves in a
straight line from (x = 0.55, y = 1.9711, z = 0) R
E
to
(x = 0.55, y = 1.5289, z = 1) R
E
with a time interval of
500 s.
SHEN ET AL.: TECHNIQUE A11207A11207
16 of 19
[78] The difference of the above two equations is
0¼X
N
a¼1
Faq1ðÞ
aq2ðÞ
a

:
Because F
a
(a=1,2,,N) may be arbitrary, it is obvious
that q1ðÞ
aq2ðÞ
a¼0, or q1ðÞ
a¼q2ðÞ
a. Therefore, the reciprocal
vectors for a spacecraft array are unique. For the four
spacecraft array, the reciprocal vectors q
a
(a=1,2,,4)as
shown in (D4) are equivalent to those in Chanteur [1998].
[79] If the spacecraft array becomes planar, w
3
equals
zero. Then the gradient in the spacecraft plane may be
expressed as
rFðÞ
P¼X
2
l¼1
~
rlF

^
klðÞ 1
Nw1X
N
a¼1
Fa~
ra1^
k1ðÞþ1
Nw2X
N
a¼1
Fa~
ra2^
k2ðÞ
;
ðD5Þ
and the corresponding N reciprocal vectors are
qa¼1
Nw1
~
ra1^
k1ðÞþ1
Nw2
~
ra2^
k2ðÞ
;a¼1;2;;N:ðD6Þ
Figure C2. Comparison between the test results and the exact analytic results. (a) The time series of the
measured magnetic field strength and the Bx, By and Bz components along the trajectory of the mesocen-
ter. (bj) The components of the magnetic gradient
x
B
x
,
y
B
x
,
z
B
x
,
x
B
y
,
y
B
y
,
z
B
y
,
x
B
z
,
y
B
z
and
z
B
z
, respectively, along the trajectory of the mesocenter, where, the thicker red lines illustrate the results
inferred from the new method and the thinner blue lines demonstrate the exact analytic results.
SHEN ET AL.: TECHNIQUE A11207A11207
17 of 19
[80] As demonstrated above, for the case of three space-
craft array, the reciprocal vectors q
a
(a= 1, 2, 3)from the
new approach as expressed by (D6) are equivalent to the
formula (D1) of Vogt et al. [2009]. Especially, for the situ-
ation when the three satellites form a regular triangle as
shown in Figure B1, applying the formula (D6) based on our
method or the formula (D1) of Vogt et al. [2009], we may
get the same reciprocal vectors as the follows.
qa¼2
3dra¼2
3d
1
2^
qþffiffi
3
p
2^
f

;ðD7Þ
qb¼2
3drb¼2
3d
1
2^
qffiffi
3
p
2^
f

;ðD8Þ
qc¼2
3drc¼2
3d
^
q:ðD9Þ
Obviously, in Appendix B, the two methods will yield the
same magnetic gradient with the assumed spacecraft array
configuration.
[81] Therefore, from the new method proposed in this
paper, we may obtain the general formulae of the reciprocal
vectors for various spacecraft arrays with any number and
any shape.
[82]Acknowledgments. This work was supported by Ministry of
Science and Technology of China Grant 2011CB811404, the National
Natural Science Foundation of China grants 40921063, 41231066, 40974101
and 41104114, China Postdoctoral Science Foundation Funded Project
(20100480446, 2012T50132), Chinese Academy of Sciences (CAS) visiting
Professorship for senior international scientists grant 2012T1G0018, and the
Specialized Research Fund for State Key Laboratories of the CAS. The authors
are thankful to Cluster II FGM team and ESA Cluster Active Archive for
providing Cluster data, appreciate the THEMIS team for providing the public
THEMIS scientific data, and also thank Q. H. Zhang for the valuable
suggestions.
[83]Masaki Fujimoto thanks the reviewers for their assistance in eval-
uating the paper.
