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**Annales Geophysicae**
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- Abstract
- Introduction
- Summary of the onset timing study using ground Pi2s at the Equator
- Pre-onset intervals leading to field line dipolarization
- Excitation of slow magnetoacoustic waves
- Field line dipolarization in the vicinity of geosynchronous orbit
- Coupling of magnetosphere and ionosphere in association with field line dipolarization
- Triggering mechanisms of low-latitude Pi2s
- Discussion and summary
- Appendix A
- Code and data availability
- Competing interests
- Acknowledgements
- Review statement
- References

**Regular paper**
07 Apr 2020

**Regular paper** | 07 Apr 2020

The increase in the curvature radius of geomagnetic field lines preceding a classical dipolarization

- Office Geophysik, Ogoori, 838-0141, Japan

**Correspondence**: Osuke Saka (saka.o@nifty.com)

Abstract

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Based on assumptions that substorm field line dipolarization at geosynchronous altitudes is associated with the arrival of high-velocity magnetotail flow bursts referred to as bursty bulk flows, the following sequence of field line dipolarization is proposed: (1) slow magnetoacoustic wave excited through ballooning instability by enhanced inflows in pre-onset intervals towards the equatorial plane; (2) in the equatorial plane, slow magnetoacoustic wave stretching of the flux tube in dawn–dusk directions resulting in spreading plasmas in dawn–dusk directions and reduction in the radial pressure gradient in the flux tube. As a consequence of these processes, the flux tube assumes a new equilibrium geometry in which the curvature radius of new field lines increased in the meridian plane, suggesting an onset of field line dipolarization. The dipolarization processes associated with changing the curvature radius preceded classical dipolarization caused by a reduction of cross-tail currents and pileup of the magnetic fields.

Increasing the curvature radius induced a convection surge in the equatorial plane
as well as inductive westward electric fields of the order of millivolts per meter (mV m^{−1}). Electric
fields transmitted to the ionosphere produce an electromotive force in the E
layer for generating a field-aligned current system of Bostrom type. This is
also equivalent to the creation of an incomplete Cowling channel in the
ionospheric E layer by the convection surge.

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How to cite.

Saka, O.: The increase in the curvature radius of geomagnetic field lines preceding a classical dipolarization, Ann. Geophys., 38, 467–479, https://doi.org/10.5194/angeo-38-467-2020, 2020.

1 Introduction

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Substorms are spatially localized and temporarily variable processes in the
nighttime magnetosphere. It is often difficult to determine the onset timing of
substorm processes such as magnetotail flow burst, field line
dipolarization and particle injections. To resolve the timing
uncertainties, auroras in global satellite images (Nakamura et al., 2001;
Miyashita et al., 2009), intensifications of auroral kilometric radiation
(Fairfield et al., 1999; Morioka et al., 2010) and dispersionless particle
injection in geosynchronous orbit (Birn et al., 1997) were used. Geomagnetic Pi2 micropulsations observed on the ground are another useful tool for determination of the substorm timing
(Sakurai and Saito, 1976; Nagai et al., 1998; Baumjohann et al., 1999).
Particularly, Pi2s in the equatorial region exhibited a small phase difference
(*m**<*1, *m* denotes azimuthal wave number) across widely separated
stations in the equatorial countries (Kitamura et al., 1988), minimizing the
timing uncertainties arising from delays in longitudinal propagations. This
enabled an accurate onset timing study of substorms using magnetometer data
from two remote locations, geosynchronous altitudes and ground stations of
the equatorial countries (Saka et al., 2010).

In this study, the focus is on the dipolarization events in geosynchronous orbit from the growth to the expansion phase. Triggering mechanisms of the field line dipolarization in the vicinity of geosynchronous orbit are our major concern. In this paper, the onset timing study of substorms using magnetometer data from equatorial countries is summarized in Sect. 2. In Sect. 3, a pre-onset scenario leading to the dipolarization onset is presented. In Sect. 4, the excitation of slow magnetoacoustic waves is discussed for triggering field line depolarization. The focus will be on the field line dipolarization in the vicinity of geosynchronous orbit in Sect. 5. A coupling of magnetosphere and ionosphere associated with this dipolarization scenario will be presented in Sect. 6. In Sect. 7, a triggering mechanism of low-latitude Pi2s that enabled the Pi2-based epoch analyses is presented. Summary and discussion of this scenario is given in Sect. 8.

2 Summary of the onset timing study using ground Pi2s at the Equator

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This section comprises a summary of the field line dipolarization occurring at the
geosynchronous orbit based on the statistical results obtained by Saka et
al. (2010). The authors used magnetometer data from geosynchronous
satellites (GOES-5 and GOES-6) and those at ground equatorial stations
(Huancayo, Peru, 1.4^{∘} N in geomagnetic latitudes) in the conjugate
meridian. GOES-5 was located at a higher latitude, 10.3^{∘} N in dipole
coordinates, and GOES-6 was closer to the Equator, 7.9^{∘} N in dipole
coordinates. This difference was caused by the separated meridians of the
satellites (2.2 h of local time). The dipole coordinates used are
equivalent to the HDV coordinates; *H* is positive northward along the dipole
axis, *V* is radial outward, and *D* denotes dipole east. The field line
dipolarization at the geosynchronous orbit can be characterized either by a
step-like or impulsive increase in the inclination angle of the geomagnetic
field lines. The inclination angle is measured positive northward from the
dipole equator. The step-like dipolarization was observed by GOES-5 located
at higher latitudes, while the dipolarization pulse was observed by GOES-6 at
latitudes closer to the equatorial plane.

