392 Brazilian Journal of Physics, vol. 27, no. 3, september, Theoretical Methods in the Design of the Poloidal

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1 39 Brazilian Journal of Physics, vol. 7, no. 3, september, 1997 Theoretical Methos in the Design of the Poloial Fiel Coils for the ETE Spherical Tokamak Gerson Otto Luwig Laboratorio Associao e Plasma Instituto Nacional e Pesquisas Espaciais S~ao Jose oscampos, SP, Brazil Receive September 11, 1996 This paper escribes the theoretical moels an the metho use in the esign of the poloial el coils system for the ETE (Experimento Tokamak Esferico) small-aspect-ratio tokamak. The metho is illustrate with the equilibrium congurations obtaine for ETE. I. Introuction require equilibria. As it is usually one in the esign of the poloial el coils for tokamaks, the plasma cross section shape is the given input an the coils currents an positions are the esire output. The solution of this problem requires a mixe approach, combining synthesis an analysis of iverse magnetostatic el sources. In the esign of the ETE tokamak, presently uner construction in our laboratory, a minimal set of coils was aopte to attain the small-aspect-ratio plasma equilibrium con- guration. It consists of: (1) the plasma magnetizing coils system, forme by the ohmic heating solenoi an two pairs of compensation coils () a pair of equilibrium el coils an (3) a pair of elongation coils. Fig. 1 illustrates the equatorially symmetric poloial el coils system for ETE. Since the numerical solution of the Gra-Schluter- Shafranov equation, that escribes the plasma equilibrium, is both computer time consuming an brings iculties in the treatment of small-aspect-ratio congurations, we use an extension of the semi-analytic irect variational metho [1] to solve the equilibrium equation. This metho is appropriate for use in personal computers an was implemente with the Mathematica package []. In the following sections we will rstly escribe the technique use in the optimization of the magnetizing coils system, which oes not require a solution of the plasma equilibrium equation, an then the integrate esign of all the poloial el coils for the Figure 1. Illustration of the poloial el coils system for the ETE tokamak. II. Magnetizing coils system The purpose of the magnetizing coils system is to prouce the poloial magnetic ux which is necessary to establish the toroial plasma current by transformer action. During the raise of the plasma current the plasma temperature increases as a result of ohmic heating. In ETE a long central ohmic heating (OH) solenoi prouces most of the require ux. However, for successful initial ionization of backgroun neutrals an in orer to avoi interference with the plasma position an shape uring the ischarge, the resiual magnetic el prouce by the OH solenoi must be reuce to a minimum in the region where the plasma is forme an

2 G. O. Luwig 393 sustaine. The creation of a region of suciently small magnetic el near the plasma center is accomplishe by means of the two pairs of compensation coils which constitute, with the OH solenoi, the magnetizing coils system. Since the compensation coils in ETE are in series with the OH solenoi (passive compensation), the reuction of the error el can be attaine only by ajusting the coils positions an the integer number of winings per coil. Employing a multipole moment expansion for the magnetizing ux on the geometrical center of the plasma cross section, we can calculate irectly the free parameters of the compensation coils that lea to cancelation of the moments to a prescribe orer an, therefore, to a reuce error el near the center. Constraints on the problem are impose by accessibility for iagnostics plus space an engineering limitations, involving the size of the coils an available power supplies. The multipole expansion for the poloial ux M of the magnetizing coils system on the geometrical center R 0 (a) of the plasma cross-section is given by (R an Z are the cylinrical coorinates) M (R Z) = 0 + M 0 + M 1 + M + ::: + R ;R 0 (a) R M 0 (a) 0 (R ;R ; 0 (a)) ;4R Z R 4 M 0 (a) 1 (R ;R + 0 (a)) 3 ;1R (R ;R 0 (a))z +8R Z 4 R 6 M 0 (a) + ::: where 0 is the ux ue to an ieal (innitely long) ohmic solenoi. The ux ue to the nite OH solenoi was obtaine by the superposition of the ux prouce by an innite equivalent current sheet minus the uxes ue to two semi-innite current sheets that represent the eect of the solenoi ens. The inner pair of compensation coils was also moele by two equivalent current sheets while the outer pair was moele by circular current loops. In this way the coecients M 0, M 1, M,... were calculate in terms of algebraic an elementary transcenental functions of the coils geometrical parameters (with strengths proportional to the current per turn in the system). Taking into account all the esign constraints, it was foun that a satisfactory compensation coul be obtaine by cancelling M 0, M 1 an M, respectively the ipole, quarupole an hexapole moments in the multipole expansion (this involves optimizing only three of the free geometrical parameters). Actually, higher orer compensation woul require an outer pair of coils locate too far from the toroial el coils an an unattainable precision in the coils positions. Fig. shows the compensate ux contours on the poloial plane an Fig. 3 shows the corresponing vertical error el on the equatorial plane, near the conition of maximumux swing operation of the magnetizing system in ETE. Fig. shows also the plasma bounary, vacuum vessel an toroial el coils outlines. Figure. Magnetizing ux contours on the poloial plane. The heavy contour inicates a poloial ux of 0.5Wb (7.8MA-turns in the OH solenoi for ouble-swing operation) with a % ux increment between contours. The plasma bounary, vacuum vessel an toroial el coils outlines are also isplaye. Figure 3. Vertical error el on the equatorial plane, near the conition of maximum ux swing operation of the magnetizing system in ETE.

