J. Plasma Fusion Res. SERIES, Vol. 2 (1999) 184-187

Effects of Periodic Helical Deformation occurrenceson the Flux Generation (Dynamol in RFP
MAEYAMA Mirsuaki and HOTTA Eiki'Saitama University, Urawa, Saitama 338-8570, Japan
lTolcyo Instituteof Technology, Nagatsuda, Midori-ku, yokohama, 226-8502, Japan(Received: 2l December 1998 / Accepted: 6 May 1999)

Abstract
Effects of periodic helical deformations on the dynamo mechanism in RFP are examined using asimple electric circuit model. Modeling the plasma to toroidal and poloidal current channels, we use themethod which switches mutual inductance matrices discontinuously on the condition conserving
magnetic flux to simulate periodic onset and extinction of helical deformations.
And even if helical deformations occur periodically, phenomena of continuous energy transports
from a poloidal external power source to a toroidal circuit and toroidal flux sustainment with appropriate@ value are shown to be demonstrated using the proposed circuit model.

Keywords:
RFP, helical deformation, dymano, electric circuit model

1. Introduction

2. Electric Circuit Model for Helical
Deformed RFP Plasma

The mechanism for self-generation of magneticflux (dynamo effect) has been studied experimentallyand theoretically as for sustainment and improvement of

The electric circuit model is useful for analysis ofan energy transport between toroidal and poloidal circuit

RFP plasma confinement. Most studies on this effect,
such as magnetic reconnection, have concentrated thoseincluding external power sources and comparisons ofwaveforms with experimental data.
Used toroidal RFP plasma model is shown in Fig.

attentions on one time deformation behavior of RFP

plasma for a toroidal flux generation. But many
experimental observations, such as ramped current
discharge atZT-40M [] and spontaneous generation of
toroidal flux at toroidal Z-pinch [2], suggest the strongl, where @, and @, are magnetic flux of toroidal andpoloidal direction, respectively. Figure 2 showsparameters specifying helical deformed plasma where ro

is a minor radius of plasma, r5 is a displacement ofplasma column from the center of discharge chamber

coupling between toroidal and poloidal plasma currents
and the continuous energy transport from poloidal to(r, z) = (Ro, 0), and 01 is an angle from r-axis and istoroidal circuit.

expressed using toroidal helical mode number n as 0n =Therefore, using simple electric circuit model, wen0.

investigate effects of the periodic occurence of helicalIn this model, we consider two current components

deformations on the generation and sustainment of amagnetic flux. In Sec. 2, electric circuit model isderived, in Sec. 3 analyzed results and comparison thoseof toroidal and poloidal directions, where in ansymmetric case, those current are independent, and inhelical deformed case those are coupled through mutualwith experimental data are described. Discussions andconclusions are made in Sec. 4.

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research184

Maeyama M. et aL, Effectsof

periodic Helical Deformation Occurrences on the Flux Generation (Dynamo) in RFPinductance. In table l, we show approximate equationsof self- and mutual inductances.
The time scale from start to its final steady state ofhelical deformation must be much smaller than that ofnormalized by those specific values.
Figure 4 shows time histories of toroidal magnetic
flux @, and poloidal current 1o on condition of steadyand periodic occurence of helical deformations, wherethe duration of symmetric ?1 and helical deformed
plasma 12 are O.l and 0.01, and ZorlRpr = l, LpzlRpz =0.12. In periods of @, rising and 1o dropping, plasma is
in a helical form. From this figure, toroidal flux is

resistive diffusions. So, motions of helical deformations
are treated as if they occur instantaneously, conserving
magnetic flux. To simulate these conditions, we adopted

the electric circuit in Fig. 3, where in Fig. 3(b)shown to be generated and conserved in case of periodic

deformation process from symmetric to helical plasma is
treated as changing values of circuit parameters,

helical occurrences as well as in case that plasma
remains in a helical form. And this phenomena is

discontinuously. Moreover, for simplicity, poloidal one

turn voltage is set to zero, although it is applied tosustain negative toroidal field, and toroidal current isinterpreted as follow;
Table 1 Approximate forms of self and mutual
inductance of Plasma.

treated as current source 1,.

