1. The role of magnetic equilibria in determining ECE in MAST J. Preinhaelter 1), V. Shevchenko 2), M. Valovič 2), H. Wilson 2), J. Urban 1), P. Pavlo 1), L. Vahala 3), G. Vahala 4) 1) EURATOM/IPP.CR Association, Institute of Plasma Physics, 182 21 Prague, Czech Republic 2) EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK 3) Old Dominion University, Norfolk, VA 23529, USA 4) College of William & Mary, Williamsburg, VA 23185, USA This work was partially funded by the United Kingdom Engineering and Physical Sciences Research Council and by EURATOM.

2. ECE and EBW in MAST  Extensive ECE data are available for MAST in the frequency range 16-60GHz  The low magnetic field and high plasma density in a spherical tokamak do not permit the usual radiation of O and X modes from the first five electron cyclotron harmonics  only electron Bernstein modes (unaffected by any density limits) - converted into the electromagnetic waves in the upper hybrid resonance region - can be responsible for the measured radiation Cutoffs and resonances in MAST cold plasma for normal incidence

3. MAST ECE antenna system 1 st mirror 2 nd mirror (adjustable) lens horn MAST window  At the 1 st waist (w 01 ) the beam is detached from the horn  The 2 nd waist (w 02 ) is the projection of the 1 st waist by the lens  ECE from MAST is transmitted only by those rays which can penetrate through the window and are not blocked by the vessel wall

4. 3D MAST plasma model density, electron temperature and magnetic potential profiles for the shot #7798, time 240ms UHR = 36.46GHz  A realistic 3D model of the MAST plasma has been developed for the simulation  The magnetic field is reconstructed by splining of the two potentials determined by the EFIT and SCENE codes, assuming toroidal symmetry  The temperature and density profile are obtained from the Thomson scattering measurements, beyond the LCFS exponentially decaying profiles are used

5. The Gaussian beam is replaced by rays  Intersection of the antenna beam with the LCFS (last closed flux surface) determines the position of the spot

6. Plane stratified slab and mode conversion efficiency estimation  Full wave solution of Maxwell’s equations in the cold plasma slab is used for determination of the EBW-X-O conversion efficiency

7. Conversion efficiency using the adaptive finite elements method  The 2 nd order ODE’s in the cold plasma model are solved assuming weak collisions  The absorbed power in the UHR equals the power of the converted EBW  Adaptive mesh is refined in regions of large local errors Sample contour map of the conversion efficiency projected to the waist plane The red dots represent individual rays, the blue line is the projection of the MAST window rim

8. Ray-tracing  A ray-tracing code is used to determine the radiation temperature from the Rayleigh- Jeans law  The rays with Z~0 propagate deep into the plasma and are absorbed close to the second electron cyclotron harmonic

9. Radiative temperature and EBW absorption  Ray equations describe the motion of EBW packet  the evolution equation for the power has to be integrated simultaneously with the ray dP/dt=-2  (t)P  Non-local reabsorption of the radiation is described by the radiative transfer equation dP/dt=  P which must be solved simultaneously with the ray evolution equation  The emitted power can be expressed by the Rayleigh-Jeans law with T rad instead of local temperature T P ~   T rad where T rad = 0    t ´ )T(t ´ ) exp{- 0  t ´  t ´´ )dt ´´ }dt ´

10. ECE intensity  The intensity of ECE detected by the antenna can be expressed as where - Gaussian weight ( w 0 is the waist radius) - conversion efficiency - Rayleigh-Jeans black body radiation law - power transmission coefficient of the MAST window - relative visible area ( w is the Gaussian beam radius at the plasma surface)  The integration is taken over the intersection of the waist and the projection of the vessel window rim

11. Polarization effects  Only linearly polarized waves can be detected  The polarization changes at the mirrors and at the MAST window (slight elliptical polarization)  The polarization has a very weak effect on the final ECE spectrum During the reflection at a mirror the linear polarization is changed: E ref =-E inc + 2N m (E inc.N m ) where N m is the normal of the mirror

12. We found that the best fit between the measured and simulated ECE can be obtained for a different beam direction to that which follows from antenna adjustment Possible explanation:  beam direction is determined with precision    diffraction of beam in a rarefied plasma in SOL  magnetic equilibrium differs from that determined by EFIT Effects of the beam direction ECE from MAST, shot #7798 time 240ms, reference frequency 23.14GHz

13. Comparison of ECE simulation for EFIT and SCENE equilibria #7798 L-mode, ECE simulation fits well to detected signal for L-modes, SCENE and EFIT give similar results Waves with f<23GHz are converted in SOL where plasma density strongly fluctuates and our model of ECE does not catch this situation properly The detected ray is inclined at  dev from the equatorial plane upward and the angle between its projection onto the equatorial plane and the vertical plane going through the tokamak axis and the antenna position is  long Contour map of square deviation of intensity of simulated ECE from measured values from #7798 at t=240ms

14. ECE from MAST, shot #4958, t=120ms  With the appropriate beam angles the agreement with the experiment is good for both SCENE and EFIT equilibria but the decreases of ECE at the beginning of the second and the third EC band are not described well by any of the simulations. This is typical for ELMy H-modes.

