Generalized Thermostatistics and Wavelet Analysis of the Solar Wind and Proton Density Variability

Abstract

In this paper, we analyze the probability density function (PDF) of solar wind velocity and proton density, based on generalized thermostatistics (GT) approach, comparing theoretical results with observational data. The time series analyzed were obtained from the SOHO satellite mission where measurements were sampled every hour. We present in the investigations data for two years of different solar activity: (a) moderate activity (MA) period (1997) and (b) high activity (HA) period (2000). For the MA period, the results show good agreement between experimental data and GT model. For the HA period, the agreement between experimental and theoretical PDFs was fairly good, but some distortions were observed, probably due to intermittent characteristics of turbulent processes. As a complementary analysis, the Global Wavelet Spectrum (GWS) was obtained allowing the characterization of the predominant temporal variability scales for both the periods and the stochastics aspects of the nonlinear solar wind variability are discussed.

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Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851
Generalized thermostatistics and wavelet analysis of
solar wind and proton density variability
Maurı´cio Jose ´ Alves Bolzana,?, Reinaldo Roberto Rosab,
Fernando Manuel Ramosb, Paulo Roberto Fagundesa, Yogeshwar Sahaia
aInstituto de Pesquisa e Desenvolvimento, Universidade do Vale do Paraı ´ba, Sa ˜o Jose ´ dos Campos, Brazil
bLaborato ´rio Associado de Computac -a ˜o e Matema ´tica Aplicada, Instituto Nacional de Pesquisas Espaciais, Sa ˜o Jose ´ dos Campos, Brazil
Available online 26 August 2005
Abstract
In this paper, we analyze the probability density function (PDF) of solar wind velocity and proton density, based on
generalized thermostatistics (GT) approach, comparing theoretical results with observational data. The time-series
analyzed were obtained from the SOHO satellite mission where measurements were sampled every hour. We present in
the investigations data for two years of different solar activity: (a) moderate activity (MA) period (1997) and (b) high
activity (HA) period (2000). For the MA period, the results show good agreement between experimental data and GT
model. For the HA period, the agreement between experimental and theoretical PDFs was fairly good, but some
distortions were observed, probably due to intermittent characteristics of turbulent processes. As a complementary
analysis, the global wavelet spectrum (GWS) was obtained, allowing the characterization of the predominant temporal
variability scales for both periods and the stochastic aspects of the nonlinear solar wind variability are discussed.
r 2005 Elsevier Ltd. All rights reserved.
Keyword: Solar wind; Turbulence; Statistical analysis; Intermittency; Generalized thermostatistics; Wavelets
1. Introduction
Due to the important role of the solar wind properties
in the solar–terrestrial plasma relations and magneto-
spheric physics, the study of its statistical properties and
their relations to those in the geomagnetic indices has
attracted growing interest (Kova ´ cs et al., 2001; Lui,
2002; Hnat et al., 2002). Statistical behavior of velocity
field fluctuations recorded in wind tunnels and those
obtained from solar wind observations exhibit striking
similarities (Hnat et al., 2002), where a common feature
found in both fluctuations is the presence of statistical
intermittency (Burlaga, 1991; Marsch and Tu, 1994;
Marsch and Tu, 1997; Burlaga and Foreman, 2002). The
intermittency phenomena in the framework of turbu-
lence theory has been investigated by many authors
through laboratory and numerical experiments (e.g.
Anselmet et al., 1984; Frisch, 1995; Ramos et al., 2001a)
and the investigation of turbulent hydrodynamical flows
has been developed considering many different ap-
proaches: Reynolds-stress models; subgrid-scale models
for large-eddy simulations (LES); spectral models; and
probability density functions (PDF) models (Frisch,
1995). On the other hand, the turbulence modeling of
intermittent magnetohydrodynamical (MHD) flows are
based on: (i) the She–Leveque approach that describes
the observed scaling structure function (Biskamp and
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www.elsevier.com/locate/jastp
1364-6826/$-see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jastp.2005.01.015
?Corresponding author at: Instituto de Pesquisa e Desenvol-
vimento, Universidade do Vale do Paraı´ba, Sa ˜ o Jose ´ dos
Campos, Av. Shishima Hifumi, 2.911-Urbanova, CEP 12244-
000, Sa ˜ o Jose ´ dos Campos, SP, Brazil. Fax: +551239471149.
