J. Plasma Fusion Res. SERIES, Vol.4 (2001) 349-353

Revisit to the Helicity and the Extended Generalized
Self-organization Theory
KONDOH Y., TAKAHASHI T. and MOMOTA H.'
Dept. of Electronic Engineering, Gunma University, Kiryu 376-8515, Japant(Jniversity of lllinois at tlrban-Champaign, Illinois, USA(Received: 5 December 2000 / Accepted: 18 August 2001)

Abstract
It is clarified that the "helicity conservation law" is never "the conservation equation ofthe helicity
K itself", but is merely "the time change rate equation of K". It is shown that since the total helicity K
can never be conserved in the real experimental systems, the conjecture of the total helicity invariance is
not physically available to real magnetized plasmas. With the use of auto-correlations for physicalquantities, a novel extended generalized self-organization theory is presented, that neither based on thevariational principle nor the energy principle. The self-organized states of every quantities may berealized during their own phases, and the dynamical system may evolve repeatedly out of phases selforganizations among quantities, depending on boundary conditions and input powers.

Keywords:
magnetic helicity, equation ofchange rate, conservation law, variational calculus, self-organization, autocorrelation, minimum change rate auto-correlation

1. Introduction

2. Theoretical Thought Analysis

Since Dr. J.B. Taylor published his famous theory[1] to explain the appearance of the reversed field pinch

With the use of the thought analysis [6], we show
here that although the energy conservation law is always(RFP) configuration [2], the magnetic helicity K has
been believed to have important role as a globalinvariant in the self-organization process or the relaxation one of magnetized plasmas [3,4]. On the other
hand, one of the authors (Y.K.) has been proposed thepartially relaxed state model (PRSM) of the RFP inorder to explain experimental data [5]. Without using
the concept of the magnetic helicity, an energy integral
was derived to deduce the PRSM and the modetransition point of the self-organized state in order tophysically correct, the "helicity conservation law" isnever "the conservation equation of the helicity I<itself".
Both of the energy conservation law and theexplain experimental data on the RFP [8.9].
In this paper, we study again the meaning of theP=-.i* Y xH,
dtQ)

Y.D=p,(3)V.B =0,(4)helicity conservation one are derived from the following
axiom set of physical laws of Maxwell's equations
written in the MKSA unit used in the usual text books:dBotmagnetic helicity itself from the thought analysis [6],because of many evidences showing no invariance ofthe total helicity in simulations [7] and various
experiments [9,10].
*Corresponding author's e-mail: kondohy@ el. gunma-u.ac.jp
349=-vxr,(1)@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchKondoh Y. et aI., Revisit to the Helicity and the Extended Generalized Self-organization Theory

We can get Poynting's energy conservation law, usingdw'

ar=- Jv| tqj.j+(jxB).rldv

Eqs.(l)-(a), as follows*,=

Le+*Eftdv,-mI f ta"ur'as.

(5)

Y=-!,, uor-hf, (ExB).ds. (6)T=-h[. nj.Bdv.frf, (ExA- 0B ).dS.

The two physical laws of Eqs.(1) and (4) are
rewritten equivalently by the following two equations0)

B=V xA.

(8)

AtThe magnetic helicity K is basically defined by thefollowing equation [1]:t.nav.
K=+lFo Jv(9)Here, we emphasize that even if we include "theexternal helicity", taking account of the gauge
invariance [], the following argument is still essentiallycorrect and applicable. After the partial derivative of thedefinition Eq.(9) with respect to t, and using only two
physical laws of Eqs.(l) and (4) with their equivalentEqs.(7) and (8), we obtain the following equation for"the time change rate of K",

ideal plasmas".

However, the part of (A) declares only that thevalue of K defined by Eq.(9) does not change along thetime variable r in "the trivial case" of n - 0 plasmas

filling fully within the ideally conducting wall.
From the following simple thought experiment, weaK_ 2 [ B.EdV

i_-F; J,.+f, (ExA- oB ).ds.

can easily find that the total helicity K is never "the timeinvariant inside the ideally conducting wall". We(10)consider a case, where some vacuum field regions with
4 = - separate slightly the 4 - 0 plasmas from theideally conducting wall. We then have to come back toEq.(10), and we can put E = 0 in the plasma but have toleave E in the vacuum field region. In this simple case,Assuming (E.D)12 << (H.B)12 in the plasma confinement experiments, we obtain the following equation
for the field energy I{1,W,=l'Jv22(E'^D *n+)dv=w^,(1the value of dKlOt is passively and resultantly
determined by the volume integral of B'E in Eq.(10).
The total helicity K can never be conserved in thedynamical system in this simple case. The simple
thought experiment shown above may lead us to al)

where W- is the magnetic field energy component oflVr. From Eqs.(l l) and (6), we obtainay,=drconclusion that the helicity conservation during therelaxation of magnetized plasmas does stay "Taylor's(t2)[i udv-+$
I'o Js rr"rr.os.
Jv"conjecture" forever.
On the other hand, the energy conservation law ofEq.(la) declares that even if 4 = g or 4 = 0 and E = 0 atUsing the simplified Ohm's law Eq.(13), we obtain theconservation laws of W1 and K as follows,

Ohm'slaw: E+vxB=r\i.

