Plasma Fusion Res. SERIES, Vol.2 (1999) 188-l9l

Interpolation of Magnetic Surface Functions
fo" an Axi-Symmetric Plasma
Ya'4AGUCHI Taiki and MAEYAMA Mitsuaki
Saitama University, Urawa, Saitama 3j8-8570, Japan(Received: 8 December 1998 / Accepted: 6 Mav 1999)

Abstract
Informations of the magnetic surface functions of magnetically confined plasma are indispensablefor equilibrium, stability and transport analyses.
In this paper, in order to identify a realistic surface functions and compare those with ones which areintroduced from Taylor's relaxation theory, we propose a code to interpolate these surface functions foran axi-symmetric plasma from experimentally measured data.
To confirm our code, we used the data which were analyzed from known functions given as ameasured data. As a result, we have developed a code which can derive surface functions I and P. Effectsof measurement error on those functions are also examined.

Keywords:
magnetic surface function, relaxation theory, equilibrium analysis, Grad-shafranov equation, reversed
field pinch, least square method

1. Introduction
A reversed field pinch (RFP) has a property tharAs numerical method, we use the least-square
method [1]. In $2 we explains equilibrium analysis codeused in the least-square method. In g3 the algorithm ofplasma itself makes magnetic fields which are necessary

to confine plasma. The self-organization phenomenon isexplained by Taylor's relaxation theory.

this code is described. In g4 and <5, numerical results
are shown. In $6 the conclusions are summarized.

From theoretical considerations or models of thisrelaxation processes, forms of the magnetic surfacefunctions I = rBt and P, where B, is a toroidal magnetic
field and P is a kinetic pressure, are derived. Giving
these surface functions, Grad-Shafranov equation can besolved for equilibrium, stability and transport analyses .

2. Equilibrium Analysis
In the cylindrical coordinate (r, Q, z) (Fig. l), MHD
equilibrium of an axi-symmetric plasma is described by
the Grad-Shafranov equation:(1)

We are developing a code to interpolate magnetic
surface functions from experimentally measured data.
The merits of this code are as follows:
r We identify surface functions from experimental
data and compare those with theoretical ones.
. We can analyze experimentally obtainedequi I ibrium configurations.
o We can obtain the time history of the helicity (K= /uo A.Bdv; which is a crucial parameter of thedP(w\ I(tt/) dl(wlr -------:--- + -fi ------i- in Plasma Vo,{dVLFor dvt

in Vacuum {a)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchrelaxation theory.

188

Yamaguchi T. et al., lnterpolation of Magnetic Surface Functions for an Axi-Symmetric Plasma

B=ryxeo+Leovacuum vessel(3)

where eois an unit vector of Q, JEis a current density
following @ direction, "/" is that of external field controlcoils.
The code 'EAFP' [2] can solve the Grad-Shafranov
equation on conditions of arbitrary functions 1(tg) andP(ry) and geometry of the limiter and conducting shell
as a free boundary problem.
Fig. 1 The cylindrical coordinate in toroidal geometry.

3. Algorithm for Least-Square Method
To use a least-square method code, we must
establish forms of model functions and choose types ofmeasured data.

For convenience the magnetic surface functions are

simply shown:

I=f(V,at,a2,..')(4)

P=g(V,bt,b2,-..)(5)

where a;, ,i are unknown parameters and f, g are functions of a model.
As for measured data, we use poloidal flux yr atsome points of vacuum vessel surface, toroidal flux t44,F-value, @-value and Bs of a B value at a plasma center.
An example of equilibrium which is analyzed by EAFP

Fig.2 A typical equilibrium by EAFP and arrangement offlux loops o.

and positions ofpoloidal flux coils are shown in Fig. 2.
First of all, to confirm our code we used the data

yr+ which are obtained from the EAFP analysis asmeasured data. For this EAFP analysis we use magnetic
surface functions [3] such asf (w\= t * l( vt + (l -'lf.')
cr+l )\(6)

P* (v)= l.S1ov $ - v2 | 3)a)where y is a normalized value of rAs, L, a and Bs are
assumed parameters. In this analysis, we use )'= 1.2, a= 4 and Fo = 0.1.
A flowchart of this code is shown in Fig. 3.
Because of nonlinearity of this problem, we use

Fig. 3 Flowchart of least-square method.

where k shows the number of flux loops and Cr, Cr, "',C, are weight coefficients which depend on a varianceof data. In this analysig, we simply use reciprocals ofsquares of measured data for weight coefficients. Finallythe unknown parameters and magnetic surface functions
I and P are obtained by iterative calculations.

