ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-769-2015In situ magnetotail magnetic flux calculationShukhtinaM. A.mshukht@geo.phys.spbu.ruhttps://orcid.org/0000-0001-7892-8392GordeevE.St. Petersburg State University, Earth Physics Department,
Ulyanovskaya 1, Petrodvoretz, St. Petersburg 198504, RussiaM. A. Shukhtina (mshukht@geo.phys.spbu.ru)18June201533676978113October201417May201526May2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/769/2015/angeo-33-769-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/769/2015/angeo-33-769-2015.pdf
We explore two new modifications of the magnetotail magnetic flux
(F) calculation algorithm based on the
(PR96) approach of the tail
radius determination. Unlike in the PR96 model, the tail radius
value is calculated at each time step based on simultaneous
magnetotail and solar wind observations. Our former algorithm,
described in , required that the “tail
approximation” requirement were fulfilled, i.e., it could be applied only
tailward x∼-15RE. The new modifications take into account
the approximate uniformity of the magnetic field of external
sources in the near and middle tail. Tests, based on magnetohydrodynamics
(MHD) simulations, show that this approach may be applied at smaller
distances, up to x∼-3RE. The tests also show that the
algorithm fails during long periods of strong positive interplanetary magnetic field (IMF) Bz.
A new empirical formula has also been obtained for the tail
radius at the terminator (at x=0) which improves the
calculations.
Magnetospheric physics (magnetotail)Introduction
Magnetotail magnetic flux is one of the key parameters in
magnetospheric physics .
However, this global parameter is difficult to measure using local observations. Only about 2 decades ago did the
global auroral images of Polar and Image spacecraft
become
available, providing the opportunity to calculate the polar cap (PC)
area
. As the lobe magnetic field threads the polar
caps, the PC area is proportional to the open magnetic flux.
However, the PC area determination has its own problems, associated
with the threshold level of the registered luminosity and with
dayglow contamination. Note that since 2006 there
have been no space missions with the purpose of global auroral imaging.
Another method of PC area determination is based on
identifying the location of region 1 field-aligned currents in
global estimates of the Birkeland current system provided by the
Active Magnetosphere and Planetary Electrodynamics Response
Experiment (AMPERE) .
proposed another approach to the
magnetotail magnetic flux calculation. Their method is based on
the idea of a flaring magnetopause and on the pressure balance at
the magnetopause:
0.88Pdsin2α+Bsw2/2μ0+nswkTsw=BL2/2μ0,
where Pd is the solar wind dynamic pressure, Bsw
is the interplanetary magnetic field magnitude and nsw and Tsw are
the observed solar wind plasma density and temperature.
Assuming the pressure balance in the tail, we
hereafter use the equivalent tail lobe magnetic field, determined
from the equation
BL2/2μ0=B2/2μ0+nkT.
Based on a large data set of simultaneous solar wind and tail lobe
measurements, PR96 constructed an empirical model, giving the
magnetotail radius RT value as a function of solar wind dynamic
pressure Pd, interplanetary magnetic field (IMF) Bz
and an x coordinate in the tail. However, the magnetotail radius
value depends not only on the solar wind/IMF parameters, but also
on the magnetospheric state. Thus
proposed a modification of the PR96 method, in which the radius is
calculated based on in situ measured lobe field (or total
pressure) values. Following PR96, the method presumes magnetopause
flaring, so that tanα=dRT/dx, where
α is the flaring angle of the axially symmetric
magnetopause (see Fig. 1 adapted from Fig. 2 of PR96). Thus
RT(x)=RT0+∫0xtanα(x)dx with RT0 for the radius value at terminator (at
x=0). As in PR96, the α(x) dependence is taken as
sin2α=A2exp(Cx). According to the statistical
study by , the average C value is C=0.0714. All formulas are presented in the geocentric solar
magnetospheric (GSM) coordinate system.
The scheme, illustrating the procedure of RT
calculation.
Assuming BL isolines perpendicular to magnetopause, a
correction Δx should be made for the x value at
magnetopause, Δx=(RT-(y2+z2)1/2)sinαcosα, where x,y,z are the coordinates of the observation
point. So the x value at the magnetopause, corresponding to the
given BL, is x∗=x+Δx, giving a new A∗ value in
the sin2α(x) dependence, etc. After several iterations
the procedure quickly converges, and finally,
RT(x)=RT0-2/C(arcsin(A∗exp(Cx/2))-arcsin(A∗)).
According to PR96:
RT0=14.63(Pd/2.1)-1/6.(2a)
In the present study this empirical dependence is reexamined, and
now it looks as follows:
RT0=16.37Pd-1/6.02(1.004-0.0054IMFBz).(2b)
Formula (2b) will be explained in more detail in
Sect. 4 and in the Appendix. As is shown there, formula (2b)
suits the tests made in this study better than (2a), so
hereafter we everywhere use it in our calculations.
