We studied the temporal evolution of fractality for geomagnetic activity, by calculating fractal dimensions from the Dst data and from a magnetohydrodynamic shell model for turbulent magnetized plasma, which may be a useful model to study geomagnetic activity under solar wind forcing. We show that the shell model is able to reproduce the relationship between the fractal dimension and the occurrence of dissipative events, but only in a certain region of viscosity and resistivity values. We also present preliminary results of the application of these ideas to the study of the magnetic field time series in the solar wind during magnetic clouds, which suggest that it is possible, by means of the fractal dimension, to characterize the complexity of the magnetic cloud structure.

There is a nontrivial magnetic interaction between Sun and Earth, coupled by the solar wind, leading to a rich variety of phenomena, which has attracted interest to the study of space plasmas for decades, and more recently to the possibility of forecasting space weather, an issue of large relevance in our increasingly technology-dependent society.

Various models and techniques have been developed to study the plasma
behavior in the Sun–Earth system. Of these, the study of complexity
has been of great interest, as it is capable of providing new
insights regarding universal behavior related to
geomagnetic activity, turbulence in laboratory plasmas or the solar
wind, to name a few

In our work we use the box-counting fractal
dimension

These ideas were implemented by us in

The results mentioned above were robust, in the sense that they were observed across a wide range of timescales, which suggests that any model describing the dynamics of geomagnetic activity should reproduce a similar fractal behavior. This is our motivation to study a shell model for magnetohydrodynamic (MHD) turbulence within this framework.

Evidence of turbulence in the Earth's magnetosphere has been found by
various spacecraft observations

At an intermediate level between these models and first-principles
approaches we find shell models, consisting of a set of coupled
equations which are similar to the spectral Navier–Stokes
equation, but which are also low dimensional models. They have been
successfully used to describe turbulence in magnetized fluids, being
able to deal with large Reynolds numbers without the associated
computational cost of simulations based on first principles and nonlinear
fluid equations

In fact, it has been shown that dissipative events in shell models can
be taken to represent solar flares and that their distribution
follows the same power-law statistics as
observed in turbulent magnetized
plasmas

In a previous work

In this paper we review our results in this field, where complexity in
magnetic field time series is measured by means of the fractal dimension.
Thus, we characterize events such as geomagnetic storms by means of analyzing
the Dst time series on various timescales (described in
Sects.

We are interested in estimating the fractal dimension to various time
series for magnetic data. We now explain the method, using the hourly Dst time
series as an example (World Data Center for Geomagnetism,

Fractal dimensions can be defined in various ways in general, and for
a specific time series in particular ways as well

Then, the scatter diagram is divided into square cells of a certain
size

Further details and discussion on the method can be found
in

The method as stated above was applied to the Dst time
series where, given the width of the data windows used (the criterion
is discussed in
Sect.

Scatter diagram for Dst time series, using data from 6
to 20 March 1989, containing a large geomagnetic storm. (Taken from

However, when resolution is larger, as is the case with
simulation and solar wind data, it is possible to consider different
time delays.
Thus, the scatter plot can be built by plotting the

Some studies

Following

In the following, states within a year are labeled by integer numbers
starting from 1. A fractal dimension is then calculated for each storm
and each quiet state in the same year. Results for 1989 are shown in
Fig.

Similar plots for 5 years of high geomagnetic activity were
obtained

We have also studied variable width windows around a storm and moving windows across storms, and results have been consistent with the findings discussed.

Storm and quiet states in the Dst time series for 1989,
also indicating the average value of the Dst index (horizontal
line), and the threshold used to identify storms (dashed line). Red (black)
arrows indicate storm (quiet) states. (Taken from

In effect, as a window is widened around a storm, more “quiet” data
are considered, and thus the fractal dimension of the data inside the
window should increase. This is actually the case, as shown for
instance in Fig.

Box-counting dimension

Box-counting dimension

Regarding the moving windows analysis, results are illustrated for the
13 March 1989 storm in Fig.

As shown in

In

Box-counting dimension

Given the intrinsic difficulties in using direct numerical simulations
to describe turbulent flows, especially for large Reynolds numbers,
shell models have been used for years in order to reproduce the
nonlinear dynamics of fluid systems in large dynamical ranges, but
with fewer degrees of freedom

In this work, we use the MHD shell model proposed by Gledzer, Okhitani,
and Yamada (GOY shell model), which describes the
dynamics of the energy cascade in MHD turbulence

The model consists of the following set of ordinary differential
equations:

The forcing terms are calculated according to the Langevin equation

The magnetic energy dissipation rate is defined as

In our simulation, we set

We numerically integrate
the shell model Eqs. (

Figure

Previous works have compared the statistics of bursts in turbulent systems
with the statistics of dissipative events in the
shell model

We now apply the same techniques used to study the Dst index, as
described in Sects.

