Complexity in plasma: From selforganization to geodynamoTetsuya Sato and the Complexity Simulation Group Citation: Physics of Plasmas (1994-present) 3, 2135 (1996); doi: 10.1063/1.871666 View online: http://dx.doi.org/10.1063/1.871666 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/3/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear Phenomena and SelfOrganization in Dusty Plasmas AIP Conf. Proc. 1397, 44 (2011); 10.1063/1.3659738 Electroluminescence from self-organized “microdomes” Appl. Phys. Lett. 84, 4696 (2004); 10.1063/1.1760592 SelfOrganization and Coupling of Waves in a Plasma AIP Conf. Proc. 669, 630 (2003); 10.1063/1.1594009 Overview: Synchronization and patterns in complex systems Chaos 6, 259 (1996); 10.1063/1.166172 Scenario of selforganization AIP Conf. Proc. 345, 335 (1995); 10.1063/1.49050

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Complexity in plasma: From self-organization to geodynamo *Tetsuya Sato† and the Complexity Simulation Groupa)Theory and Computer Simulation Center, National Institute for Fusion Science, Nagoya 464-01, Japan

~Received 6 November 1995; accepted 11 January 1996!

A central theme of ‘‘Complexity’’ is the question of the creation of ordered structure in nature~self-organization!. The assertion is made that self-organization is governed by three key processes,i.e., energy pumping, entropy expulsion and nonlinearity. Extensive efforts have been done toconfirm this assertion through computer simulations of plasmas. A system exhibits markedlydifferent features in self-organization, depending on whether the energy pumping is instantaneous orcontinuous, or whether the produced entropy is expulsed or reserved. The nonlinearity acts to bringa nonequilibrium state into a bifurcation, thus resulting in a new structure along with an anomalousentropy production. As a practical application of our grand view of self-organization a preferentialgeneration of a dipole magnetic field is successfully demonstrated. ©1996 American Institute ofPhysics.@S1070-664X~96!92905-X#

I. INTRODUCTION

The way, i.e.,modus operandi, of Modern Science is toreduce a system into its elements and to reveal the funda-mental function of every element. The underlying philosophyis that the mechanism of a system be fully restored by as-sembling all the fundamental functions of the elements. El-ementary particle physics is, in this sense, the extremity ofModern Science.

This Science has already obtained a rather high level ofmaturity. Then, one may ask, if all fundamental interactionsof a system have been revealed, can one really predict whatwill happen in a system? The answer will be ‘‘no’’ in mostcases. Even if a system is described by a complete set offundamental equations, one cannot get a full comprehensionof its dynamical behavior. This means something is missingin the underlying philosophy.

Speaking in terms of mathematics, the very root of un-predictability lies in the nonlinearity and openness of thesystem. Since any system in nature can never be isolated in astrict sense, it inevitably suffers unpredictable, internal andexternal, fluctuations. Furthermore, it exchanges information,good or bad, with an external world. Because of the nonlin-earity and the openness of the system, the behavior of thesystem becomes very complex and mathematically intrac-table.

Present-day supercomputer technology has vastly in-creased the possibility of handling a complex system and hasgiven us a hope, at least technically, that the behavior of acomplex system can be elucidated.

Science is directed toward a deeper and deeper stratumof nature by cutting off the mutual interactions among thelinked strata. Consequently, we have already acquired anabundant knowledge of the fundamental laws governing theelementary processes in physics and chemistry. Based on theaccumulated fundamental knowledge, it is time to turn oureyes from a lower to a higher stratum where the interactions

with lower strata are never ignored. The interactions are, as amatter of course, so complicated and complex that the meth-odology of computer simulation may be the most viable oneat present. By thus revealing the complex behaviors for avariety of natural phenomena and making mutual compari-sons of the revealed processes, a common and universal re-lationship that governs a complex system can be sought.

