Models for JET Polarimeter Data, Faraday Effect and constraints on Equilibrium reconstruction

1. 1st International Conference Frontiers in Diagnostic Technologies Satellite Meeting on Polarimetry Measurements Frascati 24 nov 2009 Models for JET Polarimeter Data, Faraday Effect and constraints on Equilibrium reconstruction Francesco Paolo Orsitto ENEA Frascati FDT1 / POLARIMETRY

2. Contributions • S Segre (Univ RomaII, Italy), • P Gaudio , M Gelfusa(Univ Roma II, Italy), • A Boboc ( Ukaea, JET Facility), • A Murari( RFX, Padova Italy), • C Mazzotta and E Giovannozzi (ENEA Frascati,Italy) • Acknowledgements : V Drozdov , E Solano , M Brix(UKAEA JET Facility) , L Appel (UKAEA) FDT1 / POLARIMETRY

3. Talk Plan • Part I : Introduction to Tokamak polarimetry • Part II : Modelling polarimetry and comparison with measurements FDT1 / POLARIMETRY

4. Part I :Introduction to Tokamak polarimetry • Tokamak : the magnetic confinement machine for fusion • Magnetic equilibrium of a plasma • JET ( Joint European Torus): the EU Tokamak • Polarimetry : what for? FDT1 / POLARIMETRY

5. The Tokamak • In a plasma contained in a toroidal device with axial magnetic field : a current is induced by atransformer • A magnetic field results with elical field lines which close after a certain number of turns on surfaces called ‘rationale surfaces’ FDT1 / POLARIMETRY

6. Magnetic Equilibrium in a tokamak • Plasma exists in a tokamak if there is equilibrium between the magnetic force and the force due to the plasma pressure Thus for a plasma in equilibrium the field lines and current lines lie on isobaric Surfaces (p=const.) ; These surfaces , generated by the field lines, are called magnetic surfaces and havetoroidal topology . The magnetic surfaces form a family of nested tori . The innermost torus degenerates into a curve called magnetic axis. FDT1 / POLARIMETRY

7. Plasma equilibrium ,flux function and magnetic surfaces • In a plasma there are two components of the total magnetic field :→corresponding to the poloidal magnetic field there is the associated flux Ψ • it is therefore constant on that surface. The Grad-Shafranov equation determines The flux function Ψ ( which follows from the force balance eq.): FDT1 / POLARIMETRY

8. Solution of G-S equation: exp meas into code EFIT • The inverse problem ( finding ) is determined using the following measurements : FDT1 / POLARIMETRY

9. Solution of G-S eq.: minimization and constraints • The problem is defined as a least-square minimization , the χ2 function is given by: J1 is the Faraday constraint, that will be discussed In the part II. FDT1 / POLARIMETRY

10. Result of EFIT calculations : EFIT SURFACE evaluation at two times in a discharge FDT1 / POLARIMETRY

11. JET ( Joint EU Torus) FDT1 / POLARIMETRY

12. JET interior of vacuum vessel no plasma with plasma JET: the largest tokamak device working With Tritium plant generation to operate in DT FDT1 / POLARIMETRY

13. Polarimetry : what for? • The measurements of polarimetry in tokamak plasmas can give important information on plasma current and density • In a plasma, in presence of a magnetic field, the polarization plane of a laser beam propagating along the magnetic field rotates (Faraday effect). • Whereas, if the laser beam is propagating perpendicular to the magnetic field there is a change in the ellipticity of the polarization (Cotton-Mouton effect). • In a beam propagating vertically along a line in a poloidal plane of a tokamak , both effects are presents : the polarization becomes elliptical and the plane of the polarization rotates. • In first approximation • Faraday rotation angle = electron density * plasma current • Cotton-Mouton Phase shift = electron density * toroidal magnetic field squared. FDT1 / POLARIMETRY

14. Part II : Modelling polarimetry and comparison with measurements • Models available for polarimetry • Present status of the comparison of polarimetry models and measurements for vertical and horizontal channels • Calculation of the equilibrium EFIT : i)present mathematical form of constraints on faraday measurements • ii)channels usually included in the constr. • NEW Models for Faraday to be tested for EFIT constraints. FDT1 / POLARIMETRY

15. JET polarimeter Primary measurements by the polarimeter are : Amplitudes of Wave components Ex,Ey, phase shift fbetween Ex and Ey FDT1 / POLARIMETRY

