1. Improving Predictive Transport Model C. Bourdelle 1), A. Casati 1), X. Garbet 1), F. Imbeaux 1), J. Candy 2), F. Clairet 1), G. Dif-Pradalier 1), G. Falchetto 1), T. Gerbaud 1), V. Grandgirard 1), P. Hennequin 3), R. Sabot 1), Y. Sarazin 1), L. Vermare 3), R. Waltz 2) 1) CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France 2) General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA 3) Laboratoire de Physique et Technologie des Plasmas, CNRS-Ecole Polytechnique, 91128 Palaiseau Cedex, France

2. Guideline Goal: To improve predictions on turbulent fluxes need physics based transport models Context: –Nonlinear gyrokinetic electromagnetic simulations still too costly in terms of computing tim –Interestingly, quasi-linear approximation seems to retain the relevant physics Work on quasi-linear fluxes in two parts: –quasi-linear weight : phase and amplitude, follows well non-linear predictions –electrostatic potential: based on both non-linear simulations and turbulence measurements Integrated in QuaLiKiz where flux agrees with non- linear one when ranging from Ion Temperature Gradient (ITG) to Trapped Electron Modes (TEM)

3. general approach for quasi-linear model, QuaLiKiz [Bourdelle PoP07] fluctuating distribution function linearly responds to the fluctuating electrostatic potential through Vlasov equation computed by eigenvalue code Kinezero [Bourdelle NF02] Example for particle flux: No information on the saturation of the fluctuating electrostatic potential in terms of its amplitude or on its spectral shape versus the wave number and the frequency

4. Accounting for the « non-resonant terms » Resonance Broadening Theory: non negligible finite +i0 + =+i linked to irreversibility through mixing of the particles orbits in the phase space. Moreover in the limit →0 the particle fluxes are not ambipolar intrinsic frequency spectral shape of the fluctuating potential In QuaLiKiz, =0 + and : equivalent to RBT where =  k and Nevertheless shape and width choices arbitrary. ongoing measurements vs nonlinear simulations

5. Frequency spectrum: non-linear simulations vs measurements  k =  k + Cst*k    with  reproduce widths of the frequency spectra observed from GYRO simulations and measurements. Ongoing…  =2.3  =2.2 Antar PPCF 1999 GYROBackscattering on Tore Supra

6. Saturation rule: mixing length In QuaLiKiz, flux = sum over all unstable modes each weighted by corresponding  k as [Jenko, Dannert, Angioni 2005] adding 

7. k r spectrum: non-linear simulation vs measurements nonlinear GYRO compared with fast-sweeping reflectometer [Casati TTF08] #39596, r/a=0.7  r,exp = -2.8  r,sim = -3.0

8. k  spectrum: non-linear simulations vs measurements nonlinear GYRO compared with Doppler reflectometer [Casati TTF08] #39596, r/a=0.7

9. k spectrum isotropy Isotropy found in some GYRO simulations Ongoing… Apparent (k ,k r ) anisotropy due to Doppler instrumental integration domain Hence, actual choice: from 0 to k max : and from k max to infinity:

10. quasi-linear weights in the case of an eigenvalue approach, the fluxes can not be unequivocally divided by Therefore, discussion limited to most unstable mode no simple tool allowing testing the validity of the quasi- linear approach for subdominant modes yet developed

11. Amplitude of the weight: QL/NL~1.5 local and global simulations : systematic over- prediction QL vs NL around 1.5 QL/NL ratio stays reasonably constant when changing plasma parameters, especially at low k  scales Reason of this over prediction to be assessed adiabatic electrons, r/a=0.4, R/L Ti =8.28,  *=1/256

12. Phase of the weight: OK for ITG, fails for ITG-TEM Test introduced for TEM by [Jenko 2005-2008] extended to ITG and ITG-TEM cases Good QL/NL phase matching for ITG cases: particle and energy But close to ITG/TEM transition QL phase from most unstable mode fails for particle whereas energy OK R/L Ti =9 R/L Te =9 R/L n =3 R/L Ti =6 R/L Te =9 R/L n =3

13. quasilinear fluxes vs nonlinear predictions test quasi-linear fluxes computed by actual version of QuaLiKiz versus nonlinear GYRO ion and electron energy fluxes and particle fluxes for various parameter scans ranging from ITG to TEM dominated cases only one renormalisation factor, C 0, has been used in order to get the best fit to the nonlinear fluxes

14. R/L T scan Ion energy electron energy particle effective diffusivities GYRO (diamonds) QuaLiKiz (lines) for R/L Ti= R/L Te scan with R/L n =3

15. * scan Based on Tore Supra * experiment In agreement with experimental obs. GYRO (diamonds) QuaLiKiz (lines) Ion energy electron energy particle r/a=0.5 R/L Ti =8 R/L Te =6.5 R/L n =2.5

16. T i /T e scan DIII-D T i /T e scan PRL Petty 99 Qualitative agreement with experiment GYRO (diamonds) QuaLiKiz (lines) Ion energy electron energy particle r/a=0.3 R/L Ti =6.5 R/L Te =4.6 R/L n =1.4

17. Summary Assuming a linear response of the transported quantities to the fluctuating potential works rather well: phase OK if one unstable mode, amplitude over- estimated Moreover, when coupling the choices for electrostatic potential with the quasi-linear response, find quasi- linear fluxes agreeing well to nonlinear predictions for energy and particle fluxes over a wide range of parameters, from ITG to TEM dominated cases

18. Discussion A number of challenging issues remain to be tackled: –quasi-linear approach known to fail : far from the threshold, onset of zonal flows, etc. Hence, domain in which it can be applied should be better understood –choices for the electrostatic potential deserve more comparisons with nonlinear simulations and experimental measurements. In Tore Supra, presently comparing density fluctuations k and frequency spectra from Doppler and fast-sweeping measurements versus GYRO and GYSELA –Finally, only integration of QuaLiKiz in a transport code such as CRONOS will allow testing in situ the predictive capabilities

19. R/L Ti =9 R/L Te =9 R/L n =3 R/L Ti =6 R/L Te =9 R/L n =3