A new paradigm from reduction to integration is dawn-ing. This is the Science of Complexity. As the vocabulary ofComplexity, Nicolis and Prigogine1 describe the following:nonequilibrium, instability, bifurcation, symmetry-breaking,and long-range order. Crudely speaking, the first two terms,nonequilibrium and instability, describe the initial evolutionof a complex system and the last three concepts, bifurcation,symmetry-breaking and long-range order, appear in the sub-sequent evolution. In short, the first ones are the cause andthe last are the effect. The cause comes essentially from thenonequilibrium and the effect arises as a result of the non-linearity of a system. Therefore, Complexity may well bedescribed by the two key concepts of a system, i.e., the far-from-equilibrium and the nonlinearity.

Fusion plasmas and space plasmas are always nonuni-form, thus they are in a nonequilibrium state. To release thefree energy, therefore, the system becomes unstable. Becauseof its nonlinearity the evolution becomes highly complex andresults in bifurcation, symmetry-breaking and long-range or-der. In this sense, Plasma Physics is potentially a Physics ofComplexity. Most of the fundamental laws are already estab-lished in classical electrodynamics. Therefore, plasmas willbe a good target for the study of Complexity.

II. CENTRAL PROBLEM OF COMPLEXITY—SELF-ORGANIZATION

Self-organization is a generic process which describes aspontaneous formation of an ordered structure. It may not bean exaggeration to say that the central concern and problemof our universe is the creation of an ordered structure againstthe second law of thermodynamics. It is a mystery how or-derliness is created in nature. Is there any universal law forcreating orderliness? This is one of the central problems ofthe Science of Complexity.

*Paper 2IA2, Bull. Am. Phys. Soc.40, 1666~1995!.†Invited paper.a!S. Bazdenkov, B. Dasgupta, S. Fujiwara, A. Kageyama, S. Kida, T. Ha-yashi, R. Horiuchi, H. Miura, H. Takamaru, Y. Todo, K. Watanabe, and T.H. Watanabe.

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The primary purpose of this paper is to extract a grandview of self-organization through an extensive computersimulation methodology. One may have no doubt that self-organization occurs in an open system where an information~or energy! exchange can take place with an external world.When new information is introduced into a system, the sys-tem reacts upon it. When the amount of introduced informa-tion exceeds a certain threshold, the reaction may becomesignificant and the system may experience a sizable changein its structure or state, i.e., the onset of a structural instabil-ity. A new ordered structure will thereby be created. Duringthis process of structural change, or, phase transition, somechaotic state will be realized in an intermediate stage andnatural odds and ends~entropy! will be produced, as oneusually experiences in one’s daily life, for instance, whenone cooks a meal, builds a house and makes any product. Increating a clear-cut new structure, therefore, the cleaning up,or disposal, of those odds and ends is an important function.In addition to these interactions with the external world, themost important element is the nonlinearity of the systemwhereby operations of creating an ordered structure are prac-tically conducted.

In this paper, first of all, we investigate how differentlyself-organization evolves depending on the feeding processof free energy from an external world. We consider two ex-treme cases, namely,~1! a system into which external energyis instantaneously pumped and no further supply is given,and ~2! a system which makes contact continuously with anexternal energy reservoir. Comparison of the two extremecases will give us the difference in self-organization depend-ing on whether the supply rate of the external energy is fasterthan the dynamic response time of the system or slower.

Second, we examine how the disposal of the producedodds and ends~entropy! can affect the self-organization pro-cess. We compare the case where the superfluous entropy isreserved in the system with the one where it is filtered out atthe boundary with the external world.

Third, we describe the essential nonlinearity whereby anordered structure is created.

III. CASCADING SELF-ORGANIZATION BYINSTANTANEOUS ENERGY PUMPING

In a plasma the self-organization process has long beenconsidered to be a process that an unstable equilibrium re-laxes into a minimum energy state. This concept emergedwhen Taylor proposed in 1974 a variation theory which isbased on the Lagrange multiplier method to obtain a mini-mum magnetic energy state by imposing a constraint so thatthe magnetic helicity is conserved.2,3 In an ideal magnetizedor magnetohydrodynamic~MHD! plasma, there are two glo-bal invariants, namely, the total magnetic energy(*B•BdV) and the total magnetic helicity (*A•BdV), whereA is the vector potential given byB5¹3A. Taylor hypoth-esized that the magnetic helicity is a more robust conservedquantity than the magnetic energy in a weakly dissipative~resistive! system. This process is called selective dissipationof the magnetic energy4 and it is demonstrated by computersimulations that the energy spectrum experiences a normalcascade, while the helicity spectrum does an inverse cascade

in this type of self-organization.5 This simple theory couldexplain the so-called self-reversal process of the toroidalmagnetic field in the reversed field pinch which is one of themagnetic fusion concepts.6–11 A similar discussion is appli-cable to a two-dimensional fluid where the total entropy(*V•VdV) is minimized under the constraint that the totalenergy*v•vdV be conserved,12 whereV5¹3v ~vorticity!andv is the fluid velocity.