References
Angelopoulos, V. (2009), The THEMIS Mission, in The THEMIS Mission,
edited by J. L. Burch and V. Angelopoulos, pp. 534, Springer,
New York, doi:10.1007/978-0-387-89820-9_2.
Auster, H. U. et al. (2009), The THEMIS Fluxgate Magnetometer, in The
THEMIS Mission, edited by J. L. Burch and V. Angelopoulos, pp. 235264,
Springer, New York, doi:10.1007/978-0-387-89820-9_11.
Balogh, A., et al. (2001), The Cluster magnetic field investigation: Over-
view of in-flight performance and initial results, Ann. Geophys.,19,
12071217, doi:10.5194/angeo-19-1207-2001.
Bryant, D. A., S. M. Krimigis, and G. Haerendel (1985), Outline of the
Active Magnetospheric Particle Tracer Explorers (AMPTE) mission,
IEEE Trans. Geosci. Remote Sens.,GE-23, 177181, doi:10.1109/
TGRS.1985.289511.
Chanteur, G. (1998), Spatial Interpolation for four spacecraft: Theory, in
Analysis Methods for Multi-spacecraft Data, edited by G. Paschmann
and P. W. Daly, pp. 349370, ESA Publ. Div., Noordwijk, Netherlands.
Chanteur, G., and C. C. Harvey (1998), Spatial Interpolation for four space-
craft: Application to magnetic gradients, in Analysis Methods for Multi-
spacecraft Data, edited by G. Paschmann and P. W. Daly, pp. 371394,
ESA Publ. Div., Noordwijk, Netherlands.
Curtis, S. A. (1999), The magnetospheric multiscale mission: Resolving
fundamental processes in space plasmas, NASA Tech. Memo. NASA/
TM-2000-209883, NASA, Greenbelt, Md.
Dunlop, M. W., and T. I. Woodward (1998), Multi-spacecraft discontinuity
analysis: Orientation and motion, in Analysis Methods for Multi-
spacecraft Data, edited by G. Paschmann and P. W. Daly, pp. 271306,
ESA Publ. Div., Noordwijk, Netherlands.
Dunlop, M. W., D. J. Southwood, K.-H. Glassmeier, and F. M. Neubauer
(1988), Analysis of multipoint magnetometer data, Adv. Space Res.,8,
273277, doi:10.1016/0273-1177(88)90141-X.
Dunlop, M. W., A. Balogh, D. J. Southwood, R. C. Elphic, K.-H. Glassmeier,
and F. M. Neubauer (1990), Configuration sensitivity of multipoint mag-
netic field measurements, in Proceedings of the International Workshop
on Space Plasma Physics Investigations by Cluster Regatta,Graz,
Feb. 2022,Eur. Space Agency Spec. Publ.,ESA SP-306,2328.
Dunlop, M. W., A. Balogh, K.-H. Glassmeier, and P. Robert (2002), Four-
point Cluster application of magnetic field analysis tools: The Curl-
ometer, J. Geophys. Res.,107(A11), 1384, doi:10.1029/2001JA005088.
Escoubet, C. P., C. T. Russell, and R. Schmidt (Eds.) (1997), The Cluster
and Phoenix Missions, Kluwer Acad., Dordrecht, Netherlands.
Escoubet, C. P., M. Fehringer, and M. Goldstein (2001), The Cluster mis-
sion, Ann. Geophys.,19, 11971200, doi:10.5194/angeo-19-1197-2001.
Friis-Christensen, E., H. Lühr, and G. Hulot (2006), SWARM: A constella-
tion to study the Earths magnetic field, Earth Planets Space,58, 351358.
Haaland, S., B. U. Ö. Sonnerup, M. W. Dunlop, A. Balogh, H. Hasegawa,
B. Klecker, G. Paschmann, B. Lavraud, I. Dandouras, and H. Rème
(2004a), Four spacecraft determination of magnetopause orientation,
motion and thickness: Comparison with results from single-spacecraft
methods, Ann. Geophys.,22, 13471365, doi:10.5194/angeo-22-1347-
2004.