3 394 Brazilian Journal of Physics, vol. 7, no. 3, september, 1997 III. Equilibrium el an elongation coils The equilibrium el coils provie the raially inwar force that balances the outwar force prouce by the interaction of the plasma current with its self- el. The ynamics of the plasma requires that the current in the equilibrium coils opposes an varies with the plasma current. Similarly, the elongation coils prouce a preominantly vertical force that provies some control of the plasma cross section shape. To stretch the plasma cross section, the currents in these coils must run parallel to the plasma current. In orer to optimize the esign of the complete set of poloial el coils, the plasma equilibrium has to be solve both for a given bounary shape an for speci- e parameters such as the plasma current, the external toroial inuction an the peak pressure at the plasma center. Then, the coils currents can be ajuste to t the vacuum poloial ux function consistently with the assume plasma bounary an positions of the coils. These positions can also be ajuste to some extent, but are essentially etermine by accessibility an engineering constraints. The magnetic ux contribution of all poloial coils has to be taken into account in these calculations, since there is no clear separation between the coils contributions to equilibrium. An approximate solution to the plasma xe bounary equilibrium problem can be eectively obtaine using variational techniques an a spectral representation of the ux surfaces [3][4]. Furthermore, the problem can be greatly simplie by the introuction of trial functions for the spectral amplitues, allowing the use of irect variational methos [1]. The starting point is given by the variational principle which states that the internal energy of the plasma U[ P ] = RRR V (a) = R a 0 " K() P B P! + B T ; B T 0 + p 0 0 ; L() I + p()v # 3 r is stationary uner virtual isplacements of the topological raius, = + for xe bounary conitions: (0) = (a) =0: In this expression P () anv () are, respectively, the poloial magnetic ux an the plasma volume enclose by a magnetic surface enote by, a is the minor raius of the plasma, p() is the plasma pressure pro- le, an B P, B T are the poloial an toroial components, respectively, of the magnetic inuction (B T 0 is the external el contribution) I() is the total poloial current whichows through a isk centere on the symmetry axis (the poloial plasma current is I P () = I(0) ; I()) L() is the inuctance of the toroial solenoi which coincies with a magnetic surface an K() istheinverse kernel to calculate the internal inuctance of the plasma loop [5][1]. The Euler equation for the functional U[ P ] leas to the equilibrium equation (ux-surface average Gra-Schluter- Shafranov equation) [5] P K() P = ; L I()I ; V p while the integral forms of Ampere's law give the relations between the toroial plasma current prole I T () an P () I T () =K() P an between the toroial magnetic ux T () ani() T = I()L : We next represent the neste magnetic surfaces by the truncate Fourier expansions for the inverse mapping ( )! (R Z) [6] T () R( ) = R 0 ()+cos ; sin Z( ) = E() 1 ; T () cos sin where is the poloial angle coorinate. The Fourier coecient R 0 () correspons to the geometric centers of the ux surfaces, E() to the elongation an T ()tothe triangularity. It can be shown that the geometric coecients V (), L() an K() of the equilibrium equation can be calculate analytically for an arbitrary number of terms in the spectral representation for R( ) an Z( ), an for an arbitrary epenence of the spectral amplitues on, eectively reucing the xe bounary equilibrium problem to a one imensional variational problem.

