NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-407-2017Characterization of high-intensity, long-duration continuous auroral
activity (HILDCAA) events using recurrence quantification analysisMendesOdimodim.mendes@inpe.brhttps://orcid.org/0000-0002-3353-2666DominguesMargarete OliveiraEcherEzequielHajraRajkumarhttps://orcid.org/0000-0003-0447-1531MenconiVarlei EvertonSpace Geophysics Division (DGE/CEA), Brazilian Institute for Space Research (INPE), São José dos Campos, São Paulo, BrazilAssociated Laboratory of Computation and Applied Mathematics (LAC/CTE), Brazilian Institute for Space Research (INPE), São José dos Campos, São Paulo, BrazilLaboratoire de Physique et Chimie de l'Environnement et de l'Espace (LPC2E), CNRS, Orléans 45100, FranceOdim Mendes (odim.mendes@inpe.br)1August201724340741720October201628November201626May20176June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/407/2017/npg-24-407-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/407/2017/npg-24-407-2017.pdf
Considering the magnetic reconnection and the viscous interaction as the
fundamental mechanisms for transfer particles and energy into the
magnetosphere, we study the dynamical characteristics of auroral electrojet
(AE) index during high-intensity, long-duration continuous auroral
activity (HILDCAA) events, using a long-term geomagnetic database
(1975–2012), and other distinct interplanetary conditions (geomagnetically
quiet intervals, co-rotating interaction regions (CIRs)/high-speed streams
(HSSs) not followed by HILDCAAs, and events of AE comprised in global
intense geomagnetic disturbances). It is worth noting that we also study active
but non-HILDCAA intervals. Examining the geomagnetic AE index, we apply a
dynamics analysis composed of the phase space, the recurrence plot (RP), and
the recurrence quantification analysis (RQA) methods. As a result, the
quantification finds two distinct clusterings of the dynamical behaviours
occurring in the interplanetary medium: one regarding a geomagnetically quiet
condition regime and the other regarding an interplanetary activity regime.
Furthermore, the HILDCAAs seem unique events regarding a visible, intense
manifestations of interplanetary Alfvénic waves; however, they are similar to
the other kinds of conditions regarding a dynamical signature (based on RQA),
because it is involved in the same complex mechanism of generating
geomagnetic disturbances. Also, by characterizing the proper conditions of
transitions from quiescent conditions to weaker geomagnetic disturbances
inside the magnetosphere and ionosphere system, the RQA method indicates clearly
the two fundamental dynamics (geomagnetically quiet intervals and HILDCAA
events) to be evaluated with magneto-hydrodynamics simulations to understand
better the critical processes related to energy and particle transfer into
the magnetosphere–ionosphere system. Finally, with this work, we have also
reinforced the potential applicability of the RQA method for characterizing
nonlinear geomagnetic processes related to the magnetic reconnection and
the viscous interaction affecting the magnetosphere.
Introduction
A complicated electrodynamic region populated by plasmas and ruled by the
Earth's magnetic field – designated in a classical definition as
magnetosphere – exists surrounding our planet
. This region is exposed to
influences of the space environment and submitted to several interplanetary
forcings. Initially, a summary view of the physics scenario involved is
briefly described in the two following paragraphs.
In electrodynamic terms, three main solar agents
((i) electromagnetic radiation, (ii) energetic particles,
and (iii) solar magnetized structures) act upon the Earth's
atmosphere, which is permeated by a magnetic field created in the interior of
our planet .
(i) Electromagnetic radiation both heats the planet globally and
ionizes the atmosphere. This ionization gives basis to a terrestrial plasma
environment. (ii) Also, the incidence episodes of solar energetic
particles increase the ionization in a much more localized manner.
(iii) Furthermore, escaping in a continuous way from the Sun, the
solar wind, superposed sometimes by coronal mass ejection structures and
other peculiar solar structures (e.g. solar fast-speed streams and
heliospheric current sheet), transports intrinsically the solar magnetic
field to the orbit of the Earth and beyond .
