J. Plasma Fusion Res. SERIES, Vol.2 (1999) 32-34

Micro- and Macro-Scale Self-Organization
in a Dissipative Plasma
Sronti Milos M.r'2, SATo Tetsuyar, MALUCKov Aleksandra3 and JovANovIi Moma S.atTheory and Computer Simulation Center, National InstituteFusionToki-shi, 509-5292,
forScience,Japan

2Vinfa Institute of Nuclear Sciences, P.O.B. 522, 11001 Belgrade, Yugoslavia
3The Graduate Llniversity

for Advanced Studies, Hayama, Kanagawa 240-01, JapanaDepartment of Physics, University of Nii, P.O.B. 91, 18001 Nii,Yugoslavia(Received: ll December 1998 / Accepted: 5 August 1999)

Abstract
We study a nonlinear three-wave interaction in an open dissipative model of stimulated Ramanbackscattering in a plasma. A hybrid kinetic-fluid scheme is proposed to include anomalous kinetic
dissipation due to electron trapping and plasma wave breaking. We simulate a finite plasma with open
boundaries and vary a transport parameter to examine a route to spatio-temporal complexity. An interplay
between self-organization at micro (kinetic) and macro (wave/fluid) scales is revealed through quasiperiodic and intermittent evolution of dynamical variables, dissipative structures and related entropy
rates. At this point, a consistency with a general scenario of self-organization is found.

Keywords:
complexity, self-organization, stimulated Raman, scattering

1. Introduction

Skori6 et al. l4l.

Self-organization (SO) is a generic process which
describes a spontaneous formation of an orderedstructure in a nonlinear far-from equilibrium system.
Energy pumping, nonlinear instability, entropy
production and expulsion are key governing processes.
In this paper, an attempt is made to study kinetic selforganization [] through a process of stimulated Ramanbackscattering (SRBS) in an underdense plasma [2].
Stimulated Raman scattering is a resonant three-wave(3WI) parametric instability which corresponds to thedecay of an incident electromagnetic pump wave into ascattered wave plus an electron plasma wave (epw).
Important effects, especially in the context of laser
fusion schemes [3], are input energy loss, plasma
heating and generation of suprathermal (hot) electrons.
To emulate these effects we apply a hybrid three-wave
phenomenological kinetic model of SRBS, proposed by2. A Hybrid Model
The basic equations are coupled three-wave
equations for slowly varying complex amplitudes of apump (as), backscattered wave (a1) and epw (a),respectively:
dondon+ ,,% --:-::=- a,a, .
"c,xdt
da, .. do,dt -'t Ai =-ooo' 'do, -. do,I t2^)-a! + fa, + iolarl- a, = P'sarai+U-ox. (l)Parameter f is composed of collisional damping (f|1),'linear' (Landau) damping by hot - resonant electronsdt@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchC o r r e s p o ndin g aut ho r' s e - mail : s ko ric @ nifs. ac.j pJLSkorid U.U. et al., Micro- and Macro-scale Self-organization in a Dissipative Plasma(yr- n,), where nr,(t) stands for spatially averaged hotelectron density:

expect that original spatio-temporal profiles, found indnn(t) no(L, t\n*"i'tt'" ^ ,structures, self-organized at macro-scale levels. As an
illustration, we plot the plasma wave profiles (Fig. l), inparticular, to reveal a genuine spatio-temporal nature of?

=Tvne)nnft)

f,dv -a'!::'--:!::-: ,)simulations, should correspond to large dissipativeQ)

un- vo (L. r)

an intermittent regime as compared to regular dynamical(n6 is bulk electron density,/6 is bulk distribution function, v5 is velocity of hot electrons which equals the epw
phase velocity, vo is trapping velocity and a is particle
transport parameter) and nonlinear damping due totrapped resonant bulk electrons (in the thermalMaxwellian):mn,(L.r,regimes ofthe steady-state and quasi-periodic type [3,4].

Spatio-temporal complexity of quasi-steady andtravelling wave patterns with regular and chaotic
features is found in different states of self-organization.

Further, in Fig. 2 we plot the entropy rate dS(t)/dt

un*uF(L'')

in time together with a spatio-temporal profile of thescattered wave energy. For an intermittent regime,

2y^tw(t)=':::!:::t I ur.fr(y)dy, (3)2L ,n-!,,r.,,where IV(r) is spatially averaged epw energy.
Open boundaries and re-emittance of fresh ambient
electrons [5] is assumed. The particles (bulk and hotelectrons) and energy (wave and particle) are being(b) k - 0.06

exchanged between a plasma and an environment
through open boundaries with a conservation of particle
number and total energy in the system. Accordingly,
heat balance equatio (effect of plasma heating; E,,, En
and @,o, + 4 + 4, + @o are the average bulk and hotelectron energy or corresponding energy flux of thebulk, hot and return ambient electrons, respectively)
takes form:dw(t)dtd (4(t) + EhQ))

dl+k' Q.,IL@)Fig. 1 Spatio-temporal profiles of the electron plasma
wave for varying transport parameter k values.
Different dissipative structures are seen on theroute to complexity, from the steady-state via
quasi-periodic to intermittent regimes.

