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.

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

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

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

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

This work proceeds as follows.
Section

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

Cross
Recurrence Plot Toolbox 5.21 (R31b) by the Interdisciplinary Center for
Dynamics of Complex Systems, University of Potsdam
(

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.

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

The RP is based on Poincaré's recurrence theorem from

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

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.

For the present work, we have considered an updated list of

From the list, the first

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 geomagnetically quiet intervals.

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

CIRs/HSSs not followed by HILDCAA.

AE in global intense geomagnetic disturbances.

The analyses of the results allow a comparison of the dynamical characteristics of signals.

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

From the AE plots, the differences in the amplitudes between the HILDCAA
interval (peak about

Figure

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.

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 (

Related to the cases at the beginning of this section,
Fig.

The phase space representation for the geomagnetically quiet period
example shown, between the vertical dotted lines, in
Fig.

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

The RP for the HILDCAA example. The interval shown by
the vertical dotted lines in Fig.

The RQA results considering two typical cases.

Finally, Fig.

The RP for the geomagnetically quiet period example.
The interval shown by the vertical dotted lines in Fig.

Normalized representation of the RQA parameters for auroral
electrojet (AE) indices in HILDCAA events (

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.

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

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.

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.

Margarete Oliveira Domingues and Odim Mendes thank the MCTIC/FINEP (CT-INFRA grant