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FINAL REPORT 


NASA GRANT NUMBER: NAG5-12942 

PRINCIPAL INVESTIGATOR: DR. TOM CHANG 


GRANT TITLE; 

INVESTIGATION OF THE DYNAMICS OF COHERENT STRUCTURE, BBF, 
AND INTERMITTENT TURBULENCE IN EARTH’S MAGNETOTAIL - 
A STUDY OF COMPLEXITY IN NONLINEAR SPACE PLASMAS 


Submitted to 
DR. Craig Pollock 
NASA HQ, 300 E. St., SW, Code SR 
Washington, D.C. 20546-0001 


May 25, 2005 


TABLE OF CONTENTS 


Cover 1 

Table of Contents 2 

l. Synopsis 3 

II. Publications 4 

m. Activities 5 

IV. Complexity and Intermittent Turbulence in Space Plasmas 7 


2 


Investigation of the Dynamics of Coherent Structure, BBF, and 
Intermittent Turbulence in Earth's Magnetotail - 
A Study of Complexity in Nonlinear Space Plasmas 

I. SYNOPSIS. 

We have achieved all the goals stated in our grant proposal. Specifically, these 
include: 

• The understanding of the complexity induced nonlinear spatiotemporal coherent 
structures and the coexisting propagating modes. 

• The understanding of the intermittent turbulence and energization process of the 
observed Bursty Bulk Flows (BBF’s) in the Earth’s magnetotail. 

• The development of “anisotropic three-dimensional complexity” in the plasma sheet 
due to localized merging and interactions of the magnetic coherent structures. 

• The study of fluctuation-induced nonlinear instabilities and their role in the 
reconfiguration of magnetic topologies in the magnetotail based on the concepts of 
the dynamic renormalization group. 

• The acceleration of ions due to the intermittent turbulence of propagating and 
nonpropagating fluctuations. 

In the following, we include lists of our published papers, invited talks, and 
professional activities. A detailed description of our accomplished research results is 
given in Section IV. 


3 


II. PUBLICATIONS. 


T. Chang, S.W.Y. Tam, and C.C. Wu, Complexity Induced Anisotropic Bimodal 
Intermittent Turbulence in Space Plasmas, Physics of Plasmas, 11, 1287, 2004. 

T. Chang, S.W.Y. Tam, C.C. Wu, and G. Consolini, Complexity, Forced and/or Self- 
Organized Criticality, and Topological Phase Transitions in Space Plasmas, Space 
Science Reviews, 107, 425, 2003. 

S. W.Y. Tam, T. Chang, P. Kintner, and E. Klatt, “Intermittency Analyses on the SIERRA 

measurements of the electric field fluctuations in the auroral zone”, Geophysical 
Research Utters, 32, LO5109, doi:1029/2004GL021445, 2005. 

T. Chang, S.W.Y. Tam, and C.C. Wu, “Complexity if Space Plasmas”, in Special Issue 

on Multiscale Coupling of Sun-Earth Processes, Journal of Atmospheric and Solar- 
Terrestrial Physics, edited by A.T.Y. Lui, G. Consolini, Y. Kamide, Elsevier, 2005. 

C.C. Wu and T. Chang, “Intermittent Turbulence in 2D MHD Simulation”, in Special 
Issue on Multiscale Coupling of Sun-Earth Processes, Journal of Atmospheric and 
Solar-Terrestrial Physics, edited by A.T.Y. Lui, G. Consolini, and Y. Kamide, 
Elsevier, 2005. 

S. W.Y. Tam and T. Chang, “Energization of Ions by Bimodal Intermittent Fluctuations”, 

in Special Issue on Multiscale Coupling of Sun-Earth Processes, Journal of 
Atmospheric and Solar-Terrestrial Physics, edited by A.T.Y. Lui, G. Consolini, and 
Y. Kamide, Elsevier, 2005. 

T. Chang, S.W.Y. Tam, and C.C. Wu, "Complexity and Intermittent Turbulence in Space 

Plasmas", Chapter 2, Nonequilibrium Transition in Plasmas, edited by A.S. Sharma 
and P. K. Kaw, Springer, Dordrecht, Netherlands, p. 23, 2004. 

G. Consolini, T. Chang, and A.T.Y. Lui, “Complixity and Topological Disorder in the 
Earth’s Magnetotail Dynamics”, Chapter 1 , Nonequilibrium Transition in Plasmas, 
edited by A.S. Sharma and P. K. Kaw, Springer, Dordrecht, Netherlands, p. 51, 2004. 


4 


III. ACTIVITIES. 


• Co-Guest Editor: Special Issue on “Complexity in Earth's Magnetotail”, Annales 
Geophysicae, to be published in 2003. 

• Co-Convener, Symposium on Solar System Physics, Joint EGS-AGU-EUG 
Assembly, Nice, France, 2003. 

• Member of Program Committee, Conference on Sun Earth Connection - Multiscale 
Coupling of Sun Earth System, Big Island, Hawaii, 2004. 

• Editor: Nonlinear Processes in Geophysics, European Geophysical Society, 2004- 
2007. 

• Visiting Professor, Catholic University of Louvain, Belgium, 2004. 

Invited Lectures: 

• World Space Environmental Forum 2005, Schloss Seggau, Graz, Austria, 2005. 

• Conference on Sun Earth Connection - Multiscale Coupling of Sun Earth System, Big 
Island, Hawaii, 2004. 

• Per Bak Memorial Session on Nonlinear Processes in Space Plasmas, Joint EGS- 
AGU-EUG Assembly, Nice, France, 2003. 

• Session on Solar System Physics, Joint EGS-AGU-EUG Assembly, Nice, France, 
2003. 

• Opening Lecture, Session on Nonlinear Processes in Space Physics, Singapore, 2004. 

• Invited Open Lecture on Intermittent Turbulence in Plasmas, Physics Department, 
Katholieke Universiteit Leuven, Leuven, Belgium, 2004. 


5 


• Invited Lecture on Complexity in the Space Environment, Astrophysics Division, 
University of Louvain la Neuve, Louvain La Neuve, Belgium, 2004. 

Visiting Scientists and Students: 

Sandra Chapman [University of Warwick]. 

Nick Watkins [British Antarctic Survey]. 

George Rowlands [University of Warwick] 

Andrew Yau [University of Calgary]. 

Tom March [University of Warwick]. 

James Merrifield [University of Warwick]. 

