Granetz, R.S.

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Magnetic Field Diffusion Throughthe Alcator A LinerR.S. GranetzPFC/RR-79-18August 21, 1979Francis Bitter National Magnet LaboratoryaridPlasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139IntroductionIn order to measure poloidal magnetic field fluctuations, Alcator Ais fitted with sets of discrete pickup coils located just outside thestainless steel vacuum vessel. Although the bellows sections of theliner are only 0.5 mm thick, the pickup coils are positioned on asmooth steel sleeve, 1/16 " thick, located between two bellows sections.For future experiments involving feedback stabilization of magneticisland instabilities, it is necessary to know the amplitude and phaseof these magnetic perturbations inside the stainless steel linerrelative to that measured by the coils outside the liner.TheoreticalThe diffusion equation for the magnetic field is obtained fromtwo of Maxwell's equations- (1(2)We make the following assumptions:V(k) (3)(4)7> (Ohm's law) (5)and . in stainless steel. (6)Equations (1) and (2) become:(8)-2-Substituting (8) into (7) gives:-\7 C\7.-0____D (10)When dealing with audio frequency magnetic fields diffusing into thesurface of a good conductor, H = 0.(H /H ~ / ). In plane geometry,the vector Helmholtz equation separates completely with the result:whereg 7--~ J skin depth (12)and x = distance into stainless steel.Since in the case of interest, the stainless steel liner isquite thin compared to its cross section radius, we expect this solutionto be valid. However, a bit more precise result can be obtained bysolving the Helmholtz equation is cylindrical geometry. Unfortunately,it does not separate completely in these coordinates 2:V 14,= 0Vli+H ( i )L =+A~ c'Since th& pickup coils only measure 0  and Hr H_ a 0 anyway, we areonly interested in the second equation, with the Hr term dropped.+. !M t (14)subject to the periodicity conditions:+ 4(15)-3-where m and n are the standard MHD mode numbers characterizing themagnetic islands 3,4. Another boundary condition is:lim4 -Z) 0 (16)Equation (14) can be solved by assuming that:H ((17)and using the method of aeparation of variables. We get (see appendix):H ( 9W = N )<(18)where H e(r) satisfies the equation:+-4) 0 (19)and *4 (20)p2= 1 + M2  (21)This is the simple Bessel's equation, and the complete solution isnormally written5:7!) J q (22)where C1 , C2 , and J are complex because k is complex. Unfortunately,J (Z) blows up for IZI , which corresponds to w + o. This doesn'tmean that solution (22) is theoretically incorrect, because we canadjust C1 and C2 to satisfy condition (16). But the sum becomes com-putationally impossible to handle.Instead we have to use another pair of independent solutions6:(23)As w + , I p blows up, while K vanishes. Thus C 0 and the entiresolution inside the stainless steel is:(24)-4-Since we are only interested in low m and n numbers, we canfurther simplify equation(24). In the audio frequencies we areconcerned with, 0.3 < 6 < 0.6 cm. Thus> >(R=54 cm)Therefore kc i (exact for n = 0) and the solution isTvirtually independent of toroidal mode number n (for low n). The. 7function Kp(i 'V. t) can be written in terms of ker and kei functions(25)K~Cii'~f o' ket-~ ('kf)- kop~Ij)(5This form can also be written in terms of special magnitude and phasefunctions: kerp (d ) - i keip (2j)=Np ( 4f3 )e ( (26)Since the liner has 12.60 <r < 12.75 cm, the argument rZ ~45,and the asymptotic expansions of N and 0 can be used:(27)Notice that the asymptotic expansions are independent of the poloidalmode number, m (for low m).The final result is:R (28)Notice that the only difference between this result and the flatplane is the factor . Since the liner is thin, the range of r issmall and the extra factor hardly changes anyway.Putting in the numbers gives the following attenuation and phaseshifts for He through the stainless steel liner (for low m and n mode):-5-Attenuation0.7750.7000.6470.605Phase