2022

[1,2]Mirko Stumpo

[1]INAF - Istituto di Astrofisica e Planetologia Spaziali, Roma, Italy

2]Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, Roma,
Italy

# Dynamical information flow within the magnetosphere-ionosphere system during
magnetic storms

<EMAIL_ADDRESS>Simone Benella<EMAIL_ADDRESS>GiuseppeConsolini
<EMAIL_ADDRESS>Tommaso Alberti<EMAIL_ADDRESS>*  [

###### Abstract

The direct role of successive intense magnetospheric substorms in
injecting/energizing particles into the storm-time ring current is still
debated and controversial. Whereas in the recent past it has been observed the
absence of a net information flow between magnetic storms and substorms,
previous in-situ satellite observations have evidenced that ionospheric-origin
ions dominate the population of the ring current during the main phase of
geomagnetic storms. As a matter of fact, the controversy arises mainly by the
use of sophisticated data-driven techniques somewhat contradicting in-situ
measurements. In this framework, the main aim of this work is to attempt an
adaption of the powerful information-theoretic approach, i.e., the transfer
entropy, in a consistent way with physics modeling and observations and to
explore the possible motivations behind the underlying contradictions that
emerge when these techniques are used. Our idea is to characterize the
dynamics of the information flow within the magnetosphere-ionosphere system
using a database of geomagnetic storms instead of considering a long time
series of geomagnetic indices. This allows us to consider local non-stationary
features of the information flow and, most importantly, to follow the
transition from quiet to disturbed periods and vice-versa.

###### keywords:

Solar wind-Magnetosphere-ionosphere system, Information flow, Space weather,
Magnetic storms, Magnetospheric substorms

##  1 Introduction

When the interplanetary magnetic field (IMF) is characterized by a nearly-
southward orientation for a sufficiently long time, the near-Earth
electromagnetic environment, i.e., the plasma circulation and the
magnetospheric/ionospheric current systems, undergoes some dynamical changes
to dissipate the excess of energy, momentum and mass coming from the
surrounding interplanetary medium (akasofu1961ring, ; akasofu1974auroral, ).
As a consequence, magnetic storms and substorms develop, being the macroscopic
manifestation of these dynamical processes of
dissipation(gonzalez1994geomagnetic, ).

In recent years, understanding the physical mechanisms behind the development
of these events is becoming more and more important since magnetic
disturbances may be tremendously hazardous for telecommunications, satellites
preservation and also responsible for exposing astronauts to abnormal
radiation level (Malandraki, ). The fingerprint of a magnetic storm is the
depression of the horizontal component of the magnetic field caused by the
enhancement of the ring current, usually monitored by the Disturbance Storm
Time ($D_{st}$) index (1 hour resolution by definition) or the SYM-H index,
which is its 1-minute resolution equivalent. It is now recognized that the
efficiency of energization is, in turn, primarily controlled by a long-lasting
southward component of the IMF. However, whether the southward IMF is the
unique driver of both particle energization and injection or if other
magnetospheric-ionospheric internal processes are responsible for large scale
injection into the storm-time ring current is still an open problem
(borovsky2021our, ).

Since the pioneering works by Akasofu and Chapman (akasofu1961ring, ),
magnetospheric substorms (i.e., violent electrojet activity) have been
identified as a possible source of particle injection, mainly because of their
statistical association with the occurrence of the magnetic storm main phase
(kamide1998current, ). In particular, based on investigating particle
injection at geostationary orbit it was concluded that substorms drive
particle acceleration (akasofu1974auroral, ). Soon after, it has been shown
that the constituents of the storm-time ring current are more energetic than
those energized directly by substorms (williams1987ring, ). On the other hand,
predictions of the $D_{st}$ index, by using the auroral index AL alone, are
well in agreement with observations (burton1975empirical, ;
kamide1971analysis, ; gonzalez1994geomagnetic, ). Furthermore, AMPTE and CRRES
missions (krimigis1982active, ; wilken1992magnetospheric, ) also reported the
direct observation of ionospheric-origin ions dominating the population of the
ring current during the main phase of a storm (hamilton1988ring, ;
daglis1997role, ), suggesting a direct causal link with substorms. Indeed the
upward acceleration along the magnetic field lines of these ionospheric ions
may be associated to successive occurrence of intense substorms.