Haaland, S., B. U. Ö. Sonnerup, M. W. Dunlop, E. Georgescu, G. Paschmann,
B. Klecker, and A. Vaivads (2004b), Orientation and motion of a disconti-
nuity from Cluster curlometer capability: Minimum variance of current den-
sity, Geophys. Res. Lett.,31, L10804, doi:10.1029/2004GL020001.
Harvey, C. C. (1998), Spatial gradients and the volumetric tensor, in Anal-
ysis Methods for Multi-spacecraft Data, edited by G. Paschmann and
P. W. Daly, pp. 307322, ESA Publ. Div., Noordwijk, Netherlands.
Hasegawa, H., M. Fujimoto, T.-D. Phan, H. Rème, A. Balogh, M. W.Dunlop,
C. Hashimoto, and R. TanDokoro (2004), Transport of solar wind into
Earths magnetosphere through rolled-up KelvinHelmholtz vortices,
Nature,430, 755758, doi:10.1038/nature02799.
Li, L. H., and T. Chen (2008), Shear rate determination of the flow near
magnetopause, Chin. J. Space Sci.,28, 273282.
Lundquist, S. (1950), Magneto-hydrostatic fields, Ark. Fys.,2, 316365.
McFadden, J. P. et al. (2009), The THEMIS ESA plasma instrument and
in-flight calibration, in The THEMIS Mission, edited by J. L. Burch and
V. Angelopoulos, pp. 277302, Springer, New York, doi:10.1007/978-
0-387-89820-9_13.
Ogilvie, K. W., T. Von Rosenvinge, and A. C. Durney (1977), International
sun-earth explorer: A three-spacecraft program, Science,198, 131138,
doi:10.1126/science.198.4313.131.
Panov, E. V., et al. (2010), Multiple overshoot and rebound of a bursty bulk
flow, Geophys. Res. Lett.,37, L08103, doi:10.1029/2009GL041971.
Rème, H., et al. (2001), First multispacecraft ion measurements in and near
the Earths magnetosphere with the identical Cluster Ion Spectrometry
(CIS) experiment, Ann. Geophys.,19, 13031354, doi:10.5194/angeo-
19-1303-2001.
Robert, P., M. W. Dunlop, A. Roux, and G. Chanteur (1998a), Accuracy of
current density estimation, in Analysis Methods for Multi-spacecraft
Data, edited by G. Paschmann and P. W. Daly, pp. 395418, ESA Publ.
Div., Noordwijk, Netherlands.
Robert, P., A. Roux, C. C. Harvey, M. W. Dunlop, P. W. Daly, and K.-H.
Glassmeier (1998b), Tetrahedron geometric factors, in Analysis Methods
for Multi-spacecraft Data, edited by G. Paschmann and P. W. Daly,
pp. 323348, ESA Publ. Div., Noordwijk, Netherlands.
Russell, C. T., and R. C. Elphic (1978), Initial ISEE magnetometer results:
Magnetopause observations, Space Sci. Rev.,22, 681715, doi:10.1007/
BF00212619.
Shen, C., and M. W. Dunlop (2008), Geometric structure analysis of the
magnetic field, in Multi-spacecraft Analysis Methods Revisited,ISSI
Sci. Rep., vol. 8, edited by G. Paschmann and P. W. Daly, chap. 3, pp.
2732, Kluwer Acad., Dordrecht, Netherlands.
Shen, C., X. Li, M. Dunlop, Z. X. Liu, A. Balogh, D. N. Baker, M. Hapgood,
and X. Wang (2003), Analyses on the geometrical structure of magnetic
field in the current sheet based on cluster measurements, J. Geophys.
Res.,108(A5), 1168, doi:10.1029/2002JA009612.
Shen, C., X. Li, M. Dunlop, Q. Q. Shi, Z. X. Liu, E. Lucek, and Z. Q. Chen
(2007), Magnetic field rotation analysis and the applications, J. Geophys.
Res.,112, A06211, doi:10.1029/2005JA011584.