Finally, the magnetic flux value is calculated as follows:
FT=0.5πRT2BL,
assuming BL uniformity in the given cross section; the plasma sheet width is neglected.
The algorithm is described in more detail in
, where it was tested by global magnetohydrodynamics
(GMHD) simulations using the Open Geospace General Circulation Model
model . Comparison of the algorithm with
the imager-based methods of polar cap area estimate is presented
in . Both MHD tests and comparison
with polar cap images give rather good results during substorm
growth phase and steady magnetospheric convection (SMC) events,
but the result is worse after substorm onset (especially when the
spacecraft is in the plasma sheet) due to the break of the
one-dimensional pressure balance. According to both tests the
FT values, calculated by the algorithm, exceed those given by
direct integration in MHD simulation and those, obtained from
auroral images, by ∼0.2 GWb, which is partly due to ignoring
the plasma sheet thickness.
As the algorithm is based on pressure balance, it was applied in
the region where the “tail approximation” is valid, i.e.,
approximately at x<-15RE. It is however desirable to apply
the algorithm to spacecraft measurements earthward from
-15RE, which would significantly extend the ability to
monitor the tail flux. This is the aim of the present work.
All comparisons are based on GMHD simulations made in the
Community Coordinated Modeling Center (CCMC; http://ccmc.gsfc.nasa.gov/). In spite of the well-known problems
mainly due to absence of kinetic effects, the first-principle-based GMHD
approach provides a reasonable global configuration and dynamics
of magnetospheric system .
We believe such tests are perhaps the most straightforward and reliable
at present, as they provide an opportunity to calculate the reference
magnetotail flux value with high precision for further comparison with
our estimates.
Algorithm modifications
Results of the simulation BATSRUS_Gordeev_110309_1
in the meridional
magnetosphere cross section y=0 for two simulation moments: (a) 255 min and (b) 280 min correspondingly. Black lines are the
Bext field lines; Bext values are shown by
color. Total pressure (in nPa) isolines (magenta) are also
shown. The solid black line indicates the reference magnetopause
(fluopause), whereas the red triangles and red squares designate
the RT and RText position.
Results of the simulation BATSRUS_Gordeev_110309_1.
Top panel: IMF Bz variation, shifted by convection to
x=-15RE. Bottom panel: FT0 (solid lines), FT1 (dashed
lines) and FT2 (circles) values, calculated based on the
“measurements” at different x distances (y=0, z=-10RE)
compared with FTMHD value, obtained by direct integration through
the cross section x=-15RE (thick blue line).
To broaden the working area, we make further modifications to the
method. Notice that the open magnetic flux is formed by the
“external” part of magnetic field Bext=B-BIGRF.
Unlike the strongly non-uniform dipole field, the
external field is approximately uniform and thus may be determined
from a measurement in a single point. It is illustrated by
Fig. 2 which shows the Bext field lines and
Bext magnitude distribution (by color) together with total
pressure (BL) isolines and magnetopause position before (a,t=255 min) and after (b,t=280 min) substorm onset for the
cross section y=0 for the MHD simulation
BATSRUS_Gordeev_110309_1. The substorm onset is defined here as
the magnetotail flux unloading (Fig. 3). The magnetopause
(fluopause, see ), obtained from the
simulation data, is in black, whereas the magnetopause, built
according to Eq. (2) for the virtual spacecraft with coordinates x=[-25:0]RE (with a step 1 RE), y=0,z=11RE is
designated by red triangles. Figure 2 demonstrates that
(1) the BL isolines are approximately perpendicular to
magnetopause in the tail lobes at ∼x<-10RE,
justifying the RT calculation procedure for this domain;
(2) the “external magnetic field” is approximately uniform in
the entire volume of the tail lobes up to x∼-2÷-5RE,
its isolines being perpendicular to magnetopause in this region.
Analogously to BL we introduce the quantity BLext:
BLext2/2μ0=(Bext)2/2μ0+nkT,
which is also approximately uniform in a vast volume of the
magnetotail. Then the proxy of the magnetic flux of external
sources is
FT1=0.5πRT2BLext.
Note that the procedure of BLext calculation is reasonable
only if the vectors BIGRF and Bext are more
or less collinear. So BLext is defined only if the angle
between BIGRF and Bext is less than 90∘. Otherwise (e.g., in the plasma sheet, where
Bext and BIGRF are antiparallel) BLext
is not defined. Most correctly (4) and the following formula (5) may be applied for the observations in the tail lobes.