15. Resonance topology for #8694, t=280ms, ELM-free H-mode Radial profiles of characteristic resonances at beam spot (  dev =  long =12 o ) demonstrate clearly the difference in equilibria SCENE*) EFIT *) SCENE – Simulation of Self-Consistent Equilibria with Neoclassical Effects H.R. Wilson: SCENE, UKAEA FUS 271 (1974), Culham, Abingdon, UK`

16. ELM-free H-mode (#8694) simulation based on EFIT equilibrium Simulated and detected signal do not require additional beam aiming adjustment (new antenna calibration works well) Magnetic field at UHR predicted by EFIT is too low (periodicity of the detected ECE requires f ce =11 GHz, but EFIT gives f ce <10 GHz) Shapes of the peaks in the simulated EFIT signal in higher bands resemble well the detected signal

17. ELM-free H-mode (#8694) simulation based on SCENE equilibrium  Surface currents considered in SCENE enhance magnetic field at UHR, but f ce =12 GHz is too high  Shapes of peaks of simulated signal do not correspond to the detected ones.  Only four bands do not correspond to five band in detected signal

18. Ray-tracing can explain the peaks shapes in EFIT simulation  Detailed evolution of central rays was studied for frequencies slightly below and slightly above the plasma surface electron cyclotron harmonics Time development of ray, shot #8694, f=20.84 GHz, N=2. ECE is radiated from the 2 nd harmonic. N || strongly oscillates and the absorption is highly non-local. The absorption on the 3 rd harmonic is negligible. f=49.44 GHz, N=4. Even if f<5f ce, waves are emitted from the 5 th harmonic. Because the factor |(  -5  ce )/N || v T | decreases faster then |(  -4  ce )/N || v T |. |N || | increases monotonically and reaches 1 at the absorption region.

19. Other support of EFIT adequacy follows from and N || (f), C EBW-O-X (f) and T rad (f) Value of N || at full absorption of central EBW ray, #8694, t=280ms, EFIT. Waves with f slightly below Nf ce at the plasma boundary are absorbed with | N || |~1, which is 3 times higher than at the boundary (| N || |~0.36) Conversion efficiency C EBW-X-O for the central rays do not depend on frequency Frequency dependence of T rad for the central rays #8694, t=280ms, EFIT

20. Space dependence of characteristic resonances in MAST Waves are emitted from well of the electron cyclotron resonances  #8694, t=0.280s,  dev =12 o,  long =12 o. We depicted situation at the end of EBW ray, when |N || |=1 for waves having f slightly bellow Nf ce at the plasma boundary and |N || |=0.36 for waves having f slightly above Nf ce at the plasma boundary. Broadening of Nf ce is given by the factor 1/(1±3N || v T /c)

21. Comparison of EFIT and SCENE magnetic field profiles in equatorial plane  SCENE predicts the very high paramagnetic effect so the total magnetic field increases to the plasma center too sharply. As a consequence, the shape of simulated ECE peaks do not fit with detected signal.

22. Comparison of plasma current profiles produced in EFIT and SCENE equlibria  For ELMy H-mode (#4958) slightly different hollow profiles of toroidal current J  are produce both by EFIT and also SCENE equlibria  For L- mode (#7798) we obtain the simple parabolic profiles of J  from both EFIT and SCENE   very good fit)  For ELM-free H-mode we obtain totally different profiles of plasma current (a simple parabolic profile of J  for SCENE and for a hollow profile of J  for EFIT) Equatorial profiles of plasma current #8694, t=280ms, EFIT Equatorial profiles of plasma current #8694, t=280ms, SCENE

23. CONCLUSIONS Current theoretical model incorporates nearly all the details of the MAST ECE antenna and plasma model based on experimental data. For L-mode, agreement between calculated and experimental EBW emission is good. For ELMy H-mode, agreement is good but model does not explain the smaller signal at lower frequency part within each harmonic bands. For ELM-free H-mode, simulation based on EFIT equilibrium agrees with experiment at higher harmonics while using SCENE equilibrium provides higher magnetic field at the plasma surface and better agreement at lower harmonics. These results show that EBW emission can provide an additional constraint for equilibrium reconstruction.

24. References Shevchenko V et al., 15 th RF Power in Plasma, Moran, 2003,edit. C. Forest, AIP 694, 359. Laqua H.P., et al., review, 15 th RF Power in Plasma, Moran, 2003, edit. C. Forest, AIP 694, 15. H.R. Wilson: SCENE, UKAEA FUS 271 (1974), Culham, Abingdon, UK Preinhaelter J. e al, 15 th RF Power in Plasma, Moran, 2003, edit. C. Forest, AIP 694, 388. Preinhaelter J. e al, Review of Sci. Instr. Vol, 75, No 10, Oct. 2004 Urban J.: Adaptive Finite Elements Method for the Solution of the Maxwell Equations in an Inhomogeneous Magnetized Plasma, Czech. J. Phys 54 (2004) Suppl. C, C109 Preinhaelter J., Kopecky V., J. Plasma Phys. 10, 1 (1973), part 1. Pavlo P., Krlin L., Tluchor Z., Nucl. Fusion 31, 711 (1991). Goldsmith P.: Quasioptical systems: Gaussian Beam Quasioptical Propagation and Applications, Wiley-IEEE Press (1997)

25. 3D Ray trajectories, # 8694, t=289ms f=20.84GHz f=49.44GHz

26. Adaptive method convergence properties  Typical error dependence of the global and local error are shown for /f=0.001  For common precision requirements (0.005-0.001) the method is fast  The error estimates correspond with each other Dependence of various global error estimates on the total number of nodes the error decreases approx. as ~n -4.5 Evolution of local error and node density with mesh refinement

27. Effects of the beam direction  Strong dependence of the EBW-X-O conversion efficiency can be expected  The transmission coefficient of the power of the O-mode to the X-mode at the plasma resonance depends on the detected beam direction

28. Effects of the beam direction