# Picturing global substorm dynamics in the magnetotail using low-altitude
ELFIN measurements and data mining-based magnetic field reconstructions

###### Abstract

A global reconfiguration of the magnetotail characterizes substorms. Current
sheet thinning, intensification, and magnetic field stretching are defining
features of the substorm growth phase and their spatial distributions control
the timing and location of substorm onset. Presently, sparse in-situ
observations cannot resolve these distributions. A promising approach is to
use new substorm magnetic field reconstruction methods based on data mining,
termed SST19. Here we compare the SST19 reconstructions to low-altitude ELFIN
measurements of energetic particle precipitations to probe the radial profile
of the equatorial magnetic field curvature during a 19 August 2022 substorm.
ELFIN and SST19 yield a consistent dynamical picture of the magnetotail during
the growth phase and capture expected features such as the formation of a thin
current sheet and its earthward motion. Furthermore, they resolve a V-like
pattern of isotropic electron precipitation boundaries in the time-energy
plane, consistent with earlier observations but now over a broad energy range.

Geophysical Research Letters

University of California, Los Angeles, Los Angeles, CA, USA Applied Physics
Laboratory, Johns Hopkins University, Laurel, MD, USA

Xiaofei<EMAIL_ADDRESS>
Electron and ion isotropy boundaries (IBe,i) inferred from ELFIN particle data
are compared with IBs derived from empirical magnetic field

ELFIN and empirical magnetic field IBs consistently move equatorward in the
growth phase and diverge in latitude after the substorm onset

ELFIN and empirical magnetic field IBe reveal similar V-like patterns
associated with local accumulation of magnetic flux in the tail

## Plain Language Summary

The solar wind strongly stretches the magnetic field on the night side of our
planet. This stretching slowly accumulates and rapidly relaxes during special
periods called substorms. These variations are difficult to investigate
because of the extreme sparsity of spacecraft observations. The problem can be
solved by properly sorting historical data from similar substorm phases to
form swarms of synthetic probes which are then used to reconstruct the
magnetic field configuration. The low-altitude ELFIN mission provides another
means of probing the magnetic field configuration by measuring electrons and
ions, which become scattered as the night side magnetic field is stretched.
Here, we demonstrate that both approaches yield a consistent picture of the
night side magnetic field and how it changes during a substorm that occurred
on 19 August 2022.

##  1 Introduction

Substorms are one of the most energetic phenomena in Earth’s magnetosphere,
responsible for a large-scale reconfiguration of the magnetotail, charged
particle acceleration and injection into the inner magnetosphere, as well as
the formation of strong field-aligned currents that couple the magnetosphere
to the ionosphere [Baker . (1996), Angelopoulos, McFadden . (2008), Sitnov,
Birn . (2019)]. During the substorm growth phase, the magnetotail magnetic
field stretches significantly and a thin current sheet forms [<]e.g.,¿[and
references therein]Sergeev12,Runov21:jastp. Magnetic reconnection in this thin
current sheet enables acceleration of charged particles and their injection
into the inner magnetosphere [<]see reviews by¿[and references
therein]book:Gonzalez&Parker,Sitnov19,Birn21:AGU. In-situ spacecraft
measurements provide important information for investigating substorm
dynamics. However, given its substantial volume, even multi-spacecraft
missions cannot resolve the large-scale magnetotail structure and its
evolution during substorms [<]see discussion in¿Stephens19,Sitnov19:jgr.
Nevertheless, two general approaches exist that can resolve the global-scale
structure of the magnetosphere from data. The first approach leverages machine
learning algorithms to reconstruct magnetospheric plasma quantities from
sparse in-situ datasets [<]e.g.,¿Sitnov08,Bortnik16. The second involves
remote sensing the magnetosphere using ground-based observations
[<]e.g.,¿MenkWaters13 or low-altitude orbiting spacecraft [Dubyagin . (2002),
Coxon . (2018)]. In recent years, both approaches have been employed to
progress the understanding of large-scale substorm dynamics. For instance, by
applying a data mining (DM) technique to more than two decades of space
magnetometer data, Stephens19 reconstructed the dynamics of the magnetotail
magnetic field during substorms. Termed SST19, their algorithm revealed the
formation of multi-scale current sheets, the stretching and dipolarization of
the tail magnetic field, the 3D structure of the substorm current wedge, and
the location of magnetic reconnection verified by in-situ observations of the
ion diffusion region [Stephens . (2023)].

Remote sensing can be exemplified by the Electron Losses and Fields
Investigation (ELFIN), a pair of twin CubeSats launched in September 2018 into
a polar low-Earth orbit to observe the precipitation of charged particles with
magnetospheric origins [Angelopoulos . (2020)]. Low-altitude spacecraft, such
as ELFIN, allow the global magnetotail spatial profile to be inferred
[<]e.g.,¿Sergeev11,Sergeev18:grl,Sergeev23:elfin. This approach relies on the
isotropization of proton and electron distributions when the magnetic field
curvature becomes comparable to or smaller than the particle gyroradius
resulting in chaotization of their orbits and pitch angle scattering [Sergeev
Tsyganenko (1982)]. It reveals more of the large-scale magnetotail structure
and its reconfiguration during substorms than magnetic field times-series
measurements from (approximately stationary) equatorial satellites which lack
spatial information of the magnetic field profile
[<]e.g.,¿Sergeev12:IB,Sergeev15, Dubyagin02,Artemyev22:jgr:ELFIN&THEMIS. The
basic elements of such a reconfiguration include the formation of a thin
current sheet with magnetic field line stretching and earthward current sheet
motion [Wanliss . (2000), Kozelova Kozelov (2013), Petrukovich . (2007),
Artemyev . (2016)]. Current sheet thinning scatters energetic ions and
electrons into the loss cone causing them to precipitate into the ionosphere.
Low-altitude measurements of precipitating particles of various species and
energies can localize the latitude of this scattering (from which the location
of the thin current sheet, its Earthward-most edge, and the magnetic field
radius of curvature which depend on the particles’ energy and species can be
inferred) thus tracing the thin current sheet’s dynamics [Yahnin . (1997),
Sergeev, Nishimura . (2012), Sergeev . (2018), Sivadas . (2017)].

So far, comparisons between empirical magnetic field reconstructions and low-
altitude measurements have been made using event-oriented models
[<]e.g.,¿Kubyshkina09,Kubyshkina11,Sergeev23:elfin. In that approach, a
statistical model of the magnetic field, fit to a large archive of historical
magnetometer data [<]e.g.,¿Tsyganenko95, was additionally tweaked to achieve
better consistency with low-altitude precipitation patterns using a small
number of lucky (and hence rare) observations made during the event and in the
region of interest, that is close to IBs. In contrast, the SST19 DM approach
employed in this study reconstructs the magnetic field using an “event-
oriented” subset of the archive ($\approx~{}1\%$), which is still large enough
($\sim 9\cdot 10^{4}$ synthetic probes) to allow for a far more flexible
magnetic field architecture and an increased sensitivity to storm and substorm
variations. Here we compare the DM-based magnetic field reconstruction of the
magnetotail during a 19 August 2022 substorm to ELFIN measurements of
energetic ions and electrons [Angelopoulos . (2020)]. This is a unique
substorm event in that ELFIN passed through latitudes that map to the near-
Earth ($r\approx 5$–$20R_{E}$) magnetotail about midnight magnetic local time
(MLT) six times. This comparison demonstrates that SST19 and ELFIN reveal a
consistent global dynamical picture of the magnetotail during substorms.