Two primary electrodynamic interactions are possible from this incidence of
the magnetized solar wind plasma upon the Earth's magnetosphere. These
interactions result in a transfer of energy and particles into the
magnetosphere boundary. The most intense is through the magnetic reconnection
process , when the
interplanetary magnetic field (IMF) presenting a predominantly southward
orientation, in the geocentric solar magnetosphere reference system, merges
into the geomagnetic field at the outer boundary and produces strong
modification in a large region formed by the magnetosphere and the
ionosphere – the latter is a region from about 100 to 2000km of
altitude presenting the highest quantity of ionized particles. Another
competitive process is the Kelvin–Helmholtz viscous interaction
. Most of the time
this second process is in operation when the magnetosphere acts as a closed
physical system, concerning the incident frontal solar wind, due to an IMF
with northward orientation. A macroscopic fluid dynamics developed by the
plasma sliding at the flanks of the magnetosphere creates a kind of viscous
interaction, which produces the mixing of the solar plasma inside the
magnetosphere and the occurrence of ULF waves
affecting the interior regions. The former process is more efficient in
energy and particle transfer than the latter one.
In a global sense, during events of solar wind transporting IMF parallel
(northward) to the frontal geomagnetic field, a regime of low magnetic
disturbance on the ground is noticed. However, when the IMF is strongly
southward directed, anti-parallel to the geomagnetic field, intense regimes
of disturbances are recorded on the ground. Nevertheless, there is a peculiar
interplanetary process related to manifestations of Alfvén
waves , presenting alternation of the magnetic
component orientation (in the southward–northward direction), which produces
an intermediate level of geomagnetic disturbance with the typical duration of
days. These nonlinear Alfvén waves are known to be the main origin of
high-intensity long-duration continuous auroral electrojet (AE) activity
(HILDCAA) events on the Earth
.
As presented in , the AE is a geomagnetic index
related to the quantification of the geomagnetic disturbance produced by
enhanced ionospheric electric currents flowing below and within the auroral
region (https://www.ngdc.noaa.gov/stp/geomag/ae.html).
The primary mechanism for these HILDCAA events is the high-speed solar wind
streams (HSSs) emanating from solar coronal holes accompanied by embedded
Alfvén waves , which are
characterized by significant IMF variability (see
). The sporadic magnetic reconnection formed between the southward component of the Alfvén
waves and the Earth's magnetopause fields leads to intense
substorm/convection events comprising HILDCAAs ,
which are shown to last from days to weeks . The HILDCAA events carry a large amount of
solar wind kinetic energy input into the magnetosphere affecting the polar
ionosphere . More than 60 % of this
energy is dissipated in the magnetosphere–ionosphere system. Another importance of these events is the accelerated relativistic electrons,
known as killer electrons, in the Earth's radiation belts
for their hazardous
effects on orbiting spacecraft .
The variations of AE during HILDCAAs show the nonlinear dynamics of the
physical processes involved. Therefore, a dynamical characterization is of
fundamental interest for a deeper insight into the electrodynamic coupling
between the solar wind and the related magnetosphere.
The aim of this work is to highlight dynamical characteristics related to the
HILDCAA events revealed by the AE index in the context of the
electrodynamic coupling processes.
With this purpose, we apply phase space analysis, the
recurrence plot (RP) technique, and the recurrence quantification analysis
(RQA) method . They
constitute proper tools to treat such nonlinear, non-stationary signals as in
geophysics processes. Such analysis method is applied to the HILDCAA events,
for the first time to our knowledge, allowing a comparison of dynamical
characteristics. By applying the nonlinear tools, this work investigates
AE under some distinct physical conditions of the interplanetary medium:
Alfvénic fluctuations followed by HILDCAA, Alfvénic fluctuations not
followed by HILDCAA (also related to co-rotating interaction regions (CIRs)
and high-speed streams (HSSs)), other disturbed interplanetary conditions, and
geomagnetically quiet time.
This work proceeds as follows.
Section describes the methods for analysis.
Section presents the geomagnetic database and how we apply the methodology.
Section shows the results and interpretations.
Finally, Sect. summarizes the conclusions.
Method of analysis
Information theory structures a branch of powerful mathematical tools to
analyse nonlinear systems of signal as proposed in the seminal paper of the
mathematician Claude E. Shannon . An analogy with the
concept of entropy from physics gives basis to these tools. As reviewed and
discussed in detail by , the entropy H used as
basis for the methods can be expressed by
H(X)=-∑P(x)log(P(x)),x∈X,
where X is the set of all messages {x1,…,xn} that
X could be, and P(x) is the probability of some x∈X.
In this work, we use quantification methods associated with this theory,
precisely the method developed by of RQA that is built from the RP, as
introduced in , and the proprieties of the phase
space, provided in the Cross Recurrence Plot Toolbox.