Despite such important observations, it remains extremely difficult to
disentangle the role of substorms in the energization of the storm-time ring
current directly from geomagnetic indices data. First attempts have been made
by using prediction filters and, more recently, by using sophisticated
techniques based on information theory (de2011information, ;
stumpo2020measuring, ; manshour2021causality, ). All these works share the
common idea that causation is associated to the notion of predictability,
instead of correlation. If the knowledge of the time series $Y$ reduces the
error in predicting the time series $X$ it is said that an information flow
(IF; or, equivalently, predictive causality) exists from $Y$ to $X$. This is
the reason why predictions are associated to evidence of asymmetric couplings
between the components of a physical system (in this case the magnetosphere-
ionosphere system).

The main drawback in using data-driven techniques is conceptual. Can
geomagnetic indices as $D_{st}$ (or SYM-H) and AL capture the processes of
injection and energization (mcpherron1997role, )? Other issues arise when
conditional statistics are used. Indeed, recently Runge et al.
(runge2018common, ) and Manshour et al. (manshour2021causality, ) have shown,
using conditional transfer entropy, that if one removes the influence of the
southward IMF, the IF from AL/AE to SYM-H (and vice-versa) previously found by
De Michelis et al. (de2011information, ) and Stumpo et al.
(stumpo2020measuring, ) becomes negligible. These studies are carried out,
respectively, by using 20 minutes time averaged SYM-H and 5 minutes resampled
SYM-H, although McPherron (mcpherron1997role, ) suggested that high-resolution
indices are required for unveiling the role of substorms because otherwise
fast dynamics may be lost in average. On the other hand, other authors
suggested that a magnetic storm is not a trivial superposition of intense
substorms, but that the outflow of ionospheric ions is controlled by an
efficiency function $\alpha=\alpha(B_{z},t)$ which depends on the southward
IMF. In particular, according to Kamide et al. (kamide1992substorm, ) and
Gonzalez et al. (gonzalez1994geomagnetic, ), the energy balance equation for
$D_{st}$ (justified in physics ground by the Dessler-Parker-Sckopke relation)
can be written as

$\frac{d}{dt}D_{st}(t)=\alpha(B_{z},t)\text{AL}(t)-\frac{D_{st}(t)}{T},$ (1)

where $T$ is the relaxation time of $D_{st}$, i.e., the duration of the
disturbance, which in general is not constant but depends on the specific
concentrations of ions species in the ring current. Now, IF the southward
$B_{z}$ controls the efficiency, i.e., the energy supply, when $B_{z}$ is
removed from the model, the IF from AL to $D_{st}$ may disappear. Equation (1)
suggests also that the IF from AL to $D_{st}$ is non-negligible only during
southward $B_{z}$, also according to observations of particle injection
discussed above. This fact implies that the IF is strongly non-stationary and,
as such, it is different during storm and non-storm times. Thus, unraveling
the IF by using long time series to estimate the average of transfer entropy
may be misleading, since $\alpha(B_{z},t)$ is reasonably different from zero
only for short times. As a result, the use of long time series most likely
tends to unbalance the contributions of quiet and disturbed geomagnetic
periods, thus leading to bias the IF towards values mainly due to non-storm
time intervals. This argument may be at the origin of the vanishing IF
observed by Runge et al. (runge2018common, ) and Manshour et al.
(manshour2021causality, ).

In this framework, the aim of this paper is to shed some light in the debate
about the influence of magnetospheric substorms on the storm-time ring current
development. The main task is to suit the powerful information-theoretic
approach to the study of this issue in a consistent way with physics modeling
and in-situ observations. To achieve this we consider a dataset of storms and
substorms extracted from SuperMAG database (gjerloev2009global, ) and we
perform the estimation of the transfer entropy between high-latitude, low-
latitude geomagnetic indices and IMF $B_{z}$ component by using a sliding
window technique. Since small time windows contain non-stationary data we
consider a set of storm events, i.e. an ensemble of independent time series,
that allows us to investigate the dynamics of the IF between different
physical quantities during the storm events. In fact, the main advantage of
this approach is that quiet and disturbed periods are not averaged in the
analysis and thus the ensemble enables to quantify how the IF varies during
the different phases of the storm.