Listing, 2000 Outstanding Scientists of the 20th Century. 


6 


IV. COMPLEXITY AND INTERMITTENT TURBULENCE 


IN SPACE PLASMAS 


7 



COMPLEXITY AND INTERMITTENT 
TURBULENCE IN SPACE PLASMAS 


Tom Chang and Sunny W.Y. Tam 

Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139 

USA 

Cheng-chin Wu 

Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 USA 

Abstract Sporadic and localized interactions of coherent structures arising from plasma 
resonances can be the origin of "complexity" of the coexistence of non- 
propagating spatiotemporal fluctuations and propagating modes in space 
plasmas. Numerical simulation results are presented to demonstrate the 
intermittent character of the non-propagating fluctuations. The technique of 
the dynamic renormalization-group is introduced and applied to the study of 
scale invariance of such type of multiscale fluctuations. We also demonstrate 
that the particle interactions with the intermittent turbulence can lead to the 
efficient energization of the plasma populations. An example related to the 
ion acceleration processes in the auroral zone is provided. 

Key words Complexity, Space plasmas, Dynamic renormalization group, Forced and/or 
self-organized criticality, Topological phase transitions, Intermittency, 
Particle-Fluctuation interactions, Auroral ion acceleration. 



2 


Chapter 


Introduction 


In situ observations indicate that the dynamical processes in the space 
plasma environment generally entail anisotropic and localized intermittent 
fluctuations. It was suggested by Chang (1998a,b,c; 1999) that instead of 
considering this type of turbulence as an admixture of waves, such patchy 
intermittency could be more easily understood in terms of the development 
and interactions of coherent structures. Results of two-dimensional MHD 
simulations (Wu and Chang, 2000a, b; 2001) including the calculated 
fluctuation probability distribution functions and local intermittency 
measures (LIM) based on wavelet transforms seem to validate the suggested 
characteristics of the intermittent stochastic processes. 

On the other hand, plasma waves are also generally observed in 
conjunction with the nonlinear, non-propagating fluctuations. It has been 
demonstrated that the coexistence of both propagating and non-propagating 
fluctuations in a plasma is a natural consequence of three-dimensional 
complexity for dynamical plasmas (Chang, 2003). 

For nonlinear dynamical systems near criticality, the correlations among 
the fluctuations are extremely long-ranged. The dynamics of such systems 
are notoriously difficult to handle either analytically or numerically. . It has 
been suggested that the technique of the dynamic renormalization-group 
(Chang et al., 1992) might be capable of addressing such difficulties. 
Illustrative examples will be provided to demonstrate the utility of this 
technique in handling dynamical complexity of space plasmas. 

It has been suggested that the intermittent non-propagating and 
propagating plasma fluctuations can interact efficiently with the charged 
plasma particles (Chang, 2001; Chang et al., 2003; 2004). This idea will be 
applied to the energization of ionospheric ions to magnetospheric energies in 
the auroral zone. 


1. Plasma Resonances and Coherent Structures 

Most field theoretical discussions begin with the concept of propagation 
of waves. For example, in the MHD formulation, one can combine the basic 
equations and express them in the following propagation forms: 

pdV/dt = B VB+ , dB/dt = B-VV + - (1.1) 

where the ellipses represent the effects of the anisotropic pressure tensor, the 
compressible and dissipative effects, and all notations are standard. 



. COMPLEXITY AND INTERMITTENT TURB ULENCE IN SPA CE 
PLASMAS 


3 


Equations (1.1) admit the well-known Alfven waves. For such waves to 
propagate, the propagation vector k must contain a field-aligned component, 
i.e., B • V — > <k • B ^ 0. However, at sites where the parallel component of the 
propagation vector vanishes (i.e., at the resonance sites), the fluctuations are 
localized. Around these resonance sites (usually in the form of curves), it 
may be shown that the fluctuations are held back by the background 
magnetic field, forming Alfvenic coherent structures (Waddell et al., 1979; 
Tetreault, 1992a; Chang, 1998a.b.c; 1999; 2001; 2003, Chang et al., 2003). 

Coarse-Grained Helicity. Let us now consider the geometry of the 
Alfvenic coherent structures. For an ideal MHD system, it has been 
suggested by Taylor (1974) that in a relaxed state such a structure would be 
approximately force- free (i.e., J x B = 0 ) due to the approximate 
conservation of the coarse-grained helicity defined as K = jA-BdV 
integrated over the coherent structure, where J and B are the current density 
and magnetic field and A is the vector potential. 

To obtain some physical insight of these structures, let us consider the 
special situation for the auroral region and/or the solar wind and make the 
reasonable assumption that the perturbed magnetic field fluctuations are 
much smaller than and essentially transverse to the mean magnetic field B 0 
(which will be temporarily assumed to be uniform for the current 
discussion). Thus, let us write B = (SB x ,5By,B 0 ) , where z is in the 
direction of the mean magnetic field, and (x, y) are orthogonal coordinates 
normal to z. The force-free condition for constant B 0 and V • J = 0 then 
leads approximately to the scalar condition B • VJ Z = 0 , obtained by taking 
the z-component of the curl of JxB = 0 (Rutherford, 1973; Tetreault, 
1992b; Chang, 1998a,b,c). It can be shown that, with the inclusion of the 
kinetic effects through the anisotropic pressure terms and the generalized 
Ohm’s law, the above results are still approximately valid. We have, then, 
approximately, 


B 0 dJ z / dz = ~{8B x d / 8x+ SByd /dy)J z +■■■ (1.2) 

where the ellipsis represents the other modifying effects. For convenience, 
let us introduce the flux function y/ by writing (d y/ldy ,-d y/td x) 
= ( SB x ,SB y ) for the perturbed transverse components of the magnetic field 
in the (x,>») directions such that V-B = 0 is satisfied. Then, J z and yr are 
governed by Eq. (1.2) and the Ampere’s law (neglecting the modifying 
effects represented by the ellipsis). 

A simple example of the flux function and axial current density satisfying 
the above conditions would be the class of circularly cylindrical solutions of 
y/{r) and J z (r ) . Generally, the solutions would be more involved because 


4 


Chapter 


of the variabilities of the local conditions of the plasma and the three- 
dimensional geometry. Moreover, the dynamic coherent structures with the 
inclusion of plasma pressure and other modifying effects (including electron- 
inertia terms) would be even more complicated. However, we expect these 
structures to be usually in the form of field-aligned flux tubes, Fig. 1 - 1 . 