Cross
Recurrence Plot Toolbox 5.21 (R31b) by the Interdisciplinary Center for
Dynamics of Complex Systems, University of Potsdam
(http://tocsy.pik-potsdam.de/CRPtoolbox/.).
Initially, these methods
are used to analyse dynamical systems from a theoretical point of view.
Nevertheless, since the late 1990s, they have been extended to experimental
data to characterize nonlinear complex behaviour
.
Below we summarize
the phase space, the RP, and the RQA approaches.
Phase space
A phase plot is a geometric representation of the trajectories of a dynamical
system in the phase plane. It is a fundamental starting point of many
approaches in nonlinear data analysis, which is based on the construction of
a phase space portrait of the considered system. A review of that can be
found, for instance, in N. Marwan's
tutorial.
The state of a system can be expressed by its state variables x1(t),
x2(t), …, xd(t) – for instance, the state variables density, pressure,
momentum, and magnetic field for a magneto-hydrodynamics system. The d
state variables at time t establish a vector in a d-dimensional space
which is called phase space. The state of a system changes in time, and,
consequently, the vector in the phase space describes a trajectory
representing the time evolution, i.e. the dynamics of the system.
Accordingly, the appearance of the trajectory retains information about the
system.
Therefore, the phase space is formed by coordinates that represent each
significant variable of the system to specify an instantaneous state
.
In practice, observations of a real process do not unveil all state
variables, or they are not known, or they cannot be measured. Nevertheless,
due to the couplings between the system components, we can reconstruct a
phase space trajectory from a single observation by a time delay embedding
.
It yields to the so-called Takens' embedding theorem, which states that a
reconstruction of the phase space trajectory x(t) from a time
series uk, with a cadence Δt, allows us to present a proper dynamics
of a system.
In order to do that, an embedding dimension m and a time delay τ must
be identified, related to the following reconstruction:
x(i)=xi=(ui,uii+τ,…,ui+(m-1)τ),
where t=iΔt. Here, m is found by using the false nearest
neighbour method and τ by the mutual information method
.
The idea behind this approach is to identify the influence of increasing the
embedded dimension m in the number of neighbours along a trajectory of the
system.
Recurrence plot
The RP is based on Poincaré's recurrence theorem from
1890, as discussed in . It states that a dynamic
system returns to a state arbitrarily close to the initial state after a
particular time.
Mathematically the RP is obtained by the square matrix
Ri,j=Θ(ϵi-∥xi-xj∥),
where ϵi is a predefined cut-off distance, ∥.∥ is
the norm (in our case, the Euclidean norm), and Θ(x) is the Heaviside
function . The binary values 0 and 1 in this
matrix are represented by white and black creating visual patterns.
The characteristic typology (related to macro patterns) and texture (related
to micro details) presented in the RP are the key points of the
interpretation. However, the visual interpretation of RPs requires some
training experience, usually done from standard systems or data libraries.
For instance, as described in and on the RP and RQA
website http://www.recurrence-plot.tk:
Stationary processes are associated to homogeneous distribution of points in
RP.
Periodic processes present
cycle patterns where the distance between periodic patterns
corresponds to the period.
Long diagonal lines with different distances to each other reveal a quasi-periodic
process.
Non-stationary processes can present interruption on the lines; they can also indicate
some rare state, or RP fading to the upper left and lower right corners indicating also trend or
drifts.
Single isolated points demonstrate heavy fluctuation in the process – in particular,
if only isolated points occur, an uncorrelated or anti-correlated random process is
represented.
Evolutionary processes are illustrated by diagonal lines – then the evolution of
states is similar at different times. However, if it has parallel lines
related to the main diagonal, the system is deterministic (or even chaotic,
if they occur beside single lines), and if the diagonal lines are orthogonal
to the main diagonal, or the time is reversed or the choice of embedding is
insufficient.
Long bowed line structures express evolution states that are similar at
different epochs although they have different velocity (the dynamics of the system could be
changing).
Vertical and horizontal lines/clusters are evidence that a state
has no or slow change for some time, which points to a laminar state.
The establishment of quantifiers to express the characterization of the
processes described in RP was a significant advance in the popularization of
this tool, because it can help to express in a concise and objective way a
description on the dynamics of the processes, as discussed in
and references therein. Therefore, quantification
from RP comes primarily from the recurrence patterns, and presents for example as
point density, diagonal structures, and vertical structures in the RP. In the following text, we present four of these quantifiers to study
the behaviour of physical conditions such as geomagnetically quiet intervals
and HILDCAA cases.