We first notice that, in general, setting parameters

Time series of

Now, we need to define “active states” and quiet states. For
the Dst case

Regarding the width of an active state, Fig.

We now analyze the output of the simulation for given values of

For all values of

Notice that for the smallest value of

Box-counting fractal dimension for

As observed in Fig.

Box-counting fractal dimension for quiet and active states, as a
function of

In

Time series for the magnetic field for the two magnetic cloud events
analyzed. The different stages in the data are identified: solar wind (SW),
sheath (S), and flux rope (FR).

As a way to illustrate how the ideas described so far could be used to
characterize structures in space plasmas, we apply the method to study
the time series for the magnetic field during magnetic
clouds

Box-counting fractal dimension for two magnetic cloud events during
the four stages of the time series: first the solar wind, then the sheath,
then the flux rope, and finally the solar wind again. Several values for data
sampling

Two events were selected: an event occurring on 12 July 2012 (MC1) and another on 11 July 2013 (MC2). Resolution for the magnetic field time series for these events is 16 s, covering a time span of 8 days for MC1 and 6 days for MC2, of which about 2 days correspond to the cloud events themselves.

Figure

It is found that the
calculated fractal dimension evolves in a distinctive way through the
various stages of the event as it passes by the spacecraft (namely
surrounding solar wind, sheath, and flux rope). Given the high
resolution of the data, it is possible to calculate the box-counting
dimension for several delays, given by

It can be noted that the fractal dimension, as calculated here, is
indeed able to characterize magnetic cloud structures. The sheath
state has a large dispersion of fractal dimension values as

The results above suggest that, from the point of view of the time series, the level of multifractality is large in the sheath, consistent with its more turbulent nature, intermediate in the solar wind, and that the flux rope magnetic field is essentially monofractal, consistent with the organized, smoother structure of the magnetic field expected in this region. We plan to carry out other multifractal analyses to complement these findings in a future publication.

Also, these results suggest that the fractal approach discussed in this paper may be useful to characterize the various stages of magnetic clouds, and in particular to set up a system to automatically identify similar magnetic structures in spacecraft data.

In this paper, we have reviewed recent results obtained by us, regarding the evolution of complexity in magnetized plasmas, as described by geomagnetic data, simulation results for MHD turbulence, and spacecraft data in the solar wind.

This has been done by calculating a box-counting fractal dimension
for time series of magnetic field data for the Dst
geomagnetic index

In general, it is found that
the fractal dimension

A similar behavior is found for the MHD shell model
(Sect.

Also, a more systematic test for the correlation between burst events
in the shell model and the
decrease in fractal dimension was performed, by means of the Student's

As an application of these ideas, we take two magnetic cloud events in the solar wind, and use the techniques described here to study the corresponding magnetic field time series. Our results, although preliminary, suggest that this method can characterize the various stages of the magnetic cloud structure.

Given the rich and complex dynamics governing the evolution of magnetized plasmas, we would not expect that a single index would be able to capture all their relevant information. In fact, multifractal analysis should be conducted in order to represent the dynamics of the systems studied more accurately, and such an analysis is currently being prepared for future publication. However, the findings summarized here suggest that some relevant correlations can be observed, and that the dimension used here, although simple, may give some insight into the evolution of complexity of plasmas in the Sun–Earth system and MHD turbulent states.

Dst data can be downloaded from the website of the World
Data Center for Geomagnetism,

The authors declare that they have no conflict of interest.

This article is part of the special issue “Nonlinear Waves and Chaos”. It is a result of the 10th International Nonlinear Wave and Chaos Workshop (NWCW17), San Diego, United States, 20–24 March 2017.

This project has been financially supported by FONDECYT under contract nos. 1110135, 1110729, and 1130273 (Juan Alejandro Valdivia), nos. 1080658, 1121144, and 1161711 (Víctor Muñoz), and no. 3160305 (Macarena Domínguez). Macarena Domínguez is also thankful for a doctoral fellowship from CONICYT, and a Becas-Chile doctoral stay, contract no. 7513047. We are also thankful for financial support by CEDENNA (Juan Alejandro Valdivia), and the US AFOSR grant FA9550-16-1-0384 (Juan Alejandro Valdivia and Víctor Muñoz). We thank the ACE/MAG instrument team and the ACE Science Center for providing the ACE data. Edited by: Gurbax S. Lakhina Reviewed by: three anonymous referees