The paper is organized as follows: in Section 2 we will review the concept of
IF and the definition as well as the derivation of the transfer entropy. Next
we will discuss the motivation behind the use of this tool as well as
advantages and disadvantages of the ensemble approach for geomagnetic studies.
In Section 3 we present the dataset and we discuss the synchronization of
signals involved in the ensemble. Finally in Section 4 we show the results,
while in Section 5 we discuss them in terms of interpretation and
contextualization in literature.

##  2 Methods

Generally, the study of causation is dominated by the notion of predictability
(sugihara2012detecting, ). In this context, if we have two time series $X$ and
$Y$, we say that $Y$ drives the dynamics of $X$ if the information about the
state of $X$ can be recovered from the past states of $Y$ and not vice-versa.
In the framework of non-parametric statistics, the formalization of this
concept can be retrieved in the notion of IF.

The step forward of the IF with respect to the popular concept of correlation
is that the former removes any redundant or shared information between current
$X$ and its own past. For example, if we have two processes, $X$ and $Y$, such
that $Y$ drives $X$ and not vice-versa, we need to consider that the process
$Y_{t}$ incorporates intrinsically the information about $X_{t}$’s past,
otherwise the cross correlation (or, equivalently, the delayed mutual
information) in the direction from $X$ to $Y$ would not be zero even if the IF
is absent. For further details see (bossomaier2016transfer, ) and
(schreiber2000measuring, ).

Therefore, if we want to compute the IF from $Y$ to $X$, the idea is to remove
the redundancy introduced by the past history of $X$. To account this, the
most simple formulation of transfer entropy is given in terms of conditional
mutual information (CMI), i.e.

$T_{Y\rightarrow X}^{(k,l)}(\tau)=I(X_{t}\mathchar
24635\relax\;\mathbf{Y}_{t-\tau}^{(l)}|\mathbf{X}_{t-1}^{(k)}),$ (2)

where in general
$\mathbf{Y}_{t-\tau}^{(l)}=\left(Y_{t-\tau},Y_{t-\tau-1},...,Y_{t-\tau-l}\right)$
and $\mathbf{X}_{t}^{(k)}=\left(X_{t},X_{t-1},...,X_{t-k}\right)$ are the
multivariate reconstruction of $Y$ and $X$ past histories, respectively, if we
assume they are $l$-th and $k$-th order Markov processes. From a probabilistic
point-of-view, the transfer entropy is essentially the distance in probability
from the validity of the generalized Markov condition, i.e.

$p(X_{t}|\mathbf{X}_{t-1}^{(k)}\mathchar
24635\relax\;\mathbf{Y}_{t-\tau}^{(l)})=p(X_{t}|\mathbf{X}_{t-1}^{(k)}),$ (3)

which is fulfilled if and only if $X_{t}$ depends (conditionally) only on its
own history. Using conditional Kullback Leibler Divergence (cKLD), we can
compute the distance between l.h.s and r.h.s of eq. (3) and obtain the
explicit formula for the transfer entropy (schreiber2000measuring, )

$T_{Y\rightarrow
X}^{(k,l)}(\tau)=\sum_{X_{t},\mathbf{X}_{t-1}^{(k)},\mathbf{Y}_{t-\tau}^{(l)}}p(X_{t},\mathbf{X}_{t-1}^{(k)},\mathbf{Y}_{t-\tau}^{(l)})\log\frac{p(X_{t}|\mathbf{X}_{t-1}^{(k)},\mathbf{Y}_{t-\tau}^{(l)})}{p(X_{t}|\mathbf{X}_{t-1}^{(k)})}.$
(4)

To compute eq. 4 the KSG estimator is used for its optimality in terms of
systematic errors and biases due to finite sample effects
(kraskov2004synchronization, ; kraskov2004estimating, ; wibral2014transfer, ).

Note that whereas in general for non-Markov process both
$(k,l)\rightarrow(\infty,\infty)$, for purely Markov processes, the past
history of e.g. $X_{t}$ is completely embedded into $X_{t-1}$ alone, so that
the redundancy of $X$’s own past can be elimated by simply conditioning on
$X_{t-1}$ and the multivariate vector $\mathbf{X}_{t-1}^{(k)}$ collapses to
the univariate signal $\mathbf{X}_{t-1}^{(k)}\rightarrow X_{t-1}$. At this
point it is also worth noticing that $\mathbf{X}_{t-1}^{(k)}$ and
$\mathbf{Y}_{t-\tau}^{(l)}$ can be thought as an embedding reconstruction of
the phase space according to Taken’s theorem (takens1981detecting, ), although
the equivalence with the real phase-space is not guaranteed for stochastic
systems (kantz2004nonlinear, ). Furthermore, when the noise-level is high, or
when the underlying dynamics is stochastic at coarse grained scales, the
parameters of the reconstruction $k$ and $l$, cannot be recovered by using
methods, such as False Nearest Neighbours technique (kennel1992determining, ),
because they are suited for deterministic dynamics (ragwitz2002markov, ;
kantz2004nonlinear, ).