4 G. O. Luwig 395 Following the approach for irect variational problems (the Ritz proceure), we introuce trial functions for the Shafranov shift, elongation an triangularity proles. The trial function for the geometric centers of the ux surfaces has the simple parabolic form R 0 () = R m ; [R m ; R 0 (a)] (=a) an the triangularity coecient has a linear epenence on T () = T (a)(=a) : The elongation coecient is approximate by the binomial form " E() E m E(a) (1+) = 1 ; (E(a)=E m ) 1 ; E m ; (1 ; E(a)=E m)(e(a)=e m ) i h1 ; (1 ; E(a)=E m )(=a) which reprouces many ierent proles for various values of. It was foun that the results for small an large aspect ratio tokamaks are best reprouce by values of 50 that o not allow large variations of the elongation. These approximations improve previous results [1] an lea to a problem with two variational parameters, namely, the position R 0 (0) = R m of the magnetic axis an the value E(0) = E m of the elongation at the axis. In this way, the variational proceure consists in the etermination of a stationary point for the plasma internal energy U as a function of the parameters R m an E m. This semi-analytic approach allows simple computations of all the ux surface quantities, such as the safety factor an the macroscopic plasma quantities relate to the specie pressure an current ensity proles. In particular, the plasma equilibrium parameters of ETE liste in Table 1 were calculate for the proles p() = p(0) I T () = I T (a) 3 5 h 1 ; (=a) i p 1+ a 1 ; 1 I I I a with p = an I =1=. Work in progress inicates that more appropriate forms for the elongation coecient can be attaine, an better ttings obtaine by the introuction of quarangularity corrections in the ux surfaces. These corrections must lea to consistent expansions of the spectral amplitues near the magnetic axis when the elongation is varie. The irect variational proceure gives an approximate global solution of the plasma xe bounary equilibrium problem. A further avantage in the present problem is that the equivalent surface current ensity on the plasma bounary has a simple analytic form, suggesting a irect application of the vector analogue of Green's theorem [7] to calculate the external el for equilibrium (this is equivalent to the virtual casing principle [5]). The surface current ensity can be calculate from the solution of the internal problem accoring to the formula (bn enotes the normal irection to the plasma surface) ;! K = bn ;! B (;) = 0 which leas to the expression for the toroial component K T = jrj P 0 R = 1 h IT () p 0 g K() where h is the metric h an p g is the Jacobian of the transformation ( )! (R g Accoring to the Green's theorem, the internal poloial ux in the vacuum region (prouce by the plasma current) is given by theintegral over the plasma surface current ensity I int ( ;! r )=; 0 K T ( ;! r 0 )G( ;! r ;! r 0 )`( 0 ) an the external ux (prouce by the poloial el coils) is given by the sum over the coils currents ext ( ;! X r )= M ; 0 I k G( ;! r ;! r k ) where G is the Green's function of the Gra-Schluter- Shafranov equation ( is the toroial angle coorinate) * + G( ;! r ;! r 0 RR 0 )=; ;! r ; ;! 0 r Finally, the total poloial ux at the plasma ege is given by k 0 : P (a) = int ( ;! r (a)) + ext ( ;! r (a))

5 396 Brazilian Journal of Physics, vol. 7, no. 3, september, 1997 where P (a) isknown from the solution of the xe bounary equilibrium problem. This explicit expression for the poloial ux function allows the calculation of M an I k by means of a least squares approximation an without any iterative proceure, simplifying the etermination of the external currents istribution necessary to sustain the given plasma shape. In the present paper we aopte the usual representation of the Green's function in terms of elliptic integrals [8]. Alternatively, an expansion in terms of toroial multipoles [9] can be utilize which, couple with the spectral representation for