Generally, there exist various types of propagation modes (whistler 
modes, lower hybrid waves, etc.) in a magnetized plasma. Thus, we 
envision a corresponding number of different types of plasma resonances 
and associated coherent structures that typically characterize the dynamics of 
the plasma medium under the influence of a background magnetic field. 

Generally, such coherent structures may take on the shapes of convective 
forms, nonlinear solitary structures, pseudo-equilibrium configurations, as 
well as other types of spatiotemporal varieties. Some of them may be more 
stable than the others. These spatiotemporal structures, however, generally 
are not purely laminar entities as they are composed of bundled fluctuations 
of all frequencies. Because of the nature of the physics of complexity, it will 
be futile to attempt to evaluate and/or study the details and stabilities of each 
of these infinite varieties of structures; although some basic understanding of 
each type of these structures will generally be helpful in the comprehension 
of the full complexity of the underlying nonlinear plasma dynamics. 



Figure Field-aligned spatiotemporal coherent structures. 

These coherent structures will wiggle, migrate, deform and undergo 
different types of motions and interactions under the influence of the local 
plasma and magnetic topologies. In the next section, we will consider how 


. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPA CE 
PLASMAS 


5 


the coherent structures can interact and produce the type of intermittency 
generally observed in a complex dynamical plasma. 

2. Complex Interactions of The Coherent Structures 

When coherent magnetic flux tubes of the same polarity migrate toward 
each other, strong local magnetic shears are created, Figs. 2-1 and 2-2. It has 
been demonstrated by Wu and Chang (2000a, b; 2001) that existing sporadic 
non-propagating fluctuations will generally migrate toward the strong local 
shear region. Eventually the mean local energies of the coherent structures 
will be dissipated into these concentrated fluctuations in the coarse-grained 
sense. 



Figure 2-1 . Cross-sectional view of coherent structures of the same polarity. Contours are 
y/= constants and arrows indicate directions of magnetic field in the (jc, y) -plane. 
Blackened area is an intense current sheet. 

Such enhanced intermittency at the intersection regions has been 
observed by Bruno et al. (2001; 2002) in the solar wind using the tools of 
wavelet analyses and local intermittency' measure (LIM). The coarse- 
grained dissipation will .then initiate "fluctuation-induced nonlinear 
instabilities" (Chang, 1999; Chang et al., 2002); and, thereby reconfigure the 
topologies of the coherent structures into a combined lower local energetic 
state, eventually allowing the coherent structures to merge locally. On the 
other hand, when coherent structures of opposite polarities approach each 
other due to the forcing of the surrounding plasma, they might repel each 
other, scatter, or induce magnetically quiescent localized regions. Under any 
of the conditions of the above interaction scenarios, new fluctuations will be 
generated. And, these new fluctuations can provide new resonance sites; 
thereby nucleating new coherent structures of varied sizes. 



6 


Chapter 1 



Figure 2-2. 2D MHD simulation of coherent structures (left panel) and current sheets (right 
panel) generated by initially randomly distributed current filaments after an elapsed time of 
t = 300 units. (For reference, the sound wave and Alfven wave traveling times through a 
distance of 2 71 are approximately 4.4 and 60, respectively.) 

All such interactions can occur at any location of a flux tube along its 
field-aligned direction, and the phenomenon is fully three-dimensional. In 
order to gain some insight of the physical picture of the overall dynamics of 
the interactions of the coherent structures, let us again consider the auroral 
zone or the solar wind as an illustrative example. We make the plausible 
assumption that some aspects of the plasma dynamics may be approximately 
understood in terms of the formulation of reduced magnetohydrodynamics 
(RMHD) (Seyler, 1988; Matthaeus et al., 1990). In this approximation, we 
assume that the mean magnetic field is much larger than the transverse 
fields, and the field-aligned fluctuations of the magnetic and velocity 
components are much smaller than their transverse counterparts. As a 
consequence, the density of the plasma is uniform. Writing the equations in 
SI units with p - 1 and ju 0 = 1 , we have (Strauss, 1976; Biskamp, 1993): 

dy/ 1 dt = B z d<p/dz , deal dt-B-Vj (2.1) 

where ys(x,y) is the transverse flux function defined by 
B = e z xVi^ + 5 z e z , is the transverse stream function defined by 

v ± =e z xV0, and is the vorticity, j = Vl^ is the field-aligned 

current density with d / dt = d/dt- l-v*V. The equations are written in the 
moving frame along the mean magnetic field direction z, and (x, y) are the 
transverse orthogonal directions. 

From Equations (2.1), we note that the primary nonlinear interactions 
occur generally in the transverse direction to the mean magnetic field. And, 
the coupling in the field-aligned direction is essentially linear. Thus, 
fluctuations generated by the transverse nonlinear interactions will scatter 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPA CE 
PLASMAS 


7 


and evolve nonlinearly primarily in the transverse direction. At this point, 
we realize that the RMHD formulation is too restrictive as some of the 
interactions of the flux tubes may become more oblique and thereby 
allowing the fluctuations to attain a broader range of values of k\\ than 
otherwise would have been admitted by the RMHD approximation. Thus, a 
significant amount of the fluctuations generated by the interactions can 
become commensurate with the plasma dispersion relation and propagate in 
the field-aligned direction as Alfven waves due to this three-dimensional 
complexity-induced enhanced transport. Eventually a dynamic topology of a 
complex state of coexisting propagating and non-propagating magnetic 
fluctuations is created. In the auroral region, the plasma may be electron- 
inertia dominated and the above discussion can be easily generalized to 
include such kinetic effects. 

3. Invariant Scaling and Topological Phase Transitions 

In the above sections, we provided some convincing arguments as well as 
numerical and observational evidences indicating that space plasma 
turbulence is generally in a state of topological complexity. By "complex" 
topological states we mean magnetic topologies that are not immediately 
deducible from the elemental (e.g., MHD and/or Vlasov) equations 
(Consolini and Chang, 2001). Below, we shall briefly address the salient 
features of the analogy between topological and equilibrium phase 
transitions. A thorough discussion of these ideas may be found in Chang 
(1992, 1999; 2001, and references contained therein). 