Our definitions in eqs. (2) and (4) incorporate directly the time lag $\tau$
between $X$ and $Y$. This is because the interaction may be delayed in time
more than $\tau=1$ as in the original definition by Schreiber
(schreiber2000measuring, ). In this case, only $Y$ is lagged forward in time,
while the past of $X$ remains untouched. This choice, as demonstrated
explicitly by Wibral et al. (wibral2013measuring, ), is the only one, among
the most popular definitions of information transfer (e.g.
(pompe2011momentary, ), (paluvs2001synchronization, )), allowing to restore
Wiener’s principle of causality and to recover the correct transfer delays
(bossomaier2016transfer, ), (wibral2013measuring, ).

Practically, the interpretation of the transfer entropy is straightforwardly
related to the difference in the uncertainties of $X$’s future before and
after the knowledge of $Y$’s past, recovering again the notion of
predictability in the statistical inference of IF. Following
(bossomaier2016transfer, ) this interpretation can be put in a formal way by
decomposing the total uncertainty, i.e. the Shannon entropy, $H(X_{t})$ as

$H(X_{t})=I(\mathbf{X}_{t-1}^{(k)}\mathchar
24635\relax\;X_{t})+T_{Y\rightarrow
X}^{(k,l)}(\tau)+H(X_{t}|\mathbf{X}_{t-1}^{(k)}\mathchar
24635\relax\;\mathbf{Y}_{t-\tau}^{(l)}).$ (5)

Hence, we can extract three contributions in the uncertainty on the future
state of $X$: the first accounts for the information contained in the past
$\mathbf{X}_{t-1}^{(k)}$, the second term is the IF from the $Y$’s past, and
the third term is the residual uncertainty after the knowledge of both $X$’s
and $Y$’s histories. From eq. (5) it is clear that the consideration of
$\mathbf{X}_{t-1}^{(k)}$ instead of $X_{t-1}$ controls the balance between
stored and transferred information. An inadequate reconstruction of the k-th
Markov process may lead to confusion between stored and transferred
information (bossomaier2016transfer, ).

In general, when the information-theoretic approach is applied to real data,
we have single realizations of $X$ and $Y$ as time series processes. In this
case, the PDFs in eq. (4) are estimated assuming stationarity and, naturally,
also the IF in this case should be stationary. The consequence is that
transient dynamics and local non-stationarity is completely neglected. This
point is crucial when the aim is to study the role of intense substorms in the
energization of storm-time ring current. For this reason, as mentioned in the
Introduction, we need a time-resolved estimation of the transfer entropy over
a sliding window as presented below.

Let $W=\\{\hat{t}-\delta,\hat{t}-\delta+1,...,\hat{t}+\delta\\}$ be the time
window centered around $\hat{t}$. Then, we can restrict the time series
$X_{t}$, $\mathbf{X}_{t-1}^{(k)}$ and $\mathbf{Y}_{t-\tau}^{(l)}$ to $W$ and
compute eq. (4) to get the time windowed transfer entropy, i.e.

$\mathcal{T}_{Y\rightarrow X}^{(k,l)}(\hat{t},\tau)=T_{Y_{W}\rightarrow
X_{W}}^{(k,l)}(\tau).$ (6)

In principle, in the limit $N\to\infty$ and $\delta\to 0$, the average
$\langle\mathcal{T}_{Y\rightarrow X}^{(k,l)}(\hat{t},\tau)\rangle_{\hat{t}}$
converges to eq. (4). As can be seen from this definition, one of the main
limitation of the method when we deal with empirical observations, is
represented by the need for a sufficient statistics, which cannot be always
guaranteed. Indeed, geomagnetic indices are sampled at a maximum cadence of 1
minute and the used time window must be large enough to include a sufficient
number of data points in the statistics. On the other hand, this size cannot
be too large either since we want to resolve transient dynamics, which could
be suppressed by averaging on wide time windows.