3. Dissipative Structures and Entropy Rate
The simulation is performed via the centraldifference numerical code [2], where the simulation
parameters dtai ns = 0.1 n.,, Zro = 0.5 keV and pumpintensity, Fo = 0.0253. Openness of a system, ft (0-l)
was chosen as a bifurcation parameter. Self-organizationin strongly nonlinear far-from-equilibrium systems leads
to a creation of ordered states that reflect an interactionof a given system with its environment, These novel

dd

dynamical structures or patterns, named dissipative
structures to stress the crucial role of dissipation in theircreation, have become a central theme of the science ofcomplexity [1,2,4]. On the other hand, there is afundamental role of the entropy, in particular, the rate ofentropy change in an open system. The rate of entropy

production and its removal basically governs selfFig. 2 Intermittent dissipative backscattered wave
structures versus the corresponding entropy rateorganization features of a system.

First, we focus at self-organized dissipativein time. Positive entropy jump coincides with an
onset of chaos, while a negative burst indicates atransition from a chaotic to a laminar phase of SO
at macro-scales.

structures developed at the macro-scale. Indeed, in ourmodel, basic wave and fluid density variables wereassumed to vary slowly in space-time. Therefore, weJJ

Skori6 M.M. et al., Micro- and Macro-scale Self-organization in a Dissipative Plasmafeaturing an interchange between chaotic and laminarphases, we find a clear evidence of structural transitions
conesponding to the maximum (positive) and minimum(negative) entropy rate. As a striking example of selforganization in an open system we find a rapid entropy
jump which coincides with an onset ,;i a chaotic phase.
Subsequent anomalous dissipation and entropy growth ishalted by a sudden entropy expulsion into theenvironment. Negative burst in entropy rate indicates abifurcation from a chaotic, back to a laminar quasiperiodic phase. Intervals of near zero entropy rate during

a laminar phase, mean a net balance between theentropy production and its expulsion. This appears to beFig. 3 Three-dimensional view of the electron velocity
distribution in time for different saturated Ramanan example of a stationary nonequilibrium state possiblyrealized in a strongly nonlinear open system Il].
A hybrid nature of our model allows us to recoverkinetic properties of self-organization. In Fig. 3 we see aregimes, as indicated by values of parameter k

three-dimensional view of the electron velocity

Micro-kinetic scale self-organization of varying
complexity is revealed in both thermal andsuprathermal (hot) regions of the electron

distribution in time for different saturated Raman

distribution.

regimes, as indicated by values of parameter ft. Kinetic

self-organization of varying complexity is revealed inthermal and suprathermal (hot) regions of the electron

k - 0.N7(o)(b)distribution.

0.08

0.3

I

4. Summary

9

In summary, we believe that our findings appear to
be a first indication of a generic intermittent scenario inoa kinetic self-organization of anomalous Raman

25000

50000

t (o;,)

010000t (o;')

20000

Fig. 4 Raman reflectivity versus time [aro-'l for k= 0.007

instability. At this point we may note that one is able toclaim a consistency with the working hypothesis andgeneral scenario of self-organization in plasmas [1,4].
As a further step, we expect an important justification of
our hybrid-modeling of saturated Raman complexity by
the novel open boundary particle simulation code,from the hybrid model simulation (a) and thecorresponding data (b) obtained by a 1f relativistic
particle-in-cell code (after Miyamoto ef a/. [6]). The
pump strength was Bo = 0.0253 and the plasma
parameters: no = 0.1 I1o, T. = 0.5 keV, L = 100 cl@o.

currently under development [5]. As an early
illustration, we show in Fig. 4b, recent particle-in-cellsimulation data for a model of an isolated plasma slab ina vacuum [6]. For same plasma parameters, particle
simulations (Fig. 4b) show an evident support of above
Raman reflectivity pattern, obtained for a closed (/< =References

tll T. Sato and the Complexity Simulation Group,Phys. Plasmas 3, 261 (1996).

I2l M.M. Skorii, M.S. Jovanovi'c and M. Rajkovii,Phys. Rev. E 53, 4056 (1996).

0.007) system (Fig. 4a).

[3] W.L. Kruer, The Physics of Laser Plasma

Acknowledgements
We acknowledge the partial support and thet4l M.M. Skorii, T. Sato, A. Maluckov and M.S.

Interactions, (Addison, N.Y., 1988).

Jovanovii, NIFS Report, No.549 (1998); Phys.

visiting professorship granted to one of us (M.M.S.) bythe Ministry of Education, Science and Culture of Japan.

Rev. E (to appear).

[5] H. Takamaru, T. Sato, R. Horiuchi, K. Watanabe

The hospitality of the National Institute for Fusion
Science is gratefully acknowledged. This work wassupported in parts by the Ministry of Science andTechnology of the Republic of Serbia, Contract OlEl1.

and Complexity Simulation Group, J. Phys. Soc.
Jpn. 66, 3826 (1997).

[6] S. Miyamoto, K. Mima, M. Skorii and M.S.
Jovanovii, J. Phys. Soc. Jpn. 67, lZSl (1998).
J+