We start with a Taylor self-organization process whichtakes place in an instantaneously pumped-up system. Giveninitially an unstable equilibrium state far beyond a minimumenergy state, we solve its nonlinear evolution. We consider acylinder of rectangular cross section in which the sideboundaries of the cylinder are made of conductors and theaxial boundaries are periodic. Figure 1 shows a structuralevolution of the magnetic field.5 The final spiral structure isthe self-organized one. The evolutions of the magnetic helic-ity and energy are given in Fig. 2. It is observed that therelaxation occurs in two steps for the magnetic energy butnot for the magnetic helicity. The two-step relaxation reflects

FIG. 1. A typical example of a magnetohydrodynamic~MHD! self-organization when the system is instantaneously pumped up to a high energystate~far-from-equilibrium!. The six panels show the time evolution of themagnetic structure. The structure represents an isosurface of the axial mag-netic field. The initial state consists of five straight flux tubes~t50!.

FIG. 2. Time evolutions of the magnetic energy~W!, the magnetic helicity~K! and the ratio~W/K) for the same simulation shown in Fig. 1.

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the fact that there was another equilibrium~unstable! statebetween the given initial state and the final minimum energystate.

We emphasize here that each stepwise relaxation occursin about ten units of the Alfve´n transit timetA , which isquite a fast process compared with the classical diffusiontime ~ten thousands of the Alfve´n transit time!. During eachrelaxation process the magnetic energy is rapidly convertedinto the thermal energy~entropy!. We here assume that theproduced entropy is instantaneously expulsed from the sys-tem. The main process involved in this energy conversion isthe ohmic process. However, the initial magnetic Reynoldsnumber~normalized electrical conductivity! was 1024. Thus,we can conclude that some nonlinear process, which enor-mously enhances the conversion rate~a thousand times inthis case!, must have happened. This is realized by the drivenmagnetic reconnection which is caused by strongly excitedkink flows.13

In this self-organization process one notices an importantfact regarding Taylor’s conjecture. A careful examination ofthe helicity evolution in Fig. 2 reveals that the helicity isnever conserved in a strict sense, though the conservationhas been firmly believed. On looking at the helicity curve inFig. 2, one can readily recognize that the helicity dissipationexhibits a critical slowing-down in accordance with the step-wise relaxation of the magnetic energy. In the initial phasethe magnetic helicity experiences a rather strong dissipation,which is the same as that of the magnetic energy. The dissi-pation rate of the helicity is slowed down critically at the firststepdown relaxation of the magnetic energy (t'23tA). Thedissipation rate keeps an almost constant value for a whileand then it experiences again a critical slowing-down at thesecond stepwise relaxation (t'46tA). Thereafter, the dissi-pation rate is slower but constant. The dissipation rate of themagnetic energy also becomes much slower than the initialone. These slow dissipation rates for both the magnetic en-ergy and helicity indicate that the system has reached to astable self-organized state. We note here that both the criticalslowing-down of the magnetic helicity and the stepwise re-laxation of the magnetic energy are explained by the drivenreconnection process.

These features of stepwise critical-slowing down of thehelicity and energy dissipation rates can be easily understoodas follows.

The magnetic helicityK and the magnetic energyW in adecaying equilibrium state are given by

]K

]t522hE J•BdV, ~1!

]W

]t522hE J•JdV, ~2!

where the resistivityh is assumed to be constant for simplic-ity. These equations can be rewritten as

]K

]t522hl iE B2dV, ~3!

]W

]t522hl i

2E B2dV. ~4!