In order to overcome this limitation, firstly we find a suitable trade-off for
the size of the time window used in the analysis and then we introduce the
estimation of the transfer entropy over an ensemble of independent
realizations of magnetic storms in a way similar to the method proposed by
Gómez-Herrero et al. (gomez2015assessing, ). This enables us to study how the
IF varies during the evolution of the magnetic storm. We remark that the
ensemble approach is somewhat different from computing transfer entropy
between individual trials averaging the single transfer entropies a
posteriori. This would not be an ensemble approach. In contrast, we merge
together all the time series in the specific time windows and compute directly
the total transfer entropy.

Note that the need for a sufficient statistics is also strongly affected by
the so-called curse of dimensionality. Indeed, in general the unbiased
estimation of the transfer entropy requires the $k$-th and $l$-th
reconstruction of Markov processes as explained above, but the number of data
points needed for the correct sampling of PDFs scales non-linearly with the
dimension, i.e. with both $k$ and $l$.

Another crucial point is that the transfer entropy in eq. (2), when the IF is
absent, is equal to zero only theoretically. When the sample size is finite
and the transfer entropy is empirically measured, a bias is always present,
regardless the estimator we use. In this framework, a key question is whether
or not the values found for the transfer entropy are statistically
significant, especially if we do not know a priori the underlying PDFs. To
perform such a test, we need forming the null hypothesis $H_{0}$ that the IF
is zero and the relevant distribution of the transfer entropy would be if
$H_{0}$ was true. Practically it means that we need surrogate time series
$\mathbf{Y}_{t-\tau}^{(l)}$ such that $p(X_{t}|\mathbf{X}_{t-1}^{(k)}\mathchar
24635\relax\;\mathbf{Y}_{t-\tau}^{(l)})=p(X_{t}|\mathbf{X}_{t-1}^{(k)})$. In
order to achieve this we create surrogate trials by only shuffling the source
time series $\mathbf{Y}_{t-\tau}^{(k)}$ and leaving $X$ untouched. Indeed if
$X$ was shuffled, any correlation would results to be destroyed and the Markov
condition in eq. (3), i.e. our null hypothesis, may not be fulfilled anymore.
Finally, a threshold confidence is fixed at 0.95 and the corresponding
critical value of the transfer entropy $\hat{T}_{Y\rightarrow
X}(\hat{t},\tau)$ is computed in each window, so that if our measurements of
the IF are greater than $\hat{T}_{Y\rightarrow X}(\hat{t},\tau)$ we can argue
statistical significance. For this preliminary study we use only two
suggorates to fix the background of transfer entropy values.

##  3 Data preparation

In order to investigate the IF between external driving, auroral electrojet
activity and ring current dynamics during magnetic storm events, we use the
Super-MAG high-latitude index SML and low-latitude index SMR, which are a
generalization of the traditional AL and SYM-H, respectively. They are, as
usual, derived from deviations with respect to the average value of the
horizontal (H) component of the geomagnetic field measured from a network of
nearly-auroral/equatorial ground-based magnetometers (gjerloev2009global, ).
The choice of Super-MAG indices is motivated by the fact that, since the
auroral oval moves towards lower latitudes during severe magnetic storms,
classical high-latitude geomagnetic indices have some limitations in
estimating the correct value of auroral electrojet current intensity. In this
framework the Super-MAG collaboration introduced the generalized AE-indices,
i.e. SML, SMU and SME, computed using more than 300 different stations.
Furthermore, the Super-MAG collaboration has 98 magnetometers in the range of
latitudes currently used for constructing SYM-H and $D_{st}$ indices. So that,
this sub-network is used to build up the SuperMAG equivalent of the ring
current proxies, namely the SMR index (newell2012supermag, ).

In this framework, the typical fingerprint of a magnetic storm is monitored
through the SMR index, which exhibits a sudden depression towards negative
values. On the other hand, the polar substorm activity, i.e. the magnetic
disturbance caused by the auroral electrojet current flowing in the auroral
region, is investigated by means of the SML index, which is mainly
representative of the geomagnetic tail dynamics (gjerloev2004substorm, ;
davis1966auroral, ; kamide2004physical, ). Finally, we use the z-component of
the IMF $B_{z}$ collected from OMNI database to infer the IF from the solar
wind to internal magnetosphere-ionosphere system.