For nonlinear stochastic systems exhibiting complexity, the correlations 
among the fluctuations of the random dynamical fields are generally 

extremely long-ranged and there exist many correlation scales. The 

dynamics of such systems are notoriously difficult to handle either 

analytically or numerically. On the other hand, since the correlations are 
extremely long-ranged, it is reasonable to expect that the system will exhibit 
some sort of invariance under coarse-graining scale transformations. A 
powerful technique that utilizes this invariance property is the method of the 
dynamic renormalization group (Chang et al., 1978; 1992; and references 
contained therein). • The technique is a generalization of the static 

renormalization group introduced by Wilson (Wilson and Kogut, 1974). 

As it has been demonstrated by Chang et al. (1978), based on the path 
integral formalism, the behavior of a nonlinear stochastic system far from 
equilibrium may be described in terms of a "stochastic Lagrangian L ", such 
that the probability density functional P of the stochastic system is 
expressible as: 


8 


Chapter 


P(q>(x,/)) = }D[x]exp{-/JL((p,(p,x)^} (3.1) 

where <p(x,/) = 0,(/ = l,2,...,)V) are the stochastic variables such as the 
fluctuating magnetic, velocity and electric fields, and X(x,t) = 
X/(i = 1,2, ...,N) are the conjugate stochastic momentum variables that may 
be rigorously derived from the underlying stochastic equations governing (p 
(Chang et al., 1978; Chang, 1992; Chang et al., 1992). 

Then, the renormalization-group (coarse-graining) transformation may be 
formally expressed as: 


dL / dt = RL (3.2) 

where R is the renormalization-group (coarse-graining) transformation 
operator and i is the coarse-graining parameter for the continuous group of 
transformations. It will be convenient to consider the state of the stochastic 
Lagrangian in terms of its parameters { P„ }. Equation (3.2), then, specifies 
how the Lagrangian, L, flows (changes) with £ in the affine space spanned 
by {Rn}, Fig. 3-1. 


L,(PJ 



Figure 3-1. Renormalization-group trajectories and fixed points. 

Forced and/or Self-Organized Criticality. Generally, there exists a 
number of fixed points (singular points) in the flow field, at which 
dLl di = 0. At each such fixed point (L* or L** in Fig. 3-1), the correlation 
length should not be changing. However, the renormalization-group 
transformation requires that all length scales must change under the coarse- 
graining procedure. Therefore, to satisfy both requirements, the correlation 


. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


9 


length must be either infinite or zero. When it is at infinity, the dynamical 
system is then at a state of forced and/or self-organized criticality (FSOC) 
(Bak et al., 1987; Chang, 1992), analogous to the state of criticality in 
equilibrium phase transitions (Stanley, 1971). To study the stochastic 
behavior of a nonlinear dynamical system near such a dynamical critical 
state (e.g., the one characterized by the fixed point L*), we linearize the 
renormalization-group operator R about L *. The mathematical consequence 
of this approximation is that, close to dynamic criticality, certain linear 
combinations of the parameters that characterize the stochastic Lagrangian L 
will correlate with each other in the form of power laws. These include, in 
particular, the ( k,co ), i.e. mode number and frequency, spectra of the 
correlations of the various fluctuations of the dynamic field variables. 

Such power law behavior has been detected in the probability 
distributions of solar flare intensities (Lu, 1 995), in the AE burst occurrences 
as a function of the AE burst strength (Consolini, 1997), in the global auroral 
UVI imagery of the statistics of size and energy dissipated by the 
magnetospheric system (Lui et al., 2000), in the probability distributions of 
spatiotemporal magnetospheric disturbances as seen in the UVI images of 
the nighttime ionosphere (Uritsky et al., 2002), and in the probability 
distributions of durations of Bursty Bulk Flows (Angelopoulos, 1999); 
although some of the above interpretations of observed data may, however, 
also be amenable to alternative explanations (Boffetta et al., 1999; Freeman 
et al., 2000; Watkins, 2002). 

In addition, it can be demonstrated from such a linearized analysis of the 
dynamic renormalization group that generally only a small number of 
(relevant) parameters are needed to characterize the stochastic state of the 
system near criticality (Chang, 1992) justifying the recent work suggesting 
that certain dynamic characteristics of the magnetotail could be modeled by 
the deterministic chaos of low-dimensional nonlinear systems (Baker et al., 
1990; Klimas et al., 1992; Sharma et al., 1993). 

Illustrative Examples. The intermittency description for plasma 
turbulence of fluctuations may be modeled by the combination of a localized 
chaotic functional growth equation of a set of relevant order parameters and 
a functional transport equation of the control parameters (Chang and Wu, 
2002; Chang et al., 2003). Below, we shall provide two simple 
phenomenological models, which may have some relevance to the auroral 
zone or the solar wind (Chang, 2003). 

Model I. Assuming that the parallel mean magnetic field B 0 is 
sufficiently strong and the magnetic fluctuations dominate in the transverse 
directions, we introduce the flux function y/ for the transverse fluctuations 
as follows, 


10 


Chapter 


B = e z xV^ + fi 0 e 2 (3.3) 

The coherent structures for such a system are generally flux tubes 
approximately aligned in the mean parallel direction (Chang, 2001). 
Conservation of helicity (e.g., under the RMHD approximation) indicates 
that the integral of ip over a flux tube is approximately constant. Instead of 
invoking the RMHD formalism, however, here we simply consider \p as an 
order parameter. As the flux tubes merge and interact, they may correlate 
over long distances, which, in turn, will induce long relaxation times near 
FSOC (Chang, 1992). Let us assume that the transverse size of the system is 
sufficiently broad compared to the cross sections of the coherent structures 
(or flux tubes), such that we may invoke homogeneity and assume the 
dynamics to be independent of boundary effects. We may then model the 
dynamics of flux tube mergings and interactions, in the crudest 
approximation, in terms of the following order-disorder intermittency 
equation: 


dr k /dt = -r k dF/dvr. k +f k (3.4) 

where V k are the Fourier components of the flux function, T k an analytic 
function of k 2 , F(ip k >k) the state function, and f k a random noise which 
includes all the other effects that are not included in the first two terms of 
this crude model. 