(15)

The misunderstanding on the "helicity conservationlaw" has been established from the following argument,using "the time change rate equation of K written byEq.(15)" tll. (A) At first, we consider "the ideal case"where the whole region of plasmas inside the boundary
is filled with the ideally conducting plasma and theboundary surface is the ideally conducting wall, i.e., 4 =0 and E = 0 and B.dS = 0 at the ideally conducting
wall. We then get from Eq.(15) in this "ideal case" that"the time change rate of K" becomes as dKldt = 0. (B)
From this result, we may conclude as follows: Since thevalue of K is constant along the time variable /, the total
helicity K is conserved and therefore it must be "thetime invariant in the dynamical system in the case ofwith the use of the scalar and the vector potentials;

d4 =_vp_n,(r4)the ideally conducting wall, the left-hand si.de dW1/dt
always balances with the volume integral term of (7 xB).v, which is called the dynamo term, in the right-hand

(13)

350

Kondoh Y. et al., Revisit to the Helicity and the Extended Generalized Self-organization Theory

conservation equations for qi(t, r) as follows,side. The value of the helicity K has never been conserved in the computer simulations by R. Horiuchi andT. Sato [7], and also in all experiments on the reversed
field pinch (RFP) by many authors [2,91, on the toroidal)l l dy
I f'dt-a !2 q,(t.x)q,(t.x''Jrf{ a,Q 'x) GiIcll dv.
=Z-pinch by K. Sugisaki [10] and on merging twoJu(r7)Spheromaks into one field reversed configuration (FRC)

From the standpoint of observation on over aII timeor one Spheromak by Y. Ono, Katsurai et al. [1].
Especially, in the case of the toroidal Z-pinch ex-evolution of the dynamic system, we can identify ordefine "the self-organized state" as "the state of themost unchangeable structure". The definition may beperiments, the total helicity K increases to finite values

from zero initial value within a few tens of its [0].
the conjecture of the total helicity invariance by Dr. J.B.

mathematically expressed by using auto-correlations,q;(t, x)qiQ + Lt, x), between the time, t, and slightly
transferred time, / + A/, with a small Ar in the following

Taylor is not physically available to real magnetizedway, i.e., self-organized state is defined by

These experimental results have been demonstrated thatplasmas.

. ,lq,(t.xlo,(t+N.x)dV .I state. (18)- Iminl 1aij-j:-1!rj-:-_]-*'' ! q,(t , x) q,(t , x) dV

As is well known, the variational principle and therelated or resultant dynamic equations are physicallyequivalent, i.e., we can start with either the variational
principle or the related set of dynamic equations in orderto analyze the systems. This fundamental physicalthought is also the same for the energy principle and therelated dynamic equations. Without using the concept ofthe helicity K, we can derive the Taylor state V x B =)aB from the equation of the relaxed state of MHD
plasmas written as V x (4V x B) = )8, which includes
the Taylor state of V x B = AaB as a special case where
4 is spatially uniform [12].

I

Substituting the Taylor expansion of q,(t + Lt, x) = q,(t,x) + ldq;(t, x)ldtlLt + ... into the definition Eq.(18), andtaking account of the arbitrary smallness of Ar, we
obtain the following equivalent definition for of the selforganized state from the first order of Ar in thedefinition Eq.(18):I q,(r. x tld q,rt. xY& I av-inl#lstate.
' q,(t x) q,(t x) dVI,(19),Substituting the original dynamic equation, Eq.(16), intoEq.(19), we obtain the following final condition for theself-organized state

3. Generalized Self-organizataon Theory
We develop here a novel extended generalized selforganization theory that is an extension of the last reportin [2]. It should be emphasized here that the gen-min

I q,Q ,x) G,lqldVI

I q,1t , x; q,1t , x) dv

I state. QO)

Since we have substituted the original dynamic

eralized self-organization theory with the use of autocorrelations for physical quantities is not fundamentally
based on neither the variational principle nor the energy

equations into the definition of the self-organized state,

we can recognize that "whole properties of the dynamic

principle, and the auto-correlations is never timesystem is essentially embedded in the process ofinvariants.
systems of interest shall be expressed as q(t, x) = {qt(t,calculations to derive the self-organized state from thefinal condition of Eq.(20)".
The mathematical expressions with the use of thephysical quantities having n elements. We consider adissipative nonlinear dynamic system which may bevariational calculus for the definition of Eq.(19) andfurther the final condition Eq.(20) are written as follows,defining a functional F with use of a Lagrange
multiplier 2;:

Quantities with n elements in general dynamicx), q2(t, x), ..., q"(t, r)). Here, r is time, r denotes mdimensional space variables, and q represents a set ofgenerally described by= G,tql,
Yot

(16)

where G;[q] denotes linear or nonlinear dynamicoperators, which may include no dissipative and/ordissipative terms. After taking the product of qi(t, x) andboth sides of Eq.(16), and integrating both sides of theresultant equation over the volume V, we obtain the351

F=[ {qt(t.x)G,Iq]
Jv ' -''+ ),, q,(t,x)qt!,r) ) dV.

Qr)

6F=0,Q2)

Kondoh Y. et al., Revisit to the Helicity and the Extended Generalized Self-organization Theory

62F>0,(23)E-( jxB\.v-Q"ij.j'*u

9r.= [dr'Jvffi"'"where dF and 62F are respectively the first and thesecond variations of F "with respect to the variation
64(x) only for the spatial variable r". Comparing
Eqs.(17) and (21), we can find that the conservation
equations concerning with the quantities qi(t, x) for the
dynamic system of interest are naturally included in thepresent formulation of the generalized self-organization
theory. The implicit assumption in this theory is that the
dynamical system evolves all possible area in state*h i tvP" - (m,/m)Z,YP,)l dv. Q9)

According to Eq.(21), we obtain the functional for thefield energy F1, for the kinetic energy Fu, and for thecurrent F., respectively, as follows,

F,=' |Jv l-j.E*1,f07'E
2phases.

*B--L1ydv

2ltn"-hf,(ExB)'ds,

When we apply the extended generalized self-(30)organization theory to fusion plasmas, we use here thefollowing three more physical laws, i.e., the conservation laws of the mass, Eq.Q$, and the momentum,,r=[, t+, vv.(p^v)-+ | P.E' v+ ( j x B )' v - P^v' V( f" + P,ll

Eq.(25), and the generalized Ohm's law, Eq.(26).

.(p^v).
$=-vdt+).,P^v.vldV,'2Q4)
O,=O^! =- O^(v 'V)v+ [P.E + jxB_V(p"+p, )1.

Ie2n-Ju *',"y=+rL,+ rJrN r" - (m m,) Z,Y P, 11 .
"/(3 1)li' E - ( ix B \' v - 4"; i'il*;ii. tvP" - (m./m,)Z,YP,l+ )"" i.i av.

Qs)* + tE v x B - i - +( ixBp^v .t(y .V)y l(32)In general, we take variations with respect to 6E, 68,6v, 5j, 6p^, 6p", 6P., 6Pi, 5n., 6ni, and 64";. From theEuler-Lagrange equations for the solutions of Eq.(22),)

Q6)

we will get new various equilibrium configurations of
the self-organized states with the plasma flow, the shear
flow, the space charge, the space potential, the deviation
between the ion and the electron density profiles, theresistivity profile, and so on, depending on the boundary
conditions and the external input sources such as thevarious energy injections and the particle beams. The
resultant equilibrium configurations are far beyond theconventional MHD equilibrium ones by the Grad-These three equations come from the Boltzmann kinetic

equations for electrons and ions. Therefore, we startwith an axiom set of seven physical laws, and the charge
conservation law is included in Maxwell's equations,Eqs.(l)-(a). Conservation laws of W1, the kinetic energy
1421, and rhe current defined by W"= lrlttZyi.idV are
obtained as follows:

Shafranov equation based on the equation ofj x B =Yp.
The results of the present calculation will appear

Y=-1,, Edv-hf,(ExB).dS. Q7)

elsewhere.

Dividing Eq.(29) by e2n"/m". we find the resultant
equation becomes a power balance equation.

= f t-l v.vY.(p.y)
9tar"Jv2- p^v .[(r, .V)r ]+lp"E.v+( jxB).v-p^v.V(P.+P,)lldy.

4. Concluding Remarkseg)

Analyzing the logical and mathematical structuresof the derivation process for the "helicity conservation
law", and using the simple thought experiment for thecase, where some vacuum field regions with 4 = separate slightly from the 4 = 0 plasma from the ideally352

Kondoh Y. et al., Revisit to the Helicity and the Extended Generalized Self-organization Theory

conducting wall, we have clarified that "the so-calledhelicity conservation" is never "the conservation
equations of the helicity K", but is merely the equations
for "the time change rate of K". The total helicity K cannever be conserved in the real experimental dynamicsystems, as was observed in the toroidal Z-pinch experiments [0], the conjecture of the total helicity invarianceby Dr. J.B. Taylor is not physically available to real

give us more dynamic equations added to Eq.(16) for
the system of interest.

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