##  2 Spacecraft Observational dataset

We use the low-orbit-altitude ($\sim 450$ km) twin ELFIN CubeSats (A and B) to
measure energetic electron ($50$keV–$6$MeV) and ion ($250$keV–$5$MeV) fluxes
[Angelopoulos . (2020)]. ELFIN moves along a polar orbit with a period of
$\sim 1.5$ hour and measures particle fluxes with an angular resolution of
$\sim 22.5^{\circ}$ and a temporal resolution of 2.8 s (spin period)
[Angelopoulos . (2020)]. In this study, we use two types of ELFIN data
products: energy spectra of locally precipitating (within the loss-cone) and
locally trapped (outside the loss-cone) fluxes; only measurements with more
than five counts per bin (energy, time) are included [<]see details of data
products in¿Angelopoulos23:ssr.

We use the SML and SMR indices from the SuperMag project [Gjerloev (2009)] to
monitor substorm and storm activity. The Supporting Information (SI) also
contains a comparison of the low-altitude ELFIN measurements with near-
equatorial energetic particle fluxes from the THEMIS [Angelopoulos, Sibeck .
(2008)] and Magnetospheric Multiscale (MMS) [Burch . (2016)] missions. The
THEMIS Solid State Telescope (SST) [Angelopoulos, Sibeck . (2008)] provides
electron fluxes of $30-700$ keV at a $3$ s time resolution. The MMS Fly’s Eye
Energetic Particle Spectrometer (FEEPS) [Blake . (2016)] provides electron and
ion fluxes of $50-650$keV at a $2.5$ s time resolution.

Figure 1: Overview of the substorm activity from 05:00–11:00 UT on 19 August
2022. (a) Interplanetary magnetic field and solar wind velocity measured by
ARTEMIS-P1 (THEMIS probe B) which was located within the solar wind; (b)
SuperMag substorm, SML (black), and storm, SMR (blue), indices; (c–h) ELFIN’s
observations of electron energy spectra for locally trapped fluxes for six
orbits intersect the midnight sector (the crossing times are indicated by the
red dashed lines in panels (a) and (b)), and the isotropic regions are shaded
on top. Panels (c, e, g) are observations from ELFIN A while panels (d, f, h)
are from ELFIN B.

Figure 1(a) shows the solar wind magnetic field ($B_{z}$ in the Geocentric
Solar Magnetospheric system or GSM) and the solar wind velocity from 05:30 to
11:00 UT on 19 August 2022 [<]measured by THEMIS B;
see¿Angelopoulos11:ARTEMIS: there is a prolonged interval ($\sim$06:00–08:00
UT) with $B_{z}<0$, that corresponds to the magnetosphere loading during
substorm growth phase. Figure 1(b) shows the SML and SMR indices. The
consistently moderate negative value of SMR, $\sim-20$ nT, suggests the
presence of a ring current ion population in the inner magnetosphere. This
negative value remains relatively stable, indicating continuous replenishment
of this population by new injections [<]e.g.,¿Gkioulidou14. The dip of SML
around 08:40 UT indicates a substorm onset, and thus before this moment, there
should be magnetotail current sheet thinning (the growth phase begins with the
southward turning of the IMF around 06:00 UT). Six ELFIN passes intersect the
near-midnight sector during this interval; red vertical lines in Figure 1(a,
b) mark those times. Figures 1(c–h) show ELFIN’s observations of electron
energy spectra for locally trapped fluxes ($J_{\perp}$), with the shading
indicating where the fluxes are nearly isotropic (where the trapped flux is
comparable to the precipitating flux, $J_{\perp}\sim J_{\parallel}$). The
energy spectra exhibit typical characteristics of various magnetospheric
regions [<]see¿Mourenas21:jgr:ELFIN,Angelopoulos23:ssr. In Figure 1(c), for
example: (1) from 05:42:00 to 05:43:50 UT ELFIN crossed the outer radiation
belt which is characterized by fluxes of relativistic electrons ($>500$ keV)
with strong flux anisotropy ($J_{\perp}\gg J_{\parallel}$). (2) From 05:43:50
to 05:44:35 UT, a decrease in energy levels is observed for electrons with
$J_{\perp}>10^{3}/cm^{2}/s/sr/MeV$, while the fluxes are mostly isotropic
($J_{\perp}\sim J_{\parallel}$). This is the so-called isotropy boundary (IB)
[Imhof . (1977), Sergeev Tsyganenko (1982), Wilkins . (2023)], the transition
region between the outer radiation belt and the plasma sheet. The energy-time
spectra there exhibit the energy/latitude (time) dispersion characteristic of
the IB, with the latitude of the minimum energy of isotropization increasing
with decreasing energy (the beginning of this transition region is indicated
by the vertical line with the magnetic latitude, MLAT, shown). (3) From
05:44:35 to 05:46:00 UT, fluxes are isotropic and limited to within $<200$ keV
energy; this is the plasma sheet region [Artemyev . (2022)]. The electron
isotropy boundary (IBe) separates the plasma sheet and the outer radiation
belt and can be considered as an inner (equatorward) edge of the magnetotail
plasma sheet. Note that the location for ion IB (IBi) and electron IB (IBe)
are quite different, with IBi equatorward of the IBe due to the much larger
gyroradius (and field-line curvature radius responsible for isotropization) of
ions than electrons of the same energy [Sergeev, Nishimura . (2012)].

The location and shape of IBs vary during a substorm
[<]e.g.,¿Sergeev12:IB,Wilkins23. Figure 1(c–h) reveals the dynamics of the
IBe. During the growth phase (Figure 1(c–f)), the magnetic latitude
$|\textit{MLAT}|$ of the IBe’s equatorward edge decreases, indicating that the
IBe moves equatorward. After substorm onset, (Figure 1(h)), the
$|\textit{MLAT}|$ of the IBe’s equatorward edge increases, indicating that the
IBe moves further poleward. The ELFIN orbit shown in Figure 1(g) crossed the
magnetotail shortly after substorm onset, but the $|\textit{MLAT}|$ of the IBe
kept decreasing. To explain why we examine THEMIS observations near the plasma
sheet (Figure S1): from 06:00 to 07:10 UT, there is a decrease in the
equatorial $B_{z}$ and an increase in $|B_{x}|$, indicating magnetotail
current sheet thinning during the substorm growth phase [Sergeev . (2011),
Artemyev . (2016)], and between 07:10 and 08:50 UT, the THEMIS satellites
recorded a sequence of dipolarizations, characterized by $B_{z}$ increases
(and perturbations) and $|B_{x}|$ decreases (Figure S1a). However, in between
such dipolarization, THEMIS observed periods where $|B_{x}|$ increased and
$B_{z}$ decreased, indicating there was continuous current sheet thinning but
interrupted by (or interleaved with) several dipolarizations. Such short time-
scale and likely localized dynamics of the magnetotail current sheet are not
resolved in the SML profile (a global index) that shows a single substorm
onset around 08:40UT. Thus, this is a complex substorm with enhanced
dissipation occurring in the plasma sheet before the major onset (see
discussions of similar phenomena in [Shukhtina . (2014), Yahnin . (2001)]),
but includes multiple ELFIN passes near midnight, which is infrequently
observed, making this event particularly interesting. Such complex substorm
dynamics does not allow to identify if the ELFIN orbit from Figure 1(g)
crossed the stretched or dipolarized magnetotail configuration.

We also estimate ELFIN’s position by comparing its particle fluxes during
intervals of isotropic flux measurements, with the flux data from MMS and
THEMIS spacecraft (Figure S1), the method tested recently in [Artemyev .
(2022), Shen . (2023)]. This comparison demonstrates that the IBe observed by
ELFIN maps to near $10R_{E}$, the likely transition region between the plasma
sheet and the outer radiation belt [<]in agreemnet with¿Sergeev12:IB at that
time (more details are provided in the SI).