Model II. In the above model, we have neglected both the effects of 
diffusion and convection. We next construct a phenomenological model that 
includes the transport of cross-field diffusion. We now assume the state 
function to depend on the flux function ip and the local pseudo-energy 
measure £ . Thus, in addition to the dynamic equation (3.4), we now also 
include a diffusion equation for £ . In Fourier space, we have 

d£ k / dt = -Dk 2 dF / + h k (3.5) 

where £ k are the Fourier components of £, D(k) is the diffusion 
coefficient, and the state function is now F{\p k ,4k>F) » and h k is a random 
noise. By doing so, we separate the slow pseudo-energy transport due to 
diffusion of the local pseudo-energy measure % from the noise term of (3.4). 
We note that an approach similar to these ideas have been considered by 
Klimas et al. (2000). 

Dynamic Renormalization-Group Analysis. We have performed 
renormalization-group analyses as outlined above for the two kinetic models 


. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


11 


described above. We note that under the dynamic renormalization-group 
(DRG) transformation, the correlation function C of Wk should scale as: 

e a ‘ ( C(k, co) = C(ke e , coe a<ut ) (3.6) 

where co is the Fourier transform of t, ( the renormalization parameter as 
defined in the previous section, and ( a c ,aw ) the correlation and dynamic 
exponents. Thus, Clco a<:laa> is an absolute invariant under the DRG, or 
C - co ~ X , where A = -a c / a<y . DRG analysis of Model I with Gaussian 
noise yields the value of A to be approximately equal to 2.0. DRG analyses 
performed for Model II for Gaussian noises for several approximations yield 
the value for A to be approximately equal to 1.88 to 1.66. 

Interestingly, for both models, DRG calculations give an approximate 
value of -1.0 for the co -exponent for the trace of the transverse magnetic 
correlation tensor. Matthaeus and Goldstein (1986) had suggested that such 
an exponent might represent the superposition of discrete structures emerged 
from the solar convection zone; thereby giving some credence to the above 
modeling effort. Also, the corresponding ^-exponent is found to be 
approximately equal to -2 for both models. These results compare rather 
favorably with the results of our 2D MHD numerical simulations, Fig. 3-2. 



Figure 3-2. Fourier spectrum of B 2 (k) at t = 300 (dotted) and 600 (solid). Solid straight 
line indicates a slope of -2. 

Symmetry breaking and Topological Phase Transitions. As the 
dynamical system evolves in time (autonomously or under external forcing), 
the state of the system (i.e., the values of the set of the parameters 




12 


Chapter 


characterizing the stochastic Lagrangian, L) changes accordingly. A number 
of dynamical scenarios are possible. For example, the system may evolve 
from a critical state A (characterized by L**) to another critical state B 
(characterized by L*) as shown in Fig. 3-1. In this case, the system may 
evolve continuously from one critical state to another. On the other hand, 
the evolution from the critical state A to critical state C as shown in Fig. 3-1 
would probably involve a dynamical instability characterized by a first- 
order-like topological phase transition (fluctuation-induced nonlinear 
instability) because the dynamical path of evolution of the stochastic system 
would have to cross over a couple of topological (renormalization-group) 
separatrices. For such a situation the underlying magnetic topology and its 
related plasma state will generally undergo drastic changes. Similar ideas 
along these lines have been advanced by Sitnov et al. (2000) and simulated 
based on the cellular automata calculations of sandpile models (Chapman et 
al., 1998; Watkins et al., 1999). 

Under either of these above scenarios, the spectra indices will generally 
change either continuously or abruptly. Such type of multifractal 
phenomena is commonly observed in the magnetotail, the auroral zone, and 
the solar wind (Lui, 1998; Hoshino et al., 1994; Milovanov et al., 1996; 
Andre and Chang, 1992; Chang, 2001; Bruno et al., 2001; 2002; Tu and 
Marsch, 1 995; and references contained therein.) 

Alternatively, a dynamical system may evolve from a critical state A to a 
state D (as shown in Fig. 3-1) which may not be situated in a regime 
dominated by any of the fixed points; in such a case, the final state of the 
system will no longer exhibit any of the characteristic properties that are 
associated with dynamic criticality. As another possibility, the dynamical 
system may deviate only moderately from the domain of a critical state 
characterized by a particular fixed point such that the system may still 
display low-dimensional scaling laws, but the scaling laws may now be 
deduced from straightforward dimensional arguments. The system is then in 
a so-called mean-field state. (For general references of symmetry breaking 
and nonlinear crossover, see Chang and Stanley (1973)', Chang et al. (1973a; 
1973b); Nicoll et al. (1974; 1976).) 

Experimental observations of plasma fluctuations in the Sun-Earth 
connection region generally yield broken power law spectra similar to those 
displayed in Fig. 3-2 of the 2D numerical simulation results. Such abrupt 
changes of scaling powers of the k-spectra are signatures of symmetry 
breaking. The broken symmetries may be due to the abrupt change of the 
degree of intermittency of fluctuations from large to small scales, or due to 
the change of the underlying physics (e.g., from MHD to kinetic processes), 
or variations of external forcing, or finite boundaries and other effects. 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPA CE 1 3 

PLASMAS 

4. Intermittency 

Nearly all fluctuations in space plasmas exhibit intermittency. For 
turbulent dynamical systems with intermittency, the transfer of energy (or 
other relevant scalars and tensors) due to fluctuations from one scale to 
another deviates significantly from uniformity. A technique of measuring 
the degree of intermittency is the study of the departure from Gaussianity the 
probability distribution functions of turbulent fluctuations at different scales. 
To demonstrate this point, let us refer to the 2D numerical simulation results 
described in Section 2. For example, we may generate the probability 
distribution function P{8B 2 ,8) of SB 2 (x, 8) = B 2 (x + 8) - B 2 (x) at a 
given time t for such simulations, where d is the scale of separation in the 
x-direction. Figure 4-1 displays the calculated results of P{8B 2 ,8) from a 
numerical simulation for several scales 8 . 



Figure 4-1. 2D MHD simulation result at t = 300 of PDF's of B~ at scales of 2 (dotted), 8 
(solid), and 32 (dashed) units of grid spacing £ . 

From this figure, we note that the deviation from Gaussianity becomes 
more and more pronounced at smaller and smaller scales. In an interesting 
paper by Hnat et al. (2002), they demonstrated that such probability 
distributions for solar wind fluctuations exhibit approximate mono-power 
scaling according to the following functional relation: 


P(8B 2 , 8) = 8~ S P S (8B 2 8~ s , 8) 


(4.1) 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


15 


parameters in general and may sometimes, for example, be represented by 
the K - or Castaing distributions (Sorriso-Valvo et al., 1999; Jurac, 2003; 
Forman and Burlaga, 2003; Weygand, 2003; Castaing et al., 1990). Their 
scaling properties are more subtle and will not be considered in this brief 
review. 