Marquardt-method [4]. At first we establish initial valueof unknown parameters. Using those parameters, EAFPcalculates V, Vo, F, O and Bs. The unknown parameters
are improved to minimize the functional:
kt =2. C,, (Vi - V)' + Cz(Vo - VD'
l=l

4. Interpolation+q(F-F")2 +Co(O-O.)'+ Cr(Fo- Ft)'

We analyzed using two forms of model functions
as follow.

(8)
r89

Yamaguchi T. et al., Interpolation of Magnetic Surface Functions for an Axi-symmetric Plasma

Model L: In this analysis we analyzed using model
functions 11, Plfor least-square whose interpolation
functions are same to those of 1* and p*. The
unknown parameters are L and Bo.

6'o.r
0N

F 0.69o.r

Model 2z We used model functions 12, P2 for leastsquare whose forms are different from 1*, p*:

0.2

MI, (V)= | +2 a,0 - v/\t0(e)NP"(U/)=2 U,v'

r_r'T_T:_lI

0 0.2 0.r 0.6 0.t* (normalized)

0 0.? 0.1 0.6 0.E

rry' (normalized)

Fig. 4 Profiles of measured data /*, P* and converged /r,P, without an artificial noise.

(10)

where M = 5, N = 3 and a,, b; are unknown
0.2

parameters. In this analysis, we append constraints

on P that P has a maximum value and dPldvr iszero at magnetic axis.

F - ---4.

Oo. rs.8

As a result using Model l, we obtained parameterswhich were equal to the exact parameters L= 1.2 and Fo

EEo.retr= 0.1. Next, the result using Model 2 is shown. Next,Fig. 4 shows 1*, P* and 12, P2 at the converged point
using Model 2. And we could get surface functions*0. t4

which equal to those of exact functions.
When we analyze a practical data, models of 1 andP are unknown indeed. One solution of this problem is
to use higher order power series as model functions, but

4,u,/--t--e--measured' ":,aact solution

data with SVo nob
converged result

50 poloidal 100
angle ['l0150Fig. 5 Poloidal flux in noise-free case, with 5olo noise andat the converged result.

the use of large value of M and N requires manymeasurement coils and yields a problem oforthogonality between unknown p.uameters [4]. So it isnecessary to analyze using many types of modelfunctions, in case of analysis using real data. And using
the constraints which relates P, if exact model is notchosen, we can obtain surface functions of a certain
degree of approx i mation.

0.

1

IoRo. orIo

Eo. 05dcift-0. or_o _pz

0.02

5. The Effects of Measurement Errors
Adding random error to measured data, we0investigate these effects.
Figure 5 shows poloidal flux which are exact data,measured data with 57o error and converged result.
Figure 6 shows exact functions 1*, P* and 12. P2 which
are converged ones. It was found that the maximum

0.2. 9.. 0.Q. 0.1 |rp lnormauzed)

090

0.2 ,0,. 0.6,.0.1,,

t/ (normalzeoJ

Fig.6 Profiles of l*, P* and 12, P2 provided 5% noise.

6. Conclusions

difference between exact solution 1* and converged
result 1, was within the amount of artificial errors. But
the difference between P* and P2 often becomes muchlarger. Therefore, to obtain confident function P, it isnecessary to use measured data which relate to pressure
at some points in plasma.

From analysis using Model 1, it is checked that if
the forms of surface functions are equal to those ofreference functions, we can obtain original surfacefunctions. As the result of analysis using Model 2, it isconcluded that if we select model functions which canapproximate /x and P* sufficiently, we can obtain thesurface functions. And it was found that in order to set
190Iyamaguchi T. et aI., Interpolation of Magnetic Surface Functions for an Axi-Symmetric Plasma

Referencethe suitable model, it is necessary to analyze using manytll M. Brusati et al., Compnter. Physics. Reports 1,forms of model functions.
From analysis of measured data with error, it isfound that function 1 is obtained accurately and P is345 (1984).

t2l M. Maeyama and E. Hotta, T. IEE Japan 115-A, 6(lee5).

easily influenced by measured error.
In the future, we plan to analyze magnetic surfacefunctions of experimentally obtained data using many

t3l M. Maeyama et al., J. Phys. Soc. Jpn.62' 1829
(r993).

[4] T. Nakagawa and Y. Koyanagi, Experimental datatypes of function models.

analysis by least-square method, University ofTokyo Press, (1995).

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