The aforementioned ensemble of magnetic storms is now introduced. In detail,
we started with a 23-year dataset (from 1995 to 2018) of $B_{z}$, SML and SMR
from which we selected a set of magnetic storm periods for which
$\text{SMR}\leq-150$ nT by considering a period of 10 days before and after
the minimum of SMR during each storm event. In order to complete the ensemble
we use the same periods of time for SML and $B_{z}$. As a last step, the
double peaked storms have been removed from the ensemble by visual inspection
since such complex events could introduce spurious effects when considered in
our ensemble-based analysis. The final dataset consists of $N_{r}=30$
independent storms that are reported in Figure 1.

Figure 1: From top to bottom, SMR, SML and $B_{z}$. The time series are
collected at 1 minute resolution.

##  4 Results

Figure 2: Top: Contour plot of the transfer entropy from $B_{z}$ to SML with
respect to the time window and to the time delay $\tau$. In order to compare
the IF in terms of storm phases, the averaged track of SMR is depicted in
black solid-line. Bottom: Contour plot of the transfer entropy from $B_{z}$ to
SMR with respect to the time window and to the time delay $\tau$. In order to
compare the IF in terms of storm phases, the averaged track of SMR is depicted
in black solid-line.

The main objective of this work is to provide a novel approach to inspect the
IF within impulsive and strongly non-stationary processes, such as magnetic
storms and magnetospheric substorms. As a first step we aim to evaluate the IF
from the $z$-IMF component, which is representative of the driver, to both
high-latitude and low-latitude geomagnetic activity by means of SML and SMR
indices, respectively. The top panel of Figure 2 shows the contour plot of the
ensemble-based TE $T(B_{z}\to\text{SML})$ as a function of the time window and
the time delay $\tau$. We report the ensemble-averaged trend of the SMR index
inside the figures, since this index is the one we used in the event
synchronization and moreover it serves as a guide for the eye in identifying
all the different phases of the magnetic storm. As is clear from the top panel
of Figure 2, there are different enhancements in the IF from $B_{z}$ to SML
that are not related to the occurrence of the magnetic storm. Furthermore, the
highest values of the TE are reached in proximity of the storm, i.e. during
the pre-storm period and within the recovery phase, whereas a sudden decrease
in the IF is observed during the storm main phase. Conversely, if we consider
$T(B_{z}\to\text{SMR})$, reported in the bottom panel of Figure 2, a
significant enchancement of the IF is only present during the storm main
phase. In this framework, these results emphasize the different role that the
driver ($B_{z}$ in this case) plays in contributing to the dynamics of storms
and substorms.

Figure 3: Top: Contour plot of the transfer entropy from SML to SMR with
respect to the time window and to the time delay $\tau$. In order to compare
the IF in terms of storm phases, the averaged track of SMR is depicted in
black solid-line. Bottom: Contour plot of the transfer entropy from SMR to SML
with respect to the time window and to the time delay $\tau$. In order to
compare the IF in terms of storm phases, the averaged track of SMR is depicted
in black solid-line.

In terms of internal dynamics, the characterization of the IF within the
magnetosphere-ionosphere system is in general much more complex, since such
flow is not unidirectional and feedback processes may be present as well. In
this case it is crucial to elucidate the dynamics of the IF. In previous
works, the investigation of the IF within the magnetosphere-ionosphere system
has been carried by using a one-year dataset and the TE has been estimated
over the whole signals neglecting possible time variations of the IF, e.g. in
terms of intensity or direction. The ensemble TE from SML to SMR and vice-
versa are reported, respectively, in the top and bottom panels of Figure 3. By
looking at $T(\text{SML}\to\text{SMR})$ it is clear how the maximum transfer
of information from SML towards SMR is strongly localized around the minimum
of $\langle\text{SMR}\rangle$, i.e. during the storm main phase. Moreover, the
time lag $\tau$ at which the maximum is located is $\sim 0$. By looking at
$T(\text{SMR}\to\text{SML})$, which is representative of the IF from the ring
current to the westward auroral electrojet current system, we observe a quite
different scenario. The first enhancement of the TE approaching the storm is
located just before the depression of SMR, whereas the maximum TE values are
reached in the recovery phase. Contrary to what is observed for
$\text{SML}\rightarrow\text{SMR}$, a sudden decrease of the IF between SMR and
SML is observed at the onset of the mean main phase.

