1. Full-wave test of analytical theory for fixed frequency fluctuation reflectometry Motivation Full-wave codes Analytical theory (1D) compared Fluctuation profile reconstruction Conclusion M.Schubert CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France 05/05/2009 TORE SUPRA

2. 2 Formula (1) valid, if d n rms / n c is small d n rms is homogeneous Problems : k eff = ? the case k eff 0 Local fluctuation measurement ? B A A ≠ B d  nonlocal (Eq.1)

3. 3 Example: fast-hopping reflectometer on Tore-Supra (simulation) f1f1 f2f2 f2f2 t f1f1 Profile shape not well recovered if using (1) for d  rms dn rms Using to obtain the mapping k 0 ( f i ) x. (input) Simulation result :

4. 4 Arbitrary profiles of the fluctuation level x c cut-off of vacuum wavenumber k 0 ( f ) in the average N 2 profile G(x) weighting function: turbulence properties and N 2 ( x ) profile Gusakov et al. PPCF 44 (2002) 2327 (Eq.2)

5. 5 Numerical Tools (developped at LPMIA and IST) 1D, time-independent (“Helmholtz”) –Very fast, good to do statistics d  k dn k –Stationary state: fixed frequency, snapshot of turbulence –used on TS (DREFLUC), e.g. theses L.Vermare, T.Gerbaud Spatio-temporal –Wave and turbulent dynamics –Dedicated to Ultra-fast sweep (projet ANR) –Simulation of dynamic phase measure- ment in presence of secondary cut-off 2D (space) + time –Beam diffraction, Doppler reflectometry –O-mode OK, X-mode to come soon …

6. 6 plasma density random superposition Heuraux et al. RSI 74 (2003) 1501. Modelling Turbulence wavenumber resolution 2  /  k longest wavelength of turbulence mode i snaphots of turbulence (~ t )  i  random phases [0.. 2  ], i = 1 …M

7. 7 use analytic expression, given l C is known Restrictions : linear gradient of n e (O-Mode) with characteristic length L = L N 2 homogeneous turbulence, small correlation length l C << L small fluctuation level, no secondary cut-offs Bragg backscattering negligible ( k 0 2 / L ) 1/3 >> ( l C ) -1 Application of 1D code “Helmholtz” Gaussian spectrum Gusakov et al. PPCF 44 (2002) 2327

8. 8 Effect of long wavelength fluctuations scan of  k 47 GHz L = 58 cm l C = 0.75 cm d n/n = 0.001 ( Gaussian spectrum, = const. ) Convergence !

9. 9 Full-scale comparison : theory – code result Cross hair f = 47 GHz, L = 58 cm, l C = 3 cm, dn/n = 0.001 Scan single parameters, keeping the others const. x Gusakov numeric + l C = 1, 4.5 cm dn/n = 0.0005... 0.1 f = 24, 58 GHz L = 32, 100 cm

10. 10 Inhomogeneous fluctuation profile For certain spectra G can be derived accurately : e.g. Gaussian wavenumber spectrum, O-Mode. If N 2 changes linearly near the cut-off, G near is exact. modified Bessel function () | x - x c | < l c : | x - x c | > l c : (Eq.2)

11. 11 1.Input profiles fluctuation level 2.Select set of f i evaluate cut-off positions x ( f i ) 3.Full wave code :  2 ( f i ) o (~ “measurement“) 4.Make a guess for fluctuation level profile d n ( x i ) 5.Use Eq.2, evaluate i + 6.Minimize least squares, d n ( x i ) are free parameters

12. 12 Result of Least-squares minimization -High fluctuation level (non-linear scattering) -N 2 does not change linearly near cut-off -Grid of d n ( x i ) is not fine enough Remaining deviation :

13. 13 Numerical study of error propagation using Eq.2 : Large error bar on “measurement“ at the plasma edge (30% rel.err) Effect on the reconstructed fluctuation profile: error propagates. Measurement in the plasma center is robust.

14. 14 Fluctuation measurement in the plasma center of Tore-Supra : 2D Simulation Single path transmission Profile of fluctuation amplitude Single frozen pattern of the turbulence Can the 1D (“Helmholtz”) result be applied ? Large edge fluctuation level: What happens to Gaussian beam (2D slab) probing the center ?

15. 15 Launch 40 GHz, FWHM ~ 7cm Fluctuation level at plasma edge (x = 0) ~10% Contour plot of electric field Phase coherence is intact Proc. 3 rd France-Russia Seminar, Metz (2007)

16. 16 Summary Successful validation of analytical formula for the absolute density fluctuation, using full-wave code. Sucessfully tested an inversion algorithm for the fluctuation level profile, based on integral expression for the phase fluctuations. 2D simulations in Tore Supra geometry : Beam shape and phase fronts remain intact when crossing the strongly fluctuation plasma edge. Validation is being extended to X-mode.

17. 17 Contributors : F.Clairet, R.Sabot (CEA Cadarache) E.Gusakov, A.Popov (Ioffe Institute St.Petersburg) F.Da Silva (IST Lisbon) T.Gerbaud (JET) S.Heuraux (LPMIA / IJL Nancy) Support: RFBR grants 06-02-17212, 07-02-92162-CNRS, ANR grants FI-071215-01-01, 06-blan0084

18. 18 Motivation / possible applications Understanding of plasma turbulence, particle and energy transport fluctuation measurement Systematic investigation of the density fluctuation level as function of external parameters (grad T, grad n, I P,  * etc) Measurement of the propagation velocity of a turbulent wave front generated by heat pulse Comparison : S k  S kr

19. 19 Motivation - I M.Schubert, LPMIA (2007), code by F.daSilva, IST source lens plasma Strong fluctuations (plasma density) propagation in vacuum Validate the reflectometry measurement in case of : strong fluctuations of n e on the beam path non-linear scattering

20. 20 Motivation - II x nene L  t + tt tt k0k0 Drawing: C.Fanack, UHP Nancy (1997) The Theory of Plasma Waves, T. Stix (1962)  L k 0 Exact solution usual approximation large fluctuation (H-mode/ELM, Blob) L k 0 0 non-linear phase shift from cut-off reflection

21. 21 If there are : strong fluctuations of n e on the beam path (non-linear scattering) strong fluctuations at the cut-off (non-linear phase shift of reflected wave) Only full-wave codes can validate the reflectometry measurement.

22. 22 Far from the cut-off G can be approximated as where can take into account an arbitrary density profile. Fig.2: Weighting factor G, linear density gradient L = 50 cm, turbulence with Gaussian spectrum, correlation length l c =1cm. x -axis: distance to the cut-off, normalised to l c Black: exact solution G near Red: approximation G far. modified Bessel function ()