The possibility of measuring magnetic fluctuations in a fusion plasma is considered by examining the O→X  mode conversion. Under certain conditions and with good angular resolution, this mode conversion can be attributed to the presence of magnetic fluctuations even though the level of these fluctuations is much lower than that of density fluctuations. Some nonideal effects such as mode polarization mismatch at the plasma edge are also discussed

Bretz, N.

Vahala, G.

Vahala, L. L.

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Old Dominion University

Old Dominion UniversityODU Digital CommonsElectrical & Computer Engineering FacultyPublications Electrical & Computer Engineering1990Measurement of Magnetic Fluctuations by O-XMode ConversionL. L. VahalaOld Dominion University, lvahala@odu.eduG. VahalaN. BretzFollow this and additional works at: https://digitalcommons.odu.edu/ece_fac_pubsPart of the Electrical and Computer Engineering Commons, and the Plasma and Beam PhysicsCommonsThis Article is brought to you for free and open access by the Electrical & Computer Engineering at ODU Digital Commons. It has been accepted forinclusion in Electrical & Computer Engineering Faculty Publications by an authorized administrator of ODU Digital Commons. For more information,please contact digitalcommons@odu.edu.Repository CitationVahala, L. L.; Vahala, G.; and Bretz, N., "Measurement of Magnetic Fluctuations by O-X Mode Conversion" (1990). Electrical &Computer Engineering Faculty Publications. 41.https://digitalcommons.odu.edu/ece_fac_pubs/41Original Publication CitationVahala, L., Vahala, G., & Bretz, N. (1990). Measurement of magnetic fluctuations by O→X mode conversion. Review of ScientificInstruments, 61, 3022-3024. doi: 10.1063/1.1141714Measurement of magnetic fluctuations by 0.-. X mode conversion linda Vahala Department of Electrical and Computer Engineering, Old Dominioll University. NOifolk, Virginia 23529 George Vahala Department of Physics. College of William & Mary, Williamsburg, Virginia 23185 N. Bretz Plasma Physics Laboratmy, Princeton University, Princeton, New Jersey 08544 (Presented 8 May 1990) The possibility of measuring magnetic fluctuations in a fusion plasma is considered by examining the o-x mode cOllversion. Under certain conditions and with good angular resolution, this mode conversion can be attributed to the presence of magnetic fluctuations even though the level of these fluctuations is much lower than that of density fluctuations. Some nonideal effects such as mode polarization mismatch at the plasma edge are also discussed. . I. INTRODUCTION It is of considerable importance to develop diagnostics that can detect magnetic fluctuations in a fusion plasma, especially because of their expected interconnection with transport. I Such a diagnostic must overcome a universal fea-ture present in tokamaks: the level of magnetic fluctuations, 8B lBo, is much lower than that of the electron density fluc-tuations, DniNo. Typically, 8nl No is determined by electron drift wave fluetuations: 2 8nINo;:::;3pJL Il , wherep, is basi-cally the electron gyroradius and LII the density gradient length scale. On the other hand, magnetic fluctuations are driven by magnetohydrodynamic processes which have a low-frequency, long-wavelength coherent component and a high-frequency, short-wavelength incoherent part. Scatter-ing experiments typically measure large 111 (;:::; 100) fluctu-ations, where m is the poloidal mode number. These magnet-ic fluctuations have been measured in the edge region with Mirnov loops,3 and, in very small devices, with interior mag-netic probes.4 High-frequency probe measurements·1 give MJ / Bo;:::; 10-'\-10-- 5 in the interior of Microtor forfrequen-cies ,;;;30 kHz, while oB IBo;:::; 10 ~-1O (, at the edge of To-karnak Fusion Test Reactor (TFTR)5 in the frequency range lOO-IS0 kHz. Thus, for the interior of TFTR one " might expect a fluctuation ratio ( (bE 2 )1 B 2 ) I 0 ~ 10 4_10 5. 