##  3 Data mining-based magnetic field reconstruction

Here we compare the ELFIN observations, particularly the inferred IB
locations, with those derived from an empirical magnetic field constructed
using the DM-based algorithm SST19 [Stephens . (2019)]. SST19 differs from
conventional empirical geomagnetic field models [<]e.g.,¿[and refs.
therein]Tsyganenko&Sitnov05 in two critical aspects. First, it describes the
magnetospheric magnetic field using several sets of basis functions rather
than custom-made modules. The number of such basis functions, used for the
description of the magnetic field generated by equatorial and field-aligned
currents as well as their shielding currents on the magnetopause, can be
increased to resolve important morphological features such as the eastward
ring current [Stephens . (2016)] or the Harang discontinuity [Sitnov .
(2017)]. Second, SST19 employs a DM algorithm [Sitnov . (2008)] to fit such a
flexible and multi-parameter magnetic field architecture to data. This
exploits the recurrent nature of storms and substorms to augment the handful
of space-borne magnetometer observations available at the moment of interest
with a much larger set of observations (the nearest neighbors or NNs) made
when the magnetosphere was in a similar storm/substorm configuration, based on
the geomagnetic indices, their time-derivatives, and the strength of the solar
wind driving. The number of NNs, $k_{NN}$, must be large enough, $k_{NN}\gg
1$, to avoid over-fitting, while at the same time small enough, $k_{NN}\ll
k_{DB}$ where $k_{DB}\sim 10^{7}$ is the whole database of historical
magnetometer records since 1995, to make the reconstructions sufficiently
sensitive to the specific phases of substorms and storms. The distinct
features of SST19, compared to its storm-time predecessor, TS07D [Tsyganenko
Sitnov (2007), Sitnov . (2008)], are the use of substorm indices AL or (most
recently) SML [Gjerloev (2012)] and their time derivatives, as well as two
independent basis function descriptions for thick and thin (presumably ion-
scale) current sheets. The buildup and decay of the latter is a key feature of
the magnetospheric reconfiguration during substorms [Sergeev . (2011)]. More
specifically, here we utilize the “merged resolution” version of SST19, which
concurrently resolves both the inner magnetosphere and the magnetotail
[Stephens Sitnov (2021)]. SST19 has been extensively validated by comparing
its reconstructed magnetic field to the field observed by in-situ spacecraft
[Stephens . (2019), Sitnov, Stephens . (2019), Stephens Sitnov (2021), Sitnov
. (2021), Stephens . (2023)]. The only notable modifications to SST19 employed
here are an updated formulation of the thin current sheet spatial structure
and an increased quantity of MMS magnetometer data. More details of SST19 are
provided in the SI.

Figure 2: SST19 data mining-based reconstructions of the 19 August 2022
substorm, the left column is during the substorm growth phase (07:40 UT) and
the right is during the recovery phase (09:10 UT): (a) Color-coded equatorial
distributions of the electric current density, $j$, with arrows overplotted to
indicate the direction of vector current density, $\mathbf{j}$. (b) Color-
coded equatorial distributions of the z-component of the magnetic field,
$B_{z}$, with grey dots overplotted to indicate the locations, projected to
the $x$-$y$ plane, of the spacecraft magnetometer observations identified
using the $K_{\mathrm{NN}}$ procedure and used to fit the analytical
description of the magnetic field. (c) Color-coded equatorial distributions of
the stretching factor $G^{*}=B_{z}^{2}/(\mu_{0}j)$. The critical values of
this parameter corresponding to the IB for 520 keV electrons and protons are
shown by the orange and green contours respectively. The location of ELFIN
mapped to the magnetic equator when it observed the 520 keV IBi and IBe are
overplotted in green and orange circles respectively. The corresponding error
bars quantify the uncertainty of the IB determination using $J_{||}/J_{\perp}$
distributions as described below in Figure 3. Grey lines show projections of
the ELFIN orbits to the equatorial plane.

The ELFIN-observed IBs for species $\gamma=e,p$ and the associated particle
rigidities $G_{\gamma}=m_{\gamma}V_{\gamma}/e$ (where $m_{\gamma}$ are the
masses of electrons and protons, and $V_{\gamma}$ their velocities), can be
compared to those derived from empirical magnetic field reconstructions by
mapping their positions to the magnetic equator and computing there the
equivalent parameter $G^{*}=B_{z}^{2}/(\mu_{0}j)$, termed the stretching
factor ($B_{z}$ is the northward component of the equatorial magnetic field
and $j$ is the current density), as is demonstrated in Figure 2. According to
Sergeev&Tsyganenko82 and Sergeev18:grl, the transition to isotropy due to
chaotization of particle orbits occurs when $G^{*}<8G$. Figures 2a–2c show the
equatorial distributions of $j$, $B_{z}$, and $G^{*}$ respectively for two
moments during the substorm growth (07:40 UT) and recovery phase (09:10 UT)
corresponding to the times of the fourth and sixth ELFIN passes indicated by
the vertical red lines shown in Figure 1 (all six passes at 05:45, 06:10,
07:15, 07:40, and 08:45 UT are shown in Figures S2 and S3). Over-plotted on
these panels are the mappings of the ELFIN’s position when it observed IBe and
IBi at the energy 520 keV (the middle energy channel in the range of 0.1-1
MeV). Note that these mappings nicely match the corresponding IB contours
derived from SST19.

Figures S4 and S5 show the equatorial $B_{z}$ and $G^{*}$ mapped to ELFIN’s
altitude on an MLT-MLAT grid. These plots facilitate the subsequent comparison
of ELFIN-observed IBs with those inferred from the SST19 magnetic field
reconstruction. Also, to allow the comparison of these reconstructions with
similar results from other missions [<]e.g.¿Sergeev18:grl, the analogs of
Figures S4 and S5 in coordinates $MLT$ and $AACGM$ [Shepherd (2014)] are
provided in Figures S6 and S7. In addition, Figure S8 presents validation of
the SST19 reconstructions using the observed magnetic field from THEMIS and
MMS.

Figure 2 reveals several important substorm features resolved by SST19. During
the growth phase, a strong ($>5$ nA/m2) current forms in the near-tail from
$r\approx 5$–$13R_{E}$ (Figure 2a1) accompanied by a $B_{z}$ minimum
($B_{z}=4.5$ nT) around $11R_{E}$ (Figure 2b1). This stretches the near-tail,
as indicated by the non-monotonically decreasing $G^{*}$ distribution, which
possesses a local minimum about $11R_{E}$ in the pre-midnight sector (Figure
2c1). The tail configuration dramatically changes during the expansion phase,
which persists into the recovery phase, signified by the collapse of the
cross-tail current (Figure 2a2) and a dipolarization of the magnetic field
(Figure 2b2). This inflates the value of $G^{*}$ across most of the tail
(Figure 2c2), pushing the reconstructed IBe and IBi to larger radial
distances. Note that the domain over which the SST19 reconstructions are
presented here is limited to $r\leq 20R_{E}$. Beyond this limit the mapping
may be strongly complicated by magnetic reconnection [Stephens . (2023)].