Recurrence quantification analysis
addressed the problem of quantifying the structures
that appear in the RPs and used them to analyse experimental data. This
approach is useful to reveal qualitative transitions in a system. The
corresponding measurements capture the dynamical characters of the system as
represented by the signal. Therefore, RQA provides a qualitative
description of a system regarding complexity measures .
We refer to , and for a detailed
discussion on this subject. Notably, the diagonal structures in the RP and
the recurrence point density are used to measure the complexity of a physical
system .
In the present work we restrict our analysis to four characteristic parameters described below:
Recurrence rate (RR): This denotes the overall probability that
a certain state recurs and is obtained from the RP byRR=∑i,j=1NRi,j(ρ)N2.Larger values mean more recurrence.
Determinism (DET): this represents how predictable a system is, and is expressed by
the ratio of recurrence points that form diagonal lines of the RP of at least length ℓmin to all recurrence points, i.e.DET=∑ℓ=ℓminNℓP(ℓ)∑ℓ=1NℓP(ℓ),where P(ℓ) denotes the probability to find a diagonal line of length ℓ in the RP.
Laminarity (LAM): this measures the occurrence of laminar states and
is related to intermittent regimes – namely, it is the ratio between the
recurrence points forming the vertical lines and the entire set of recurrence
points computed byLAM=∑ν=νminNνP(ν)∑ν=1NνP(ν),where P(ν) denotes the probability to find a vertical line of length ν in the RP.
LAM does not describe the length of laminar phases. However, if this measure decreases the
RP consists of more single recurrence points than vertical structures.
This measurement is relatively more robust against noise in signals.
Entropy (ENT): this reflects the complexity of the deterministic structure in
the system referred to as Shannon entropy ; namely,ENT=-∑ℓ=ℓminNp(ℓ)ln(p(ℓ)),where p(ℓ)=P(ℓ)/Nℓ.
This measure reflects the complexity of the RP concerning the diagonal lines.
In this form computed from RP, the interpretation of these values
differ from traditional Shannon entropy – i.e. larger values are
related to low entropy compared to physics analogy .
Database and methodology procedure
For the present work, we have considered an updated list of 136 HILDCAA
events occurring between 1975 and 2012, compiled by .
The events were detected from the geomagnetic AE and middle- to low-latitude disturbance Dst indices by using the four strict HILDCAA criteria
: (i) the events have peak AE intensities
greater than 1000nT, (ii) the events last for more than 2 days,
(iii) high auroral activity lasts throughout the interval, i.e. AE never
drops below 200nT for more than 2h at a time, and
(iv) the events take place outside of the main phase of a geomagnetic storm.
For a better understanding, the main phase is determined by the depression in
the horizontal component, from middle to low latitudes, in the geomagnetic
field. This behaviour is identified and quantified using the hourly value
equatorial Dst index, which represents ideally the axially symmetric
disturbance magnetic field at the dipole equator on the Earth's surface. This
index is derived by monitoring the equatorial ring current variations
(http://wdc.kugi.kyoto-u.ac.jp/dstdir/dst2/onDstindex.html). The
AE data set is provided by the OMNIweb Service
(http://omniweb.gsfc.nasa.gov/) by NASA and Dst from World Data
Center for Geomagnetism, Kyoto Dst index service
(http://wdc.kugi.kyoto-u.ac.jp/dstdir/).
From the list, the first 16 events were eliminated due to incomplete
information. Among the remaining events, 33% were preceded by geomagnetic
storm main phase (Dst<-50nT). Thus, 80 events were analysed in
this work, because we selected the events classified as pure HILDCAAs, i.e.
events not preceded by any geomagnetic storm main phase.
As data sets, the high-time-resolution (1 min) AE indices were
analysed to study the dynamical characterization of the HILDCAA events by the
RQA method. To eliminate any marginal influences, we considered a 2280 min
interval centred at the middle point of a HILDCAA event. This number of
records was determined by the least interval among the events.
For a quantitative comparison of disturbance geomagnetic regimes, we also
performed the same RQA during the geomagnetically quiet period listed in
Table . The quiet days follow the criteria: Kp≤30,
Dst≥-50nT, and AE≤300nT.
The planetary 3 h range Kp index was introduced by J. Bartels in
1949 and designed to be sensitive to any geomagnetic disturbance affecting
the Earth
(http://www.gfz-potsdam.de/en/section/earths-magnetic-field/data-products-services/kp-index/explanation/).
