# 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

ARTICLE IN PRESS

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).

Page 2

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.

ARTICLE IN PRESS

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.

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

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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

ARTICLE IN PRESS

Fig. 2. Same as in Fig. 1, but for the year 2000.

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

1845

Page 4

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,

ARTICLE IN PRESS

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

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

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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

ARTICLE IN PRESS

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

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

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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

ARTICLE IN PRESS

Fig. 5. Parameter q for different scales for both variables and for the 2 years (1997 and 2000).

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

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Page 7

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.

ARTICLE IN PRESS

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.

M. Jose ´ Alves Bolzan et al. / Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005) 1843–1851

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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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