##  4 Comparison of ELFIN observations with DM reconstructions

Figure 3: An overview of the SST19 magnetic field reconstructions and ELFIN
observations for six sub-intervals. The SST19 results show equatorial (Column
1) and Y-GSM components (Column 2) of the cross-tail current density,
respectively, projected on the GSM planes shown in the axes. The equatorial
projections of the $520$ keV electron and ion isotropy boundaries (IB)
determined by ELFIN (traced to the equator using SST19 magnetic field
reconstruction) are shown by colored dots (the horizontal bars show the
uncertainty of IB observations). Various times are depicted in various rows
(a-f). $J_{\parallel}/J_{\perp}$ for electrons (Column 3) and ions (Column 4)
measured by ELFIN around the times corresponding to rows (a-f) in the left two
columns. Blue dotted curves represent the IB locations as determined by ELFIN
observations. The orange and green dots indicate the positions of the $520$
keV IB, with horizontal bars illustrating the associated uncertainties. Yellow
and green curves depict the electron and ion isotropy boundaries (IBe and IBi)
from the SST19 reconstructions, respectively. Gray ellipses in Panels (a3, b3)
highlight V-like IBe patterns arising from SST19 reconstruction that bears
similarity to the pattern of precipitating-to-trapped flux ratio observed by
ELFIN at that time.

Figure 3 provides an overview of the SST19 magnetic field reconstructions and
ELFIN observations for all six ELFIN orbits: before the substorm growth phase,
05:40-05:45UT (panels a1-a4); during the substorm growth phase, and
06:06-06:10 UT, 07:10-07:15UT and 07:37-07:40 UT (panels b1-b4, c1-c4 and
d1-d4); during the expansion phase of the substorm, 08:40-08:45UT (panels
e1-e4); and during the recovery phase, 09:10-09:15 UT (panels f1-f4). The two
left columns display the SST19 results, illustrating the equatorial and
meridional distributions of the cross-tail current density. On the right-hand
side, the two columns show the $J_{\parallel}/J_{\perp}$ ratios for both
electrons and ions measured by ELFIN during the same time sub-intervals. The
pre-substorm time is characterized by a weak cross-tail current density in the
center plasma sheet ($<4$nA/m2, panel a1) and a thick current sheet (panel
a2). The IBe is located around $|\textit{MLAT}|\sim 68^{\circ}$ with the
$\Delta|\textit{MLAT}|\sim 0.5^{\circ}$ range for $[100,1000]$ keV, whereas
IBi is around $|\textit{MLAT}|\sim 66^{\circ}$. During the substorm growth
phase, the cross-tail current density increases to $>5$ nA/m2 (panel c1) and
is concentrated within the thin current sheet (panel c2). The IBe is moved
earthward to $|MLAT|\sim 66.5^{\circ}$ (panel c3), and shrinks to
$\Delta|\textit{MLAT}|\sim 0.1^{\circ}$ range for $[100,1000]$ keV. Thus, for
sub-intervals shown in panels (a, c) ELFIN observations of IBe and IBi are
consistent with the SST19 reconstructions: stronger current density and
thinner current sheet are associated with a smaller distance between IBe and
IBi, and earthward motion of both boundaries [<]see¿[for discussion of similar
earthward IBs motions derived from POES measurements during the growth
phase]Sergeev12:IB.

Figures 3e and 3f describe a very short expansion phase and the following
recovery phase. SST19 exhibits cross-tail current density distributions
consistent with the expansion and recovery phases, characterized by weak
current density and a broader current sheet (as seen in panels e1, f1 and e2,
f2 compared to panels c1 and c2). ELFIN shows the IBe at $|\textit{MLAT}|\sim
65^{\circ}$ and $|\textit{MLAT}|\sim 67^{\circ}$ (panels e3 and f3), and the
latter returns to its value in the early growth phase at 06:10 UT, while IBi
at $|\textit{MLAT}|\sim 62^{\circ}$ (panels e4 and f4), is found well
equatorward of its growth phase location.

The third and forth columns of Figure 3 also compare the ELFIN precipitation
ratios $J_{\parallel}/J_{\perp}$ with the IBs derived from SST19 and marked by
yellow and green lines for IBe and IBi, respectively. The latter closely
follow the ELFIN IBs seen as sharp transitions to dark red pixels
(isotropization) marked by blue dotted lines. To determine the IB position, we
select all pairs of bins (two latitudinal bins) for fixed energy ($E$) where
the flux ratio crosses from $\geq 0.6$ to $<1.0$. These groups of points in
the energy and latitudinal space are then fitted by a power-law function
$|\textit{MLAT}|=a\cdot E^{b}$ [<]the power-law $MLAT-E$ fitting underlines a
nonlinear relation between particle energies and equatorial magnetic field
(radial distance) in the equation of pitch-angle scattering rate,
see¿Birmingham84,Delcourt94:scattering. This function indicates the IB
position, while the standard deviation of the fitting describes the
uncertainty range of the IB position. The resulting discrepancies between IBs
derived from the merged resolution SST19 model and ELFIN data (except IBes in
the expansion phase) are much smaller than $1^{\circ}$ MLAT, typical errors
reported for statistical and adaptive models [Shevchenko . (2010)]. Note that
the latter, which somewhat outperform the former, cannot be applied to our
event because of the absence of any real probes in the IB source region.

Interestingly, in Figures 3a3 and 3b3, the SST19 IBes closely follow similar
ELFIN IBes forming the characteristic V-like patterns (first discussed in
[Artemyev . (2023)]). They suggest the appearance of the dipolarized (less
stretched) magnetic field region tailward of the left (lower-latitude, high-
energy) IBe, where the electron precipitation is suppressed by the enhanced
equatorial magnetic field (increasing the curvature radius). Similar non-
monotonic profiles of the tail stretching parameter ($G^{*}$) were reported by
Sergeev18:grl based on the POES data for selected energies (30 and 100 keV).
With ELFIN measurements we resolve such non-monotonic profiles within a wider
energy range and show their consistency with similar SST19 features. More such
IBe transitions at higher latitudes (in the plasma sheet, far tailward from
IBe) are observed by ELFIN but not resolved by SST19, suggesting that ELFIN
has a potential to detect even more complex patterns of alternating stretched
and dipolarized regions in the tail [Artemyev . (2023)], which may have
implications for magnetotail stability and dynamics $B_{z}$ [Erkaev . (2007),
Pritchett Coroniti (2010), Sitnov Schindler (2010), Birn . (2018)].

Figure 4: Dynamics of isotropy boundaries for electrons and ions derived from
the SST19 reconstructions (grey-shaded) and ELFIN measurements (orange for
electrons and green for ions). The energy range used to determine the MLAT
range of isotropy boundaries is from 100 keV to 1 MeV.

To further quantify the comparison of ELFIN observations of IBs with SST19, we
plot the latitudinal range of IBs for all six ELFIN orbits in Figure 4. During
the entire interval, 05:40–09:30, the SST19 IBe traces well the dynamics of
IBs from ELFIN observations: IBe (orange) moves equatorward (its equatorial
projection moves earthward) during the growth phase. The overall dynamics of
the IBi (green) and its inference from SST19 magnetic reconstruction are
similar to those of the IBe during the growth phase. After the substorm onset,
IBe and IBi diverge in latitude. Notably, SST19 traces well the motion of IBs
during the substorm growth phase and after substorm onset.

To ensure the reliability of the reconstruction algorithm, we provide in
Figure S9 (in a format similar to Figure 2) the reconstruction of three more
ELFIN observed IB events, one of which is another V-like pattern event shown
in Figure 8 of Artemyev2023 (2022-08-11). Yet another event (2022-08-07) shown
first in Figure 8 of [Artemyev . (2023)], occurred in the period of high
geomagnetic activity when the consistency of ELFIN IBs with those derived from
SST19 is not expected because precipitation is strongly modified by plasma
waves.