shift14.2020.1024.6028.50ExperimentThe poloidal magnetic diffusion was actually measured using a sparesection (60*) of the Alcator A bellows. Two identical magnetic pick-upcoils similar to those on the machine were attached at a single locationon the liner, one on the inside surface and one on the outside surface.A power amplifier was used to drive several amps (A.C.) through wiresrunning inside the liner, parallel to its axis. (Thus the toroidalmode number, n, was zero.) By using the proper number of wires, fieldswith poloidal mode numbers, m, of 0,1, and 2 were generated. Theresults are tabulated here:v (kHz) Attenuation Phase shift5 0.74 + 0.05 15 + 5010 0.63 16015 0.54 17020 0.50 2005 0.58 + 0.03 340 + 3*10 0.40 47015 0.29 50020 0.24 520v (kHz)5101520m=0m=1-0-v (kHz) Attenuation Phase shift5 0.59 + 0.03 320+ 3010 0.42 49015 0.32 57020 0.26 630m 2ConclusionExcept for the lower frequency m=O measurements, the theoreticalresults do not match the measured values well at all. It has been8suggested that errors might be due to, the fact that the experimentalbellows was of finite length, although if it wasn't (such as in areal tokamak, where the bellows. connects with itself), the measuredm=0 values should more closely match the m=1 and m=2 cases. Thiswould.mean the theory would not match any measured values at all.Possibly the discrepancy is due to the ripples in the bellows.There are several other important differences between the experi-ment done here and the actual Alcator tokamak- First of all, thesemeasurements were done at room temperature, whereas Alcato- operatesnear liquid nitrogen temperature. The conductivity of its stainlesssteel liner doubles under these cryogenic conditions, and therefore themagnetic field attenuation should be greater.Secondly, the Alcator vacuum vessel is closely surrounded by athick copper shell, which should compress the poloidal field againstthe outside of the bellows, thus effectively, the magnetic field will notbe attenuated as much. This effect was measured at room temperature bypostioning a thick piece of copper just over the outside pickup coil. Anincrease in the external B-field of 2411 % was seen for m=0 (10-20 khz) and-7-of 60 ± 5 % for m=2 (20 khz). No measurable increase was seen on theinside coil, and the phase was independent of the copper. At cryogenictemperatures, the conductivity of copper goes up by a factor of 7, soif anything, the compression outside the bellows will be greater inAlcator.REFERENCES1) Jackson, J.D., Classical Electrodynamics, Wiley, N.Y. 1962.2) Morse, and Feshbach, Methods of Theoretical Physics, Part I,McGraw-Hill, Kogakusha, Japan, 1953.3) Mirnov, S.V. and Semenov, I.V., in Plasma Physics and ControlledNuclear Fusion Research (IAEA), Vienna, 1971, Vol. II.4) Hosea, J.C.,Bobeldtjk, C., and Grove, D.J., in Plasma Physics andControlled Nuclear Fusion Research (IAEA), Vienna, 1971, Vol.II.5) Hildebrand, F.B., Advanced Calculus for Engineering, Prentice-Hall,Englewood Cliffs, N.J., 1962.6) Molvig, K., Private Communications,6/79, M.I.T. Plamsa Fusion Center,Cambridge, MA. 02139.7) Abramowitz and Stegun, Handbook of Mathematics Functions, Dover, N.Y. ,1970.8) Parker, R.R., Private Communication, 7/79, M.I.T., Cambridge, MA. 02139.APPENDIXSubstitute equation (17) into (14) and divide by HOZ to get:Rearrange to isolate Z(z):Th"Thus Z(;;/, I~A___ =randin order to satisfy the periodicity condition on z (equation 15),2Substitute this back into (A2), multiply by r , and isolate 0(0):/~1 U / 2 ---//z (h'LjThusand in order to satisfy the periodicity condition on 6 (equation 15),Substitute this back into (A5), multiply by H, and rearrange to get:A- - (A8)L-C", C rH r1 9// (Al)(A2)(A3)(A4)(A5)(A6)(A7)( 8)(14- Pn IH= D4- t

Magnetic field diffusion through the Alcator A liner

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Magnetic field diffusion through the Alcator A liner

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