Proceedings of ITC/ISHW2007

Self­similarly evolving, minimally dissipated, and dynamically stableself­organized states obtained by a universal theory
Y. Kondoh, A. Okada, N. Kobayashi, R. Nakajima, T. Takahashi and J. W. Van Dama
Dept. of Electronic Engineering, Gunma University, Kiryu, Gunma 376­8515, JapanaInstitute for Fusion Studies, Univ. of Texas at Austin, Texas, USA(Received: 15 Oct. 2007 / Accepted: 15 Oct. 2007)

With the use of a universal theory of self­organization, a novel set of simultaneous eigenvalue equations having dissipativeterms are derived to find self­similarly evolving, minimally dissipated, and dynamically stable states of plasmas realized
after relaxation and self­organization processes. Typical spatial profiles of plasma parameters, electric and magnetic fields
and dissipative factors are presented, all of which are determined self­consistently with each other by physical laws andmutual relations among them, just as in experimental plasmas.
Keywords: self­organization, self­similarly evolving state, minimally dissipative state, dynamically stable, shear flow

1. Inroduction[13, 14] observed and reported so far.

By using model equations such as the Grad­Shafranov
equation, stability of equilibrium configurations in a stationary
state is analytically or numerically judged by using suitablestability criteria in the traditional algorithm to find stable states[1]. However, all dynamic quantities in any experimental
systems are never stationary, but are continuously evolving
along time. Reasonable judgment on experimentally stable states
can be done only when the configurations have come into aphase of self­similarly evolving and dynamically stable states. Arecent universal theory of self­organization for finding
eigenvalue equations to obtain self­similarly evolving anddynamically stable states [2­7] has been shown to incorporate aprevious theory [8] for obtaining the minimum dissipative stateof magnetic energy, by means of which “the so­called Taylorstate” [9] can be derived. The concept of selective decay together
with that of helicity invariance in the traditional theories [9­11] isanalytically proved in [4­6] to have theoretically unrelated with“relaxed states”. The fusion plasma is known to be described for
simulations by a set of charge, mass, momentum, and energy
conservation laws, and Maxwell's equations to follow its
dynamic evolution and to analyze relaxation processes andrelaxed states. However, replacement of any element in the set
by “the so­called helicity conservation law” [9­11] makes the
dynamic evolution untraceable due to this nonphysical law. This
fact also means that the all theoretical basis of the theories[9­11] has no theoretical and physical connection withsimulation results [12] and experimental ones of relaxed plasmas―――――――――――――――――――――――
author’s e­mail : kondohy@el.gunma­u.ac.jp

2. Theories and numerical results
Applying a principle of the minimum change rate of global
auto­correlations for a generalized dissipative dynamic open orclosed systems with N dynamical quantities of M dimensionalvariables and using the variational calculus to the second
variation, we obtain the following N simultaneous eigenvalue
equations, which is used as a universal theory to findself­similarly evolving and dynamically stable self­organizedstates and is applicable to various dynamical systems [3, 7, 15];
¶q i

j#

[x ] ¶x = D [U ] = t
kj#ji-1idf

L im U im [x k k ¹ j ] .

(1)Here, D i j [ q ] represents dissipative or non­dissipative, linear or
nonlinear operators for the change of a dynamical quantity q i bya variable ξ j. Applying Eq.(1) directly to all equations of mass,
momentum and energy conservation laws and Maxwell'sequations with the displacement current neglected for thetwo­fluid model of fully ionized, compressible and dissipative
fusion plasmas and using usual normalization, we obtain thefollowing set of simultaneous eigenvalue equations;¶ ne ¶ t = - Ñ × ( ne u e ) = L n e ne ,(2)¶ ni ¶ t = - Ñ × ( ni ui ) = L ni ni ,(3)¶ ue ¶ t = - ( ue × Ñ ) ue - [ f me1 Ñ ( neTe ) + f me2 Ñ × Π e + f me3 n e ( E+ ue ´ B ) - f me4 ( ne + Z ni )( 0 .5h ^ j// + h ^ j^ )] / ne = L u e u , (4)¶ ui ¶ t = - ( ui × Ñ ) ui - [ f mi1 Ñ ( niTi ) + f mi2 Ñ × Π i - f mi3 ni ( E+ ui ´ B ) + f mi4 ( ne + Z ni )( 0 .5h ^ j// + h ^ j^ )] / ni = L ui ui , (5)3¶ Te ¶ t = (1 - g )Te Ñ × ue - Ñ Te × ue + {f ene1 å , j Π e i , j ¶ u ei ¶ x j+ f ene2 [k e//0 Ñ × (k e// Ñ //Te ) + k e ^ 0 Ñ × (k e ^ Ñ ^ Te ) + f ene3 ( Z ui ne1

Proceedings of ITC/ISHW2007

(a)

(b)

(c)

Fig.1 . Typical self­similarly evolving and dynamically stable self­organized configurations of the RFP in a simplified cylindrical.