16. JET Interfero - Polarimeter FDT1 / POLARIMETRY

17. Physics of polarimeter Usual paradigm of polarimetry Faraday rotation ≈ne Bz dz Cotton-Mouton ≈ne BT2 dz The present talk is about 1.the limits of this physical scheme and 2. how polarimetry can be useful in giving directly correction to the Calculation of the magnetic equilibrium FDT1 / POLARIMETRY

18. Propagation of polarization ( vertical channels) W = ( W1, W2 , W3 ) Z = z / ( k a ) k = elongation a = minor radius c = velocity of light= 3 108 m/s w/2p =laser frequency= c / 195mm wp/2p=plasma frequency=8.8(ne)0.5. wc/2p=cycl frequency= 28 * BT GHz By = B toroidal Bx = B radial Bz = B poloidal Cold plasma approximation FDT1 / POLARIMETRY

19. Relation between Stokes vector and Faraday and Cotton-Mouton FDT1 / POLARIMETRY

20. Input to the Stokes equations • Bx,By,Br are given by EFIT ( only magnetic measurements) • Ne(z,t) , Te(z,t) given by LIDAR Thomson Scattering ( and when available ) by HRTS( High Resolution TS) projected on the vertical line of sight using equilibrium. FDT1 / POLARIMETRY

21. Type I approx W1=Ω1 dz=C1 e BT2 dz W3=Ω3 dz=C3 ne Bz dz ( W12 and W32<<1) FDT1 / POLARIMETRY

22. Type II approx Stokes eqs. In a tokamak : FDT1 / POLARIMETRY

23. Meaning of the results of type II approx • The Type II solution can be interpreted saying : • in a tokamak , the Faraday and Cotton-Mouton effects cannot be treated separately when W3 is large. • In practice for Faraday rotation angles ψ≤12o, 1≥cos (W3)≥0.9 and tanφ≈W1, within an approximation of 10%. • For Faraday angles higher than 12o the Cotton-Mouton increases due to the enhancement linked with Faraday rotation( W3≥π/15=0.2). FDT1 / POLARIMETRY

24. Summary of results • i) It turns out that the Faraday rotation measurements on JET can be reproduced in any condition only by the numerical solution of Stokes equations and • ii)by a suitable shift of the magnetic surfaces, a study shows that the comparison between model calculations and measurements lead to a more refined identification of teh positions of teh magnetic surfaces as predicted by EFIT equilibrium code • iii) a rigorous approach to the interaction between Faraday and Cotton-Mouton, ( studied in recent papers, see ref [2] for details): it is demonstrated that at high density and current, the Cotton-Mouton must be calculated including the dependence by Faraday rotation. FDT1 / POLARIMETRY

25. Modelling of Faraday rotation angle: low density shot Faraday rotation. comparison between the models(symbols + ad o) and measurements( blue lines) is presented for the shot #60980, channel #3. From the top the comparison of measurements with the Type II approximation, the 'linear' W3 approximation, and the Guenther Model A. Faraday rotation measurement (blue — continuous line, shot #60980, channel #3) is plotted together with the calculated values ( ' * ' symbol ) using the numerical solution of Stokes equations. FDT1 / POLARIMETRY

26. Modelling Faraday Ch #3 (R=3.04m)for a High Density shot (neL19=29)a shift of the magnetic surfaces is needed to obtain agreement between data and model DR=0.04m DR=0 FDT1 / POLARIMETRY

27. Modelling Faraday Ch #4 (R=3.78m)for a High Density shot (neL19=29) • The value of faraday angles is y4=0.45rad • Only Type II and numerical solution are in agreement with measurements, • while type I ( linear) =W1≈1 underestimate the measurement. FDT1 / POLARIMETRY

28. Interaction between Faraday and Cotton-Mouton(I) Some hints can be extracted from Type II approximation : • the Faraday interact with Cotton-Mouton effect : • i) for large Faraday effects (W3≈1) the Cotton-Mouton increases strongly with respect to the ‘linear ‘ form ( tan f = W1) ( see figure where W1 is not enough for the estimation of Cotton-Mouton); • ii) the Faraday is not depending from Cotton-Mouton at this level of approximation. FDT1 / POLARIMETRY

29. Departure from linearity of cotton-mouton for high density shot • Plot of • tanφ(Cotton-Mouton)/W1 versus W3 • demonstrates that already for W3>0.4 the ratio of cotton-mouton and W1 ( the linear evaluation) FDT1 / POLARIMETRY