E-mail address: bolzan@univap.br
(M. Jose ´ Alves Bolzan).

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Mu ¨ ller, 2000); (ii) Fokker–Planck equation considering
the Castaing distribution (Castaing et al., 1990; Hnat
et al., 2002). Such approaches are used in the isotropic
inertial subrange of turbulent fluctuations assuming the
Kolmogorov hypothesis and contain the energy cascade
phenomenology (Kolmogorov, 1941, 1962).
Usually, the properties of turbulent flows, despite the
nature of the underlying physics, are studied from the
probability density functions (PDFs) of fluctuating
quantities (velocity differences, for example) at different
separation scales. It is a well-known property of
turbulent flows that, at large scales, these PDFs are
normally distributed. However, at increasingly smaller
scales, they become strongly non-Gaussian and display
tails flatter than expected for a Gaussian process. This is
interpreted as the signature of the intermittency: the
emergence of strong bursts in the kinetic energy
dissipation rate. Within this framework, intermittency
and non-extensive turbulence are linked by the entropic
parameter q from generalized thermostatistics theory
(Ramos et al., 1999; Beck, 2000; Arimitsu and Arimitsu,
2000; Ramos et al., 2001a,b; Bolzan et al., 2002; Ramos
et al., 2004). Recently, characterization of intermittent
turbulence in the solar wind velocity was performed
using a generalized distribution from the non-extensive
statistics approach (Burlaga and Vin ˜ as, 2004). The non-
extensive parameter q represents a measurable quantity,
flow independent and robust to variations in the
Reynolds number, that can be used to quantify the
occurrence of intermittency in turbulent flows. More-
over, the existence of possible coherent structures in the
time–frequency domain related to intermittent turbulent
fluctuations can be well-characterized by means of the
global wavelet spectra (GWS) (Rosa et al., 2002). In this
paper, analyzing the solar wind and proton density data,
we show that this new approach provides interesting
insights on MHD turbulence in a complex environment
such as the solar–terrestrial plasma.
2. Data
The data of solar wind and proton density analyzed in
this work were observed by the SOHO satellite and
provided by the University of Maryland database
(http://umtof.umd.edu/pm/crn/). The time-series used
were measured in the years 1997 and 2000, and were
sampled at the rate of 1 measurement per hour. Fig. 1
shows both the time-series (solar wind velocity and
proton density) for 12 months of 1997 and Fig. 2 shows
similar data sets for 12 months of 2000.
The solar wind for 1997 has moderate amplitude when
compared with the similar data set for 2000. The plot of
solar wind for 2000 represents a characteristic scenario
in the evolution of the solar cycle, notably the impulsive
change of the velocity amplitude in the month July. This
enhancement in the solar wind velocity is associated with
the very strong solar disturbance that occurred in July.
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Fig. 1. Time-series for the year 1997. The top plot corresponds to the solar wind velocity and the lower plot corresponds the proton
density.
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Also, for the proton density time-series, for both the
moderate activity (MA) and high activity (HA) periods,
we found significant differences in temporal variability,
notably a higher amplitude for 2000 compared with
1997.
3. Theory
Recently, Kova ´ cs et al. (2001) have suggested that,
disregarding storms and/or substorms as the main
sources of the evolution of geomagnetic disturbances,
the fluctuating nature of the field can be interpreted in
the present context as manifestation of turbulent
phenomena that take place within the plasma of the
magnetosphere. It has long been accepted that turbu-
lence evolves through cascade processes that involve a
hierarchy of coherent vortex structures belonging to a
wide range of spatial scales. Kolmogorov (1962)
proposed the inhomogeneous flow-down (cascade) of
energy from system-size scales to dissipative ones. The
inhomogeneity involves the singular behavior of energy
distribution in physical space resulting in strong
gradients, or intermittency in the time-series of energy-
related physical quantities of the system, e.g. velocity
(Ramos et al., 2001a), temperature (Bolzan et al., 2002)
or magnetic fields (Kova ´ cs et al., 2001; Lui, 2002).