2 2 ~ (on )INo ",;',>] (1) Here we explore the possibilities of electromagnetic scattering of an ordinary (0) --+ extraordinary (X) mode as a possible diagnosticS in a high-temperature plasma. The inci-dent 0 mode approaches a cutoff near the local electron plas-ma frequency while the X mode will propagate through this cutoff layer and he detected. In particular, for mode propa-gation perpendicular to the magnetic field B, the 0 mode is linearly polarized while the X mode is elliptically polarized. If the incident and scattered angles are chosen to be both perpendicular to the B field, then density fluctuations, being scalar in nature, cannot force this 0 .... X mode conversion. However, the tensorial magnetic fluctuations can facilitate in this mode conversion. These conclusions are strictly true only for exactly per-pendicular propagation to B. Because the density fluctu-ation levels are so much higher than those of the magnetic fluctuations, Eq. (1), the effects of finite beamwidth and finite angular resolution should also be considered. These effects are examined here by following a bundle of modes using a toroidal ray tracing code and examining the refrac-tion of this bundle. For an appropriate choice of scattering parameters, we find in Sec. II that refractive effects on the mode bundle are very small. This then allows us, in Sec. III, to use the scattering formalism of Sitenkd' for a locally ho-mogeneous plasma to derive the differential scattering cross section for an Oc>.X mode conversion. Finally in Sec. IV, we show that the generation of X modes due to incident polar-ization mismatch with B at the plasma edge and the subse-quent X ->X scattering by density fluctuations will not mask the O~X conversion due to magnetic fluctuations. It RAY TRACING To examine the refractive effects on a bundle of 0 modes incident nearly perpendicular to B, from either the low or high magnetic field side, we employ the TORCH ray tracing code. 7 The 0 modes (with incident frequency 0);) will be reflected near the cutoff position rc where the incident fre-quency becomes equal to the local electron plasma frequen-cy, (Ui = OJ],<, (re ), as shown in Fig. 1 (a). On the other hand, in the neighborhood of re , a corresponding bundle of X modes will propagate out of the plasma perpendicular to B and with little refraction, as shown in Fig. 1 (h). Moreover,5 the 0 modes and X modes that propagate perpendicular to B retain their identity in that the polarization of these modes remains invariant relative to the local magnetic field in the eikonal approximation. This is also consistent with the ob-servationS of tokamak emission at 2(oco in which the perpen-dicularly polarized mode retained its orientation relative to B in the emitting layer as well as to B at the plasma edge. One can thus conclude from Fig. 1 that under appropri-ate scattering orientations, the O-X mode conversion around rc can be examined by treating the system as locally homogeneous. III. SCATTERING THEORY Consider the Sitenko" formalism for scattering from fluctuations in a locally homogeneous magnetized plasma. 3022 Rev. Sci.lnstrum. 61 (10j, October 1990 0034-6748/9011 03022-03$02.00 ~c) 1990 American Institute of Physics 3022  Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  128.82.253.131 On: Mon, 15 Feb 2016 15:31:1180 40 ] 0 N -40 -80 POLOIDAL CROSS SECTION: incident O-mode [45 GHzj 100 200 300 400 (a) R (em) POLOIDAL CROSS SECTION: sca!lercd X-mode [45 GH7] 80 40 e- O ~ " -40 80 IOfJ 2\X) 300 400 (b) R «'m) FIG.!. (a) Result of ray tracing in the poloidal plane (R,Z) for a bundle of 45 GHz 0 modes (eight modes with a total angular width 4' about 17/2) incident on the high field side in TFTR from below. The 0 modes approach a cutoff at Y, and are reflected (with some refraction) out afthe plasma. (b) A bundle of 45 GHz X-mode ray trajectories in the pa\oidal plane emanat-ing from r,., perpendicular to B. Refractive cftccis in the poloidal plane arc very small, and near the cutoff r,. there is 110 refraction in the X modes since tlle refractive index here is unity. The incident plane electromagnetic wave satisfies the wave equation 1 a 2E inc VXVXEine +1£;'--,-=0, (.'