Herel i represents thei -th eigenvalue of the equilibrium, or,the i -th discrete energy state.

Equation~3! states that the helicity dissipation rate at thei -th energy state is proportional to the eigenvalue. Thismeans that the lower the energy state is, the slower the dis-sipation rate is. Thus, the observed stepwise slowing down ofthe helicity dissipation rate is explainable. On the other hand,the energy dissipation rate is proportional to the square of theeigenvalue. Thus, as the energy state goes to a lower state,the dissipation rate of energy becomes more drastically re-duced than that of helicity, which is consistent with the ob-servation in Fig. 2. Remember, however, Eqs.~3! and~4! cannot be applied to the transition phase. The drastic increasesof the energy dissipation rate in the transition phases aroundt/tA;20 and; 40 are attributed to highly nonlinear evolu-tions of driven reconnection which causes structural transi-tion ~bifurcation!.

IV. INTERMITTENCY IN SELF-ORGANIZATION BYCONTINUOUS ENERGY PUMPING

One may consider that the example given above is atypical self-organization process in the sense that a systemrelaxes in a transient fashion into a steady minimum energystate starting from an initial nonminimum energy state. Aquestion then arises as to how such an initial far-from-equilibrium state can be realized without suffering any insta-bility on the way to it. The answer will be that the far-from-equilibrium state is realized if the external pumping ofenergy into the system is a reasonably fast action and thesystem cannot swiftly react upon the pumping; in otherwords, if the pump-up time scale is shorter than the dynamicresponse time of the system. We call such a self-organizationprocess an instantaneously pumped self-organization.

In the instantaneously pumped self-organization processthe system relaxes in a cascading fashion but monotonouslyinto a steady ordered structure. The process is a transientfrom the initial state to the minimum energy state. In nature,however, the energy pump in a local system from an externalworld is not always a fast process. It often happens that anenergy gradually and continuously flows from an externalworld. In such a situation no Taylor type variational methodwould be applied. Generally, any mathematical methodologywould not be able to predict what will happen in the system.The evolution would be strongly dependent on the nonlinear-ity of the system and also would be strongly influenced bythe processes of information exchange between the systemand the external world.

Based on the above consideration, we shall present asimulation result of a new self-organization process which isstimulated by an external energy reservoir that has an abilityto supply a continuous energy flow. The thermal energy gen-erated during the evolution is assumed to be instantaneouslypumped out in the same way as in the previous example.

The simulation system is the same rectangular box as theprevious simulation. Uniform straight field lines are appliedin the axial direction. The axial boundaries are not periodic

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but are the suppliers of a continuous energy flow. Specifi-cally, at the boundaries a circular motion is steadily appliedin a limited region at both ends with the opposite polarity toeach other, so that the magnetic field lines keep twisting.14

This problem may find some application in the problem ofsolar flares.15–17

We have accumulated sufficient simulation data wherethe field lines are twisted at both boundaries under variousconditions, e.g., single-twisted flux tube, twin-twisted fluxtubes, etc. From among them we choose the case where thetwisting is given at a single circular flux tube, because everycase has an essentially similar feature. Figure 3 illustrates thetopological change of a bundle of the field lines starting fromone axial boundary. The bundle of straight field lines aretwisted and squeezed toward the center in the initial phase.As the twisting proceeds, the stored tension energy of thetwisted bundle of field lines is released in terms of a knot-of-tension type of instability. The instability develops in sucha way that the kinked part of the flux tube collides againstuntwisted field lines at the twist-untwist boundary. As a con-sequence, reconnection is driven to take place between thekinked and untwisted field lines and relaxes toward an inter-connected structure~a local minimum energy state!. As thetwisting continues, reconnection advances stepwise to an

outer untwisted region by intermittently generating knot-of-tension instability and alternating a tensioned~supersatu-rated! state with a relaxed state~local minimum state!, i.e.,local cycle. As the twisting goes further, the flux tube linkedwith distant untwisted field lines reconnects with a compan-ion twist-untwist flux tube to come back to the originalstraight topology, i.e., grand cycle. This is because thestraight structure of the field lines is energetically lower thanthe linked structure between the twisted field lines and thedistant untwisted ones, presumably the global minimum en-ergy state. Thereafter a process similar to that which we haveobserved is repeated. This indicates that the topological evo-lution exhibits a recurrence to a topology closer to the origi-nal one ~global minimum energy state! and reproduces asimilar behavior thereafter. The observed intermittency andrecurrence are observed for the interaction of two twistedflux tubes and of four twisted flux tubes, as well.