However, the usage of the F1 proxy faces some difficulties. Note
that Eq. (2) contains arcsin functions and cannot be
applied when the arcsin argument exceeds 1. Furthermore, for the
points in the inner tail the “corrected” value x∗ may become
positive (the observation point “moves” to the dayside). According
to our experience, this situation is not infrequent in the
near-Earth region with large B values. To avoid this difficulty, we
attempt one more modification of the algorithm. In the right-hand side
of expression (1) instead of BL we put BLext. As pointed
above, the BLext isolines (perpendicular in the lobes to the
field lines of external field) are approximately perpendicular to
the magnetopause in a more vast region (including the inner
magnetotail) than the BL isolines. So the procedure of radius
calculation may be applied to the BLext quantity instead of
BL up to ∼-5RE. The surface, corresponding to the
obtained radius, is not the real magnetopause, but is a surface
inside it, with the radius RText. The proxy of the open
magnetic flux confined by this surface is now
FT2=0.5πRText2BLext.
The magnetopause proxy, corresponding to the RText value, is
shown in Fig. 2 by red squares. For snapshots in Fig. 2, earthward
from -15RE the RT value is slightly larger than the
fluopause radius, whereas the RText value is smaller. The
actual magnetopause lies between RT and RText. Tailward
from -15RE the RT value is close to the actual radius,
whereas RText lies inside it. At ∼-23RE both
estimates become identical.
So we have three magnetotail magnetic flux proxies, FT0 (according to Eq. 3),
FT1 (Eq. 4) and FT2 (Eq. 5).
Note that all of them assume the lobe field to be uniform across the whole
tail cross section, neglecting the plasma sheet thickness. At the
same time the FT2 algorithm removes a part of the magnetic
flux, which offsets the excess flux in the plasma sheet.
The quantities that may be calculated in the MHD simulations are
FTMHD – the total magnetic flux through the given tail cross
section and FPCMHD – the magnetic flux, formed by the
open lobe magnetic field lines. In the middle tail (e.g., at
x=-15RE) these quantities should be close to each other, the
FTMHD value being slightly larger. We expect our proxies
FT1 and FT2 to lie somewhere between them, FT2
being closer to FPCMHD. The algorithm of
FT0
calculation was examined in
. The FT1
and FT2 calculation is less strict and requires thorough
testing. These tests are presented below.
Evaluation of the method
In this section we present the results of the FT0, FT1 and
FT2 comparison with the FTMHD values, obtained by
direct magnetic flux integration through the magnetotail
cross section x=-15RE as described in
. The comparison was carried out for
two global MHD BATS-R-US simulations : with
synthetic SW input (artificial event) and with real SW input (real
event). For the second simulation, we also consider the polar cap
magnetic flux FPCMHD quantity, available from the CCMC
website.
Artificial event (event 1)
Figure 3 presents results based on the simulation
BATSRUS_Gordeev_110309_1, which was already presented in
Fig. 2. The only variable input parameter
IMF Bz (shifted by convection time from x=33RE to
x=-15RE) is shown in the top panel, all other inputs being
fixed: IMFBx=IMFBy=0, solar wind speed Vx=-300 km s-1,
Vy=Vz=0, ion density N=20 cm-3 and temperature T= 100 000 K, zero dipole tilt. The bottom panel presents the directly
integrated magnetotail magnetic flux FTMHD (thick blue line)
together with FT0 estimates based on “measurements” of the
virtual spacecraft at x=-15RE and -7RE (y=0, z=-10RE).
The FT0 quantity behaves similarly to FTMHD, but there
is an offset, which grows when approaching the Earth (at x=-7RE
the FT0 value is twice as large as FTMHD).
Dashed lines of the corresponding color indicate the FT1
values for the same virtual spacecraft positions, whereas circles
correspond to FT2. In contrast to FT0, the FT1 and
FT2 values for different observation points are close to each
other and to the FTMHD value, on average bounding it. So
FT1 and FT2 may be considered as the upper and lower
FTMHD estimates. Note that the worst correspondence is
observed during the long (∼90 min) period of IMF Bz=+10 nT, where all estimates do not follow FTMHD and
exceed it. We shall return to analysis of this
effect in Sect. 3.2.
Figure 3 shows that the FT0 proxy is not appropriate inside
-15RE, so in the next section we test only the FT1 and
FT2 proxies on a more interesting simulation, reproducing a
real geophysical event.
Real event (event 2)
Another CCMC simulation studied is BATSRUS_Sergeev_060508_1.
The input parameters (including the dipole tilt) were taken from a
real event, 00:00–16:00 UT 5 March 2008 (Fig. 4). The event includes
different geomagnetic situations with variable IMF Bz; Pd
smoothly changes between ∼1 and 3 nPa. The laborious
FTMHD calculations were done for the period 09:00–16:00 UT,
whereas the polar cap magnetic flux value FPCMHD, formed
by the open field lines, is presented at the CCMC website for the
whole simulation period.