Shi, Q. Q., C. Shen, Z. Y. Pu, M. W. Dunlop, Q.-G. Zong, H. Zhang, C. J.
Xiao, Z. X. Liu, and A. Balogh (2005), Dimensional analysis of observed
structures using multipoint magnetic field measurements: Application to
Cluster, Geophys. Res. Lett.,32, L12105, doi:10.1029/2005GL022454.
Shi, Q. Q., C. Shen, M. W. Dunlop, Z. Y. Pu, Q.-G. Zong, Z.-X. Liu, E. A.
Lucek, and A. Balogh (2006), Motion of observed structures calculated
SHEN ET AL.: TECHNIQUE A11207A11207
18 of 19
from multi-point magnetic field measurements: Application to Cluster,
Geophys. Res. Lett.,33, L08109, doi:10.1029/2005GL025073.
Shue, J.-H., J. Chao, H. Fu, C. Russell, P. Song, K. Khurana, and H. Singer
(1997), A new functional form to study the solar wind control of the mag-
netopause size and shape, J. Geophys. Res.,102(A5), 94979511,
doi:10.1029/97JA00196.
Vogt, J., A. Albert, and O. Marghitu (2009), Analysis of three-spacecraft
data using planar reciprocal vectors: Methodological framework and spa-
tial gradient estimation, Ann. Geophys.,27, 32493273, doi:10.5194/
angeo-27-3249-2009.
Zhang, Q.-H., et al. (2011), The distribution of the ring current: Cluster
observations, Ann. Geophys.,29, 16551662, doi:10.5194/angeo-29-
1655-2011.
SHEN ET AL.: TECHNIQUE A11207A11207
19 of 19
... It plays such an important role in the magnetic field interactions with matter (Boozer 2005), magnetic reconnection (Petschek 1964), particle heating, and acceleration (Guo et al. 2014;Dahlin et al. 2017) that it has drawn considerable attention from plasma physics, space physics, and magnetospheric physics researchers. Many works have been carried out to determine the curvature of stationary magnetic field structures in the different regions of the magnetosphere, such as current sheets, the geomagnetic tail, and the ring current (Shen et al. 2003(Shen et al. , 2007(Shen et al. , 2012Yang et al. 2014). Curvature is a key factor that affects the motion of charged particles in a magnetic field. ...
... To explore the electronscale kinetic physics in the region around the reconnection site, NASA has conducted the ongoing Magnetospheric Multiscale (MMS) mission (Burch et al. 2016). The MMS consists of four spacecraft in a tetrahedral formation approximately 10-160 km apart (Burch et al. 2016), enabling researchers to calculate the accurate spatial gradient of the physical quantities we are interested in by means of multiple-point analysis (Shen et al. 2012). Shuster et al. (2021) analyzed the electron phase-space density gradients around magnetic reconnection at Earthʼs magnetopause and compared observations to a simplified Maxwellian model. ...
... The B could be directly measured by the spacecraft, but the gradient needs to be deduced from measurements. To obtain accurate spatial gradients, we employ the multiple-point analysis method put forward by Shen et al. (2012), which has been validated as performing well. ...
Article
Full-text available
This study presents statistical features of magnetic field curvature in the magnetosheath region. Two sets of high-quality field and plasma data measured by the Magnetospheric Multiscale mission are analyzed by the multiple-point analysis method. The results include the following: (a) The probability distribution function (PDF) of the curvature exhibits two different power laws consistent with previous studies; the PDF of small curvatures depends on the plasma condition and the PDF of large curvatures shows better agreement. (b) The data validate the derived relation between the current density and the guiding center current as well as the diamagnetic current. (c) The acceleration due to curvature drifts in the perpendicular direction occurs when κ/κ rms is larger than 1, which is a potential mechanism for anisotropic distribution of plasma pressure at large curvatures.