In order to see more clearly the suddenness of reconnec-tion at each burst, we plot in Fig. 4 the average radial dislo-cation of the other ends~footpoints! of the twisted field lines,which should be zero unless reconnection occurs. Suddenjumps in this figure indicate intermittent reconnections andlarge sudden drops indicate recurrent reconnections returningto the original structure. Figure 5 shows the temporal evolu-

FIG. 3. An example of MHD simulation exhibiting an intermittency in self-organization when the system is in contact with a continuous energy supplier. Aweak resistive MHD plasma is filled in a rectangular cylinder in which initially the magnetic field is uniform and directed in the axial direction. Within thecircle regions of the top and bottom conductive boundaries indicated by dotted lines the magnetic field lines are continuously twisted in the opposite directionto each other.

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tion of the kinetic energy converted from the twisted mag-netic tension energy for the same flux tube case as shown inFig. 3 and Fig. 4. One can see that, accordingly, the magneticenergy is converted into kinetic energy in bursts (1;2tA) inan intermittent fashion~local cycle! and exhibits a recurrenceto the original topology after several intermittent bursts~grand cycle!. Intermittency and recurrence are common fea-tures observed for the continuous pumping, irrespective ofwhether the twisting~helicity and energy injection! is donefor a single flux tube, twin flux tubes, or four flux tubes.

V. ENERGY PUMPING AND ENTROPY EXPULSION

The above two examples have shown that the systemmanifests an essentially different evolution depending on thedifference in the energy pumping process. When it is instan-taneously pumped to a far-from-equilibrium state, the systemrelaxes into the final global minimum energy state in a tran-sient way and keeps its state in a steady way. In contrast,when it is in contact with a continuous energy supplier, thesystem repeats intermittently an energy burst and exhibits a

recurrent behavior, changing the state from a supersaturated~tensioned! state to a local, or global, minimum energy state.

In addition to energy pumping there is another crucialelement which can change the fate of the self-organization.In the previous examples, we have assumed that the thermalenergy produced during the relaxation process is instanta-neously pumped out. Knowing that in these examples theproduced thermal energy is the entropy which is the super-fluous quantity for the created ordered structure, it is prudentat this point to clarify the role of the entropy on the self-organization.

We present two different simulations. In the first we at-tempt to examine how seriously the maintenance of the pro-duced entropy in the system influences the fate of the self-organization. For this purpose we take as an example theinstantaneously pumped MHD self-organization which waspreviously presented. Without pumping out the producedthermal energy we solve the pressure dynamics self-consistently with the plasma motion and the magnetic fieldevolution.18 The thermal conduction effect is discarded.

The simulation result has shown that the magnetic struc-ture evolves in a very similar way to the previous case wherethe pressure is discarded. Nevertheless, two important newfacts have been discovered. The first is that the structure ofthe pressure exhibits a very similar structure to that of themagnetic field intensity. An example is shown in Fig. 6.

The other discovery is that the magnetohydrodynamic~MHD! self-organized state is not the Taylor force-free mini-mum energy state, but a force-balanced state, i.e.,J3B5¹p. The currentJ consists of the perpendicular andparallel components. As the relaxation proceeds, the parallel~force-free! component decreases, while the perpendicularcomponent increases inversely. The components approachnearly the same level in the self-organized state. This resultindicates that unless some pumping out of the thermal energyor a fast perpendicular thermal conduction does exist, theTaylor relaxation does not occur. Instead, the system relaxes

FIG. 4. Dislocating evolution of the average footpoint of the twisted fieldlines shown in Fig. 3.

FIG. 5. Intermittent bursts of the kinetic energy in the self-organizationprocess of the twisted flux tube shown in Fig. 3.