The input parameters of the simulation
BATSRUS_Sergeev_060508_1. From top to bottom: ion density
(N), ion temperature (T), solar wind speed components (Vx,Vy,Vz) and IMF components (Bx,By,Bz).
Results of the simulation BATSRUS_Sergeev_060508_1.
(a) Solar wind “merging electric field” Em variation.
(b) Different F estimates: FTMHD (green), FPCMHD (black), FT1
(red) and FT2 (blue). The FT1 and FT2 quantities are
calculated for the point (-15,0,12).
(c)FT1 values, calculated at different distances x =-11 (red), -7 (blue) and -3 RE (magenta),
compared with FPCMHD (black) and FTMHD (green).
Real event simulation BATSRUS_Sergeev_060508_1.
Distributions of the average over the simulation flux
value, correlation and regression coefficients and the free term,
describing the relationship of FT0, FT1 and FT2,
“measured” at x=-7RE, with the FPCMHD value.
Black thick contour on each panel bounds the lobes with β<1.
Note the different color bar scale for the average values of
FT0 and FT1, FT2 in the first row.
Figure 5a presents the “merging electric field” Em variation
for the period studied. Here Em=VBtsin3(Θ/2), where
Bt=(IMFBy)2+(IMFBz)2, V is
the solar wind velocity and Θ is the clock angle (see,
e.g., ). Solar wind/IMF parameters are
again shifted to x=-15RE by convection. Figure 5b presents
different magnetic flux estimates for the Northern Hemisphere: in
green and black – the FTMHD and FPCMHD values, in
red and blue – the FT1 and FT2 values correspondingly.
The FT1 and FT2 values were calculated based on the
“measurements” of a virtual spacecraft at the point (-15, 0, 12)
RE. FT1 values, calculated for different x positions, are
presented in Fig. 5c. Variations of all magnetic flux proxies
repeat the main features of Em variations. The FTMHD and
FPCMHD curves are close to each other except the period of
positive IMFBz after 15:00 UT. The regression equation for the
period 09:00–15:00 UT is
FTMHD=0.87FPCMHD+0.05,
with correlation coefficient CC=0.85, demonstrating
cross-validation of the two estimates. Notice however that as
FTMHD is the total flux estimate, and the FPCMHD is
the open flux proxy, FTMHD should slightly exceed
FPCMHD. According to Fig. 5 the balance is inverse
(average FTMHD and FPCMHD values for the period
09:00–15:00 UT are 0.62 and 0.66 GWb correspondingly). Until now we
do not know the reason of such inconsistency in the GMHD
simulation.
We made a regression analysis of FT1 and FT2 quantities,
calculated for different observation points (see the legends in
Fig. 5b, c) with both FTMHD and FPCMHD for the
period 09:00–15:00 UT, when both variables were available. The results
appeared to be very close (identical taking into account the error
bars). As the FPCMHD values are available for the whole
period 00:00–16:00 UT, hereafter we compare different magnetic flux
proxies with the FPCMHD value.
Note that the major discrepancies between algorithm results and
FPCMHD appear during the periods of zero Em (strong
positive IMF Bz) 03:40–05:40 UT and 15:00–16:00 UT. The same
situation was observed in the simulation with artificial input
(Fig. 3), i.e., such periods are not adequately described by our
algorithms. Thus we exclude these periods (03:40–05:40 UT and 15:00–16:00 UT) from our regression analysis, leaving this topic for the
future.
According to Fig. 5 the variations and absolute values of
FT1 are similar to those of FPCMHD, whereas the
FT2 proxy reproduction of the reference magnetic flux is slightly
worse. The figure also demonstrates that the FT1 quantity may
serve as a rather stable estimate of the tail magnetic flux for
spacecraft positions up to x=-7RE.
However, until now we considered only separate observation points.
Below, we discuss the global distribution of different magnetic
flux estimates.
Global regression analysis for the real event
Figure 6 presents the global distribution of results of regression
analysis of all three (FT0, FT1 and FT2) magnetic
flux estimates with FPCMHD quantity for the
cross section x=-7RE for the same simulation
BATSRUS_Sergeev_060508_1. The solid black line designates the
boundary where plasma β=1.
The comparison is carried out using the equation
FTj=P1FPCMHD+P2,j=0,1,2.
Figure 6 presents the distribution of the following statistical
parameters: average over the simulation F values (upper
row), correlation CC (second row) and regression P1 (third
row) coefficients and the free term P2 (bottom row) for all
observation points in the cross section x=-7RE (note that the
intervals of large positive IMF Bz 03:40–05:40 and 15:00–16:00 UT
are excluded). The points in the Northern Hemisphere are compared
with the northern polar cap flux FPCMHD, in the Southern
Hemisphere – with the southern FPCMHD.