Thus, Figures 3 and 4 show reasonable overall consistency between the SST19 DM
reconstructions and the ELFIN data. The residual differences, especially after
the substorm onset, can be explained by an additional electron scattering
caused by whistler-mode waves, that form intense precipitation bursts
equatorward from the IBe (see, e.g. Figures 3e3 and 3f3) around
$\textit{MLAT}\in[-64,-65^{\circ}]$) and increase the uncertainty range for
IBe position determination [<]see¿[for detailed investigation of night-side
electron precipitation events associated with whistler-mode wave
activity]Tsai22,Artemyev24. Another possible cause of IBe variations are
transient meso-scale perturbations of the equatorial magnetic field [<]e.g.,¿
Lin14:hybrid_code,Panov&Pritchett18,Sorathia20. Further systematic comparative
analyses of DM reconstructions and ELFIN observations could clarify the roles
of different transient precipitation mechanisms resulting in the observed
spatial/temporal variability of the IBe.

##  5 Conclusions

In this study, we investigate the substorm dynamics of electron and ion
isotropy boundaries (IBe and IBi), which are the transition regions between
the inner edge of the electron plasma sheet and the outer radiation belt, and
between the inner edge of the ion plasma sheet (the tail current sheet) and
the ring-current, respectively. By combining low-altitude ELFIN measurements
of energetic particle (ions and electrons) spectra with the SST19 DM-based
empirical magnetic field reconstruction algorithm [Stephens . (2019), Stephens
. (2023)], we demonstrate that multiple localized reductions of energetic
electron precipitation within the IBe (seen as a V-like pattern) are
associated with quasi-steady features in the magnetic field configuration
reproduced by SST19. This comparison confirms the previously suggested
interpretation of V-like patterns as the formation of a flux accumulation
region in the near-Earth tail resulting in reduced field line stretching
tailward of the transition region [Sergeev . (2018)]. This flux accumulation
region may be the locus of a developing magnetotail instability prior to
substorm onset [Sitnov, Birn . (2019)]. The overall agreement between the
location of the ELFIN observed and SST19 reconstructed IBe and IBi are on the
order of $\textit{MLAT}<0.5^{\circ}$ over a broad range of energies
($100$–$1,000$ keV) during the substorm growth phase and includes their
equatorward (earthward) motion. Sergeev12:IB have demonstrated similar current
sheet dynamics by comparing POES IB measurements with a dynamically-adapted
magnetospheric model [Kubyshkina . (2011)] assimilating a rare conjunction of
multiple THEMIS and GOES satellites. Following Sergeev12:IB, our results
demonstrate the potential of combining low-altitude energetic particle
measurements with magnetic field reconstruction algorithms for probing
magnetotail substorm dynamics.

###### Acknowledgements.

We are grateful to NASA’s CubeSat Launch Initiative for ELFIN’s successful
launch in the desired orbits. We acknowledge the early support of the ELFIN
project by the AFOSR, under its University Nanosat Program, UNP-8 project,
contract FA9453-12-D-0285, and by the California Space Grant program. We
acknowledge the critical contributions of numerous volunteer ELFIN team
student members and support by NASA 80NSSC22K1005 and NSF grants AGS-1242918,
AGS-2019950. We acknowledge NASA contract NAS5-02099 for the use of data from
the THEMIS Mission. The work of G. K. S. and M. I. S. was supported by NASA
grants 80NSSC19K0074, 80NSSC20K1271, and 80NSSC24K0556. A.V.A and X.S.
acknowledge support from the NASA grants 80NSSC20K1788, 80NSSC23K0108,
80NSSC24K0558. V.A. acknowledges support by NSF grants AGS-1242918,
AGS-2019950 and NASA contract NAS5-02099.

## Open Research

Fluxes measured by ELFIN are available in the ELFIN data archive
https://data.elfin.ucla.edu/ in CDF format.
THEMIS dataset and summary plot are available in
http://themis.ssl.berkeley.edu
MMS dataset and summary plot are available in
https://lasp.colorado.edu/mms/sdc/public
SuperMag dataset are available in https://supermag.jhuapl.edu
The SST19 reconstruction output data is available on a Zenodo archive at
https://doi.org/10.5281/zenodo.11625067
Data analysis was done using SPEDAS V4.1 [Angelopoulos . (2019)]. The software
can be downloaded from http://spedas.org/wiki/.

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## Supporting Information

Contents of this file

  1. 1.

ELFIN, THEMIS, and MMS flux comparisons

  2. 2.

SST19 algorithm description

  3. 3.

Magnetotail stretching factor and isotropy boundaries for ions and electrons

  4. 4.

Figures S1 – S9

Introduction

The Supporting Information includes the description of THEMIS and MMS
observations and their consistency with ELFIN observations, details of the
data mining algorithm SST19, and the SST19 derived estimates for the
stretching factor and its equatorial distributions including the corresponding
locations for the electron and ion isotropy boundaries.

ELFIN, THEMIS, and MMS flux comparisons

Here, we bracket the radial location of ELFIN’s projection to the magnetic
equator by comparing its particle fluxes during intervals of isotropic flux
measurements to flux data from MMS and THEMIS spacecraft. We select ELFIN
spectra with flux at 50–100keV closest to what THEMIS and MMS observed. We use
omni-directional flux averages from THEMIS and MMS (see references with
spacecraft instruments in the main text). Figures S1(a–d) depict magnetic
field and electron flux observations obtained from THEMIS-E and MMS during
05:00–11:00 UT. In this interval, THEMIS-E was located close to the equator,
moving tailward from $8R_{E}$ to $11R_{E}$. MMS was located at a radial
distance of about $16R_{E}$ well below the equator ($|B_{x}|>B_{z}$) and
mostly near the plasma sheet boundary layer which maps to equatorial locations
much farther tailward than $16R_{E}$. Figure S1(g) compares electron flux
measurements from THEMIS at $8R_{E}$ and $10R_{E}$ (black lines), MMS at
$16R_{E}$ (red lines), and ELFIN (at the ionosphere) both near the IBe (blue
lines) and far poleward of it (gray line). The comparison between ELFIN and
THEMIS fluxes suggests that the IBe source population is outside $8R_{E}$ and
inside of, or near $10R_{E}$. The electron flux measured by ELFIN well
poleward of the IBe (gray line) at 08:45 UT is comparable to the MMS flux
measured at $\sim 16R_{E}$ and mapping to the plasma sheet equator near
(though still tailward of) $\sim 16R_{E}$. In summary, the above
ELFIN/MMS/THEMIS comparisons demonstrate that the IBe observed by ELFIN maps
to $\sim 10R_{E}$ and that the ELFIN plasma sheet measurements project to
distances tailward of $16R_{E}$.

Figure S1(e) shows the ion flux measured by MMS. For the two intervals, around
$\sim$06:00 and 10:00 UT, MMS was in the local plasma sheet and was closest to
the local equatorial plane to measure ion fluxes above the noise level. The
MMS ion spectra for these two times are shown in Fig. S1(f) for comparison
with ELFIN’s. As expected from electron spectra comparisons, MMS ion fluxes
from equatorial distances at or beyond $16R_{E}$ are well below ELFIN ion
fluxes within the electron isotropy boundary, confirming that the ELFIN IBe
was mapped to the equator well earthward of $\sim 16R_{E}$ radial distances.
Figure S1(f,g) determines the equatorial projection of the IBe (and IBi, that
is equatorward of, or maps earthward of the IBe) in the near-Earth plasma
sheet at or around $8R_{E}$ to $10R_{E}$, the likely transition region between
the plasma sheet and the outer radiation belt at that time.