In this paper, we will adopt a generalization of the
PDF model used in our previous works (Ramos et al.,
1999, 2001a,b; Bolzan et al., 2002), assuming that pqðvrÞ
is given by:
pqðvrÞ ¼ ½1 ? bð1 ? qÞðjvrj2a? C signðvrÞðjvrja
?1
where C is a small skewness correction term, and Zqis
given by
3jvrj3aÞÞ?1=ð1?qÞ=Zq,
ð1Þ
Zq¼ hjvrj0i ¼am0þ1
a
Bðf0;w0Þ,(2)
with Bðf0;w0Þ¼Gðf0ÞGðw0Þ=Gðf0þ w0Þ, f0¼ ð1 þ m0Þ=2,
w0¼1 ? f0;l ¼1=ðq ? 1Þ, m0¼ ð1 ? aÞ=a and a ¼
Neglecting the skewness correction term, we obtain
for the PDF nth moment:
ffiffiffiffiffiffiffi
l=b
p
.
jvrjn
h i ¼ amn?m0Bðfn;wnÞ
Bðf0;w0Þ,
fn¼ ð1 þ mnÞ=2,
:The parameters q and b determine the
shape of the PDF and are computed from Eq. (3)
(Bolzan et al., 2002; Ramos et al., 2004). Thus, note that
the q and b parameters are derived from the experi-
mental kurtosis for each scale.
We also used the GWS through the Morlet wavelet
transform (MWT). This mathematical tool is similar to
power spectrum density (PSD) obtained by fast Fourier
transform (FFT) and is based in the calculation of
variance in each scale, or period, obtained by MWT.
The objective of this procedure is to identify the
predominant scales (periods) driving the turbulent
process. For this, the computation consists in to sum
(3)
where
mn¼ðnþ1Þ?a
wn¼ l ? fn
and
a
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Fig. 2. Same as in Fig. 1, but for the year 2000.
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all energy associated with each scale. This can be
performed according to the following equation (Le and
Wang, 2003):
Z
where a is the scale, Wða;tÞ is the Morlet wavelet
transform applied in the time-series, and t is the
temporal size of the time-series.
MðaÞ ¼Wða;tÞ
????2dt;
(4)
4. Results and discussions
In order to validate the model described in Section 2,
we compared measured distributions, corresponding to
two different periods of solar activity, for both variables,
solar wind and proton density, with the theoretical
PDFs obtained from Eq. (1). For each data set, we
measured the variance and kurtosis, which allowed us to
compute q and b, by means of the corresponding
expressions obtained from Eq. (3). The parameter a was
chosen according to the empirical formula a ¼ 6 ? 5q as
used by Bolzan et al. (2002) and Ramos et al. (2004).
Fig. 3 presents the theoretical and experimental semi-
logarithmic plots of pqðxrÞ versus xr at four different
scales, properly rescaled and vertically shifted for better
visualization, for the solar wind velocity in 1997. The
increment scales r used were r ¼ ð2;20;200;2000Þ and
correspond to lags of r1¼ 2h, r2¼ 20h, r3¼ 8:3 days
and r4¼ 83:3 days. These scales are similar to the ones
used by Burlaga and Vin ˜ as (2004). Overall, we observe
that the theoretical results (solid lines) are in good
agreement with measurements across spatial scales
spanning three orders of magnitude and a range of up
to 5 standard deviations, including the rare fluctuations
at the tail of the distributions. We performed a simple
error analysis given by the correlation coefficient
between experimental and theoretical PDF for each
scale, as shown in Table 1. We note that there are higher
values of correlation coefficient for all the increment
scales, indicating good agreement between experimental
data and our model results. The transition from large-
scale Gaussian behavior to a stretched exponential form,
as r decreases, is quite evident and well-reproduced by
Tsallis’ distribution (Tsallis, 1988). At small scales, the
distributions have tails larger than that expected for a
normal process. This excess of large fluctuations,
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Fig. 3. Theoretical and experimental PDFs for the solar wind velocity in 1997.