FIG. 6. A non-Taylor MHD self-organization. The figure represents theself-organized magnetic structure~left! and pressure structure~right! in amagnetohydrodynamic plasma.

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to a force-balanced, non-Taylor, minimum energy state. Un-fortunately, we have not yet succeeded in extracting someselection rule that might refine the physical property of theobtained force-balanced, minimum energy state. However,there are some observed features that may characterize thisstate. Namely, the parallel current,J// , in the force-balancedminimum energy state satisfies the Taylor condition thatJ// /B5const. For the perpendicular~diamagnetic! current,J', the pressure is redistributed in such a way that its spatialstructure becomes similar to the self-organized magneticstructure.

Next, we shall attempt to look for what will happenwhen the produced entropy is filtered out. We take the ionacoustic double layer19 for this purpose. What we have ob-tained is a remarkable and striking new self-organizationphenomenon caused by filtering out the produced unneces-sary entropy from the system. It is well known by particlesimulation that weak ion acoustic double layers are gener-ated in a one-dimensional collisionless plasma where elec-trons have a shifted Maxwellian distribution and ions have anormal Maxwellian distribution. In the previous simulation,however, particles were regarded as periodic in a one-dimensional system. This periodic condition implies that theparticles leaving from the downstream boundary enter intothe system from the upstream boundary as they are and viceversa, no matter how seriously they are disturbed by thegeneration of ion acoustic double layers. In other words, thedisturbed information, or disorderliness, is retained in thesystem.

A new particle simulation code is developed in whichfresh particles continuously flow into the system from eachboundary in place of dirty particles leaving the system.20Wecan thus filter out the superfluous information~entropy!,when produced. The result is really striking. A giant doublelayer is generated. In the initial phase of the ion acousticinstability the normal ion acoustic double layers are gener-ated. They disappear after a while, as we observed in theprevious simulation. However, new ion acoustic double lay-ers emerge at different positions. They again disappear aftera while. Once again a double layer is generated. At this time,the double layer grows singly and the potential reaches to asurprisingly large amplitude~see Fig. 7!. Compared with theconventional ion acoustic double layer, the potential differ-ence of which is of the order of one electron thermal energy,the potential amplitude of this giant double layer reaches asurprisingly large level, say, larger than ten times the thermalenergy. This giant double layer has a much longer lifetimethan that of the normal weak ion acoustic double layer andexhibits a recurrent generation. In view of the fact that thedrifting electron energy was only 0.36 of the electron thermalenergy in this example, the obtained potential difference isreally huge. This giant ion acoustic double layer can be anefficient electron accelerator that can be a good candidate forthe acceleration mechanism of aurora electrons.

The implication of this simulation is of incalculablevalue. If the system contains a proper filtering function of thesuperfluous disorderliness, then a well-organized structurecan be realized and sustained for a long period. Also, thisexample supports a recurrent behavior for a continuous en-

ergy flow as we observed for the continuously twisting mag-netic flux. In this sense, this example may well be consideredas a kinetic self-organization.

Along with the first simulation of the MHD self-organization where the produced entropy is confined withinthe system, the present simulation filtering out the producedentropy leads us to an assertion that the expulsion of theproduced entropy plays a crucial role in the self-organization.

VI. ENTROPY PRODUCTION

It has already been seen that, depending upon whetherentropy produced during self-organization is pumped outfrom the system or contained in it, the system experiences adifferent evolution. This indicates that entropy, which is asuperfluous product, is produced in the course of self-organization. In this section we examine, with particular at-tention, the dynamical relationship of the entropy productionrate with creation of orderliness.

Let us begin with the MHD phenomena. The entropyproduction rate can be easily monitored by observing theohmic heating. It is observed that the production rate of en-tropy is anomalously increased in accordance with the struc-tural transition and minimized when the transitionsubsides.5,10,11,14,18

Figure 8 gives a typical example of the entropy produc-tion rate, namely, the volume-integrated ohmic heating, whenan energy ~helicity! is instantaneously pumped into asystem.5,18As one sees, the entropy production rate is maxi-

FIG. 7. A particle simulation creating a giant potential structure by filteringout the disturbed part~produced entropy! of the outgoing particles. Theupper panel shows the potential structure of normal~weak! ion acousticdouble layers generated by a shifted Maxwellian electron stream with theaverage velocity of 0.6Veth (Veth is the electron thermal velocity! under thecondition that particles are periodically circulated.