SST19 algorithm description

In this study, we employ the SST19 empirical magnetic field reconstruction
algorithm (Stephens et al., 2019, 2023; Stephens and Sitnov, 2021) to
determine the location of IBs within the magnetotail during a substorm. In
particular, the version of SST19 used here follows that of Stephens et al.,
(2023), with three modifications detailed below. First, 29 additional months
of MMS data have been added to the space magnetometer archive. Secondly, a new
formulation of the spatially varying thickness for the magnetotail current
sheet is utilized. Third, to better resolve the magnetic field in the inner
magnetosphere and the transition region, the “merged resolution” procedure
(Stephens and Sitnov, 2021) is employed.

The SST19 algorithm consists of a data mining (DM) and a fitting component.
The DM part characterizes the storm/substorm state of the magnetosphere using
a 5-D set of global parameters $\mathbf{G}(t)=(G_{1}-G_{5})$: composed of the
solar wind electric field, $vB_{s}^{\mathrm{IMF}}$, (where $v$ is the solar
wind speed and $B_{s}=-B_{z}^{\mathrm{IMF}}$ when the northward component of
the IMF is negative, $B_{z}^{\mathrm{IMF}}<0$, and $B_{s}=0$ otherwise), the
SuperMAG pressure-corrected storm, SMRc, and substorm, SML, indices, as well
as their time derivatives. These parameters are smoothed in time over substorm
and storm scales, standardized by dividing by their standard deviations, and
sampled at a 5-min cadence as is detailed in Stephens et al., (2019, 2023).

At every moment of interest, $t=t^{(q)}$, the historical archive of space
magnetometer observations is mined in the 5D state-space to select other
moments, termed nearest-neighbors (NNs), whose global parameters,
$\mathbf{G}$, are closest to the query point $\mathbf{G}^{(q)}$. The archive
of magnetometer data ($\sim 9.2\cdot 10^{6}$ records averaged to 5- and 15-min
cadences spanning the years 1995–2023) employed here is similar to that
Stephens et al., (2023) with the only difference being that the MMS portion
was extended to include data through the end of May 2023, thereby adding 29
months of MMS data. The chosen number of NNs, $k_{\mathrm{NN}}$, must be small
enough to ensure sensitivity to the specific event being reconstructed while
large enough to avoid overfitting. Here, as with prior SST19 studies,
$k_{\mathrm{NN}}=32,000$ which corresponds to $S_{\mathrm{NN}}\sim 9\cdot
10^{4}$ magnetometer records or approximately $1\%$ of the entire archive,
which can include the few records available at $t=t^{(q)}$. $S_{\mathrm{NN}}$
is not constant and is larger than $k_{\mathrm{NN}}$ as there tends to be more
than one magnetometer record for any given NN. The NN farthest from
$\mathbf{G}^{(q)}$ defines the radius, $R_{\mathrm{NN}}$, of the NN
hypersphere, and distances are computed using the Euclidean distance metric
such that $|\mathbf{G}-\mathbf{G}^{(q)}|\leq R_{\mathrm{NN}}$.
$R_{\mathrm{NN}}$ likewise is not constant and generally increases during
times with greater activity.

This instance-based set of $S_{\mathrm{NN}}$ magnetometer records is then used
to fit the analytical description of the magnetic field model represented by
the sum:
$\mathbf{B}=\mathbf{B}_{\mathrm{int}}+\mathbf{B}_{\mathrm{eq}}+\mathbf{B}_{\mathrm{FAC}}+\mathbf{B}_{\mathrm{MP}}$,
where the internal field, $\mathbf{B}_{\mathrm{int}}$, is not part of the
reconstruction but prescribed using the IGRF model (Alken et al.,  2021). The
reconstructed fields $\mathbf{B}_{\mathrm{eq}}$, $\mathbf{B}_{\mathrm{FAC}}$,
and $\mathbf{B}_{\mathrm{MP}}$ are generated by the equatorial, field-aligned
(FAC), and magnetopause currents, respectively. Since the resulting
reconstruction is specific to the query-time, $t^{(q)}$, it is not universal
and is thereby “event-oriented”. This learning process is called instance-
based, in contrast to conventional empirical reconstructions (e.g., Tsyganenko
and Sitnov, 2005, and refs. therein) which are called “model-based” using
machine learning terminology. The fitting element finds the optimal values for
the free parameters defining the model’s analytical structure by minimizing
the root-mean-square difference between the modeled magnetic field and the set
of $S_{NN}$ magnetometer records. The result is the reconstructed magnetic
field for time $t^{(q)}$: $\mathbf{B}(\mathbf{r},t=t^{(q)})$. The DM and
fitting components are repeated for each time step, at a 5-min cadence, to
resolve the dynamical evolution of the magnetospheric magnetic field.

A fundamental advantage of the SST19 approach compared to conventional
empirical reconstructions is the description of the magnetospheric currents
using basis function expansions for the corresponding magnetic fields rather
than custom-made modules with variable amplitudes. In particular, the magnetic
field of the equatorial current system, $\mathbf{B}_{\mathrm{eq}}$, is based
on the general solution for Ampère’s equation for a thin current sheet (CS) in
cylindrical coordinates $(\rho,\phi,z)$ taking the form (Tsyganenko and
Sitnov, 2007):

$\mathbf{B}_{\mathrm{sheet}}(\rho,\phi,z)=\displaystyle\sum_{n=1}^{N}a_{0n}^{\mathrm{(s)}}\mathbf{B}_{0n}^{\mathrm{(s)}}+\displaystyle\sum_{m=1}^{M}\displaystyle\sum_{n=1}^{N}(a_{mn}^{\mathrm{(o)}}\mathbf{B}_{mn}^{\mathrm{(o)}}+a_{mn}^{\mathrm{(e)}}\mathbf{B}_{mn}^{\mathrm{(e)}}),$
(1)

where $\mathbf{B}_{\alpha\beta}^{(\gamma)}$ are basis functions with axial,
odd (sine), and even (cosine) symmetry, while $a_{\alpha\beta}^{(\gamma)}$ are
the amplitude coefficients. This solution possesses a characteristic CS half-
thickness represented by the variable $D$. An example of the basis functions
used in eq. (1) can be given by the azimuthal component, $A_{\phi}$, of the
vector potential corresponding to the first group of basis functions:
$\mathbf{B}_{0n}^{(s)}$:
$(A_{\phi})_{0n}^{\mathrm{(s)}}=J_{1}(k_{n}\rho)\exp{(-k_{n}\sqrt{z^{2}+D^{2}})}$,
where $J_{1}$ is the Bessel function of the first order, $k_{n}=n/R_{0}$, and
$R_{0}$ is the radial scale set to $20R_{E}$, corresponding to the largest
mode in the radial expansion. The variables $R_{0}$, $N$, and $M$ are fixed
because they determine the adopted spatial resolution of the equatorial
current described by eq. (1). Other variables, such as the coefficients
$a_{\alpha\beta}^{(\gamma)}$ and the CS thickness $D$, are determined by
fitting the model to data. Thus, the spatial resolution of such an expansion
is determined by the number of terms in eq. (1) and can be increased to any
desired level, commensurate with the data availability. Each element in eq.
(1) is independently shielded (has its subsystem of Chapman-Ferraro-type
currents at the magnetopause contributing to $\mathbf{B}_{\mathrm{MP}}$).