It completes a set of indices to diagnose the level of geomagnetic
disturbance in a global sense. The geomagnetic indices
can be obtained from the World Data Center, Kyoto, at
http://wdc.kugi.kyoto-u.ac.jp/wdc/Sec3.html. In that way,
different physical regimes allow us to find a distinct
characterization of the signals. In our case, we investigate periods of
HILDCAA events that alter a physical regime that exists during the
geomagnetically quiet times.
The geomagnetically quiet intervals.
DateKp≤AE≤Dst≥14–18 November 200030267nT-20nT26–30 November 20013-133nT-50nT19–25 June 200420167nT0nT19–27 June 200620167nT-9nT15–23 July 200620200nT32nT1–9 December 200730200nT-5nT
For a more complete dynamical diagnosis, this work investigates AE index
under some other different physical conditions of the interplanetary medium.
Completing the earlier mentioned cases of the interplanetary Alfvénic
fluctuations followed by HILDCAA (related to CIRs and HSSs), and the
geomagnetically quiet time, cases of interplanetary Alfvénic fluctuations not
followed by HILDCAA (also related to CIRs and HSSs) and cases of intense
interplanetary conditions (characterized by simultaneous activities in the
AE, Dst and Kp indices) produced by different interplanetary causes are also
analysed.
Table presents the CIRs/HSSs not followed by HILDCAA event.
The first column shows the data set interval and the second column the 2280 min interval considered in the analysis calculations.
Table presents the events with AE index related to global intense geomagnetic disturbances.
The first column shows the data set interval and the second column the 2280 min interval considered in the analysis calculations.
CIRs/HSSs not followed by HILDCAA.
Data set intervalInterval considered2008, 012–018 (Jan 12 to 17)2008, Jan 14 (00:00)–15 (13:59)2008, 030–036 (Jan 30 to Feb 4)2008, Feb 2 (00:00)–3 (13:59)2008, 058–064 (Feb 27 to Mar 3)2008, Mar 2 (00:00)–3 (13:59)2008, 165–171 (Jun 13 to 18)2008, Jun 15 (00:00)–16 (13:59)2008, 175–181 (Jun 23 to 28)2008, Jun 26 (00:00)–27 (13:59)
AE in global intense geomagnetic disturbances.
EventInterval considered2012 (Mar 9)2012, Mar 9 (00:00)–10 (13:59)2012 (Apr 23–24)2012, Apr 23 (00:00)–24 (13:59)2012 (Jun 17)2012, Jun 17 (00:00)–18 (13:59)2012 (Jul 15)2012, Jun 15 (00:00)–16 (13:59)
The analyses of the results allow a comparison of the dynamical characteristics of signals.
Results
Initially, two typical cases are shown and analysed, one from the HILDCAA
events and another from the quiet time intervals. As examples for the
methodology application, they help to understand the analysis and its
interpretation. Figure shows AE variations including a
HILDCAA interval. The HILDCAA started at 17:34 UT on 30 May (day 150) and
continued until 09:34 UT on 2 June (day 153) of 1986, with a total duration of
about 64 h. In that figure, the double arrow horizontal line indicates
the exact interval of the event. For the RQA calculation we consider the
2280 min interval centred at the middle of the HILDCAA. Two vertical
dotted lines mark this interval.
Figure shows AE variations during a geomagnetically
quiet period. The plot shows the geomagnetically quiet period from 17 to 22
July (day 198 to day 203) of 2006 (from Table ). The region
between the two vertical dotted lines shows the same 2280 min interval
selected for the RQA study as in the HILDCAA case.
From the AE plots, the differences in the amplitudes between the HILDCAA
interval (peak about 1200nT) and the quiet time interval (peak
about 300nT) are remarkable, as expected. Both of them presents
fluctuations in the signal intensities. The application of the RQA
methodology aims to characterize the dynamical behaviour of the signals.
Figure represents the phase space plots for the
HILDCAA. As a value estimated by the earlier-mentioned mutual information
methodology, the time delay (τ) used is 34 min. The phase space
charts present snapshots of the interconnections of the records for each
case. As described by the theory in Sect. , the geometric
representation in the plot gives the trajectory of the dynamical system
involved in the AE index records. Although slightly insinuated by the
distribution of points, a proper representation is not achieved because the
noise in the signal disturbs the identification of the trajectory. Following
the same procedure, Fig. gives the representation
for the quiet interval shown earlier. The time delay (τ) found is also
34 min. Although the signal amplitude is quite different compared to
the one of the HILDCAA event, the trajectory behaviour is similar. A
question arises from the comparison – is it possible to distinguish from the
dynamical behaviour analyses the two kinds of occurrences as the AE indices
point out?