The onset of field line dipolarization preceded the initial peak of the ground Pi2 pulse by 2 min, suggesting that the onset was initiated in association with the first increase in the Pi2 amplitudes. Following the dipolarization onset, field line magnitude decreased at the geosynchronous orbit, and field lines deflected westward in the dawn sector and eastward in the dusk sector (see Fig. 1 for dawn–dusk deflection, reproduced from Saka et al., 2010). This is caused by the dawn–dusk expansion of the plasma flows occurring tailward of the geosynchronous orbit. These longitudinal expansions lasted for about 10 min and decreased the field magnitudes therein. Expansion in the dusk sector, however, continued over this characteristic 10 min interval. Asymmetries of the dawn–dusk expansion may be caused by diamagnetic drifts in the plasma sheet (Liu et al., 2013). It is suggested that classical dipolarization, caused by the reduction of cross-tail currents in the midnight magnetosphere, happened after the nightside magnetosphere experienced this characteristic 10 min interval. For this reason, the first 10 min intervals are referred to as a transitional state of substorm expansion (Saka et al., 2010).

3 Pre-onset intervals leading to field line dipolarization

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In the pre-onset intervals, a decrease in the field line inclination started
2 h prior to the dipolarization onset. It attained minimum angles
(33.6^{∘} for GOES-5 and 49.4^{∘} for GOES-6 in dipole
coordinates) right before the dipolarization onset (Saka et al., 2010; Saka, 2019).

One of the properties of plasmas in pre-onset intervals are continuing inflows of lobe plasmas towards the equatorial plane (Birn and Hesse, 1996), Poynting flux enhancement (Machida et al., 2009) and Ey (westward electric fields) penetration toward the equatorial plane (Machida et al., 2014). Corresponding plasma properties at geosynchronous altitudes may be predominant perpendicular temperature anisotropies of thermal plasmas (30 eV–40 keV) obtained from a three-dimensional temperature matrix, and they gradual decrease towards the onset (Birn et al., 1997). At the onset, however, the increase in parallel anisotropy stopped and perpendicular anisotropy increased again. Such changes of temperature anisotropy at onset were observed in a roll-angle spectrogram of energy flux of electrons in 15 eV–40 keV (Saka and Hayashi, 2017). This transition of the temperature anisotropies may be accounted for by the following scenario.

A continuing tailward stretch of the field lines in the pre-onset intervals
as depicted in Fig. 2 may increase equatorward flux by the
counterclockwise rotation of the inflow vectors (*F*_{⊥}) in the north
of the equatorial plane (clockwise rotation in the south) and produce a
parallel component as well by the relation

$$\begin{array}{}\text{(1)}& \mathit{\delta}{F}_{\parallel}={F}_{\perp}\left(\mathit{\omega}\cdot \mathit{\delta}t\right).\end{array}$$

Here, *δ**F*_{∥} denotes an increase in parallel flux per time, *δ**t*, and *ω* is the angular velocity of the rotation of*F*_{⊥} vectors
associated with the thinning of the flux tubes caused by stretching. In
pre-onset intervals lasting 90 min at geosynchronous altitudes, field line
stretching decreased the field line inclination by 7^{∘} from
40.6 to 33.6^{∘} (Saka, 2019). This gives angular velocity
of the rotation of the field line inclination in Eq. (1) of $\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ rad min^{−1}. Total parallel flux gained in *T* min may be given by
the integral of Eq. (1) with time from 0 to *T*. Substituting *T*=60 min
and $\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ rad min^{−1} for angular velocity of field line
inclination, this yields ${F}_{\parallel}=\mathrm{8.2}\times {\mathrm{10}}^{-\mathrm{2}}\cdot {F}_{\perp}$. The gain
of *F*_{∥} is about 10 % of the perpendicular flux (*F*_{⊥}). This
is consistent with the parallel temperature anisotropies gained prior to the
onset (20 % gain) in geosynchronous orbit (Birn et al., 1997).

Continuing parallel flux flows associated with the flux tube stretching in the pre-onset intervals may increase plasma pressures in the flux tube at its tailward end. This condition leads to further stretching of the flux tube (small curvature radius) (Ohtani and Tamao, 1993; Rubtsov et al., 2018) by the relation

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{\mathit{\beta}}{\mathrm{2}}}\mathit{\kappa}+{\mathit{\kappa}}_{B}+{\displaystyle \frac{\mathrm{1}}{R}}=\mathrm{0}.\end{array}$$

Here, *β* is plasma to magnetic pressure ratio, and *κ* and *κ*_{B} denote reciprocal spatial scales of radial inhomogeneity of plasma
pressure and magnetic fields in the equatorial plane, respectively. *R* is
the curvature radius of the field lines.