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mized as the transition is taking place and minimized as itsubsides. When the final minimum energy state is realized,the production rate almost vanishes.

Let us see another example where self-organization takesplace intermittently.14 The evolution of the entropy produc-tion rate is shown in Fig. 9. One sees that the production raterepeats maximization at every sudden structural transitionand minimization after transition.

This feature of maximization and minimization of theentropy production rate in accordance with structural transi-tion and creation of a new ordered state is not specific toMHD phenomena, but is also pertinent to kinetic self-organization. As one sees in Fig. 10, the entropy production,namely,** f ln fdvdx, is maximized and minimized repeat-edly in good accordance with the creation and destruction ofeach double layer.20

From these results, one can definitely confirm that when-ever a new structure is being created the entropy productionrate is anomalously increased, then minimized after the newstructure is created.

VII. DIPOLE FIELD GENERATION IN A ROTATINGSPHERE

How the earth’s dipole field is created is a challengingproblem for plasma physicists as well as geophysicists. It ispostulated that there are some heat sources in the central coreof the earth’s interior which keep the core temperature high,say, 1500 °C. A conducting mobile spherical shell mediumcovers the heat core and, on top of that, there is a cold im-mobile covering spherical shell medium that removes theheat flux from below. This situation assumes that the inter-mediate conducting mobile medium is an open system inwhich self-organization can take place.

A magnetohydrodynamic simulation is performed21 in arigidly rotating spherical shell, including Coriolis force,gravity, viscosity, resistivity and thermal conductivity. Theinner spherical surface is kept to a constant high temperatureand the outer surface to a constant low temperature. Theconvection instability develops and creates a well-organizedconvection structure, more specifically, paired cyclon-anticyclon columns along the rotation axis which stand sideby side in a circle as shown in Fig. 11. When a tiny magneticfluctuation is imposed in the system, the fluctuation developsand creates a well-organized magnetic structure, as well asthe convection columns, that is, the dipole-dominant mag-netic field structure, as shown in Fig. 12.

The present run has not revealed an intermittent or re-current behavior which might occur in the long time evolu-tion. Such a symptom did appear in the result, however, thisproblem is left for future investigation. In any case, thepresent example favors our assertion that an ordered struc-ture can be created in an open nonlinear system where freeenergy is externally supplied and produced entropy is re-moved to the outside world.

VIII. SUMMARY: SCENARIO OF SELF-ORGANIZATION

We are now in a position to draw a somewhat integratedpicture of the grand view of self-organization when a systemis pumped up to a far-from-equilibrium state. The evolutionexhibits a remarkably different feature depending on the type

FIG. 8. Time evolution of the entropy production rate for the same case asFig. 2.

FIG. 9. Time evolution of the entropy production rate for the same case asFigs. 3 and 4.

FIG. 10. Time evolution of the total system entropy for the kinetic self-organization~see Fig. 7!.

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of energy pumping. When externally pumped up faster thanits dynamical response time, the system relaxes in a transientfashion to a steady self-organized state. Upon a continuoussupply of an external energy flow, on the other hand, thesystem exhibits a recurrence in the structure and energy statewhile repeating intermittent bursts.

It is also seen that whether a superfluous entropy or dis-orderliness can be pumped out or not, the evolution and thecreated structure are largely changed. When the entropy or asuperfluous material is filtered out, a well-organized structurecan be created. The pumping action of energy and the ex-pulsing action of entropy are both closely related to the pro-cesses of information exchanges between the system and theexternal world. They are thus considered to be the crucialcontrolling external elements of self-organization.