Geomagnetic AE index from 29 May (DOY 149) to 3 June (154) 1986
includes a HILDCAA event. The HILDCAA interval is identified by the double
arrow horizontal line, and the AE interval used for the RQA is shown between
the vertical dotted lines.
Geomagnetic AE index during the geomagnetically quiet period on 17
(DOY 198)–22 (203) July 2006. The AE interval used for the RQA is marked by
vertical dotted lines.
The phase space representation for the HILDCAA example shown in
Fig. . The delay time is 34 min.
To verify whether the question deserves study effort, we use the RP
technique to allow a visual inspection of the signal features. Dealing with
the RP theory for all the cases studied, we estimated the typical values
related to these dynamical systems. The embedded dimension (m) determined
by the false nearest neighbour method was found to be around 6, and
following the time delay (τ) was around 34 min. The cut-off
distance (ϵ) was ≈30nT for the HILDCAAs, and
≈10nT for the quiet intervals. For the other interplanetary
conditions, the values were similar to the value of HILDCAAs. The estimation
of ϵ uses a value defined by the additive effects of the data
resolution and the Gaussian noise threshold.
Related to the cases at the beginning of this section,
Fig. shows the RPs for the HILDCAA and
Fig. for the quiet interval. Here we take the embedded
dimension (m) and the time delay (τ) equal to 1 for RQA
calculations. These parameter choices take into account the categorization
purpose of the present work, and those values do not alter our
characterization process
. The RPs highlight the
recurrences in the signal records showing differences in the dynamical
patterns between the HILDCAA interval and the quiet period. For both
systems, the analyses on the large-scale patterns in the plots, designated as
typology, denote that they are of the disrupted kind – i.e. with abrupt
changes in the representation of the dynamics. However, the analysis of the
small-scale patterns, designated as texture, denotes a more complex dynamics
in the HILDCAA event than the one in the quiet interval. To obtain an
objective interpretation, we need to translate this visual appreciation to
quantitative descriptors of the dynamics of the system interpreted by the
AE index. As examples of this quantification, the results of the RQA
dynamical parameters for the quiet and HILDCAA case examples are presented in
Table . We verify they are about 1 order of magnitude
smaller for the HILDCAA than the values for the quiet interval. Thus, we
have a little evidence that encourages this kind of study.
The phase space representation for the geomagnetically quiet period
example shown, between the vertical dotted lines, in
Fig. . The delay time is 34 min.
RQA measures for the geomagnetically quiet interval and typical HILDCAA cases.
To pursue a comprehensive answer, we apply the RQA methodology to all 80
HILDCAA events completed by the examination of other cases selected
(six geomagnetically quiet intervals, five CIRs/HSSs not followed by HILDCAA, and
four events of AE in global intense geomagnetic disturbances) to allow
comparisons. The values of the RQA dynamical variables (RR, DET, LAM, and
ENT) were obtained for each case.
Table shows the minimum, maximum, mean, standard
deviation, median, and mode values estimated to the HILDCAAs and the quiet
periods. As can be seen, a difference of 1 order of magnitude for each
variable exists between these cases. For minima and maxima, the differences
are between half and 1 order of magnitude. The standard deviation, median,
and mode are in agreement with normal distributions for the phenomena.
The RP for the HILDCAA example. The interval shown by
the vertical dotted lines in Fig. is used to obtain the
plot.
The RQA results considering two typical cases.
HILDCAA period Geomagnetically quiet interval ValueRRDETLAMENTRRDETLAMENTMin0.00100.0100.0140.0000.01150.2510.3970.574Max0.00560.0860.1390.2730.03070.3570.5360.766Mean0.00160.0310.0490.0910.01950.3210.4730.672SD0.00050.0120.0200.0730.00650.0460.0580.075Med0.00150.0280.0460.1040.01940.3450.4870.690Mod0.00130.0100.0140.0000.01150.2510.3970.574
Finally, Fig. shows the RQA dynamical parameters for
all events under study. For each parameter, we normalized the values for all
events concerning extreme values obtained for the parameter. The empty
circles represent the HILDCAA events, and the plus signs show the quiet
periods. A clear distinction between the HILDCAA events and quiet time
intervals may be noted from the figure. The separation of the results for the
HILDCAA event and the quiet time interval establishes a clustering of the
results, which characterize two well-defined physical regimes. Further, the
symbol x indicates the results for AE index in CIR/HSS events not
followed by HILDCAA, and * in a whole global disturbance scenario.