The most prominent feature of Fig. 6 is the scope of the
FT2 algorithm, which covers almost the whole cross section in
contrast to the other two proxies.
The upper row shows that the average FPCMHD value is best
of all reproduced, though slightly underestimated (by
10÷15 %) by FT1, whereas the FT2 proxy is smaller
than FPCMHD by 20÷25%. Note that the average flux
distribution as well as other statistical characteristics are very
uniform in space for FT2 and rather uniform for FT1. On
the contrary the FT0 characteristics strongly depend on the
observation point, making their use problematic. The CC values
(second row) are mostly ≥0.8 and often ≥0.9 for β≤1 for all three proxies. Again the distributions for FT1
and FT2 are more uniform than those for FT0. The
regression coefficient P1 (third row) is mostly around 0.7–0.8
(sometimes decreasing to 0.6) for all three proxies. Finally, the
free term P2 (the bottom row) is small (about ±0.1 GWb) for
FT1 and FT2 proxies compared to the values 0.3–0.5 GWb
for FT0.
In summary, the FT2 proxy may be
applied almost in the whole tail lobe region and demonstrates
uniform parameters' distributions, though the average flux values
are underestimated by ∼ 0.1–0.15 GWb. The FT1 proxy
reproduces the FPCMHD value slightly better than FT2 and
better than FT0, but its working area (as well as the scope of
FT0) is much smaller than that of FT2.
Improved formula for the tail radius at terminator
The magnetopause radius in the terminator plane (RT0) is the
starting point for the tail radius calculation (Eq. 2) in our
method. As the RT0 value strongly influences the tail radius
and consequently the magnetic flux value, it is important to use a
precise formula for RT0. For today, there is a set of
different empirical magnetopause models based on different data
sets and mathematical methods. All these models give RT0 in
the global magnetopause shape context, leading to different
functional forms RT0(SW,IMF). To determine this
dependence more precisely, we tried to construct an empirical magnetopause model
in the specified narrow region – terminator (x=0) plane. For this
purpose we used the data set of magnetopause crossings from the
GSFC NASA website (ftpbrowser.gsfc.nasa.gov/magnetopause.html). This data set
spans the time range 1963–1998 and uses crossings of 18 different
satellites that were analyzed by different authors. The data is
tagged by 1 h averaged SW/IMF measurements. To collect a
sufficient number of magnetopause crossings, we selected crossings
in the vicinity of the terminator plane (namely in the region
-3<x<2RE) and traced them to x=0 using the analytical
magnetopause shape given in PR96. This simple procedure enabled us
to collect 1192 crossings that further reduced to 1022 points due
to cutting off the deep tails of the SW/IMF parameter
distributions and the low-latitude crossings. The low-latitude
crossings were rejected because of not sufficiently sharp
gradients of plasma and magnetic parameters, which lead to a high
uncertainty of the magnetopause determination. Next, the
coordinates of Southern Hemisphere crossings were mapped to the
Northern Hemisphere with the dipole tilt sign inversion assuming
the north–south shape invariance. Applying multiple regression
analysis to the selected data set, we found a clear and pronounced
linear dependence of RT0 on magnetic dipole tilt angle
Ψ, a power law dependence on solar wind dynamic pressure Pd
and weak linear dependence on IMF Bz. The final empirical
formula for RT0 in the Northern Hemisphere is
RT0=(14.1+0.045Ψ)⋅1.161Pd-1/6.02⋅(1.004-0.0054IMFBz),
for Ψ in degrees, Pd in nPa, IMF Bz in nT and RT0 in Earth's radii RE.
The formula is valid for all tilt angles in the SW/IMF parameter
range 1<Pd<8 nPa and -6<IMFBz<6 nT. For more details
on the derivation of Eq. (7), see the Appendix.
Summary for methods application.
MethodApplicationEstimationSpatialCommentsscopeaccuracyuniformityFT0x<-15REsignificantly overestimated (30÷60 %)stronglynon-uniform1. Suitable only if tail approximation is fulfilled 2. Coincides with FT1 & FT2 in the midtail 3. Difficult to calibrateFT1x<-7REclose to the reference value (±20 %)rather uniform1. Applicable up to -3RE if possible 2. Coincides with FT0 & FT2 in midtail 3. Calibration: FTMHD=(0.98±0.01)FT1, x=-7,-11,-15REN=81.418, CC=0.74, σ= 0.039 GWb (σ∼10 %)FT2x<-7REclose to the reference value (±20 %)highly uniform1. Applicable up to -3RE if possible2. Coincides with FT0 & FT1 in midtail 3. Calibration: FTMHD=(1.12±0.01)FT2, x=-7,-11,-15REN=95.330, CC=0.72, σ= 0.037 GWb (σ∼10 %)
To check whether the new formula improves the result, we compared
calculations based on the PR96 terminator radius and on the new
RT0 value for the already considered simulation
BATSRUS_Sergeev_060508_1 assuming zero dipole tilt in Eq. (7).