22+ 1)(h // j // + h ^ j ^ ) - f ene4 (Te - Ti ) / t ε ei } / ne = L p eTe ,2. Concluding remarks
We have derived a novel set of simultaneous eigenvalue
equations for finding self­similarly evolving, minimallydissipated and dynamically stable states realized after relaxationand self­organization processes [cf. Eqs.(1) ­ (12) ]. The set ofsimultaneous equations is applicable to all type of magneticallyconfined fusion plasmas. Solving numerically the set ofequations in the cylindrical model, we have shown typicalself­similarly evolving and dynamically stable self­organizedconfigurations of the RFP plasma including a lot of spatial
information on related physical quantities useful for detailed
experimental investigation. It should be emphasized that allphysical quantities of interest are self­consistently determined byphysical laws and mutual relations among them.
From the present universal theory of self­organization, weshould recognize a clear fact that a paradigm shift from thetraditional concept of stationary stability to more real concept ofdynamical stability is definitely necessary for faster developmentof fusion science and technology in order to avoid near futureenergy crisis by shortening distance theories and experiments.

(6)3¶ Ti ¶ t = (1 - g )Ti Ñ × ui - Ñ Ti × ui + {f ene1 å , j Π i i , j ¶ u ii ¶ x j+ f ene2 [k i//0 Ñ × (k i// Ñ // Ti ) + k i ^ 0 Ñ × (k i ^ Ñ ^ Ti )- f ene4 (Te - Ti ) / t ε ei } / ni = L p i Ti ,¶B ¶t = - Ñ ´ E = L B B ,(7)(8)

Ñ ´ B = j , (9) Ñ × E = f ce ( Z ni - n e ) , (10) Ñ × B = 0 , (11)j º f cj ( Z ni ui - ne ue ) + j //bs ,(12)where j//bs is the bootstrap current, (fme1, fme2, fme3, fme4), (fmi1, fmi2,fmi3, fmi4) and (fene1, fene2, fene3, fene4), are factors by normalization
for the conservation laws of electron and ion momentums andenergy, respectively, and fce and fcj are those for Eqs. (10) and (12),respectively. Using dominant terms in the limiting case of
uniform conductivity σ and negligible viscosity ν and thermal
conductivity κ, Eqs. (1) ­ (11) lead to the so­called Taylor state,
just the same as in [7]. In general cases, however, these equations
can be applicable to finite beta confinement systems of theTokamak, the reversed field pinch (RFP), the field reversed
configuration (FRC), and so on. Using the cylindrical model
for simplicity and the 4 rank and 4th order Runge Kutta method
under suitable boundary conditions on measurable quantities byreferring to experimental data [14], we have numerically solved
central terms = right­hand sides of Eqs. (1) ­ (11) to getself­similarly evolving and dynamically stable self­organizedconfigurations of the RFP. A typical result is shown in Figs. 1(a),1(b) and 1(c), where all physical quantities and dissipative
factors are shown by their symbols. It is seen from the data
profiles that all physical quantities are related self­consistently
with each other, i.e. , κe⊥ andκi⊥ are determined by ne, ni, Te,Ti, and B, and Te and Ti are determined byκe⊥ , κi⊥, σ, and j,and so on, to lead to negligibly small current density at theboundary wall like as in experimental plasmas. We also findReferences[1] Y. Kondoh, Nucl. Fusion 21, 1607 (1981).
[2] Y. Kondoh, Phys. Rev. E 48, 2975 (1993).
[3] Y. Kondoh, Phys. Rev. E 49, 5546 (1994).
[4] Y. Kondoh, et. al., J. Plasma Fusion Res. SERIES, 5, 598 (2002).
[5] Y. Kondoh, et. al., J. Plasma Fusion Res. SERIES 6, 601 (2004).
[6] Y. Kondoh, et. al., J. Plasma Physics 72, Part 6, 901 (2006).
[7] Y. Kondoh, et. al., Phys. Rev. E 70, 066312­1 (2004).
[8] S. Chandrasekhar, et. al., Proc. Natl. Acad.Sci. 44, 285 (1958).
[9] J. B. Taylor, Phys. Rev. Lett. 33, (1974) 1139.
[10] L. C. Steinhauer and A. Ishida, Phys. Plasmas 4, 2609 (1998).
[11] Z. Yoshida and S. M. Mahajan, Phys. Rev. Lett. 88, 095001 (2002).
[12] R. Horiuchi and T. Sato, Phys. Rev. Lett. 55, 211 (1851).

from profiles of uep, uip,uet and uit in Fig. 1(a) that there exists theshear flow which depends on the profile of ν , i.e. , mainly onthat of Ti, and would stabilize of the self­organized RFP plasma.

[13] Y. Kondoh, et. al., J. Phys. Soc. Jpn. 62, 2038 (1993).
[14] Y.Yagi, et. al., Nuclear Fusion, 45, 138 (2005).
[15] Y. Kondoh and J. W. Van Dam, Phys. Rev. E 52, 1721 (1995).

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