As also seen in the figure, parameter behaviour is similar for CIRs/HSSs
causing HILDCAAs and CIRs/HSSs not causing them, and distinct from the behaviour
of quiet intervals. Therefore, based on this plot, one could say that the
bottom part shows the behaviour of Alfvénic solar wind intervals, CIRs and
HSSs, while the top part shows the behaviour related to the slow solar wind
interval. The analysis taking into account the AE in a whole global
disturbance scenario regarding geomagnetic behaviour shows larger spreading
values for the parameters (except by the RR parameter); nevertheless, values
are also different to the one in the quiet time regime. Based on the current
geophysical knowledge, the RQA patterns in the signals for these events help
to characterize/identify the standard physical features. Examining the
physics of every case in the active interplanetary regimes, one might point
out that the AE signature relates to HILDCAA that is connected to long-duration, large-amplitude Alfvénic fluctuations; to CIRs/HSSs not followed by
HILDCAA connected to short-term Alfvénic fluctuations and with or without a
small interplanetary southward magnetic amplitude; and to events in a global
geomagnetic disturbance scenario connected to small-amplitude southward
interplanetary magnetic field without Alfvénic fluctuations or to a large
southward interplanetary magnetic amplitude.
The RP for the geomagnetically quiet period example.
The interval shown by the vertical dotted lines in Fig.
is used to obtain the plot.
Normalized representation of the RQA parameters for auroral
electrojet (AE) indices in HILDCAA events (∘), in CIRs/HSSs not followed by
HILDCAA (x), in a global geomagnetic disturbance scenario (*), and in the
geomagnetically quieter intervals (+).
Thus, the RQA result comparisons lead us to achieve some interpretations.
The HILDCAAs seem unique events regarding visible, intense manifestations
of interplanetary Alfvénic waves; however, they are similar to the other
kinds of conditions regarding a dynamical signature (based on RQA), because
the effect of HILDCAA is involved in the same complex mechanism of generating geomagnetic
disturbances.
Allowing an interpretation of the geomagnetic disturbances, mainly the AE studied
here, the physics scenario could be properly interpreted according to a basic
view. As is well known, the fundamental mechanisms are the magnetic reconnection and
viscous interaction with a transfer of energy and particles by
electrodynamics interaction and generation of geomagnetic disturbance on
ground. Supported by the parameter clustering behaviours shown in
Fig. , the interpretation obtained from the RQA
examination of AE index is in agreement with those fundamental mechanisms.
Although describing an expected result, the quantitative study using this
method indicates in a clear way categories of phenomena (showed in
Fig. ).
On the one hand, during geomagnetically quiet conditions, the effective
interaction is the ram pressure on the solar front side of the magnetosphere
and the development of viscous interaction at flanks.
On the other hand, during HILDCAA events, the two fundamental electrodynamics
interactions (magnetic reconnection and viscous interaction) with a transfer
of energy and particles are indeed happening.
In principle, interplanetary phenomena producing both of those coupling
mechanisms, as processes examined in , concern the
mechanisms related to interplanetary Alfvén waves. In this kind of
occurrence, magnetic disturbances can be detected by magnetometers at the
polar regions as the HILDCAA events. Although they can be clearly noticed at
high latitudes, those disturbances are noticed as weak worldwide
manifestations.
CIR/HSS occurrences not followed by HILDCAA related to short-term Alfvénic
fluctuations and with or without small southward interplanetary magnetic
amplitude produce sporadic, low AE index disturbances, designated as
geomagnetic substorms.
Events in a whole global disturbance scenario related to large southward
interplanetary magnetic amplitude produce geomagnetic storms and associated
geomagnetic substorms.
Identified as distinct regimes by the RQA diagnosis, the geomagnetically quiet
intervals and HILDCAA events seem the proper conditions of transitions from
quiescent conditions to weaker geomagnetic disturbances inside the
magnetosphere and ionosphere system. Therefore, those RQA features can be useful for
other study purposes. The RQA method gives a clear indication of the dynamics to
be evaluated by magneto-hydrodynamics simulations, as developed
by or , to understand the processes
involved in a transfer of energy and particles into the
magnetosphere-ionosphere system.