30. Main results(I) • detailed analysis of Faraday measurements at JET and comparison with available models: It turns out that the Faraday rotation measurements on JET can be reproduced in any condition only by the numerical solution of Stokes equations; • this means that the linear formula used presently in EFIT • W3=Ω3 dz=C3 ne Bz dz=1/tan2ψ is of limited range of application . FDT1 / POLARIMETRY

31. Main results(II) • ii) in a rigorous approach to the interaction between Faraday and Cotton-Mouton,it is demonstrated that at high density and current, the Cotton-Mouton must be calculated including the dependence upon the Faraday rotation. • This means that if a new constraint using Cotton-Mouton phase shift measurements is introduced in EFIT the formula for the constraint must include the effect of the Faraday rotation on Cotton-Mouton. FDT1 / POLARIMETRY

32. Main results(III) • iii)analysis for channel #3 and #4 is carried out. • to reconcile the measurements of Faraday rotation and the calculations using Stokes equations it is necessary for medium- high density and high current shots to shift the position of the magnetic surfaces by ≈0.04m in direction of the high field side. • This means that the polarimetric measurements could give an additional correction to the position of the magnetic surfaces produced by EFIT. FDT1 / POLARIMETRY

33. The constraint for Faraday in EFIT • Type I approximation is used into EFIT throughout the calculations : rigorously this restricts the validity of EFIT calculations • Since Type I formulas are valid only for low –medium density and low plasma current, and not for all channels. A clear example is the analysis carrried out for channel #4. • The analysis shows that in general the Stokes model is more suitable to be used in any plasma conditions and all the channels • Alternatively the Type II approximation could be used FDT1 / POLARIMETRY

34. Conclusions • A complete analysis of the polarimetry data for all the KG4 channels leads to the conclusion that Stokes model is suitable to predict the measurements. • In this context( modelling of polarimetry measurements) the tools for the validation of data for all the polarimetry channels are available. • The present version of EFIT has built in a constraint • i) on Faraday rotation angle measurements • Ii) using Type I approximation which has been tested as not valid for all the plasma conditions and polarimetry channels. • The introduction into EFIT of the Stokes model could remove the limitation related to the plasma parameters and in addition • It could be used to inroduce a new constraint on Cotton-Mouton phase shift which lead to improvement of evaluation of electron density • The possibility of using all the channels into EFIT constraints is suggested. FDT1 / POLARIMETRY

35. Relation betweenflux ψ and B,j Axisymmetric coordinate system : R = major radius of torus Z = vertical coordinate Φ = toroidal angle FDT1 / POLARIMETRY

36. Scheme measurements( A Boboc,M Gelfusa,K Guenther) InSb He-cooled Detectors Phase sensitive Electronics Definition of parameters that describe the polarization state of ellptically polarized light. Amplitude ratio =Ey/Ex Polarization state Phase shift angle Ellipticity Stokes vector Faraday rotation angle FDT1 / POLARIMETRY

37. Solutions of G-S eq • The G-S problem is solved using a finite element method FDT1 / POLARIMETRY

38. Concept of Magnetic Confinement • plasma must be confinedandisolated • Confined: it must be contained in a stable state • Isolated : the interaction with the container must be controlled . • the aim is to minimize the interaction of plasma with the container which could change the quality of the confinement , for example increasing the metallic impurities.( with consequent emission of radiation and lost of energy) particles gyrate around the magnetic field, in trajectories with perp radius : r= m Vperp / eB FDT1 / POLARIMETRY

39. Results of analysis on horizontal channels • It turns out that the Faraday data on channels #5,6,7,8 are in broad agreement with Stokes models: • →this means that all the channels from (2)3 to 8 can be used as input data for EFIT • We find that including channel#4 in the EFIT analysis could be helpful to imporve the determination of the characterization of equilibrium at edge. FDT1 / POLARIMETRY

40. Using Stokes model into EFIT • Stokes model is a 1st order differential system which gives the possibility of evaluating the Faraday and Cotton-Mouton. • ( may be) some care must be used in the calculations of Ω1 and Ω2 because of the quadratic dependence upon the magnetic fields ( toroidal and radial). • It allows ‘naturally’ to introduce into EFIT also a constraint on Cotton-Mouton phase shift measurements. FDT1 / POLARIMETRY

41. Using Type II approximation • It is a rough estimation but it works and it is simple to evaluate numerically FDT1 / POLARIMETRY