The tilt was taken as zero because it is the assumption of our
F calculation algorithm. We performed regression analysis,
comparing the polar cap magnetic flux value from the simulation
(FPCMHD) with the estimated tail magnetic flux in each
point of the chosen cross section. The comparison was made for all
three algorithm modifications for different distances (x=-25 ,-15, -11 and -7RE).
For comprehensive comparison, we calculated differences
between the values given by new and old formulas of RT0 for
the correlation coefficient CC, slope P1 and free term
P2 for the aforementioned cross sections. The differences were
calculated as follows:
ΔCC=CCNew-CCPR96ΔP1=|P1(PR96)-1|-|P1(New)-1|ΔP2=P2(PR96)-P2(New),
to be positive in the case of the flux estimation accuracy
increase.
The values from Eq. (8) turned out to be positive for all algorithm modifications,
indicating slightly higher accuracy of the flux estimates. So the
new formula for the tail radius at terminator is preferable.
The value of the regression coefficient P1 in FT0(1,2)=P1FTMHD for different observation points: x=-15,-11,-7,-3RE,
y=0 and z=10,12RE. Results are shown for both artificial event (a) and real event (b) simulations. Vertical bars
denote the confidence level.
Discussion
To emphasize the ability of our modified FT1 and FT2
algorithms let us again turn to Fig. 6. The figure shows that the free term in the regression equations for
FT1 and FT2 is fairly small, within ±0.1 GWb, and
the regression coefficient P1 is rather uniform. Similar
results were also obtained for other tail cross sections, though
there is not enough space to present them. Instead, we briefly
illustrate the P1 volume uniformity by presenting two x profiles
(for z=10 and 12 RE) of P1 coefficient for each flux estimation method (Fig. 7).
Here we use the regression equation assuming zero free term
FTj=P1FTMHD,j=0,1,2.
The results for Events 1 and 2 are presented in panels a and b
correspondingly. The sampling points are taken at x=-15,-11,-7,-3RE, y=0 and z=10, 12 RE. Both Fig. 7a and b confirm the
high uniformity of regression coefficient corresponding to
FT1 and FT2 estimates along the tail (in addition
to cross-tail P1 uniformity shown in Fig. 6), which slightly
varies around unity and has moderate confidence bounds, supporting
the idea that the free term in the regression equation can be set
to zero. On the contrary, the P1 for FT0 estimate is
highly non-uniform and has a significantly wider confidence
interval due to ignoring the large free term.
High uniformity of the regression coefficient P1 and small
absolute value of the free term P2 in the entire middle- and
near-tail lobes give us the opportunity to make a unified
calibration of our modified algorithms FT1 and FT2 using
simple linear regression with the zero free term for the tail
lobes. The calibration formulas are presented in Table 1,
summarizing the general results of our analysis. Briefly, in the
middle tail (x<-15RE), the modifications FT1 and FT2
have values, very close to the former algorithm FT0.
Furthermore, in contrast to the original algorithm, the modified
algorithms have much wider spatial application limits extended to
the inner magnetosphere (especially FT2) with highly uniform
calibration coefficients. On the basis of the presented
analysis, for further applications we recommend using the
combination of FT1 and FT2 modifications instead of
original method.
It is important to note that the algorithm presumes zero dipole
tilt. In Event 2 the dipole tilt varied between -15 and 5∘,
which caused the asymmetries in Fig. 6, but still provided a
satisfactory result. However, the larger tilt angles can lead to
larger errors.
Conclusions
We present an algorithm for magnetotail magnetic flux FT1
calculation which may be applied for the measurements inside
x=-15RE. The algorithm is a modification of the old one
(computing the FT0 value), which could be applied tailward
x=-15RE. The modified algorithm gives a proxy of the magnetic
flux of the external magnetic field (Eq. 4) based on the
approximate uniformity of this field. Both algorithms should be
preferably applied in the tail lobes, where the 1-D pressure
balance is valid according to MHD tests. However, the working area
of both FT0 and FT1 estimates is limited because of
their dependence on the arcsin function used for the tail
radius calculation. To broaden the application domain of the new
algorithm, it was again modified (Eq. 5) by using the new radius
value RText, calculated from the balance of the “external
magnetic pressure” (the pressure of the magnetic field of external
sources) with the magnetosheath pressure. RText corresponds
to a surface, lying inside the actual magnetopause, and is the
lower estimate of the tail radius value. According to Fig. 2a and b,
RT and RText bound the actual magnetopause. Tests based on
two MHD simulations show good FT1 correlation/regression
relationships with the magnetic flux value FTMHD, obtained by
direct magnetic flux integration through the tail cross section
at x=-15RE, and with the magnetic flux of the polar cap
FPCMHD, given by the CCMC. The regressions for FT2
are slightly worse, but the scope of FT2 algorithm is
considerably wider than that of FT1; furthermore, the FT2
distribution appeared strongly uniform (weakly depending on the
observation point). The quality of all methods degrades for
periods of large positive IMF Bz, when all algorithms give
overestimated magnetic flux values.