... It plays such an important role in the magnetic field interactions with matter (Boozer 2005), magnetic reconnection (Petschek 1964), particle heating, and acceleration (Guo et al. 2014;Dahlin et al. 2017) that it has drawn considerable attention from plasma physics, space physics, and magnetospheric physics researchers. Many works have been carried out to determine the curvature of stationary magnetic field structures in the different regions of the magnetosphere, such as current sheets, the geomagnetic tail, and the ring current (Shen et al. 2003(Shen et al. , 2007(Shen et al. , 2012Yang et al. 2014). Curvature is a key factor that affects the motion of charged particles in a magnetic field. ...
... To explore the electronscale kinetic physics in the region around the reconnection site, NASA has conducted the ongoing Magnetospheric Multiscale (MMS) mission (Burch et al. 2016). The MMS consists of four spacecraft in a tetrahedral formation approximately 10-160 km apart (Burch et al. 2016), enabling researchers to calculate the accurate spatial gradient of the physical quantities we are interested in by means of multiple-point analysis (Shen et al. 2012). Shuster et al. (2021) analyzed the electron phase-space density gradients around magnetic reconnection at Earthʼs magnetopause and compared observations to a simplified Maxwellian model. ...
... The B could be directly measured by the spacecraft, but the gradient needs to be deduced from measurements. To obtain accurate spatial gradients, we employ the multiple-point analysis method put forward by Shen et al. (2012), which has been validated as performing well. ...
Preprint
This study presents statistical features of magnetic field curvature in the magnetosheath region. Two sets of high-quality field and plasma data measured by the Magnetospheric Multiscale (MMS) mission are analyzed by the multiple-point analysis method. The results include the following: (a) The probability distribution function (PDF) of the curvature exhibits two different power laws consistent with previous studies; the PDF of small curvatures depends on the plasma condition and the PDF of large curvatures is universal. (b) The data validate the derived relation between the current density and the guiding center current as well as the diamagnetic current. (c) The acceleration due to curvature drifts in the perpendicular direction will result in an anisotropic distribution of plasma pressure when $\kappa/\kappa_{rms}$ is larger than 1.
... Clearly, three of the spacecraft provide one component of J normal to that face of the tetrahedron and for very irregular spacecraft separations the relative alignment of the spacecraft configuration to the local field geometry is important, so that often only one face provides an accurate determination of the J component normal to that face (see also the note in Section 3.3 and the methodology in Shen, Rong, Dunlop, Ma, et al., 2012;Vogt et al., 2009). This partial estimate can still provide useful infor- Dunlop et al., 1988;top left); a configuration of the three THEMIS spacecraft in the ring current (from Yang et al., 2016, top right); a configuration of the three Swarm spacecraft (A, B, C) with adjacent positions (A', C') from a few seconds earlier (lower left), and a schematic showing some of the large-scale magnetospheric currents around the Earth (adapted from Kivelson & Russell, 1995, lower right). ...
... The magnetic field measurements on board multiple, formation flying spacecraft more generally allow both the gradient and curvature terms in the dyadic of the magnetic field, B, to be linearly estimated (Chanteur, 1998;Harvey, 1998aHarvey, , 1998bShen, Rong, Dunlop, Ma, et al., 2012;Vogt et al., 2008), from which the current density can be extracted; usually employing a formalism in barycentric coordinates (for J this is identical to the form in Section 2, but the error handling is slightly different). A full set of gradient terms can be obtained with an array of at least four spacecraft. ...
... A full set of gradient terms can be obtained with an array of at least four spacecraft. Key formulations of this methodology are magnetic curvature and rotation analysis (Shen et al., 2007, Shen, Rong, Dunlop, Ma, et al., 2012 and least squares analysis applied to planar reciprocal vectors (De Keyser et al., 2007;Hamrin et al., 2008;Vogt et al., 2009Vogt et al., , 2013. The performance of these related gradient methods, in general, depends on the integrity of the spacecraft array and the stationary properties (temporal dependence) of the magnetic structures, although additional constraints for some structures can be added. ...