4 Excitation of slow magnetoacoustic waves

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The continuing parallel flows may excite magnetoacoustic waves. From a set of
linearized MHD equations there is a relation between parallel displacement
along the field lines (*ξ*_{z}) and divergence of perpendicular
displacements (*ξ*_{⊥}) in the following form (see Appendix):

$$\begin{array}{}\text{(3)}& {\mathit{\xi}}_{z}={\displaystyle \frac{{C}_{\mathrm{s}}^{\mathrm{2}}}{{\mathit{\omega}}^{\mathrm{2}}}}F\cdot {B}_{\mathrm{0}}^{\mathrm{2}}{\displaystyle \frac{\partial}{\partial z}}\left(\mathrm{div}{\mathit{\xi}}_{\perp}\right).\end{array}$$

Here, *C*_{s}, *ω* and *B*_{0} are the sound velocity, angular
frequency of waves and background field magnitudes, respectively. *F* is given
by

$$\begin{array}{}\text{(4)}& F={\displaystyle \frac{{C}_{\mathrm{A}}^{\mathrm{2}}}{{B}_{\mathrm{0}}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{{C}_{\mathrm{s}}^{\mathrm{2}}-{\left(\frac{\mathit{\omega}}{k}\right)}^{\mathrm{2}}}}.\end{array}$$

*F* is positive for the slow magnetoacoustic wave and negative for the fast
magnetoacoustic wave. *C*_{A} and *k* denote Alfvén velocity and wave vector,
respectively. Equation (3) is used for the classification of slow and fast
magnetoacoustic waves. Slow magnetoacoustic waves yield perpendicular
expansion of the flux tubes at the converging point of parallel flows on the
equatorial plane. For fast waves, perpendicular shrinkage of flux tubes
occurs at the converging point of parallel flows (equatorial plane).

Equation (3) will be applied to simulate a possible effect of
magnetoacoustic waves on the pitch angle spectrogram. For this, drift
Maxwell distributions for phase space density (PSD) are used, assuming gyrotropy for
particle trajectories. PSD was composed of three parts: one drifting
parallel, another anti-parallel along the field lines, and the third part
perpendicular to the field lines. Figure 3a shows a pitch angle spectrogram
of energy flux with no drift velocities either perpendicular or parallel to
the background field lines. Energy flux is defined by (2*E*^{2}∕m^{2})*f*, where *E*, *m* and *f* are energy, mass
of particles and phase space density, respectively. Energy flux is given in
eV (cm^{2} s sr eV)^{−1}. Only parallel drift increased by
0.3, 0.6 and 1.0 *V*_{th} as shown in *B*, *C* and *D*. *V*_{th} denotes thermal velocity of the drift Maxwell distribution function. For *E*
and *F*, perpendicular drift increased to 0.3 and 0.5 *V*_{th} while
parallel drift remained at 1.0 *V*_{th}. Energy fluxes initially in quasi-trapped distribution (*A*) changed to more parallel and anti-parallel fluxes
as parallel and anti-parallel drift increased (*B*, *C* and *D*). Increasing
perpendicular drifts increased perpendicular fluxes in the pitch angle
distributions of *E* and *F*.

It was confirmed that magnetoacoustic waves produced coupling of parallel flux along the field lines and the perpendicular flux. However, slow magnetoacoustic waves were chosen for the wave mode because the flux tubes expanded (did not shrink) in the transitional interval as discussed in Sect. 2. Slow magnetoacoustic waves may be triggered through ballooning instability, when a large enough pressure gradient is reached in an earthward direction (Ohtani et al., 1989; Rubtsov et al., 2018).

The ballooning instability threshold *κ* (reciprocal
scale of radial inhomogeneity of plasma pressure) can be estimated using calculation results
given in Rubtsov et al.(2018). In a distance from *L*=5 to 10 *R*_{e},
instability threshold is given as approximately $\mathit{\kappa}=-\mathrm{1.0}\phantom{\rule{0.125em}{0ex}}{R}_{\mathrm{e}}^{-\mathrm{1}}$ (*κ* denotes a reciprocal spatial scale of radial
inhomogeneity of plasma pressure, and *R*_{e} is the Earth radius) for beta
defined by the ratio of plasma pressure and magnetic pressure exceeding 0.1.
This suggests that the ballooning instability develops at the geosynchronous
altitudes (curvature radius *R* is 2.2 *R*_{e}) when the spatial scale of the earthward
pressure gradient caused by the inflows becomes steeper than 1.0 *R*_{e}. In the following section it is shown that this theoretical consideration matched
observations.

