22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 13: PF Design II The Coil Solver

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1 .615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 13: PF Design II The Coil Solver Introduction 1. Let us assume that we have successfully solved the Grad Shafranov equation for a fixed boundary lasma equilibrium.. The second imortant tas is to calculate the location of the PF coils and the current flowing in each coil to generate the desired equilibrium. 3. This is an overdetermined roblem. A finite number of coils cannot in general match the normal and tangential magnetic field on an infinite number of oints on the lasma surface. 4. Thus, the roblem reduces to one of obtaining a best fit with the existing coils. In the fitting rocedure, we obtain an accurate estimate of the error, thereby roviding insight as to the lausibility of the original equilibrium. 5. We shall see that in designing the PF system, it is far simler comutationally to vary the coil currents (linear algebra as oosed to changing their location (nonlinear roblem. 6. There are two common classes of roblems involved in the design of the PF system a. Fixed, flat to design oints b. Dynamic, time deendent oerational analysis 7. For the fixed design oints there are two issues worth noting a. During flat to oeration, the eddy currents in the vacuum chamber can be assumed to have decayed to zero. This is a great simlification. b. The OH transformer is slightly subtle. It is not clear what value of the flux swing corresonds to the oint of interest in the flat to oeration. Recall that even though I const, I Ω is varying in time and its value will have some influence on the remaining EF currents. For a tightly couled transformer this effect is not too large, as very little of the transformer flux returns through the lasma. 8. The dynamic mode is far more comlicated to calculate as a model must be introduced to account for the vacuum chamber eddy currents. Also, circuit equations must be introduced and we must be careful to distinguish between active and assive coils. 9. We now describe how the flat to roblem can be solved by fast, efficient, and robust comutational techniques.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 1 of 8

2 Flat To Oeration 1. From the fixed boundary Grad Shafranov solver assume we now a. The shae of the surface: R R(, Z Z( b. The oloidal flux in the lasma: ψ b c. The oloidal field around the surface: ψ n. Assume we now the location of the EF coils and the OH transformer. Model each coil as a grou of filaments. 3. There is no unique way to obtain a best fit for the EF currents. The method described below is fast and robust and requires no iteration. 4. There are two stes to the rocedure. a. Assign an arbitrary current I to each PF coil. Treat the lasma as a rigid erfect conductor. Calculate the oloidal field on the surface of the lasma as a function of the I : B is linearly related to the I. b. Determine the I by minimizing the difference between B and the oloidal field calculated from the fixed boundary equilibria B. This is the best fit rocedure. The Vacuum Field 1. The magnetic field surrounding the lasma can be written as B B + B + B + Φ c Ω. B Ω is the field due to ohmic transformer, whose current is assumed to be secified: I Ω is a nown quantity.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page of 8

3 B Ω I ψ e Ψ e K R R Ω Ω K φ IΩ Ω 1 φ 1 K Ω Ψ Ω ψ K Ω 1 ( R, Z ψ ( RR 1 ( K ( E ( 4RR ( R + R + ( Z Z The filaments modeling the transformer are located at R, Z. Alternatively, and more common, is to secify the volt seconds lining the magnetic axis. This is a slightly more comlicated constraint than fixing I Ω. 3. B is the field due to a filament carrying a current I located at the geometric center of the lasma. The necessity for this term will be aarent shortly. I is assumed nown. B I Ψ e R φ ( RZ, Ψ ( RR 1 ( K ( E ( 4RR ( R + R + ( Z Z The geometric center of the lasma is located at R, Z. 4. B c is the field due to the currents flowing in the EF system. There are J such coils, each carrying a current I and modeled by a grou of filaments. B c J K e J I ψ φ Ψ e I K 1 R R 1 1 φ K 1 K 1 Ψ ψ.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 3 of 8