Also a new empirical dependence for terminator radius RT0,
which is the “boundary condition” for our algorithm,
was obtained from spacecraft data. We show that the new RT0
formula better represents magnetopause position versus its analogues
and should be used in the future.
New formulation for the magnetopause radius at terminator (plane x=0)Fitting the data
To find the analytical representation of RT0, we applied the
linear regression to three interplanetary medium parameters: Earth's
dipole tilt angle Ψ, SW dynamic pressure Pd and IMFBz. First
we extracted the RT0 dependence on Ψ (Fig. A1, top
panel):
RT0=(14.1±0.1)+(0.045±0.005)Ψ.
Top panel – scatterplot of terminator radius dependence
on dipole tilt angle. Bottom panel – demonstration of results of
the data correction procedure (Eq. A2) that eliminates the
RT0(Ψ) dependence. Red straight lines are the linear
regression lines. Black lines are the moving averages (window size
is 3∘, step size is 0.5∘).
The figure demonstrates significant RT0(Ψ) dependence
with amplitude ∼3RE between extreme angles
(∼±30∘) in good agreement with recent models
[hereafter
L10 and W13].
It is interesting to note the small positive offset of RT0
for large negative tilt angles (Ψ<-27∘, negative
Ψ corresponding to anti-sunward tilt of dipole axis in the
Northern Hemisphere) which may be associated with passage of the
sunward edge of cusp indentation across the terminator plane (see
Fig. 8 in W13, or Fig. 8d in L10).
Since we obtained a clear and strong RT0(Ψ) dependence,
the dipole tilt influence can be excluded from the following
analysis using the correction
RT0c1=RT0(measured)RT0(Ψ)⋅RT0avg,
where RT0 in the numerator is the measured radius,
RT0(Ψ) is the radius (predicted by Eq. (A1)) and
RT0avg=14.34RE is the terminator radius averaged
over the whole data set. The bottom panel of Fig. A1 visualizes
the corrected data with the Ψ dependence subtracted according
to Eq. (A2).
The relationship between solar wind dynamic pressure and
terminator radius with subtracted dipole tilt dependence. The red
curve is the regression line corresponding to Eq. (A4). The
black line is the moving average (with the window size 1 nPa,
step size 0.2 nPa).
Now we can determine the RT0 dependence on solar wind dynamic
pressure using refined data RT0c1. Considering the
theoretical background and experience of previous authors, the
power law dependence
RT0c1=aPd-b,
with constant coefficients a and b is expected
(Fig. A2). Taking the logarithm of both sides of Eq. (A3) and
applying linear regression we find the coefficients
RT0c1=(16.64±0.21)Pd-0.166± 0.012.
Dividing the RT0c1 by normalization factor
RT0avg we obtain the explicit dependence
RT0=RT0(Ψ)⋅[(1.161±0.015)Pd-0.166± 0.012].
To reveal the IMF Bz effect on the RT0, we again construct
the normalized correction RT0c2 with subtracted dependence
RT0(Ψ,Pd) given by Eq. (A5). The results of regression
analysis (Fig. A3) show moderate dependence of corrected
RT0c2 quantity on IMF Bz with ∼1RE variance
amplitude inside the used parameter range. Dividing
RT0c2 by RT0avg we get
RT0=RT0(Ψ,Pd)⋅[(1.004±0.005)-(0.0054±0.0022)IMFBz].
Relationship between IMF Bz and terminator radius
corrected to eliminate the dipole tilt and ram pressure
dependences. Red straight line – regression line. Black line is
the moving average (with the window size 2 nT, step size 0.2 nT).
Substituting the RT0(Ψ,Pd) dependence we come to
Eq. (7):
RT0=(14.1+0.045Ψ)⋅1.161Pd-1/6.02⋅(1.004-0.0054IMFBz)
for the radius of the northern terminator magnetopause (radius of the
southern side can be obtained by reversing the sign of dipole tilt angle).