Conclusions
Obtained from a diagnosis of features of a nonlinear system analysis, a
physics scenario of the auroral electrojet (AE) index is built with the aid
of the recurrence quantification analysis (RQA) information extracted from
the recurrence plot (RP) calculation. We performed this analysis using
80 HILDCAA events completed by the examination of other cases selected
(six geomagnetically quiet intervals, five CIRs/HSSs not followed by HILDCAA, and
four events of AE in global intense geomagnetic disturbances) to allow
comparisons.
Some significant RQA variables (RR, DET, LAM, and ENT) quantify and
characterize the dynamical signatures of the AE index related to HILDCAA
occurrences and other interplanetary environment conditions.
The key findings are as follows:
The quiet intervals as compared to HILDCAA intervals are characterized by
larger values of DET, LAM, and ENT, which means higher predictability,
lower entropy, and larger laminarity of the corresponding nonlinear dynamics.
There is distinct clustering, identified by RQA, of the dynamical behaviours recorded on the
ground produced by the interplanetary medium conditions: one regarding a
geomagnetically quiet condition regime and another regarding an effective
disturbed interplanetary regime.
The RQA results identify similar dynamical behaviours for HILDCAA events and the other disturbed cases.
On the one hand, the HILDCAAs seem unique events regarding the visible, intense manifestations
of Alfvénic waves; on the other hand, they are similar to the other phenomena regarding dynamical signatures (based on RQA), because they are involved
in the same complex mechanism of generating geomagnetic disturbances.
This complex mechanism is composed by the magnetic reconnection and the
viscous interaction implying ground geomagnetic effects triggered by the
southward interplanetary magnetic field.
One regime of clustering is AE index organized by geomagnetically quiet conditions, related to a
predominant interaction from the incidence of ram pressure on the solar front
side of the magnetosphere and the development of viscous interaction at
flanks, while there is a northward interplanetary magnetic field (IMF).
Another regime is AE organized by disturbed interplanetary conditions, with the presence of the southward IMF.
As the geomagnetically quiet intervals and HILDCAA events characterize the
proper conditions of transitions from quiescent conditions to weaker
geomagnetic disturbances inside the magnetosphere and ionosphere system, the RQA
method gives a clear indication of the two fundamental dynamics to be
evaluated with magneto-hydrodynamics simulations to understand in a better
way the fundamental processes related to energy and particle transfer into
the magnetosphere–ionosphere system.
With the present work, we have also demonstrated the potential applicability
of the RQA method for characterizing of nonlinear geomagnetic processes
related to magnetic reconnection and viscous interaction affecting the
magnetosphere, mainly with the aid of magneto-hydrodynamics simulations.
All data are publicly accessible; see section “Database and methodology procedure” for how to obtain the datasets.
All authors discussed the idea and the approach for the work development and took part in the preparation of the paper.
OM and MOD worked also in the application of the methodology.
The authors declare that they have no conflict of interest.
Acknowledgements
Margarete Oliveira Domingues and Odim Mendes thank the MCTIC/FINEP (CT-INFRA grant 0112052700) and
FAPESP (grant 2015/25624-2) for the financial support. Odim Mendes, Margarete Oliveira Domingues and
Ezequiel Echer thank the Brazilian CNPq agency (grants 312246/2013-7, 306038/2015-3
and 301233/2011-0, respectively). Rajkumar Hajra thanks the FAPESP 2012/00280-0 for
a postdoctoral research fellowship at INPE, and now the work supported by ANR
under financial agreement ANR-15-CE31-0009-01 at LPC2E/CNRS. Varlei Everton Menconi
thanks
the MCTIC-PCI program (grant 455097/2013-5) by the research fellowship at
INPE. The authors would like to thank the team of Interdisciplinary Center
for Dynamics of Complex Systems, University of Potsdam, for the RQA tools
(http://tocsy.pik-potsdam.de/), OMNIweb Service (http://omniweb.gsfc.nasa.gov/)
by NASA and the World Data Center for Geomagnetism, Kyoto, Japan
(http://wdc.kugi.kyoto-u.ac.jp/), where the geomagnetic indices used in this
study were collected from. The authors thank Olga Verkhogfolyadova and
two anonymous referees for constructive and useful suggestions leading to
significant improvement of the manuscript.
Edited by: Giovanni Lapenta
Reviewed by: Olga Verkhoglyadova and two anonymous referees
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