On top of these external controlling elements there areseveral important internal natures that govern self-

organization. They are instability, bifurcation~phase transi-tion!, anomalous dissipation, etc. When a system gains en-ergy, gradually or instantaneously, from an external world,beyond a threshold~marginal! point, an instability arises.When the instantaneously gained energy or the energy sup-plied from a continuous energy budget far exceeds the insta-bility threshold~a supersaturated state!, the instability devel-ops sufficiently. The nonlinearity is enormously enlargedthereby and gives rise to a large deformation of the structure.The deformation progresses in such a way that the exces-sively deposited energy is released along the shortest path. Inorder to release it in the shortest path an anomalous dissipa-tion, or an anomalous entropy production, must take place inone way or another. Simultaneously, the structure~topology!must be drastically changed, namely, a nonlinear bifurcationmust take place.

In the magnetohydrodynamic self-organization, thedriven magnetic reconnection plays these roles at once,namely, the anomalous magnetic energy conversion into thethermal energy and the topological change of the magneticfield. In a collisionless plasma with drifting electrons, cre-ation of localized anomalous resistivity plays the roles of thecurrent redistribution and the formation of an electrostaticpotential structure.

These results and others have led us to the assertion thata natural open system in a far-from-equilibrium state tends toboost itself up to a bifurcation~phase transition! point, whichthen gives rise to an anomalously enhanced dissipation rate,or an anomalous entropy production rate, by whatever pro-cess it may be realized, whereby a well-organized orderlinessis created.

1G. Nicolis and I. Prigogine,Exploring Complexity~Freeman, New York,1989!.

2L. Woltjer, Rev. Mod. Phys.32, 914 ~1960!.3J. B. Taylor, Phys. Rev. Lett.33, 1139~1974!.4D. Montgomery, L. Turner, and G. Vahala, Phys. Fluids21, 757~1978!; S.Riyopoulos, A. Bondeson, and D. Montgomery,ibid. 25, 107 ~1982!.

5R. Horiuchi and T. Sato, Phys. Rev. Lett.55, 211 ~1985!; Phys. Fluids29,1161 ~1986!; ibid. 29, 4174~1986!; ibid. 31, 1142~1988!.

6A. Sykes and J. A. Wesson, inProceedings of the Eighth European Con-ference on Controlled Fusion and Plasma Physics, Prague~EuropeanPhysical Society, Petit-Lancy, 1977!, Vol. 1, p. 80.

7E. J. Caramana, R. A. Nebel, and D. D. Schnack, Phys. Fluids26, 1305~1983!.

8A. Y. Aydemir and D. C. Barnes, Phys. Rev. Lett.52, 930 ~1984!.9D. D. Schnack, E. J. Caramana, and R. A. Nebel, Phys. Fluids28, 321~1985!.

10T. Sato and K. Kusano, inPlasma Physics and Controlled Nuclear FusionResearch~International Atomic Energy Agency, Vienna, 1985!, Vol. 2, pp.461–475.

11K. Kusano and T. Sato, Nucl. Fusion26, 1051 ~1986!; ibid. 27, 821~1987!; ibid. 30, 2075~1990!.

12A. Hasegawa, Adv. Phys.34, 1 ~1985!.13T. Sato and K. Kusano, Phys. Rev. Lett.54, 808 ~1985!.14H. Amo, T. Sato, and A. Kageyama, Phys. Rev. E51, 3838~1995!.15Z. Mikic, D. C. Barnes, and D. D. Schnack, Astrophys. J.328, 830~1988!.16D. Biskamp and H. Welter, Sol. Phys.120, 49 ~1989!.17K. Kusano, Y. Suzuki, and K. Nishikawa, Astrophys. J.441, 942 ~1995!.18S. P. Zhu, R. Horiuchi, and T. Sato, Phys. Rev. E51, 6047~1995!.19T. Sato and H. Okuda, Phys. Rev. Lett.44, 740 ~1980!.20T. Sato, H. Takamaru, and the Complexity Simulation Group, Phys. Plas-mas2, 3609~1995!.

21A. Kageyama, T. Sato, and the Complexity Simulation Group, Phys. Plas-mas2, 1421~1995!; A. Kageyama, K. Watanabe, and T. Sato, Phys. FluidsB 5, 2793~1993!.

FIG. 11. Well-organized cyclonic~black! and anticyclonic~white! columnsalong the rotation axis.

FIG. 12. Well-developed dipole-like magnetic field expanding in the openspace from the rotating sphere.

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