Since the degree of intermittency generally increases inversely with scale, 
it will be interesting to study the degrees of intermittency locally at different 
scales. This can be accomplished by the method of Local Intermittency 
Measure (LIM) using the wavelet transforms. A wavelet transform generally 
is composed of modes which are square integrable localized functions that 
are capable of unfolding fluctuating fields into space and scale (Farge, 
1992). Figure 4-3b is the power spectrum of a complex Morlet wavelet 
transform of the current density for one of the 2D MHD simulations 
mentioned in Section 2. We notice that the intensity of the current density 
is sporadic and varies nonuniformly with scale. 

We now define LIM(l) as the ratio of the squared wavelet amplitude 
\V{X,S)\ 2 and its space averaged value <\y/(x,5)^ > x . We note that 
LIM(l) = 1 for the Fourier spectrum. To emphasize the variation of 
intensity with scale, we also consider the logarithm of LIM(l). It has been 
suggested by Meneveau (1991) that the space average of the square of 
LIM(l), which is a scale dependent measure of the kurtosis or flatness, is a 
convenient gauge of the deviation of intermittency from Gaussianity. We 
denote this measure by LIM(2). It is equal to 3 if the probability distribution 
is Gaussian. Figures 4-3d,e and 4-4 are graphical displays of the calculated 
results of LIM(l), logLIM(l) and LIM(2) for our 2D numerical simulations 
using the complex Morlet transform. We notice that the fluctuations are 
indeed scale dependent, localized and strongly intermittent at small scales. 
Similar experimental results using the wavelet transforms have been found, 
for example, by Consolini et al. (2004) for the magnetotail and Bruno et al. 
(2001) for the solar wind. 

In the above, we considered some simplified models and numerical 
examples that may have some relevance to intermittent turbulence in space 
plasmas. Realistic models for these phenomena will generally be much 
more complicated. For example, from the RMHD formulation of (2.1), we 
recognize that there should at least be two competing order parameters. 
These are the flux function y/ and the stream function <p (which is linearly 
proportional to the electrostatic potential). Thus, the intermittency equation 
(such as (3.4)) needs to be generalized to accommodate these coupled order 
parameters. Within the RMHD formulation, there exist useful Hamiltonian 
and operator-algebra structures (Morrison and Hazeltine, 1984), which 
should prove invaluable in developing the generalized state function F of the 



14 


Chapter 


where s is the mono-scaling power. We demonstrate that mono-power 
scaling also holds approximately for our simulated results with the value of s 
equal to approximately 0.335. 

The reason for mono-power scaling for SB 2 may be understood in terms 
of the renormalization-group arguments presented in Section 3. If we 
assume that SB 2 is one of the relevant eigenoperators near a critical fixed 
point, then the probability distribution function for P(SB 2 ,S) , SB 2 , as well 
as 5 will scale linearly as follows: 

P' = Pexp(a p l) , SB 2 ' = SB 2 exp (a B 2l) , S' = Sexp(a s l) (4.2) 

where (a py a B i ,a$) are scaling powers. Thus, we obtain two irreducible 
absolute invariants: p/S aplas and SB 2 / S a ^ las . Since P = P(SB 2 y S ) , 
there must be a functional relation between these two invariants (Chang et 
al., 1973a,b). Therefore, we obtain the following scaling relation among 
(/>, SB 2 , 8) : PI S ap lag = F{SB 2 / 8“ * 2 >as ). 

Without loss of generality, we may choose a$ = L With the additional 
constraint that the probability distribution functions are normalized, we 
immediately obtain the expression of Hnat et al. as shown in (4.1), Fig. 4-2. 



Figure 4-2 . Scaled PDF's according to Eq. (4.1) with s = 0.335 . Line styles are the same as 
in Figure 4-1. 

Actually the scaling relation (4.1) is approximate in that the tails of the 
distributions in Fig. 4-2 do not exactly fall onto one curve. This is the 
intrinsic nature of the strong intermittency at small scales. Thus, 
representations of the probability density functions will involve multi- 



16 


Chapter 


a) J 2 along y =n 

10i t- 



10 1 1 ^ * * J 

0 100 200 300 400 500 

x 


b) J z Wavelet Power Spectrum 





Figure 4-3. (a) 2D MHD simulation result of current density J z along y -axis at t = 300 . 
(b) Power spectrum of complex Morlet wavelet transform of J z . (c) 2D MHD simulation 

result of B distribution along the x -axis for y - n at / = 300 . (d and e) Contour plots 
of LIM( 1 ) and LogLIM( 1 ) of B . The * -axis and scale are in units of the grid spacing € . 








. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPA CE 
PLASMAS 


17 


10 - 

eg + 



0 ! i . » * * * ‘ 1 

0 10 20 30 40 50 60 70 80 

Scale 


2 2 

Figure 4-4. LIM(2) of B for the same B distribution of Figure 4-3c. 

coupled order parameters and the state variables as well as the intermittency 
equation itself. Formulations of coupled order parameters and their related 
theoretical analyses for a variety of criticality problems in condensed matter 
physics have been considered by Chang et al. (1992; and references 
contained therein). 

In addition, the transport equation such as (3.5) for the global system 
should generally also include convection and acceleration terms in addition 
to that of diffusion (Chang et al., 2003; 2004). Thus, at the minimum our 
model transport equation must take on the form of the RMHD (3.5) with the 
addition of "coarse-grained" dissipation terms, which generally will be 
functionals of the coupled order parameters. It should also contain terms 
representing the complexity-induced enhanced field-aligned transport. 
These generalizations will not be considered in this brief review. 

5. Energization of Ions by Intermittent Fluctuations in 
The Auroral Zone 


It has long been recognized that the commonly observed broadband, low 
frequency electric field fluctuations are responsible for the acceleration of 
oxygen ions in the auroral zone. In order for the fluctuating electric field to 
resonantly accelerate the ions continuously as the ions evolve upward along 
the field lines, they must be in continuous resonance with the ions. There 
did not seem to exist a fully viable mechanism that can generate a spectrum 
of fluctuations broadband and incoherent enough to fulfill this stringent 
requirement. 