5 Field line dipolarization in the vicinity of geosynchronous orbit

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The westward electric fields in the dipolarization front (DF)
(Runov et al., 2011) embedded in the leading edge of bursty bulk flow (BBF) can be assumed as an external stimulus for triggering ballooning instability. In this case
westward electric fields in the DF temporarily amplified the parallel flux
flowing towards the end point of the flux tube in the equatorial plane and
further steepened the earthward pressure gradient. If the earthward pressure gradient exceeds instability
threshold determined by *β* and initial curvature radius *R*, slow
magnetoacoustic waves can be excited (Rubtsov et al., 2018). Once the slow
magnetoacoustic wave was excited, perpendicular fluxes spread the plasmas in
dawn–dusk directions and smoothed (or relaxed) the radial gradient of plasma
pressures in the equatorial plane (smaller *κ*). This may result in
the transition of the flux tube geometry to a new configuration, an increase in the curvature radius of the field lines (larger *R*) (see Eq. 2).

Multiple Pi2 events were observed by AMPTE CCE on 31 August 1986 (Saka
et al., 2002) and an example can be seen of relaxation of radial inhomogeneity of
plasma pressures associated with field line dipolarization in Fig. 4. The
satellite passed the midnight sector (20:00–23:00 MLT) from 3 to 7 *R*_{e} at
latitudes south of the equatorial plane ($-\mathrm{8}{}^{\circ}$ MLat) when multiple
Pi2 events (with positive bays) were observed at a low-latitude station (KUJ) at
*L*=1.2 in the midnight sector (Fig. 4a). The inclination angle of field lines
along the satellite trajectory is shown in Fig. 4b. Dipolarization
occurred as marked by vertical arrows correlating to multiple onsets of Pi2s,
1 through 4 in Fig. 4a. Ion fluxes coming from the dawn sector (*J*_{−}) and from the dusk sector (*J*_{+}) at satellite altitudes were
measured by the instruments (two energy channels, 63–85 keV and 125–210 keV)
on board AMPTE CCE (Takahashi et al., 1996). A schematic of particle
measurement is shown at the top of Fig. 5. The flux difference
(${\mathit{J}}_{-}-{\mathit{J}}_{+}>\mathrm{0}$) increased in association with the onset of
multiple Pi2s (15:05 UT) and positive bays at KUJ (Fig. 4c and d). The sudden
increase was followed by the slow decrease in flux in the 63–85 keV channel and
rapid decrease in flux in the 125–210 keV channel. The flux difference,
${\mathit{J}}_{-}>{\mathit{J}}_{+}$, may be caused either by an earthward pressure
gradient or westward convection of plasmas. The different patterns of
the flux decrease with time in two energy channels, suggesting that the
measured flux difference, ${\mathit{J}}_{-}-{\mathit{J}}_{+}$, can be attributed to
an increase in the earthward pressure gradient and subsequent relaxation. Note
that the guiding center of ${\mathit{J}}_{-}/{\mathit{J}}_{+}$ is earthward or tailward of
the satellite position as depicted in the top of Fig. 5. The different
relaxation speed in two energy channels, slower for 63–85 keV and faster for
125–210 keV, suggest that the earthward pressure gradient (assumed to be
proportional to the flux gradient) decreased with time during the multiple-Pi2 event (Fig. 5). The flux difference (50 counts per sample) was 10 % of
the background flux both for 63–85 keV (Larmor radius is 250 km for 150 nT)
and for 125–210 keV (Larmor radius is 450 km); that is, the flux level
differed by 10 % at two locations 1000 km apart in radial distance for
63–85 keV and 1800 km for 125–210 keV. This gives an *e*-folding scale of the
earthward pressure gradient being 0.98 and 1.77 *R*_{e} for 63–85 keV and
125–210 keV, respectively. The 31 August event shows that the radial pressure
gradient was relaxed in the inner magnetosphere in association with the
increase in the field line inclination (dipolarization). Although the field
line dipolarization showed a sharp onset in satellite magnetometer data, it was noted that it did not occur in ion flux data. This may be true because the
ion flux change at the onset may be obscured by the contamination from the
past onsets transported across the field lines from the adjoining sector by
the electric fields and gradient or curvature drifts. The conclusion is that the
relaxation of spatial inhomogeneity started when the spatial scale of the
radial inhomogeneity approached 1.0 *R*_{e}, consistent with the theoretical consideration of ballooning instability by Rubtsov et al. (2018).

Meanwhile, field lines in the further earthward locations may be compressed
by the inward movement of the outer field lines. This process, associated
with the dipolarization onset, may increase the parameter *κ*_{B} in
Eq. (2), which may result in transition to a new geometry of earthward
field lines, a decrease in the curvature radius *R*. Transition of the field
line geometries for onset locations and ones in earthward locations are
schematically illustrated in Fig. 6. These field line geometries in the
meridian plane matched the third harmonic and fundamental harmonic
deformations of outer and inner field lines, respectively. This is often
observed in the midnight magnetosphere in the initial pulse of Pi2s (Saka et
al., 2012). Transitions of the flux tube geometry in the magnetosphere also
correspond to the production of negative bays in higher latitudes and
positive bays in lower latitudes. If it can be assumed that negative bays switched
to positive bays at certain latitudes, for example 60^{∘} in geomagnetic coordinates, this latitude can be mapped beyond the geosynchronous orbit
(*L*∼7 *R*_{e} or further tailward) as field line dipolarization
occurs along the stretched flux tubes. Consequently, this scenario requires
that the BBFs are not necessary to reach the inner magnetosphere to trigger the
substorm onset at lower latitudes. In the inset, flux tube deformations are
illustrated in the equatorial cross section at onset locations (field lines
1 and 2). Divergence of perpendicular flows (solid arrows) produced
dawn–dusk expansion of flux tubes (2) and the shrinkage of stretched flux
tubes (1) by relaxation of the radial inhomogeneity. Flux tube deformation
from 1 to 2 tended to preserve the total magnetic fluxes in the equatorial
cross section. From the local time distribution of the dawn–dusk expansion
of the flux tubes shown in Fig. 1, most of the flux tube transition such
as from 1 to 2 may occur tailward of geosynchronous orbit. Some of the
events, however, may happen earthward of the geosynchronous orbit (i.e.,
Ohtani et al., 2018).