Article
Full-text available
We review the range of applications and use of the curlometer, initially developed to analyze Cluster multi‐spacecraft magnetic field data; but more recently adapted to other arrays of spacecraft flying in formation, such as MMS small‐scale, 4‐spacecraft configurations; THEMIS close constellations of 3–5 spacecraft, and Swarm 2–3 spacecraft configurations. Although magnetic gradients require knowledge of spacecraft separations and the magnetic field, the structure of the electric current density (for example, its relative spatial scale), and any temporal evolution, limits measurement accuracy. Nevertheless, in many magnetospheric regions the curlometer is reliable (within certain limits), particularly under conditions of time stationarity, or with supporting information on morphology (for example, when the geometry of the large scale structure is expected). A number of large‐scale regions have been covered, such as: the cross‐tail current sheet, ring current, the current layer at the magnetopause and field‐aligned currents. Transient and smaller scale current structures (e.g., reconnected flux tube or dipolarisation fronts) and energy transfer processes. The method is able to provide estimates of single components of the vector current density, even if there are only two or three satellites flying in formation, within the current region, as can be the case when there is a highly irregular spacecraft configuration. The computation of magnetic field gradients and topology in general includes magnetic rotation analysis and various least squares approaches, as well as the curlometer, and indeed the added inclusion of plasma measurements and the extension to larger arrays of spacecraft have recently been considered.
... De Keyser et al. (2007) and also Hamrin et al. (2008) applied least squares inversion with error control to more general satellite configurations over a finite period of time, with localization enforced through the choice of covariance matrices or weighting functions. Shen et al. (2012) discussed stable gradient and curl estimators in terms of the effective array geometry resulting from the nonzero eigenvalues of the volumetric tensor. This paper is concerned with a family of gradient and curl estimators that are based on a local least squares principle and formulated in terms of so-called reciprocal vectors q , = 1, 2, … , S, where S denotes the ...
... Meaningful estimates can only be constructed in the subspace spanned by the eigenvectors corresponding to the larger eigenvalues, leading to an effective error control strategy closely associated with the geometry of the spacecraft array. Earlier studies following a similar approach include the reports of De Keyser et al. (2007), De Keyser (2008), Hamrin et al. (2008), and Shen et al. (2012). ...
Article
Full-text available
Plain Language Summary The space environment of the Earth is studied using spacecraft measurements of physical variables. To resolve their variability in both space in time, constellations of several spacecraft such as Cluster, Themis, Swarm, and MMS are needed. This report combines methods to determine the spatial variability from distributed spacecraft observations in a unifying software framework. The approach works for arbitrary numbers of spacecraft in the constellation and adapts to possibly degenerate geometries that otherwise could lead to large errors. The analysis method is validated using synthetic data and then applied to magnetic measurements from the Cluster and Swarm missions. Implementation in the popular numerical software Python is shown to be compact and computationally efficient and presented in a way that facilitates easy integration in existing free and open source software.
... The geometry of planar reciprocal vectors for a three-spacecraft configuration is sketched in Fig. 4.2. The relationships of planar reciprocal vectors to the eigenvalues and eigenvectors of the volumetric tensor (1/S)R are discussed in detail by Shen et al. (2012b). The singular position tensor case is most relevant for the Swarm mission: here the gradient vector cannot be resolved fully from the measurements, and additional information has to be taken into account to reconstruct its out-of-plane component. ...
... (4.6). For a thorough discussion of volumetric tensor (1/S) R eigenvectors and eigenvalues in the three-spacecraft case, see Appendix D of Shen et al. (2012b). ...
Chapter
Full-text available
Multi-spacecraft probing of geospace allows the study of physical structures on spatial scales dictated by orbital and instrumental parameters. This chapter highlights multi-point array analysis methods for constellations of two or three spacecraft such as Swarm, and also discusses multi-scale techniques for the geometrical characterisation of auroral current structures using observations of stationary or weakly time-dependent current structures along the tracks of individual satellites. Linear estimators are based on a least squares approach which is local in the sense that only few measurements around a reference point are considered for the reconstruction of geometrical and physical parameters. Local least squares estimators for field-aligned currents are compared with non-local counterparts and also local estimators based on finite differences. Uncertainties, implementation and other practical aspects are discussed. The techniques are illustrated using selected Swarm crossings of the auroral zone.