18 


Chapter 


Assuming that the RMHD formulation holds approximately in the auroral 
zone, the electrostatic fluctuations transverse to the field-aligned direction 
are given approximately by the velocity fluctuations: vxB r e z . The ordering 
due to the stream function 0 may be important in the auroral zone and 
therefore, the electrostatic fluctuations can be quite significant there. 
Because of the small scales involved, the dynamic intermittency produced by 
the merging and interactions of the coherent structures are probably 
generated by the whistler turbulence, electron-inertia related tearing modes, 
and/or other collisionless modes (Chang, 2001). Therefore, a significant 
portion of the fluctuations would be kinetic. Nevertheless, the electric field 
fluctuations would still be predominantly transverse and electrostatic. Thus, 
the low frequency fluctuations commonly observed in the auroral zone are 
probably contributed by these non-propagating intermittent fluctuations 
intermingled with a small fraction of propagating modes. 

Below, we shall briefly discuss how such fluctuations can efficiently 
energize the oxygen ions from ionospheric to magnetospherie energies. 
Assuming the oxygen ions are test particles, they would respond to the 
transverse electric field fluctuations E± near the oxygen gyrofrequency 
locally according to the Langevin equation: 


dv ± / dt- q i E ± / m l (5.1) 

To understand the stochastic nature of the Langevin equation, we 
visualize an ensemble of ions /(v x ) and study its stochastic properties. 
Assuming that the interaction times among the particles and the local electric 
field fluctuations are small compared to the global evolution time, we may 
write within the interaction time scale: 

/(v 1) ; + A0= J/(Vx -Avj., /)/>,( Vi - Avj., AvjJ^/Av.,. (5.2) 

where P,{y L - Av x , Avj.) is the normalized transition probability of a 
particle whose velocity changes from v x -Av.,. to Vi in At, and Av x 
ranges over all possible magnitudes and transverse directions. Standard 
procedure at this point is to expand both sides of (5.2) in Taylor series 
expansions: 

—At + 0((At) 2 ) = - g“‘[(Av ± )/] 

1 d 2 

:[<A v 1 Av i )/] + 0(( Av i ) 3 ) (5.3) 

2 dv x dv x L J 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 1 9 

PLASMAS 

where 


(Av i )= lP,( v x> Av i) A M( Av i)> and 


( Av i Av i) = l^( v i’ Av i) Av i Av i^( Av 1 ) • 

If we assume the G ) ((Av i ) 3 ) terms are of order (A/) 2 or higher, then in 
the limit of At— >0, we obtain a Fokker-Planck equation (Einstein, 1905; 
Chandrasekhar 1943), where the drift and diffusion coefficients are defined 
as: D] =< Avj. >/A t and D 2 ^AvxAVj. >/2A t in the limit of At— »0. 
These coefficients may be calculated straightforwardly using the Langevin 
equation. If the transition probability P, is symmetric in Avj., then D; 
vanishes and (5.3) reduces to a diffusion equation in the transverse direction. 
We note that if the electric field fluctuations are Gaussian, then the higher 
order correlations of the fluctuations are automatically equal to zero. 

We shall come back to the discussion of the effects of general 
intermittent fluctuations on particle energization processes. For the moment, 
let us assume the approach using the Fokker-Planck formulation is valid and 
.proceed. Since we have assumed the time scale for the particle-fluctuation 
interactions is much smaller than the global evolution time of the ion 
populations, we may then write the steady-state global evolution equation 
along an auroral field line s under the guiding center approximation and 
neglecting the cross-field drift as (Chang et al., 1986; Retterer et al., 1987; 
Crew and Chang, 1988): 


d 

\ £ 

_a_ 

1 

1 

'"S 

__ I 

_LJL 

. V i V ll dB z f 

ds 


+ dV|| 

[ 2 B z ds B z _ 

V X 

_ Vl 2 B z ds B z _ 


1 d [ ^ d f 
v, dv L [ ' dv L B z 


(5.4) 


where Vy , v ± are the parallel and perpendicular components of the particle 
velocity with respect to the field-aligned direction. This expression may be 
interpreted as a convective-diffusion equation for the density of the guiding 
center ions per unit length of flux tube / / B z , in the coordinate space of 
M|,v x ). 

To evaluate D L , the gyrotropic perpendicular diffusion coefficient, from 
»2> we recognize that die fluctuations are broadband both in k ± and co . 
Therefore, at all times, some portion of the fluctuations will be in resonance 
with the ions. The resonance condition, however, is strongly dependent on 



20 


Chapter 


the localization and scale dependency of the intermittent fluctuations 
(Chang, 2001). We demonstrate below, as a simple illustrative example, 
how such resonant interactions may be accomplished by neglecting the 
Doppler shifts due to k such that only the intermittent fluctuations 
clustering around the instantaneous gyrofrequency of the ions provide the 
main contributions to the diffusion process. Standard arguments then lead to 
the following expression for the perpendicular diffusion coefficient: 

D 1 =(^, 2 /2/« ( 2 )(|£ 2 |(Q i )) r (5.5) 

where < [is 2 |(Q, ) > r is the resonant portion of the average of the square of 
the transverse electric field fluctuations evaluated at the instantaneous 
gyrofrequency of the ions, Q ,• . 

Measurements by polar orbiting satellites indicate that the electric field 
spectral density Z follows an approximate power law lT a in the range of 
the local oxygen gyroffequencies, where a is a constant. This is a natural 
consequence of the intermittent turbulence when the fluctuations are close to 
a state of forced and/or self-organized criticality. If we make the additional 
approximations by assuming that the spectrum observed at the satellite is 
applicable to all altitudes and choosing the geomagnetic field to scale with 
the altitude as s ~ 3 , we would then expect E(£ 2 ,,s) to vary with altitude s as 
s * a . Because we have made some rather restrictive resonance requirements 
for the fluctuations to interact with the ions, we expect the resonant portion 
of the average of the square of the transverse electric field fluctuations to be 
only a fraction t] of the total measured electric field spectral density. 
Therefore, we arrive at the following approximate expression for the 
diffusion coefficient: 


D l = (nnq) / 2m] )I 0 (s / ) 3a (5.6) 


We have performed global Monte Carlo simulations for Equations (5.4) 
and (5.6) for the conic event discussed by Retterer et al. (1987) with a = 1.7 
and lo = 1.9xlO _ 7 (V/m ) 2 sec/ rad . Figure 5-1 shows the measured oxygen 
velocity distribution contours (top panel) along with the corresponding 
calculated contours for 77 = 1/8 (bottom panel) at the satellite altitude of 
s 0 = 2 R e . Thus, with one eighth of the measured electric field spectral 

density contributing, the broadband fluctuations can adequately generate an 
oxygen distribution function with the energy and shape comparable to that 
obtained from observations. We have also calculated the oxygen ion 
distributions for a range of altitudes under the same conditions. Figure 5-2 is 
a plot of the average parallel energy versus the average perpendicular energy 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


21 



\ (km/s) 



Figure 5-1. Observed and calculated velocity contour plots for conic event of Retterer et al. 
(1987). 