An increase in the curvature radius, or earthward shrinkage of the flux
tubes, produces a reduction of the radial component of the field lines (*V* in
dipole coordinates) by adding positive *V* in the north of the equatorial
plane and negative *V* in the south. If amplitudes of the *V* component changed
by 10 nT in 1 min, the expected inductive electric fields (westward)
could be of the order of 1.0 mV m^{−1} when shrinkage was confined within 1 *R*_{e}
from the equatorial plane. The dawn–dusk expansion of the flux tubes may
also produce inductive electric fields (earthward and tailward in dawn and
dusk sector, respectively) of the same order of magnitude. They are Alfvén
waves, a wave mode in ballooning instability coupled with slow
magnetoacoustic wave (Rubtsov et al., 2018). The westward electric fields
produce earthward flow bursts referred to as convection surge. The inductive
electric fields produced by the dipolarization are of the same order of
magnitude observed in DF (Runov et al., 2011).

6 Coupling of magnetosphere and ionosphere in association with field line dipolarization

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The inductive electric fields may be transmitted along the field lines as
poloidally and toroidally polarized Alfvén waves (Klimushkin et al., 2004).
These electric fields produce a dynamic ionosphere in the polar region that
includes nonlinear evolution of ionospheric plasmas (poleward expansion), as
well as production of field-aligned currents and parallel potentials by
exciting ion acoustic waves under quasi-neutral conditions (Saka, 2019). It is
not the aim of this paper to describe in detail the dynamic processes in the
ionosphere, but to show a local production of currents in the ionosphere as
well as field-aligned currents by the penetrated electric fields. For this
purpose, the 10 August 1994 substorm event studied by Saka and
Hayashi (2017) was revisited. In this event, eastward expansion was observed in the field
line dipolarization region, which started at 11:55 UT (00:27 MLT) from 260^{∘} E geomagnetic longitude and expanded to 351^{∘} E in about 48 min. At the leading edge of the expansion, ground magnetometer data showed
bipolar event (quick change of the *D* component from positive to negative in
about 5 min), being confined in the expanding dipolarization front as a
substructure. The substructure in the leading edge of the field line
dipolarization will be examined as follows.

It can be assumed that magnetic signals on the ground are associated with the sum of the horizontal Hall currents in the ionosphere (Fukushima, 1971). These currents can be calculated by the relation

$$\begin{array}{}\text{(5)}& {\left(\mathrm{rot}\phantom{\rule{0.125em}{0ex}}\mathit{J}\right)}_{z}=-{\displaystyle \frac{\mathrm{1}}{{\mathit{\mu}}_{\mathrm{0}}}}{\mathrm{\nabla}}^{\mathrm{2}}{B}_{z}.\end{array}$$

The ground vertical component (*b*) was used as a proxy of *B*_{z} in the
ionosphere. The second derivative on the right-hand side of Eq. (5) is
approximated as

$$\begin{array}{}\text{(6)}& {\mathrm{\nabla}}^{\mathrm{2}}{B}_{Z}^{i}=\left({\displaystyle \frac{{b}^{i+\mathrm{1}}-{b}^{i}}{{L}_{i+\mathrm{1}}-{L}_{i}}}-{\displaystyle \frac{{b}^{i}-{b}^{i-\mathrm{1}}}{{L}_{i}-{L}_{i-\mathrm{1}}}}\right)/\left({L}_{i+\mathrm{1}}-{L}_{i-\mathrm{1}}\right).\end{array}$$

Here, *i* denotes *i*th station in the meridian chain. *L*_{i} is the
geomagnetic latitude of the *i*th station. Only meridional change
was considered. This is because the vertical component changed from negative to
positive across the meridian, while in longitudes it simply
decreased or increased in lower and higher latitudes after onset,
respectively. Hence, longitudinal variations may contribute less to the
Laplacian. The results reproduced from Saka and Hayashi (2017) are shown in
Fig. 7a. The eastward propagation of the dipolarization front crossed this
meridian (300^{∘} E) at 12:13 UT, corresponding to the interval
labeled 1. Two points arose from this figure: (1) the loop of the Hall current pair
existed, and counterclockwise rotation (CCW) can be viewed from above the ionosphere in the lower latitudes and clockwise rotation (CW)
in the higher latitudes; (2) these current patterns expand poleward. Current
patterns in the interval from 1 to 5 in Fig. 7a are illustrated in
Fig. 7b to facilitate the poleward expansion. It is clearly demonstrated
that current pair forming CW in higher latitudes and CCW in lower latitudes
expanded in time towards the pole. Bipolar change can be seen in the D
component data (not shown) when the ground station, FSIM in this case,
passes from segment 1 to 2 in Fig. 7b. As a result, the dipolarization front
expanded eastward progressively by producing the poleward expansion at each
meridian. The front left behind the current pattern comprising upward
field-aligned currents in lower latitudes and downward in higher latitudes,
or Bostrom-type current system. It is proposed that the ionosphere itself has
inherent electromotive force to drive this Bostrom-type current system. The
reasons are as follows.