Table 1
Correlation coefficients between experimental and theoretical
PDF to four increment scales for solar wind velocity time-
series, 1997
Increment r
Correlation coefficient (%)
2
20
200
2000
92.81
94.03
98.15
89.63
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compared to a Gaussian distribution, is a well-known
signature of intermittency. The spiky shape near the
origin is also a signature of intermittency (Frisch, 1995).
According to Burlaga and Vin ˜ as (2004), these high
points on the tail of the distribution in PDF for r1¼ 2h,
represents a few large jumps in the solar wind velocity
associated with shocks, stream interfaces and some
discontinuities with large shear. Furthermore, PDF for
r2¼ 20h presented significant positive skewness, similar
to results reported by Burlaga and Vin ˜ as (2004) for lags
of 16h. According to these authors, this skewness is a
consequence of stream-steepening, i.e., faster plasma
overtaking slower plasma. The PDF for r3¼ 8:3 days,
presents a transition between stretched exponential to
Gaussian form. This behavior may be associated with
the slow flows, which have temporal scales in the range
of one to several days. Similar result was also obtained
by Burlaga and Vin ˜ as (2004) for lag of 1.3 days. Now,
the PDF for r4¼ 83:3 days, presents a Gaussian format,
where according to Burlaga and Vin ˜ as (2004), this
Gaussian behavior maybe showing a variety of flows
characteristic of a particular epoch of the solar cycle
activity.
Fig. 4 presents also the theoretical and experimental
semi-logarithmic plots of pqðxrÞ versus xrat same scales
for proton density for 1997. We observe that the
theoretical results (solid lines) are in good agreement
with measurements across spatial scales spanning three
orders of magnitude and a range of up to 5 standard
deviations. Also, we performed the correlation coeffi-
cient between experimental and theoretical PDF for
each scale, as shown in Table 2. Again, we can observer
the higher values for correlation coefficient for all scales,
showing good agreement between experimental data and
our model. We can observe that all the PDFs exhibit
stronger non-Gaussian behavior than the solar wind
velocity PDFs. This distinct behavior between both
variables is due the peculiar characteristic of passive-
scalar as pointed by Warhaft (2000) and Basu et al.
(2003). This is an interesting aspect of differences
between both variables.
To study more closely this distinct behavior in both
the variables, we also estimated the variation with scale
of parameter q and plotted the parameter q by increment
r for both quantities and for both years, as shown in
Fig. 5. As a first analysis, we note that the four curves
have similar behavior, where the parameter q value
decreases as r grows. Katul et al. (1994), using a
parameter related to scale kurtosis—the wavelet flatness
factor (FF), have shown that in the inertial subrange
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Fig. 4. Theoretical and experimental PDFs for the proton density in 1997.
Table 2
Correlation coefficients between experimental and theoretical
PDF to four increment scales for proton density time-series,
1997
Increment r
Correlation coefficient (%)
2
20
200
2000
98.10
99.20
97.64
99.04
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scales, the more the separation distance r increases, the
lower are the values of FF. They have also shown that
this FF value trend is caused by the increase of
intermittency in the smallest scales of the inertial
subrange. In this sense, their results are similar to the
ones presented in Figs. 3 and 4, in which there is a clear
enhancement of q as r decreases. The important aspect
found is that the proton density data set is more
intermittent than the solar wind velocity. This peculiar
feature was observed in data set of passive scalars, like
turbulent temperature in atmospheric of Amazonia
(Bolzan et al., 2002; Ramos et al., 2004) and laboratory
flows (Warhaft, 2000). Another interesting fact is that the
intermittency level for both the variables is higher for
2000 if compared with 1997. This characteristic is
important because it shows the difference between the
two time-series for two different conditions: one time-
series that represents a moderate solar activity period and,
the other time-series that represents high solar activity.
From the analyses presented above, we note that the
proton density time-series are more intermittent than the
solar wind velocity. However, it is observed in Fig. 3, we
do not have good agreement between experimental and
theoretical PDFs of the solar wind velocity. This
behavior may be related of the particular skewness in
solar wind velocity. Many investigations have reported
the presence of skewness in the solar wind velocity time-
series (Burlaga and Foreman, 2002; Burlaga and Vin ˜ as,
2004). Furthermore, Basu et al. (2003), through the
proposed scheme to generate synthetic turbulent velocity
and passive-scalar in hydrodynamics fields, show that in
the small scales the skewness presents values of
approximately ?0.3 to ?0.4 and these small negative
values are believed to be the origin of vortex stretching
and nonlinear energy transfer from large to small scales.