- at-where £i is the plasma dielectric function. The scattered field E, is given by 1 alE, 41T aJ "i1X"i1XE\ + 1 lOs) -- (.'2 -at' c- (it-where J is the current due to the interaction of the incident wave ((ui,k j ) with plasma density and magnetic fluctuations 8n (' ({tJ,k) and DB (ru,k). From the conservation of momen-tum and energy, the scattered wave number and frequency satisfy k, = k j + k and 0)\ -= (U, + OJ. Thus [e:~~ (k'.~~',(3 _ OafJ ) + Ea/,«(us,k,)] E'./i(OJ"k,) ( OJi . {me (w,k) = E ille .!! (oJ"k i ) --[ Da'i .- Eap, (wj,k i )] -----w,. 1Vo ie Wi +---2-[8,,), -E"y(u)"k,)] n1e C wpe X E,_:;,/ [ D ;''(! - t:~fJ «(()ok )] bB ,/ «(i),k) ) , where E)'1;11 is the standard Levi-Citiva symbol, and No the background density (with summation over repeated Greek subscripts). Hence the radiation field at the observation po-sition f, is 3023 Rev. Sci. instrum., Vol. 61, No. 10, October 1990 where L' denotes the sum over all wave numbers k, that lie on the surface A = 0 such that the group velocity (in the direction V A) is parallel to the observer position rs. Cs is the normalized polarization of the scattered electric field and A is defined by __ [e2k; ( kw k',/3 .) ] A=det --.2- ---2- -b"/3 + Cnfj(o)"k,.) . (u,. \ k, K is the Gaussian curvature of the surface A = 0 at the points ks' From the average power at rs one can readily de-termine the differential scattering cross section d -'G[If:'12 (8n2)." .• +w~Ju.~ * (8B"8BO )",,k (7 -!o 2 --4- OaO(J 2 No OJ DC Bo 2{ucc O)s ( . + --7-- 1m ~a~ (u pe (2) where ~=e~a[Ea/3(Wi,ki) -baffle,.!;, aa=Eu/lye~~[£,:.-t3«(u."k;) -O;;f;J [E)'p(oJi,ki) -i5yp le i ,(" e i is the incident electric field polarization vector and (Uce is the electron gyrofrequency. G is a complicated geometric factor which only affects the overall magnitude of the cross section for a particular scattering process. It is considered in some detail by Hughes and Smith9 following the analysis of Simonich and Yeh. 10 The third term in Eq. (2), the cross correlation term, can be shown to have a negligible effect on the differential cross section dO' and so need not be consid-ered further. Our major concern is to be able to make the dominant contribution to cia come from magnetic fluctu-ations, the second term in Eq. (2). For perpendicular propagation to B (OJ = 1'1"/2 = 0" Fig. 2), one finds from Eq. (2) that an 0 -> X mode conver-sion can only arise from the presence of magnetic fluctu-ations in the plasma. However, one must incorporate the effects of finite beamwidths and finite angular resolution, especially since the level of magnetic fluctuations is general-ly much lower than that of the density fluctuations. In Fig. 3 we plot the relative contribntions of the second and first terms in da, Eq. (2), as a function of the relative intensities of magnetic/density fluctuations for various tokamaks for a scattering angle of <P::::; On. Typically, we assume a total an-gular spread of 1!l about <P = On and consider an 0 mode z Bo k I FIG. 2. Scattering geometry used in calculating the differential scattl,ring cross section dlT (here we choose 0, -_. 8, = rr/2). The optimal scattering angle is forward scattering. <I> ::::0" . High temperature plasma diagnostics 3023  Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  128.82.253.131 On: Mon, 15 Feb 2016 15:31:110 -+ X mode Perpendicular Scattering .c :;: eo .c >:l 20.0 "0 IB.O .8 CIT· low 105 16.0 c: i 14.0 12.0 c JET -low 45 0 10.0 u "" 8.0 CIT-low 150 ~'" 6.0 c err -high {2~] ., c " 4.0 150 GHz .~ i 2.0 CIT - low [2'1 1500Hz .. 0.0 >: 2.0e-5 4.00-5 6.0e-5 g.De-5 I.Oe·4 {.lB~ (sn2) 1No2 FIG. 3. Relative contribution of magnetic to density fluctuations to the dif-ferential scattering cross section da as a function of their relative intensities for an angular width of I" (unlabeled) or 2" (labeled) about n-/2. Various tokamaks are considered (TFTR:u.>, ~ 45 GHz, CIT: (U, ~, 105 and 150 GHz, JET:u.