In the E region, drift trajectories may be written (Kelley, 1989) for electrons by

$$\begin{array}{}\text{(7)}& {\mathit{U}}_{\mathrm{e}\perp}={\displaystyle \frac{\mathrm{1}}{B}}\left[\mathit{E}\times \widehat{\mathit{B}}\right]\end{array}$$

and for ions by

$$\begin{array}{}\text{(8)}& {\mathit{U}}_{i\perp}={b}_{\mathrm{i}}\left[\mathit{E}+{\mathit{\kappa}}_{\mathrm{i}}\mathit{E}\times \widehat{\mathit{B}}\right].\end{array}$$

Here, *b*_{i} is the mobility of ions defined as Ω_{i}∕(*B**ν*_{in}), and *κ*_{i} is defined as
Ω_{i}∕*ν*_{in}. Symbols Ω_{i} and
*ν*_{in} are ion gyrofrequency and ion-neutral collision frequency,
respectively. $\widehat{\mathit{B}}$ denotes a unit vector of the magnetic fields
*B*. It was assumed that ** E**×

7 Triggering mechanisms of low-latitude Pi2s

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From ground magnetometer observations in the auroral zone, it is natural to
assume that flux tubes linked to negative bays (decreasing of the *H* component) and positive bays (increasing of the *H* component) at higher and
lower latitudes, respectively, oscillated coherently at Pi2 periods.
Oscillating flux tubes associated with positive bays may produce local
compression of magnetic fields at the Equator and trigger cavity mode at low
latitudes (Takahashi et al., 1995). Oscillations, however, are short-lived
and may not establish true cavity modes. They excite cavity/waveguide modes
in the plasmasphere (Allan et al., 1996; Li et al., 1998).

At the dip equator, a singular latitude of the cavity/waveguide mode, only the isotropic mode can be excited (Allan et al., 1996). This leads to the supposition that a very large propagation velocity (or large wavelength exceeding the whole circle of the Earth) of equatorial Pi2s in the nightside sector (Kitamura et al., 1988) would be associated with the dawn–dusk asymmetries of non-propagating compressions.

Pi2 periodicity may be determined primarily by the consecutive arrival of BBF substructures referred to as a dipolarization front bundle (DFB) (Liu et al., 2013, 2014). Repeating arrival of DFB produces periodic dipolarization or oscillation of negative bays. Positive bay oscillations in the plasmasphere would follow the negative bay oscillations to excite cavity/waveguide modes for low to equatorial Pi2s at the same periodicities. To estimate the onset time of the field line dipolarization using the very low-latitude Pi2s, delays in transmission are from the magnetosphere; longitudinal delays across the meridian may not be significant.

High-latitude Pi2s may not be caused by cavity/waveguide modes but by the
oscillation of field-aligned currents comprising a Bostrom-type current system
(incomplete Cowling channel), R1 (region 1) type current system associated
with convection surge (i.e., Birn and Hesse, 1996) and R2 (region 2) type
current system of expanding flux tubes in longitudes (i.e., Tanaka et al.,
2010). In contrast to the very low-latitude Pi2s associated with the
non-propagating compression, the high-latitude Pi2s propagated on the ground
typically at 20 km s^{−1} eastward and westward in the sectors east and west of the
substorm center, respectively (Samson and Harrold, 1985). Propagation across
the meridian may cause further delays: 35 s for propagation of 1 h of
local time. Caution should be exercised when using high-latitude Pi2s for
the timing study.

The above scenario assumes that the DFBs arrived periodically in the inner magnetosphere at a frequency not very different than the cavity frequency of the plasmasphere.

8 Discussion and summary

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The definition of field line dipolarization is a configuration change from stretching to shrinkage of geomagnetic field lines in the midnight meridian of the magnetosphere. Two models have been proposed to account for the configuration change; diversion of the cross-tail currents via the ionosphere, referred to as a substorm current wedge (SCW), as first proposed in McPherron et al. (1973), and extinction of the cross-tail currents by a local kinetic instability, current disruption (CD) (Lui, 1996). These models have been adopted for many decades to account for the critical issues associated with substorm onset. It is proposed, based on the ballooning instability scenario, that field line dipolarization is caused by the relaxation of the radial inhomogeneity of plasma pressures in association with the excitation of slow magnetoacoustic waves. Dipolarization regions expand in longitudes and decrease field magnitudes by expanding flux tubes therein. This condition continued for about 10 min, and classical dipolarization caused by the reduction of cross-tail currents or pileup of the magnetic flux transported from the tail begins.