... The symmetries in plasma structures and the electromagnetic field laws can also be useful. It has been found by Shen et al., [2012a] that, for a force-free magnetic structure in which the current is field-aligned, the 3 dimensional (3-D) magnetic gradient can be completely obtained with 3 spacecraft magnetic measurements. In their derivation, Ampere's law 0 = Bj and the solenoidal condition of the magnetic field 0   B are used to reduce the equations. ...
Article
Full-text available
Plain Language Summary The magnetic field plays a key role in the dynamical evolution of space plasmas; it traps and stores plasma particles, and controls the transfer, conversion, and release of the energies. The magnetic field can form various structures, where the magnetic field lines can bend and twist. At the present time, full imaging of the magnetic field has not been achieved. Therefore, it is very important to estimate the magnetic gradients at every order, as well as the geometrical features (curvature and torsion) of the magnetic field lines (MFLs), from the in situ observations. Although we have successfully deduced the first order magnetic gradient and the curvature from multiple S/C magnetic measurements, how to estimate the high order magnetic gradients and the torsion of MFLs is still unsolved. Here, for the first time, we put forward a novel explicit algorithm, which can acquire the quadratic magnetic gradient and the torsion of MFLs with the four‐point magnetic field and current density measurements as the input. This algorithm has sound accuracies and can be effectively applied to analyze magnetospheric multiscale (MMS) observations. This method has many applications in space exploration and theoretical and applied research.
... A full set of gradient terms can be obtained with an array of at least four spacecraft. Key formulations of this methodology are magnetic rotation analysis (Shen et al., 2007(Shen et al., , 2012 and least squares analysis applied to planar reciprocal vectors (De Keyser et al., 2007;Hamrin et al., 2008;Vogt et al., 2009Vogt et al., , 2013. The performance of the methods, in general, depends on the integrity of the spacecraft array and the stationary properties (temporal dependence) of the magnetic structures. ...
Chapter
Full-text available
This chapter covers a selection of the range of multispacecraft techniques that have been initially developed to analyze Cluster data. We begin the chapter with a short introduction, following this with an account of the methods and their application. The topics are separated into those dealing with magnetic field gradients and topology (which include the curlometer, magnetic rotation analysis, and least squares approach); magnetic field reconstruction and the analysis of magnetic field nulls (which are significant for magnetic reconnection and other geometries); time series analysis, adapted for multispacecraft data (including boundary identification, dimensional, and motion analysis); and wave vector analysis methods in the Fourier domain.
Preprint
The geometrical invariants of magnetic field gradient tensors are used to classify the topological structures of magnetic field. This study presents statistical analysis on the geometrical invariants of magnetic field gradient based on high quality data measured by magnetospheric multiscale(MMS) mission in turbulent magnetosheath. The method for the classification of velocity field topologies cannot be applied to magnetic field with strong intensity directly, because the magnetic field cannot be transformed to zero by selecting co-moving reference frame in which velocity is zero. During strong magnetic field, flux ropes and tubes are the most possible magnetic structures. Statistics in the plane formed by geometrical invariants show that about 23% are force free structures consist of 20.5% flux tubes and 79.5% flux ropes. The remaining actively evolved structures are comprised of 30% flux tubes and 70% flux ropes. Moreover, the conditional average of current density and Lorentz force decomposition in geometrical invariants plane are conducted. Results show that flux ropes carried more current density than flux tubes for same geometrical invariants, and flux ropes tend to associate with magnetic pressure force and flux tubes tend to associate with magnetic tension.