>, =- 45 and 60 GHz). The advantage of using the high-field side over the low-field side is somewhat illusionary. It is due to the larger gyro-frequency at r,., but to sustain the same level of (oB, }/HZ) one requires a correspondingly higher level of magnetic fluctuations. incident along the midplane on both the low-field and high-field sides. For typical drift wave fluctuation magnitudes2 and a TFTR plasma with an incident 0 mode of frequency OJ; = 45 GHz, an appropriate level of magnetic to density fluctuations is' (oB2)/B2 1 0 ;:::;6.0X 10-5. (8n2>/N~ If the mode is incident on the low-field side, then the magnet-ic fluctuation contribution to the differential cross section for the O-X conversion is about 14 times greater than that arising from the density fluctuation contribution, However, if the 0 mode is injected on the high-field side (Fig. 1), then the ratio of magnetic to density contribution to the differen-tial cross section rises to a factor of 36. IV. DISCUSSION An important nonideal effect that remains to he dis-cussed is the 0 ..... X conversion that can occur due to mode polarization mismatch with B near the plasma edge. II This X mode can then be scattered by density fluctuations into another X mode, possibly masking the a ..... x conversion due to magnetic fluctuations. However, from a full wave calcula-tion, Brambilla and Moresco'2 have shown that the power fraction converted to an X mode from the incident 0 mode due to polarization mismatch is ;:::; 10-5 for low-shear toka-3024 Rev. SCi.lnstrum" Vol. 61, No. 10, October 1990 mak plasmas. Hence, for small forward scattering angles «I>;:::; Oll , see Fig. 2) one finds from Eq. (2) that the cross section ratio for a-x scattering by magnetic fluctuations to that for 0...." X due to edge mismatch with B and subsequent X -X due to density fluctuations (in the same frequency range as the magnetic fluctuations and an (llce/lU; ;:::;3.96) drl';.x GbB {j).~e «oBi > /B~) 5 /j --- --2 2 - X 10 . dC7(;'.x_x Gbn OJ; (on >/N~ Moreover, the corresponding geometric factor for X-X scattering is G,," 'ZOJf/OJ~c ;:::;0.064, whileforO-Xscattering the geometric factor G,\B ~ 1, due to the cutoff layer. Thus, it appears that polarization mode mismatch at the edge will not mask the effect of magnetic fluctuation scattering even for magnetic to density fluctution ratios as low as 10- 6 • Oth-er nonideal effects, such as the horn limitations on the polar-ization of the emitted wave, will be considered in more detail elsewhere. Hence it appears that millimeter 0-X mode conversion can be used as a diagnostic for magnetic fluctuations. ACKNOWLEDGMENTS This work was supported by the Department of Energy at Old Dominion University grant No. DE-FG05-90ER54093, William & Mary grant no. DE-FG05-84ER53176, and Princeton grant no. DE-AC02-76-CHO-3073. 1M. Malacarne and P. Duperrex. Nne!. Fusion 27. 2113 (1987); M. Ohaybu et al .• Phys. Rev. Lett. 58,120 (1987); A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978). 2E. MazZlleato, Phys. Fluids 21, 1063 ( 1978); T. Crowley and E. Mazzu-cato. N uel. Fusion 25, 507 ( 19X5). 'R. L. Boivin et al., Princeton Plasma Physics Report, PPPL-2564, Nov. 1988. "S. 1. Zweben, C. R. Menyuk, and R. J. Taylor, Phys. Rev. Lett. 42, 1270 (1979). 'N. Bretz., in Proceedings of the 3rd IntI. Conf. on Laser-Aided Plasma Diagnostics, UCLA 1987. see also T. Lehner, in Turbulence and Anoma-lous Transport in Magnetized Plasmas, edited by D. Gresillon and M. A. Dubois (Editions de Physique, Orsay, 1987); T. Lehner, J. M. Rax. and X. L. ZOllo Emophys, Lett. 8, 759 (1989). "1\. G. Sitenko, Electromagnetic l''iuctuations in Plasma (Academic. New York,I967). 7G. R. Smith (private communication. 1988). HI. Hutchinson, Plasma Phys. 21,1043 (1979). "T. P. Hughes and S. R. P. Smith, 1. Plasma Phys. (1990). IOD. M. Simonich and K. C. Yeh, Radio Sci. 7, 291 (1972). lID. Boyd. GA ReportGA-AI8294, Nov. 1985; F. R. Hansen. J.P. Lynov, and 1'. Michelson, Plasma Phys. Contr. Fusion 27, 1077 (1985); F. R. Hansen, J. P. Lynov, C. Maroli, and V. Petrillo, J. Plasma Phys. 39, 319 (1988) . 10M. Brambilla and M. Moresco, Plasma Phys. Contr. Fusion 29, 381 ( 1987). High temperature plasma diagnostics 3024  Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  128.82.253.131 On: Mon, 15 Feb 2016 15:31:11

Measurement of Magnetic Fluctuations by O-X Mode Conversion

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Measurement of Magnetic Fluctuations by O-X Mode Conversion

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