It is noted that BBFs with low-entropy plasmas (plasma bubbles) often penetrated the inner magnetosphere (Dubyagin et al., 2011). In numerical simulations, those bubbles localized in local time produced global dipolarization in the inner magnetosphere (Merkin et al., 2019) and generated an ionospheric current system such as the westward electrojet, Harang discontinuity and poleward expansion of aurora in the substorm expansion phase (Yang et al., 2012). These classical features of substorm expansion occurred in the first 10 min of intervals of the Pi2 onset, referred to as transitional intervals in the midnight magnetosphere. The transitional intervals may be the most active periods in the substorm phase.

The proposed scenario was deduced from the geosynchronous observation and
cannot be readily applied to the onset scenario beyond the geosynchronous
orbit. Nevertheless, dawn–dusk expansion of the flux tubes may be a
fundamental property of field line dipolarization not only at geosynchronous
altitudes but also in tailward locations (8–12 *R*_{e}) (Yao et al., 2013; Liu
et al., 2013). It is suggested that the field line dipolarization at
tailward locations is subdivided by the faster expanding (in longitudes)
dipolarization front (DF) and slower expanding dipolarization front bundle
(DFB) led by DF (Liu et al., 2015). Such substructures in field line
dipolarization are also observed at geosynchronous altitudes (Saka and
Hayashi, 2017). The geosynchronous dipolarization expanded (in longitudes)
at 1.9 km s^{−1}, while Pi2s emitted in the dipolarization region propagated one
order of magnitude faster. The fast longitudinal velocities associated with
Pi2s may be embedded within the slowly expanding region of dipolarization,
similarly to the relationship between DF and DFB. If this relationship can also
be adapted to the transitional state and subsequent field line pileup,
the dipolarization scenario in geosynchronous observations can be extended
further tailward upstream. Or, the onset scenario in 10 *R*_{e} can be applied
in geosynchronous dipolarization. In that case, the dipolarization pulse at
GOES-6 latitude (7.9^{∘} N) may represent DFs. This assumption may be
supported because electron energy flux pitch angle distributions in tailward
locations beyond 10 *R*_{e} appear parallel to perpendicular transitions, like those
in Fig. 3, at the arrival of the DF (Deng et al., 2010).

It should be emphasized that two different types of the dipolarization exist in the substorms; one is associated with the change of curvature radius of field lines in the transitional state (faster expansion in longitudes) and the other is subsequent pileup of the magnetic flux transported from the tail (slower expansion). Field line pileup caused by the flow braking processes (Shiokawa et al., 1997) may lead to tailward regression of the dipolarization region as reported in Baumjohann et al. (1999).

In the transitional state lasting for about 10 min, the inductive electric fields pointing westward were produced in the equatorial plane. They propagated along the field lines to the ionosphere to produce meridional field-aligned currents of the Bostrom type (downward in higher latitudes and upward in lower latitudes). The Bostrom-type current system was indeed observed on the ground at the front of dipolarization expanding towards the east. The magnetospheric dynamo produced by earthward electric fields in the equatorial plane (Akasofu, 2003) and the E layer dynamo in the ionosphere worked together to activate the Bostrom current system.

Appendix A

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In order to derive Eqs. (3) and (4), Kadomtsev (1979) was followed. Linearized MHD equations may be written as

$$\begin{array}{}\text{(A1)}& {\displaystyle \frac{{\partial}^{\mathrm{2}}\mathit{\xi}}{\partial {t}^{\mathrm{2}}}}={C}_{\mathrm{S}}^{\mathrm{2}}\mathrm{\nabla}\mathrm{div}\mathit{\xi}+{C}_{\mathrm{A}}^{\mathrm{2}}{\mathrm{\nabla}}_{\perp}\mathrm{div}{\mathit{\xi}}_{\perp}+{C}_{\mathrm{A}}^{\mathrm{2}}{\displaystyle \frac{{\partial}^{\mathrm{2}}{\mathit{\xi}}_{\perp}}{\partial {z}^{\mathrm{2}}}}.\end{array}$$

Here, *C*_{S}, *C*_{A} and ** ξ** denote sound velocity, Alfvén velocity and
plasma displacement, respectively. The term ($\perp ,z$) denotes perpendicular and
parallel components with respect to the background field lines.

After a few manipulations of Eq. (A1), magnetoacoustic wave equations
for finite *β* plasmas were obtained:

$$\begin{array}{}\text{(A2)}& {\displaystyle \frac{{\partial}^{\mathrm{2}}\mathrm{div}{\mathit{\xi}}_{\perp}}{\partial {t}^{\mathrm{2}}}}={C}_{\mathrm{A}}^{\mathrm{2}}\mathrm{\Delta}\mathrm{div}{\mathit{\xi}}_{\perp}+{C}_{\mathrm{S}}^{\mathrm{2}}{\mathrm{\Delta}}_{\perp}\mathrm{div}\mathit{\xi}\end{array}$$

and

$$\begin{array}{}\text{(A3)}& {\displaystyle \frac{{\partial}^{\mathrm{2}}{\mathit{\xi}}_{z}}{\partial {t}^{\mathrm{2}}}}={C}_{\mathrm{S}}^{\mathrm{2}}{\displaystyle \frac{\partial}{\partial z}}\left(\mathrm{div}\mathit{\xi}\right).\end{array}$$

Equations (A2) and (A3) present compressive properties across and along the background field lines, respectively.