Chapter
The Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission is the fifth NASA Medium-class Explorer (MIDEX), launched on February 17, 2007 to determine the trigger and large-scale evolution of substorms. The mission employs five identical micro-satellites (hereafter termed “probes”) which line up along the Earth’s magnetotail to track the motion of particles, plasma and waves from one point to another and for the first time resolve space–time ambiguities in key regions of the magnetosphere on a global scale. The probes are equipped with comprehensive in-situ particles and fields instruments that measure the thermal and super-thermal ions and electrons, and electromagnetic fields from DC to beyond the electron cyclotron frequency in the regions of interest. The primary goal of THEMIS, which drove the mission design, is to elucidate which magnetotail process is responsible for substorm onset at the region where substorm auroras map (∼10 RE): (i) a local disruption of the plasma sheet current (current disruption) or (ii) the interaction of the current sheet with the rapid influx of plasma emanating from reconnection at ∼25 RE. However, the probes also traverse the radiation belts and the dayside magnetosphere, allowing THEMIS to address additional baseline objectives, namely: how the radiation belts are energized on time scales of 2–4 hours during the recovery phase of storms, and how the pristine solar wind’s interaction with upstream beams, waves and the bow shock affects Sun–Earth coupling. THEMIS’s open data policy, platform-independent dataset, open-source analysis software, automated plotting and dissemination of data within hours of receipt, dedicated ground-based observatory network and strong links to ancillary space-based and ground-based programs. promote a grass-roots integration of relevant NASA, NSF and international assets in the context of an international Heliophysics Observatory over the next decade. The mission has demonstrated spacecraft and mission design strategies ideal for Constellation-class missions and its science is complementary to Cluster and MMS. THEMIS, the first NASA micro-satellite constellation, is a technological pathfinder for future Sun-Earth Connections missions and a stepping stone towards understanding Space Weather.
Article
For the first time, the Cluster spacecraft have collected 3-D information on magnetic field structures at small to medium scales in the Earth's dayside magnetosphere. We focus here on the first application of the Curlometer (direct estimation of the electric current density from curl(B), using measured spatial gradients of the magnetic field) analysis technique. The applicability of this multipoint technique is tested, for selected events within the data set, in the context of various mission constraints (such as position, timing, and experimental accuracy). For the Curlometer, nonconstant spatial gradients over the spacecraft volume, time dependence, and measurement errors can degrade the quality of the estimate. The estimated divergence of the magnetic field can be used to monitor (indirectly) the effect of nonconstant gradients in the case of many magnetic field structures. For others, and at highly distorted spacecraft configurations, this test may not reflect the quality of the Curlometer well. The relative scales and relative geometry between the spacecraft array and the structures present, as well as measurement errors, all are critical to the quality of the calculation. We demonstrate that even when instrumental and other errors are known to contribute to the uncertainty in the estimate of the current, a number of current signatures within the magnetosphere can be plausibly determined in direction, if not absolute size. A number of examples show consistent currents at the magnetopause, both separate from, and nearby or in the cusp region. Field-aligned currents near the polar cap boundary are also estimated reliably. We also demonstrate one example of an anomalous current arising from the effect of a highly distorted spacecraft configuration.
Article
The geometrical configuration of the magnetic field underpins important research top- ics in magnetospheric physics. The magnetic field is the skeleton of the magnetosphere and plays a crucial role in determining the plasma distribution of particles, the occurrences of various macro and micro instabilities, the triggering and evolution of substorms and magnetic storms, etc. On the other hand, the magnetic reconnection process at the magnetopause and in the tail plasma sheet alters the topological structures of the magnetosphere, producing transient magnetic structures, such as rotational discontinuities, flux ropes, plasmoids, etc. The magnetometer investigation on the multiple spacecraft Cluster mission has made it possible to reveal the three-dimensional geometrical structure of the magnetic field in the magnetosphere, at least to local first order gradients. To achieve a full analysis of the local nature of the magnetic field geometry, several new gradient and curvature based methods have been proposed to describe the topological configurations of the key regions of magnetosphere. These methods are the curvature analysis method [Shen et al., 2003], magnetic field strength gradient method [Shen et al., 2003, 2007a], and the magnetic rotation analysis [MRA, see Shen et al., 2007b].