3. Analysis of Periodic Deformed Plasma

Lt

In the following analysis, values of time, electriccurent, magnetic flux, inductances and resistances areMt

Lprpnftafln(8na/rr) - z)po(no- Jnl-rtrl

0Ltz

Lpz

M2

L11 + pan2r2nf2fu

Lo'l'fi+@f,iEF

pr21,n/2Ra

wffifl@i::f;L=-iO) Swithing circtit mcdel(et Symnetric andhclicd defomedPlaem

Fig. 3 Forms of a symmetric and a helical deformed

plasma {a} and the electric circuit model usedperiodic deformed plasma analysis (b).

Fig. 1 Definition of toroidal and poloidal current /,, /o andflux @,. @o in the cylindrical coordinate.

ltcdy h3licd pLtm.

o-2

p.dodb heLc.l Plastna

oo.5

o-o.5

-t-r.5-2

1.5

t

Fig. 4 Typical waveforms of a toroidal flux and poloidal

current in case of steady and periodic helicalFig. 2 Definition of a helical deformed plasma' (Figure isshown in case of rnlr* = 9.5.1

deformed plasma.
185

Maeyama M' et al.' Effe;c;ts of Periodic Helical Deformation occurrencesi)on the Flux Generation (Dynamo) in RFp

at first, toroidal flux is generated by a helicaldeformed toroidal current (Mzlr).

ii) at a time changing from helical to symmetricplasma, poloidal current (10) is driven to*

0.5*lo./i\conserve @r, and due to resistive dissipation,- ...'...'.ll\@, decreases gradually.

iii) again at a time changing to helical deformed

0.6
C2plasma, to conserve @, in the presence of M21.,
1o droppeds to negative value instantaneously.

Fig. 5 A dependence of the coefficie nt @, I M rl, on c, andAnd due to resistive dissipation of l1ol, @,d2.

increases further.

Table2 Comparison the proposed @ value withAnd the steady state value of toroidal flux @, isdeduced as follow/ r-ct. \@, | -----------:- |'=experimental ones.

74" 1.

DeviceHBTX-1[31HBTXlBI4I

TPE-1RM15[5]O(,tr.)

0.8 (4.5)

2.56 (1.6)

l.1r - 1.8 (2.2 - 3.6)

\l-ctr0r/

where fi1 = exp (-Rprf llLr), az exp (-Rpzrz/Lpz)=and M21, is a flux that may be sustained in case thatplasma deforms helically in steady state. Figure 5 shows
dependences of @rlMrl, on o, and a2. And onconditions of a1 - | or a2 - 0, larger toroidal flux is:osustained.

--'ti.---li----ii---ti---From above analysis, pinch paramelq @ = (8"*)/(8,) may be calculated asw'

Et Fo r* I, (l - a' ar; lto r*t:t=-

2 Q,2.6I(l - cr:) 2M,2.5

Fig. 6 Waveforms of a toroidal flux and the systemenergy W* relating to the work done e><ternally.

_(l-a,ar). t /r*\,(l -0:) ftr l 4/ 'or discontinuity of a poloidal flux, the system energy

where t = n/& is a toroidal wavenumber of a helical de-relating to work done externally is given as

formation, r* is a minor radius of the shell. Using 4r r*= 0.5 typically, because the deformation should be anW^ =: 1cDrI,+ <D,I.lainternal mode, in table 2 we show evaluated @ values ofdevices HBTX-1 [3], HBTXIB [4], TpE-tRM15 t5l,
where (1 - ara)l(l - az) = l. And derived values of,)- <Dol,,and waveforms of @, and W* on the same conditions ofFig. 4 are shown in Fig. 6. Dis-continuous drops of W*
are interpreted to correspond to the free energy making
grow helical deformations which is indeed identical tothe kink instability.

@ from our circuit model are shown to agree
approximately with experimentally measured values.

In conclusion, modeling periodical helicaldeformations of RFP plasma in simplified electric

4. Discussions and Conclusions
The direction of a helical deformation of RFp

circuit, continuous energy transport from a external
poloidal circuit source to a toroidal circuit, generation/

plasma has been recognized experimentally [6] to theone increasing the toroidal magnetic field at plasma
center. And the condition for the growth of helicaldeformation is also estimated approximately by theelectric circuit model. Because ofusing current source dsustainment of toroidal flux are shown to be
demonstrated. And from @ estimations. sustained
toroidal flux agree with experimental values.

186

Maeyama M. et al., Effects of Periodic Helicat Deformation Occurrences on the Flux Generation (Dynamo) in RFP

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