Assuming plane harmonic wave solutions, first-order quantities of density
and magnetic field compressions (*δ**N*, *δ*** B**) may be given
by the following equation.

$$\begin{array}{}\text{(A4)}& {\displaystyle \frac{\mathit{\delta}N}{{N}_{\mathrm{0}}}}=-{\displaystyle \frac{{C}_{\mathrm{A}}^{\mathrm{2}}}{{B}_{\mathrm{0}}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{{C}_{\mathrm{S}}^{\mathrm{2}}-{\left(\frac{\mathit{\omega}}{k}\right)}^{\mathrm{2}}}}\left({\mathit{B}}_{\mathrm{0}}\cdot \mathit{\delta}\mathit{B}\right)\end{array}$$

Here, *N*_{0} and *B*_{0} denote background density and magnetic fields,
respectively.

Substitution of Eq. (A4) into Eq. (A3) using $\mathrm{div}\mathit{\xi}=-\mathit{\delta}N/{N}_{\mathrm{0}}$ yields

$$\begin{array}{}\text{(A5)}& {\displaystyle \frac{{\partial}^{\mathrm{2}}{\mathit{\xi}}_{z}}{\partial {t}^{\mathrm{2}}}}={C}_{\mathrm{S}}^{\mathrm{2}}F{\displaystyle \frac{\partial}{\partial z}}\left({\mathit{B}}_{\mathrm{0}}\cdot \mathit{\delta}\mathit{B}\right).\end{array}$$

Here,

$$F={\displaystyle \frac{{C}_{\mathrm{A}}^{\mathrm{2}}}{{B}_{\mathrm{0}}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{{C}_{\mathrm{S}}^{\mathrm{2}}-{\left(\frac{\mathit{\omega}}{k}\right)}^{\mathrm{2}}}}.$$

A linearized Faraday's law under frozen-in conditions, $\mathit{\delta}\mathit{B}=\mathrm{\nabla}\times ({\mathit{\xi}}_{\perp}\times {\mathit{B}}_{\mathrm{0}})$, may be reduced to

$$\begin{array}{}\text{(A6)}& \mathit{\delta}\mathit{B}=-{\mathit{B}}_{\mathbf{0}}\mathrm{div}{\mathit{\xi}}_{\perp}+{B}_{\mathrm{0}}{\displaystyle \frac{\partial}{\partial z}}{\mathit{\xi}}_{\perp}.\end{array}$$

Substituting Eq. (A6) into Eq. (A5), final expressions relating parallel and perpendicular displacements were obtained:

$$\begin{array}{}\text{(A7)}& {\displaystyle \frac{{\partial}^{\mathrm{2}}{\mathit{\xi}}_{z}}{\partial {t}^{\mathrm{2}}}}=-{C}_{\mathrm{S}}^{\mathrm{2}}F\cdot {B}_{\mathrm{0}}^{\mathrm{2}}{\displaystyle \frac{\partial}{\partial z}}\left(\mathrm{div}{\mathit{\xi}}_{\perp}\right).\end{array}$$

Replacing $\partial /\partial t$ with −*i**ω*, Eq. (A7)
yields Eq. (3) in Sect. 4,

$${\mathit{\xi}}_{z}={\displaystyle \frac{{C}_{\mathrm{S}}^{\mathrm{2}}}{{\mathit{\omega}}^{\mathrm{2}}}}F\cdot {B}_{\mathrm{0}}^{\mathrm{2}}{\displaystyle \frac{\partial}{\partial z}}\left(\mathrm{div}{\mathit{\xi}}_{\perp}\right).$$

Code and data availability

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Code and data availability.

Satellite data from GOES-5 and GOES-6, AMPTE CCE and ground magnetometer data in Figs. 1, 4 and 7 are available upon request to Osuke Saka (saka.o@nifty.com).

Competing interests

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Competing interests.

The author declares that he has no conflict of interest.

Acknowledgements

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Acknowledgements.

The author would like to express his sincere thanks to all the members of Global Aurora Dynamics Campaign (GADC) (Oguti et al., 1988). He is also grateful to Zhonghua Yao and the anonymous referees for their critical review.

Review statement

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Review statement.

This paper was edited by Elias Roussos and reviewed by Zhonghua Yao and two anonymous referees.

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Short summary

The first 10 min interval of Pi2 onset is the most active period of substorms composed of field line deformations associated with an increase in curvature radius of flux tubes and their longitudinal expansion. The flux tube deformations were triggered by the ballooning instability of slow magnetoacoustic waves upon arrival of the dipolarization front from the tail. They preceded the classical dipolarization caused by the reduction of cross-tail currents and resulting pileup of the field lines.

The first 10 min interval of Pi2 onset is the most active period of substorms composed of field...

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