To account for the dependence on the solar wind dynamic pressure,
$P_{\mathrm{dyn}}$, the scaling coefficients $a_{\alpha\beta}^{(\gamma)}$ are
split into two groups $a_{\alpha\beta}^{(\gamma)}\to
a_{\alpha\beta}^{(\gamma)}+a_{\alpha\beta}^{(\gamma)}\sqrt{P_{\mathrm{dyn}}}$,
doubling the number of coefficients in eq. (1). To account for seasonal and
diurnal variations of the Earth’s dipole tilt angle $\Psi$, resulting in a
periodic transverse motion and large-scale deformation of the tail current
sheet as well as the CS warping and twisting effects, the equatorial magnetic
field is deformed (e.g., Tsyganenko and Sitnov, 2005, and refs. therein).
These deformations introduce three free nonlinear parameters: the hinging
distance $R_{\mathrm{H}}$, the warping parameter $G$, and the twisting
parameter $T_{\mathrm{W}}$, which are determined during the fit.

SST19 employs two independent equatorial current sheets described by eq. (1)
with the same structure but different CS half-thickness parameters $D$ and
$D_{\mathrm{TCS}}$ separated by their spatial sizes to take into account the
buildup and decay of ion-scale thin current sheets (TCS) during substorms
(Sergeev et al., 2011). Moreover, while the thicker CS adopts a spatially
constant half-thickness defined using a single free nonlinear parameter
$D=\mathrm{const}$, the structure of the TCS utilizes a spatially varying
half-thickness, $D_{\mathrm{TCS}}(\rho)$, based on the distance from Earth,
$\rho=\sqrt{x^{2}+y^{2}}$, following similar approaches in Tsyganenko and
Sitnov, (2007) and Stephens et al., (2023). Here, we employ a new flexible TCS
thickness model:

$D_{\mathrm{TCS}}(\rho)=d_{\mathrm{c}}\tanh{[[d_{\mathrm{thr}}+D_{0}\tanh(\beta\rho)+\alpha\exp{(-\varepsilon\rho/2)}\cos{(3\varepsilon\rho/2)}]/d_{c}]},$
(2)

introducing four free nonlinear parameters $D_{0}$, $\alpha$, $\beta$, and
$\varepsilon$, whereas $d_{\mathrm{thr}}=0.2R_{E}$ and
$d_{\mathrm{c}}=3.0R_{E}$ are fixed to constrain the TCS thickness within a
physically valid range.

The last element employed for the description of the magnetic field from the
equatorial current system is the “merged resolution” version of the SST19
(Stephens and Sitnov, 2021). The “merged resolution” version addresses a
shortcoming of the SST19 approach: resolving different regions of the
equatorial current systems requires different spatial resolutions for the same
set of NNs, that is, there is not a single value for $(M,N)$ that concurrently
resolves all the equatorial currents. This is caused by the disparate density
of the available spacecraft magnetometer data and the inherently different
spatial scales of the equatorial current systems. For example, adequate
resolution of the inner magnetosphere and the transition region ($r<12R_{E}$)
requires around 20 radial expansions ($N=20$ in eq. (1)) (Stephens et al.,
2016). However, applying this resolution to the near-tail region
($r>12R_{E}$), where the data density is sparser, results in signatures of
overfitting. This results in using one spatial resolution for studies of the
inner magnetosphere and the transition region, $(M,N)=(3,20)$ or $560$
equatorial expansions in eq. (1), and another for resolving the near-tail
region, $(M,N)=(6,8)$ or $416$ equatorial expansions. The “merged resolution”
version rectifies this issue by reconstructing the inner magnetosphere and the
transition region using the higher spatial resolution ($(M,N)=(3,20)$),
reconstructing the magnetotail using the lower resolution ($(M,N)=(6,8)$), and
then merging these two results into one coherent picture that concurrently
resolves both regions. The merging procedure fits a composite resolution
architecture, $(M,N)=(6,20)$, to magnetic field records constructed by
randomly sampling the two other magnetic field reconstructions within their
respective spatial domains, as is described in detail in Stephens and Sitnov,
(2021).

The magnetic field of the FAC system, $\mathbf{B}_{\mathrm{FAC}}$, is
presented in the form of multiple elementary current blocks similar to eq. (1)
and described in more detail in Sitnov et al., (2017). The number of blocks is
$N_{\mathrm{FAC}}=16$, which introduces 16 more linear amplitude coefficients.
Two additional free nonlinear parameters, $\kappa_{\mathrm{R1}}$ and
$\kappa_{\mathrm{R2}}$, spatially rescale the higher latitude and lower
latitude systems respectively, enabling the FAC ovals to expand and contract
with changing activity levels. As with the equatorial system, each FAC system
is shielded by adding additional terms to $\mathbf{B}_{\mathrm{MP}}$, as
described in previous works. Other key variables of the SST19 algorithm
include the NN weighting index, $\sigma=0.3$, used to assign different weights
to the NNs depending on their distance to the query point within the
$R_{\mathrm{NN}}$ hypersphere (see Stephens et al. (2023) for more detail),
and the maximum radial distance of magnetometer records included in the
reconstructions, $r_{\mathrm{max}}=36R_{E}$, which are the same values used in
Stephens et al. (2023).

This analytical structure is fit by minimizing the weighted root-mean-square
difference between the SST19 magnetic field and the magnetic field of the set
of $S_{\mathrm{NN}}$ magnetometer records. Specifically, the linear amplitude
coefficients are found by applying the standard singular value decomposition
method for linear regression, while the 10 free nonlinear parameters ($D$,
$D_{0}$, $\alpha$, $\beta$, $\varepsilon$, $R_{\mathrm{H}}$, $G$,
$T_{\mathrm{W}}$, $\kappa_{\mathrm{R1}}$, $\kappa_{\mathrm{R2}}$ ) are solved
using the Nelder-Mead downhill simplex method (Press et al., 1992). The linear
solver is nested within the nonlinear one such that a new set of amplitude
coefficients is found whenever the values of a nonlinear parameter are
adjusted. This process repeats for 80 iterations.

In summary, the configuration of the SST19 model employed here follows that of
Stephens et al., (2023), which itself is a modest revision to the first
Stephens et al., (2019) version, with three changes: (1) an additional 29
months of MMS magnetometer records included in the space magnetometer archive,
(2) the new formulation for the spatially varying TCS structure described by
eq. (2), and (3) the employment of the “merged resolution” description of the
equatorial current system detailed in Stephens and Sitnov (2021).

Magnetotail stretching factor and isotropy boundaries for ions and electrons

To assess the stretching of the tail magnetic field, which may cause chaos in
particle orbits and isotropize their distributions as detected by ELFIN at
low-altitudes, we calculate the parameter $G^{*}=B_{z}^{2}/(\mu_{0}j)$, here
termed the magnetotail stretching factor. Lower values of $G^{*}$ indicate a
more stretched magnetotail. The value of the stretching factor for an ideal
infinitely thin current sheet, with antiparallel magnetic field lines above
and below the sheet, is zero. In contrast, the magnetic equator of a purely
dipolar field, where the current density is zero, has an infinitely large
stretching factor. Its equatorial distributions (along with the corresponding
distributions of meridional and equatorial currents as well as the equatorial
$B_{z}$ field) for several moments in the 19 August 19 2022 substorm are shown
in Figures S2 and S3. The isotropization is expected to appear (e.g., Sergeev
et al., 2018, and refs. therein) when $G^{*}<8G$, where
$G_{e,p}=m_{e,p}V_{e,p}/e$ is the particle rigidity, $m_{e,p}$ are masses of
electrons and protons, and $V_{e,p}$ their velocities. The specific values
used in the plots correspond to 520 keV electrons and protons. In constructing
the equatorial distributions in Figures 2, 3, S2, and S3, the dipole tilt and
twisting deformations are ignored by setting $\Psi=T_{\mathrm{W}}=0$.