The resultant correlation coefficient between the predicted terminator radius
and that “measured” from the original data set is CC= 0.74 with the
standard deviation SD = 1.07 RE, which is significantly better
than what PR96 gives (CCPR96 = 0.54, SDPR96 =
1.27 RE). To verify the statistical significance of our model we
performed the Student's t test of regression
coefficients in Eq. (7) and found that each coefficient undoubtedly satisfies
the condition of significance. As an example, the calculated t statistic value tcalc= 4.83 is
most weak for the second coefficient of RT0(IMFBz) dependence; however, it is still larger than the critical value even
for the highest confidence probability (cp = 0.999)
tcrit= 3.3.
Remember that in order to “move” the original data to the
terminator plane we used the analytical magnetopause shape from
PR96. To verify how much this procedure could affect the results,
we tried another model developed by
[hereafter S98] with another analytical shape and SW/IMF
dependence. Results appeared very close: the coefficients
RT0(Ψ) and RT0(IMFBz) are almost the
same, whereas the coefficients RT0(Pd) only slightly differ:
RT0(S98tracing)=(14.19+0.046Ψ)⋅1.178Pd-1/5.48⋅(1.004-0.006IMFBz),
confirming validity of the tracing procedure. Results
obtained based on PR96 and S98 models are presented in Fig. A4.
Features of the new model
The developed empirical model of the terminator radius has a
series of interesting features. It demonstrates a strong enough
perturbation of magnetopause position in the x=0 cross section
associated with dipole tilting in the xz plane (about 3 RE
for extreme opposite tilts) in agreement with the modern
asymmetric magnetopause models L10 and W13. The coefficient of
the power law in dynamic pressure dependence was found to be close to -1/6
in accordance with the PR96 model.
Terminator radius dependence on solar wind dynamic
pressure for different empirical models (see the legend). For each
model, two profiles corresponding to IMF Bz=+6 nT and IMF
Bz=-6 nT are shown. Dipole tilt is zero. Solid and dashed red
lines show the results of the new model developed using the
PR96 magnetopause shape for tracing to x=0. Red empty circles
and squares correspond to tracing based on the
magnetopause shape.
However, the most interesting observation is the IMF Bz dependence of RT0.
First of all, we found a clear and statistically significant IMF
Bz effect (about 1 RE) in contrast to PR96 model. Note that
the RT0(IMF Bz) linear dependence is the same for the
negative and positive IMF Bz values. It seems that this is not
the result of the regression method but is a real feature supported
by the moving average values (Fig. A3). Our results demonstrate
the increase of the terminator radius for negative IMF Bz and
its decrease for positive IMF Bz. Since the cusp axis is almost
always located sunward of the x=0 plane (except during extreme
dipole tilt angles), such dependence looks natural. During
southward IMF Bz, the magnetic tubes, reconnected on the
dayside, convect in the anti-sunward direction and drape the
magnetopause tailward of the cusps, increasing the radius at
terminator. The picture is the opposite for the northward IMF – the
solar wind tubes reconnect behind the cusps adding magnetic flux
to the dayside magnetosphere and reducing the magnetic flux and
radius tailward of the cusps including the terminator plane.
However, it should be noted that the L10 and S98 models predict
the opposite RT0(IMF Bz) dependence. This discrepancy is
likely due to the analytical predefinition of the magnetopause
shape in those models which turns out to be a limitation not only
in the quantitative but also in the qualitative sense. The most recent 3-D
magnetopause model W13 was constructed using an excellent set of
crossings without any assumptions about the analytical shape. This
model predicts the same qualitative terminator radius dependence
on IMF Bz as the present study (Fig. A4) with more pronounced
(roughly by factor 2) difference of RT0 between southward and
northward IMF orientation. However, W13 has no explicit
analytical parametrization which makes its application difficult.
Several points that we used for comparison in Fig. A4 were
adopted from their Fig. 10. It is also worth noting that the
same sign of the RT0(IMFBz) dependence was obtained for the
BATS-R-US “empirical” magnetopause model . This
conformity with our new formula could be the main reason for the
estimation improvement described in Sect. 4.
Acknowledgements
We thank V. A. Sergeev for useful discussions and Marianna Holeva for her
help in the manuscript preparation. We are grateful to the topical editor
C. Owen for their objective assessment of this paper and to the referees for their constructive criticism.
Global MHD simulation results were provided by the Community
Coordinated Modeling Center at the Goddard Space Flight Center through
their public Runs on Request system (http://ccmc.gsfc.nasa.gov/).
The BATS-R-US Model was developed by the group led by Tamas
Gombosi at the Center for Space Environment Modeling, University
of Michigan.
The work was carried out as part of EU FP7 ECLAT project. Data on
magnetotail magnetic flux (F1 and F2 values) can be found on the site http://geo.phys.spbu.ru/eclat/. E. Gordeev's work was supported
by a grant from the Government of St. Petersburg, Russian
Federation (2014). The topical editor C. Owen thanks L. B. N. C. Clausen and the two anonymous referees for help in evaluating this paper.
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