To show the skewness aspect, we performed the
skewness for same four increment r to the solar wind
velocity and proton density data, for both the years.
Fig. 6 displays only the skewness of the solar wind
velocity, because the results for the proton density
show values of the skewness close to zero. The bulk
velocity is usually between 200 and 700km/s with an
average of 400km/s (Hargreaves, 1979). We note the
high values in the skewness for 2000, in all increment
scale r, if compared with 1997. We did not get negative
values for the skewness, as obtained by Basu et al.
(2003). This behavior may be indicating a different role
of the skewness between the two turbulences: hydro-
dynamic and magnetohydrodynamic. However, we note
that the solar activity has an important role in this
parameter.
For understanding the influence of the solar activity in
both the variables, we used an algorithm for GWS
presented by Torrence and Compo (1998). This algo-
rithm was applied in the solar wind and proton density
time-series for two years, 1997 and 2000. Fig. 7 shows
the GWS for proton density for both the years. We note
that there is increase of energy in some periods for 2000.
In particular, the increase of energy with a period of
approximately 26 days, corresponds to the solar rota-
tion. We also observe the increase of energy with lower
periods like 9 and 13 days. These periods can be
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Fig. 5. Parameter q for different scales for both variables and for the 2 years (1997 and 2000).
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explained through the fact that during this year many
solar disturbances of short periods occurred. For the
solar wind velocity, the GWS show great differences
between both years (Fig. 8). Again, we note the increase
of energy in all the periods for 2000, mainly in the short
periods, corresponding to approximately 9 and 13 days.
However, we did not observe the influence of solar
activity in the increase of energy in smaller periods of
less than 1 day. This subject will be investigated in near
future to understand how energy transfer occurs
between large and small scales during the high solar
activity period of the Sun.
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Fig. 7. Global wavelet spectrum (GWS) applied to the proton density time-series for the years 1997 and 2000.
Fig. 6. Increment scale variations of the skewness parameter for the solar wind velocity time-series.
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5. Concluding remarks
We have studied PDFs of solar wind velocity and
proton density for the two different periods of solar
activity, 1997 (moderate activity) and 2000 (high
activity), measured by spacecraft sensors. Our approach
was based on generalized thermostatistics theory. The
behavior of the entropic parameter q can be used to
objectively quantify intermittency buildup in turbulent
flows. From a practical point of view, the use of the
entropic parameter as a measure of intermittency is
justified by the fact that q is the key parameter that
controls the shape of the PDF, which accurately models
the statistics of turbulent solar wind and proton density.
As expected from the earlier theoretical results, we
found higher values of q in proton density for both the
years. This is due to the peculiar characteristics of the
scalar parameters. Similar results were obtained by
Bolzan et al. (2002) and Ramos et al. (2004) using
temperature time-series measured in the turbulent flow
of the Amazonian forest. Among the physical mechan-
isms which would be responsible for this behavior, we
could mention the influence of the coherent magnetic
vortices, studied by Kinney and McWilliams, 1995. As a
consequence of this influence we did not observe good
agreement between theoretical and experimental PDFs
for the year 2000 solar wind velocity time-series.
Through the analysis of the skewness parameter related
to solar wind time-series, we observed high values of this
parameter. The energy necessary to provide the increase
in skewness values was due to the increase of dis-
turbances during this year. By global wavelet spectrum
(GWS) analyses, these disturbances increase the energy
in lower periods for both the time-series in 2000, but
being much efficient in the solar wind velocity time-
series. These periods were approximately 9, 13 and 26
days. Taking into account these results, we have shown
that the generalized thermostatistics approach combin-
ing GWS analysis provides a simple and accurate
framework for modeling the statistical behavior of
MHD turbulence involved in the solar–terrestrial
plasma dynamics.
Acknowledgements
All the data used in this work were obtained from
MTOF/PM data by Carrington rotation from the
website: http://umtof.umd.edu/pm/crn/. Thanks are also
due to the referees for their valuable suggestions and
comments.
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