Figures S4 and S5 show the equatorial $B_{z}$ and $G^{*}$ mapped to ELFIN’s
altitude on an MLT-MLAT grid. These plots facilitate the subsequent comparison
of ELFIN-observed IBs with those inferred from the SST19 magnetic field
reconstruction. Also, to provide the comparison of these reconstructions with
similar results from other missions (e.g., Sergeev et al., 2018), the analogs
of Figures S4 and S5 on an MLT-AACGMLAT grid (Shepherd 2014) are provided in
Figures S6 and S7. Note, that the higher latitude portions of the ELFIN orbits
often map to either open field lines or field lines that intersect the
magnetic equator beyond the $r=20R_{E}$. Any IBs inferred using $G^{*}$ values
that map to $r>20R_{E}$ are not considered in the main text. In addition,
Figure S8 presents validation of SST19 reconstructions using the magnetic
field from THEMIS and MMS.

Finally, to make sure that our reconstruction algorithm is reliable, we
provided in Figure S9 (in the format similar to Figure 2) the reconstruction
of three more IB events, one of which is another V-like pattern event shown in
Figure 8 of Artemyev et al. (2023) (2022-08-11). Two other events describe the
IBe detected by ELFIN on June 22, 2021, when the ion data was not available,
and an event with very strong activity (2022-09-05) when IBi and IBe came
particulalrly close to each other because of the very strong tail current and
its strong radial gradient.

Figure S1: Observations from THEMIS and MMS: (a,c) Magnetic field measurements
($B_{x}$ and $B_{z}$) in GSM coordinates. (b,d) Electron fluxes between $50$
keV to $600$ keV. (e) Ion fluxes from $50$ keV to $600$ keV. (f) Ion spectra
measured by ELFIN and MMS at the times indicated. (g) Electron spectra
measured by ELFIN, THEMIS, and MMS at the times indicated. Time moments of
ELFIN, THEMIS, and MMS spectra are shown in the panels; these moments indicate
the center time for $\pm 1.5$s averaging ELFIN data and $\pm 15$min averaging
MMS and THEMIS data.  Figure S2: (Caption next page.) Figure S2: DM-based
empirical reconstructions of the magnetotail when ELFIN observed IBs during
the 19 August 2022 substorm. The times approximately correspond to the first
three lines from Figure 1 but are rounded to the nearest 5-min based on the
SST19’s time cadence. (a1–a3) Meridional slices ($y=0$) of the color-coded
y-component of the electric current density, $j_{y}$, with green (pink)
corresponding to current flowing out of (into) the page. Magnetic field lines
(black), seeded at every $2~{}^{\circ}$ MLAT, are overplotted with selected
field lines at $\textit{MLAT}=66^{\circ},68^{\circ},70^{\circ}$ highlighted.
(b1–b3) Color-coded equatorial distributions of the electric current density,
$j$, with arrows overplotted to indicate the direction of vector current
density, $\mathbf{j}$. (c1–c3) Color-coded equatorial distributions of the
z-component of the magnetic field, $B_{z}$, with grey dots overplotted to
indicate the locations, projected to the $x$-$y$ plane, of the spacecraft
magnetometer observations identified using the $K_{\mathrm{NN}}$ procedure and
used to fit the analytical description of the magnetic field. (d1–d3) Color-
coded equatorial distributions of the stretching factor
$G^{*}=B_{z}^{2}/(\mu_{0}j)$. The critical values of this parameter
corresponding to the IB for 520 keV electrons and protons are shown by the
orange and green contours respectively. The projections of the locations of
the THEMIS E and MMS1 at the indicated times are overplotted by the colored
circles in each panel. The location of ELFIN mapped to the magnetic equator at
the time when it observed the 520 keV IBi and IBe are overplotted in green and
orange circles respectively. Figure S3: DM-based empirical reconstructions of
the magnetotail when ELFIN observed IBs during the 19 August 2022 substorm.
The times approximately correspond to the last three lines from Figure 1 but
are rounded to the nearest 5-min based on the SST19’s time-cadence. The panels
are the same as Figure S1 but are at different times. Figure S4: DM-based
empirical reconstructions of the magnetotail mapped to the altitude of ELFIN’s
orbit during the 19 August 2022 substorm. The times approximately correspond
to the first three lines from Figure 1 but are rounded to the nearest 5-min
based on the SST19’s time-cadence. (a1–a3) The color-coded equatorial
distribution of the z-component of the magnetic field mapped to $328$ km
altitude in the midnight sector. (b1–b3) The color-coded equatorial
distributions of the stretching factor $G^{*}=B_{z}^{2}/(\mu_{0}j)$ mapped to
$328$ km altitude in the midnight sector. ELFIN’s orbit is indicated by the
grey line and its locations where it observed the 520 keV IBi and IBe are
shown by the green and orange circles respectively. The turquoise contours
indicate the radial distance of the magnetic equator position to which the
field lines map. Figure S5: DM-based empirical reconstructions of the
magnetotail mapped to the altitude of ELFIN’s orbit during the 19 August 2022
substorm. The times approximately correspond to the last three lines from
Figure 1 but are rounded to the nearest 5-min based on the SST19’s time-
cadence. The panels are the same as Figure S3 but are at different times.
Figure S6: DM-based empirical reconstructions of the magnetotail mapped to the
altitude of ELFIN’s orbit during the 19 August 2022 substorm. The panels are
the same as Figure S3 but the $y$-axis now uses AACGM latitudes. Figure S7:
DM-based empirical reconstructions of the magnetotail mapped to the altitude
of ELFIN’s orbit during the 19 August 2022 substorm. The panels are the same
as Figure S4 but the $y$-axis now uses AACGM latitudes. Figure S8: Validation
of the DM-based empirical reconstructions of the 19 August 2022 substorm.
(a–c) Component-wise comparison between the observed total magnetic field,
$\mathbf{B}_{\mathrm{tot}}$, (black line) averaged to a 5-min cadence and its
modeled value (red line) also at a 5-min cadence for the MMS1 spacecraft in
GSM coordinates. (d) The MMS1 ephemeris showing the $x$ (solid line), $y$
(dashed line), $z$ (dotted line), and radial distance $r$ (purple line) in GSM
coordinates. (e–h) The same as panels (a–d) except for the THEMIS A
spacecraft. (i) The SuperMAG pressure-corrected storm index SMRc (black line)
and substorm index SML (orange line). Their smoothed values, used in the
$K_{\mathrm{NN}}$ procedure, are indicated by the dashed lines. (j) The solar
wind electric field parameter $vB_{z}^{\mathrm{IMF}}$ (black line) and dynamic
pressure $P_{\mathrm{dyn}}$ (orange line). The smoothed value of
$vB_{s}^{\mathrm{IMF}}$, used in the $K_{\mathrm{NN}}$ procedure, is indicated
by the dashed line. (k–n) The same as panels (a–d) except for the THEMIS D
spacecraft. (o–r) The same as panels (a–d) except for the THEMIS E spacecraft.
Figure S9: (Caption next page.) Figure S9: SST19 reconstructions of IBs
compared to mapped ELFIN observed locations for three more events. (a) The
SuperMAG pressure-corrected storm index SMRc (black line) and substorm index
SML (orange line). Their smoothed values, used in the $K_{\mathrm{NN}}$
procedure, are indicated by the dashed lines. (b) The solar wind electric
field parameter $vB_{z}^{\mathrm{IMF}}$ (black line) and dynamic pressure
$P_{\mathrm{dyn}}$ (orange line). The smoothed value of
$vB_{s}^{\mathrm{IMF}}$, used in the $K_{\mathrm{NN}}$ procedure, is indicated
by the dashed line. The reconstructed time is shown by the vertical purple
line. (c–f) The panels are the same as Figures S2 and S3, but for different
events.