4 ( RZ, ψ 1 ( RR ( K ( E ( 4RR ( R + R + ( Z Z The th EF coil is modeled by K filaments located at R, Z and carrying a current I /K. The I s are the basic unnowns in the roblem. 5. Φis the scalar magnetic otential reresenting the remainder of the field in the vacuum and satisfies the following equation Φ 6. Φ has two imortant roerties a. Φ is regular everywhere in the vacuum region, since the singular coil currents have been exlicitly subtracted from the comlete solution. b. Φ is single valued, since the lasma current has been subtracted from the comlete solution. This is why we needed B. 7. Because of this Φ satisfies the following boundary conditions Φ( regularity ( Ω n B n B B B n Φ c where Φ and n Φ are single value eriodic functions of surface. on the lasma 8. Ultimately we need to calculated B on the lasma surface in terms of the I. To do this we will need to evaluate Φ ( on the lasma surface. We already now n Φ from the boundary condition above. 9. The function Φ ( on the lasma surface is found from Green s theorem. Alication of Green s Theorem 1. Recall Green s theorem ( a. Φ G G Φ Φ G G Φ Φ G dr ( b. Φ Gdr Φn G G n Φ ds S.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 4 of 8

5 c. Assume G δ ( r-r. Also there is only one S corresonding to the lasma surface since Φ( by regularity. d. Thus σφ ( G n Φ Φn G ds n n 1 for r in the region 1 σ for r on the surface for r out of the region e. Choose r to lie on the lasma surface ( n ( ( n Φ Φ Φ S G S S G ds f. Once we now, ( G this is an integral equation for ( S n Φ S from the boundary conditions.. Solve the integral equation ( 1 φ + φ ds R dl d R R Z d d 1 1 G 4 π r -r Φ since we now 3. Since the roblem is axisymmetric, we can integrate the Green s function over the toroidal angle φ G (, Gdφ K ( ( RR 1 π (, 4 RR ( R + R + ( Z Z.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 5 of 8

6 In these exressions R R(, Z(, R R(, Z Z ( the lasma surface. 4. The equation for Φ becomes ( Φ ( G n Φ Φn G J d ( ( 1 J J R R + Z 5. This equation can be solved by Fourier analysis corresonding to Φ M M im m a e M J im Jn Φ c + m DmI e M 1 6. The coefficients c m and D m are nown from the boundary condition on n Φ, which can be written as n Φ Ψ + Ω Ψ Ω + Ψ J J I I I 1 7. Thus im im cm ( Ψ + ΩΨΩ I I e d D m im Ψ e -im d 8. Substituting into the integral equation yields a linear algebra roblem for the unnown coefficients a m I+ A a C c + D i ( ( 1 im -im A J n G e dd mm π ( Cmm 1 π Ge im -im dd ( 1 3 i I, I, I,... I unnown currents.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 6 of 8

7 9. Solve for a a d+ E i d I+ A C c ( 1 E I+ A C D ( 1 1. Knowing a it is straightforward to evaluate B on the surface, which requires a nowledge of dφ ( ( f B ( ( b ( c + b ( d i ( f n Ψ n ΨΩ R b ( I + IΩ + imd R R J b ( ( n Ψ R + imeme R J c im m 11. The quantities ( f b and the error. Determination of i m im me ( c b are required in order to calculate and minimize 1. The vector i is determined by minimizing the difference between B ( calculated above and B ( as calculated from the fixed boundary Grad Shafranov code.. Define the error as follows ( B B ds B ds 3. A straightforward calculations yields i M i-n i+ M ( c ( c b b Jd B Jd.615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 7 of 8

8 n ( ( c ( f b B b Jd B Jd π ( f ( B b Jd B Jd 4. The best fit is obtained by minimizing with resect to the individual I. This leads to a set of linear algebraic equation for the I a. Set I b. The result is M in c. This is a standard numerical roblem. 5. Once i is found, then can be calculated. 6. For sufficiently small, the equivalent free boundary roblem would give a comarable shae as the fixed boundary. 7. For sufficiently large, no choice of EF currents can reroduce the desired shae. 8. The rocedure is robust, giving an answer even for very comlicated (e.g. clover leaf shae lasma with one EF coil. Obviously the error would be large. However, there is no need to hunt around aimlessly with free boundary codes, not nowing whether a satisfactory answer exists or not. 9. Finally, note that the quantities M and n deend only uon the shae of the lasma and the location of the PF coils. Once these are set, the lasma arameters can be varied indeendently without recomuting M and n..615, MHD Theory of Fusion Systems Lecture 13 Prof. Freidberg Page 8 of 8