0 10 20 30 40 50 

(eV) 


Figure 5-2. Solid line depicts versus W± for simulated conic events. Dashed line is the 
asymptote predicted by Chang et al. (1986). 




22 


Chapter 


per oxygen ion as the ions evolve upward along the geomagnetic field line. 
We note that as the energies increase with altitudes, the ratio of the energies 
becomes nearly a constant. 

These results are comparable to our previous calculated results based on 
the assumption that the relevant fluctuations were purely field-aligned 
propagating electromagnetic ion cyclotron waves (Chang et al., 1986; 
Retterer et al., 1987; Crew and Chang, 1988). As discussed in the previous 
sections, we generally expect the coexistence of non-propagating transverse 
electrostatic nonlinear fluctuations and a small fraction of field-aligned 
propagating waves in the auroral zone. Thus, the ion energization process in 
the auroral zone is probably due to a combination of both types of 
fluctuations. As it has been discussed in Chang et al. (1986), an asymptotic 
solution exhibiting such behavior may be obtained analytically in closed 
form. Therefore, in the asymptotic limit (i.e, at sufficiently high altitudes), it 
is expected that such an ion distribution will become entirely independent of 
its low altitude initial conditions. In fact, it has been shown by Crew and 
Chang (1988) that the ion distributions will become self-similar at 
sufficiently high altitudes and everything will scale with the altitude. 

The above sample calculations did not include the self-consistent electric 
field that must be determined in conjunction with the energization of the ions 
as well as the electrons. This is particularly relevant in the downward 
auroral current region where the electric field can provide a significant 
pressure cooker effect such as that suggested by Gomey et al. (1985) and 
demonstrated convincingly by Jasperse (1998) and Jasperse and Grossbard 
(2000) based on global evolutional calculations similar to those considered 
by Tam and Chang (1999a, b; 2001; 2002) for the solar wind and Tam et al. 
(1995, 1998) for the polar wind. These ideas will not be considered in this 
brief review. 

We now return to the discussion of the effect of intermittency on ion 
heating. Measurements of the electric field spectral density are generally 
limited by the response capabilities of the measuring instruments. The faster 
the instruments can collect data, the more refined the scales of the 
measurements. As it has been seen in Section 4, we expect the measured 
spectrum density to exhibit small-scale intermittency behavior. In fact, it is 
known that fast response measurements generally exhibit strongly 
intermittent signatures of the fluctuations. In the diffusion approximation, 
the ion energization process is limited by the amplitude of the second 
moment of the probability distribution of the fluctuations. This amplitude 
may become smaller as the scale of measurements is reduced. Thus, in the 
limit of small scales, the amplitude of the measured spectrum may decrease 
and thereby requiring a larger value of r) to accomplish the same level of 
energization. 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


23 


But, the effects of the intermittency of the fluctuations on particle 
energization may be underestimated if we stay within the diffusion 
approximation. As it can be seen from the derivation of the diffusion 
approximation above, only the second order correlations of the fluctuations 
were included in the energization process. Since for intermittent turbulence, 
the probability distributions of the fluctuations are generally non-Gaussian, 
the effects of the intermittency can manifest in the higher order correlations 
beyond the second order diffusion coefficient. This implies that the higher 
order correlations of the velocity fluctuations may be of the order of At and 
therefore cannot be neglected in (5.3). Under such circumstances, the 
Fokker-Planck and diffusion approximations of the ion energization 
processes can become inadequate. A more appropriate approach to address 
such non-Gaussian stochastic processes is to refer directly to the functional 
equation (5.2) using the non-Gaussian transition probability or the Lange vin 
equation (5.1) with the actual intermittent time series of the electric field 
fluctuations. Again, the details of these ideas will not be considered in this 
review. 


6. Summary 

We have provided a modem description of dynamical complexity 
relevant to the intermittent turbulence of coexisting non-propagating 
spatiotemporal fluctuations and propagating modes in space plasmas. The 
theory is based on the physical concepts of sporadic and localized 
interactions of coherent structures that emerge naturally from plasma 
resonances. 

The technique of the dynamic renormalization-group is applied to the 
study of forced and/or self-organized criticality (FSOC) and scale invariance 
related to such type of multiscale fluctuations. We also demonstrated that 
the particle interactions with the intermittent turbulence could lead to the 
efficient energization of the plasma populations such as auroral ions. 
Numerical examples are presented to illustrate the concepts and 
methodology. 


Acknowledgments 

This review touches upon a broad range of research areas covering the 
physics of space plasmas and complexity. The authors wish to thank their 
past and present colleagues, M. Andre, V. Angelopoulos, R. Bruno, S. 


24 


Chapter 


Chandrasekhar, S.C. Chapman, G. Consolini, B. Coppi, G.B. Crew, G. 
Ganguli, M. Goldstein, A. Hankey, J.R. Jasperse, S. Jurac, C.F. Kennel, P. 
Kintner, M. Kivelson, A. Klimas, A.T.Y. Lui, E. Marsch, W. Matthaeus, P. 
De Michelis, J.F. Nicoll, J.M. Retterer, C. Seyler, A.S. Sharma, M.l. Sitnov, 
H.E. Stanley, D. Tetreault, V. Uritsky, J. Valdivia, D. Vassiliadis, D. 
Vvedensky, N. Watkins, J. Weygand, F. Yasseen, and J.E. Young for very 
useful discussions. Our wavelet analysis employed some of the wavelet 
software provided by C. Torrence and G. Compo, which is available at URL: 
http://paos.colorado.edu/research/wavelets. This research was partially 
supported by AFOSR, NASA and NSF. 


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1993. 



. COMPLEXITY AND INTERMITTENT TURBULENCE IN SPACE 
PLASMAS 


27 


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28 Chapter 

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