the ux surfaces, leas to an analytic approximation of the ieal external el for equilibrium. This latter approach can be use with avantage in a free bounary formulation of the plasma equilibrium problem for magnetic reconstruction purposes. the vertical equilibrium el. In this way, the best t of the currents in the coils gives, besies the equilibrium el, the Ampere-turns in the magnetizing system necessary to rive the plasma current (neglecting the resistive losses). Fig. 4 shows the vacuum poloial ux contours generate by the coils an Fig. 5 shows the equilibrium ux contours for the initial phase of operation of ETE [10]. Figure 5. Equilibrium ux for a -0kA plasma current in ETE. The separatrix lies between 1.1 an 1. times the poloial ux at the plasma ege. Figure 4. Vacuum ux prouce by the poloial el coils in ETE. The metho briey escribe in the previous paragraphs was applie to the small-aspect-ratio conguration of ETE. The equilibrium an elongation coils were moele by circular current loops an the magnetizing coils system was moele by an ieal transformer, since the error el in the plasma region (accoring with the calculation in Section II) is much smaller than The minimal set of coils in ETE ts the constant ux requirement at the plasma bounary within 1.5%. From Fig. 5 we verify that this error, which is larger at the outer plasma ege, can be reuce by the introuction of quarangularity corrections in the plasma bounary shape. This improvement will be implemente in a future free bounary version of the present moel. IV. Results Table 1 lists the main plasma parameters etermine by the irect variational solution for the ETE tokamak equilibrium in the initial (ohmic) an extene (auxiliary heate) phases of operation. Table lists the geometrical parameters an currents of the poloial el coils consistent with the plasma equilibria an optimize using the metho escribe in this paper.

6 G. O. Luwig 397 Plasma parameter Initial operation Extene operation Major raius R 0 (a) [m] 0:30 0:30 Minor raius a [m] 0:0 0:0 Elongation (a) 1:6 1:8 Triangularity (a) 0:3 0:3 External toroial inuction B 0 [T] 0:4 < 0:8 Toroial plasma current I T (a) [ka] Pressure on the magnetic axis p(0) [kpa] 8 80 Internal inuctance `i 0:57 0:53 Current iamagnetism I 0:47 0:16 Current beta I 0:0 0:55 Plasma beta 0:036 0:09 Toroial beta T 0 0:047 0:118 Safety factor on the magnetic axis q(0) 0:98 0:98 Safety factor at the plasma ege q(a) 5:55 7:04 Table 1: Parameters of the ETE tokamak equilibrium congurations. The extene operation lists maximum parameters that can be attaine with auxiliary heating an near the Troyon an Greenwal limits. Coil enomination R [m] Z [m] R [m] Z [m] NR NZ I N [ka-turns] Ohmic Heating Solenoi 0: :01 1: (6760) Internal Compensation Coils 0:105 0:707 0:01 0: (50) External Compensation Coils 0:650 0:871 0:010 0: (5) Equilibrium Coils 0:700 0:390 0:040 0: (0) Elongation Coils 0:00 0:830 0:040 0: ;4 (;99) Table : Geometrical parameters an currents of the poloial el coils in ETE. The values of the currents in parenthesis correspon to preliminary results for the extene operation. References [1] G. O. Luwig, Plasma Phys. Control. Fusion 37(6) 633 (1995). [] Mathematica, version. (Champaign, IL: Wolfram Research, Inc.) 199. [3] L.L. Lao, S.P. Hirshman an R.M. Wielan, Phys. Fluis 4(8) 1431 (1981). [4] K.M. Ling an S.C. Jarin, J. Computational Phys. 58, 300 (1985). [5] L.E. Zhakharov an V.D. Zhafranov Reviews of Plasma Physics e M A Leontovich vol 11 (New York: Consultants Bureau) (1986). [6] W. Weitzner, Appenix of Ref. [3] [7] J.A. Stratton, Electromagnetic Theory (New York: McGraw-Hill) 50 (1941). [8] J.D. Jackson, Classical Electroynamics (New York: John Wiley) 141 (196). [9] F. Allaio an F. Crisanti, Nucl. Fusion 6(9) 1143 (1986). [10] G.O. Luwig, L.F.W. Barbosa, E. Del Bosco, J.G. Ferreira, A. Montes, C.S. Shibata, M. Uea Proc. 3 0 Enc. Brasil. Fs. Plasmas (S~ao Jose os Campos: INPE) 18 (1995).