##  5 Discussion and conclusion

In this study we provided a first attempt to characterize the dynamics of the
IF within the magnetosphere-ionosphere system using a database of magnetic
storms instead of considering a long time series of geomagnetic indices. This
allows us to avoid mixing the statistics of quiet and disturbed periods, as
well as, thanks to our moving-window approach to follow the transition from
quiet and disturbed conditions. However, one of the main limitation in
considering the IF as an intrinsically non-stationary measure during transient
periods, is the need for a sufficient statistics which clearly cannot be
guaranteed. In order to overcome this problem, we introduced the analysis of
transfer entropy over an ensemble of independent realizations of magnetic
storms in a way similar to the method proposed by Gómez-Herrero et al.
(gomez2015assessing, ). We emphasize again that this approach is somewhat
different from computing the transfer entropy between individual trials and
then averaging the single results a posteriori.

We presented our approach by analyzing an ensemble of 30 independent magnetic
storms. Firstly we studied the dynamics of the IF from solar wind to the
magnetosphere-ionosphere system and we found a delayed information transfer
according to previous findings (de2011information, ; alberti2017timescale, ;
stumpo2020measuring, ; runge2018common, ; manshour2021causality, ). However,
whereas the IF from $B_{z}$ to SMR, i.e., from the solar wind to the low-
latitude magnetosphere (ring current), is enhanced only during the onset of
the main phase of a magnetic storm, the IF from $B_{z}$ to SML, i.e., from the
solar wind to the polar ionosphere, enhances not only during storm-times. This
fact may be explained by the fact that auroral disturbance, i.e.
magnetospheric substorms, can also occur outside a magnetic storm
(kamide1992substorm, ). Indeed, whereas the development of a main-phase
requires a southward oriented $B_{z}$ for a sufficient long time, the
injection of solar wind particles into the polar ionosphere, i.e., the onset
of geomagnetic tail reconnection and the successive impulsive energy
dissipation through magnetospheric substorms, occur whenever $B_{z}$ is
southward-oriented.

The study of the internal dynamics, i.e., the magnetosphere-ionosphere
coupling, is in general much more complex to interpret because a large number
of current systems are involved. In this framework we found an important IF
from SML to SMR at the beginning of the depression of SMR, which can be
interpreted as the contribution of the outflow from the ionosphere. The
current systems which act as mediators between the magnetosphere and the
ionosphere in this case are the Field-Aligned-Currents (FACs). When the
minimum of the disturbance is reached, the IF drops abruptly and is enhanced
again during the recovery phase. The coupling induced by the FACs is not
unidirectional, indeed they form a closed system with a reverse IF from SMR to
SML, especially during the recovery phase. A possible explanation is that the
excess of energetic particles are re-injected into the ionosphere from the
ring current, where dissipation occurs via secondary substorms. On the other
hand, this effect may be due to the non-Markovian nature of SML index at time-
scales larger than 60 minutes as recently demonstrated by Benella et al.
(benella2022markov, ). Without an appropriate reconstruction of the $l$-th
order Markov process (see Section 2), limited essentially by the need for a
sufficient statistics, the effects of non-Markovianity may not be negligible
so that the IF may be overestimated due to SML itself in this case.

Figure 4: Top: Contour plot of the transfer entropy from SML to SMR computed
by using a window width of 1 day. Bottom: Contour plot of the transfer entropy
from SMR to SML compute by using a window width of 1 day.

The picture that during the early stages of a magnetic storm there exists a
concurrent effect between a direct driving due to the solar wind activity and
high-latitude processes is in agreement with the energy balance equation of
$D_{st}$ index written in the form of eq. (1),which enabled very good
predictions of $D_{st}$ (kamide1971analysis, ; gonzalez1994geomagnetic, ;
kamide1998current, ). In particular, eq. (1) represents the interplay between
the direct external driver (i.e., efficiency of magnetic reconnection and
efficiency of energy supply) and those internal processes triggered by the
energy input given by the external driver. From a statistical point of view,
our findings are also in great agreement with a very recent study by Alberti
et al. (alberti2022concurrent, ). By using a novel approach based on dynamical
systems theory, they computed the dimension of the reconstructed phase space
by firstly considering AL and SYM-H alone and then by considering the joint
process (AL, SYM-H). Interestingly, from this analysis figured out an
independent contribution of AL in the dynamics of SYM-H during the development
of the main-phase, in agreement with the IF between SML and SMR.

At this stage, it is also important to mention that the window width used for
computing the transfer entropy in eq. (4), naturally influences the behaviour
of the IF. This is not surprising since the window width defines the time-
scales in which the IF is measured. For example, if we compute the transfer
entropy using a window width of 2 days, we found the results shown in Figure 4
for the IF from SML to SMR (top panel) and vice-versa (bottom panel). In this
case we can see only the contribution of SML to the outflow localized just
during the development of the main phase. In the reverse direction, i.e. from
SMR to SML, we found a feedback process localized during the start of the
recovery phase. Therefore, this behaviour highlights again the dependence of
the IF on the time-scales in which it is measured. The time-scale dependence
of the coupling for the case external-internal processes has been highlighted
by Alberti et al. (alberti2017timescale, ) by using the delayed mutual
information on the filtered signals.

The works by Runge et al. (runge2018common, ) and Manshour et al.
(manshour2021causality, ), in contrast to previous findings by De Michelis et
al. (de2011information, ) and Stumpo et al. (stumpo2020measuring, ), found
that the IF from the high-latitude to low-latitude (and vice-versa) is
completely explained by the IMF, which might be the common driver. However,
these results must be carefully interpreted since they provide an average view
of the SMI system. This is related to the use of long time series to infer the
IF despite the fact that magnetic storms and substorms do represent transient
dynamics of the magnetosphere-ionosphere system. Furthermore, the conditional
transfer entropy used by Manshour et al. (manshour2021causality, ) is
implicitly averaged for both positive and negative values of $B_{z}$, i.e.,
without providing any discrimination between open and closed conditions of the
magnetosphere. Furthermore, this approach does not take into account
preconditioning features of the magnetosphere-ionosphere system. From a
phenomenological point of view and again with the help of eq. (1), it means
that periods when the coupling function of the magnetosphere-ionosphere
systems is virtually set to zero (closed magnetosphere) are averaged together
with those periods in which the coupling function is considerably different
from zero (open magnetosphere). The relative importance of these two
contributions is dependent on the total time the conditions explained above
are satisfied, so that non-storm time coupling dominates the time average of
the IF. This argument may explain the absence of the IF found by Manshour et
al. (manshour2021causality, ) and Runge et al. (runge2018common, ) and, of
course, the reason why we performed the analysis without removing the past-
history of $B_{z}$. A more comprehensive analysis including the difference of
southward and northward periods will be presented in a forthcoming paper.

In conclusion, our method provides a framework to study the time-variations of
the IF at fixed time-scale. It is particularly suitable for the study of the
relation between magnetic storms and substorms and, of course, of the
magnetosphere-ionosphere coupling. Nevertheless, in this preliminary study
some technical problems such as the reconstruction of l-th and k-th order
Markov process as well as the accurate computation of the statistical
threshold, have been considered only qualitatively. To assess must be followed
by a more comprehensive analysis. From physics side, it is interesting to
discriminate the IF during northward (closed magnetosphere) and southward
(open magnetosphere) IMF periods separately. This may reveal some interesting
features of the injection/energization processes as well as the importance of
the solar wind dynamic pressure, solar wind velocity and convection electric
field. This is only the starting point. For accounting all this ideas, further
work is needed. We remark also that in this preliminary study we used SuperMAG
data. It will be interesting to perform the same analysis on SYM-H and AL
indices. Again, this is matter of a more comprehensive work which will be
given in future.

Acknowledgments The results presented in this paper rely on data collected at
SuperMAG. We gratefully acknowledge the SuperMAG collaborators
(https://supermag.jhuapl.edu). The OMNI data were obtained from the Space
Physics Data Facility (SPDF) Coordinated Data Analysis Web (CDAWeb) interface
at https://cdaweb.gsfc.nasa.gov/index.html/. M.S. acknowledges the PhD course
in Astronomy, Astrophysics and Space Science of the University of Rome
“Sapienza”, University of Rome “Tor Vergata” and Italian National Institute
for Astrophysics (INAF), Italy. G.C. and S.B. acknowledge the financial
support by Italian MIUR-PRIN grant 2017APKP7T on Circumterrestrial
Environment: Impact of Sun-Earth Interaction.

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