1. Transport of parallel momentum induced by up-down asymmetry, role of collisions and thermoelectric pinch A.G. Peeters 1, Y. Camenen 1 C. Angioni 2, N. Kluy 2, D. Strintzi 3, F.J. Casson 1, W.A. Hornsby 1, A.P. Snodin 1,, 1 University of Warwick, CV4 7AL Coventry, UK 2 Max-Planck Institut fuer Plasmaphysik, Garching, Germany 3 National Technical University of Athens, Greece

2. Symmetry : Local approximation “If there are no equilibrium flow and gradient in this flow, … from the symmetry of the distribution function, the flux of the parallel momentum is zero, since it is an integral over the flux surface of the product of an even (the potential … ) and an uneven (the parallel velocity … ) function “ A.G. Peeters, PoP 12, 072515 (2005) Coordinate along the field

3. Current symmetry breaking up-down asymmetry of the flux surface Y. Camenen, A.G. Peeters et al, PRL 2009 Gyrokinetic results from GKW

4. Role of the symmetry (again)

5. Toroidal current ( up-down ) symmetry breaking

6. Which terms generate the flux ?

7. Role and strength

8. Localization at the edge

9. Dependence on the sign of the current and B The direction of the current asymmetry flux changes with the direction of Ip and the direction of Bt, or if the plasma is rotated up-side down Toroidal rotation is increased in counter-current direction for favorable grad B drift configuration, and in co-current direction for unfavorable grad B drift

10. Self consistent mode structure Inverse aspect ratio and collisionality scaling of the pinch Gyrokinetic results from GKW, GS2 and GYRO Coriolis drift in GS2 by N. Kluy and C. Angioni In GYRO by R.E. Waltz (PoP 2007), N. Kluy and C. Angioni

11. Coriolis pinch effect NOTE :Unlike the curvature and grad-B drift the Coriolis drift is linear in the parallel velocity. While the former couple density and temperature fluctuations, the latter couples density and temperature fluctuations to parallel velocity fluctuations  Flux of parallel momentum is generated Grad-B driftParallel velocityCurvature drift ExB driftCoriolis drift A.G. Peeters Phys. Rev. Lett 98 265003 2007

12. All u terms (rotation) are due to the Coriolis drift effect NOTE: without Coriolis drift the perturbed velocity is proportional to u’  Only a diffusive flux is obtained Origin of the momentum transport

13. Angular Momentum equation Same as in Hahm PoP 2007 and in Diamond NF 2009 Can be built combining density and parallel velocity equations

14. Fluid equations One notices that the parallel dynamics and the Coriolis drift appear in a symmetric way Same as Hahm PoP 2007, Diamond NF 2009 A.G. Peeters et al PRL 2007, submitted PoP 2008

15. Mode structure adjusts Mode structure for u = 0 is symmetric For finite rotation the mode structure adjusts Parallel dynamics has a tendency to compensate the Coriolis drift effect Strong constraints on analytic models. One cannot average over a mode structure obtained without considering the toroidal rotation Toroidal rotation is an external parameter, mode structure is determined by the eigenfunction Top mode structure for u = 0. Bottom mode structure for u = 0.2

16. Gyro-kinetic simulations Fluid result partly related to the approximations. The transformation is not possible for both electrons and ions Result, however, carries over to the gyro-kinetic regime Normalized momentum flux as a function of time. Simulation is started with u = 0, after which a restart is made with u = 0.2.

17. Inverse aspect ratio scaling If the above is true one would expect a scaling with the trapped particle fraction Indeed the pinch scales almost linearly with the root of the inverse aspect ratio Explains the low pinch in the centre of NSTX ? Relevant for comparisons among tokamaks and spherical devices Fig. Pinch as a function of the square root of the inverse aspect ratio A.G. Peeters, Self consistent mode structure, submitted PoP 2008

18. Collisionality dependence of the momentum pinch ITG (Waltz standard case) Momentum pinch does not change significantly with increasing collisionality … … unless very large collisionalities are reached, where a strong reduction of momentum pinch and a weak reduction of momentum conductivity occur

19. Collisionality dependence of the momentum pinch (2) A.G. Peeters et al, PoP subm, 2008 Momentum pinch does not change significantly with increasing collisionality … … unless very large collisionalities are reached, where a reduction of momentum pinch and momentum conductivity occur Collision operator in GKW (recently implemented) in very good agreement with GS2

20. Collisionality dependence of the momentum pinch ITG (Waltz standard case) Momentum pinch does not change significantly with increasing collisionality in the usual collisionality range At very large collisionality, strong reduction of momentum pinch and weak reduction of momentum conductivity leads to a reduction of absolute value of RV  / 

21. Thermoelectric pinch Dependence on the logarithmic temperature gradient Gyrokinetic results from GKW, GS2 and GYRO Coriolis drift in GS2 by N. Kluy and C. Angioni In GYRO by R.E. Waltz (PoP 2007), N. Kluy and C. Angioni

22. With the goal of clarifying a controversy Foreword: the only results quantitatively correct are the gyrokinetic ones. Compensation effects must be self-consistently computed through the solution of the eigenfunction. This is of course the case in all numerical GK results by Peeters and co-Authors A controversy appears in the literature. Hahm PoP 08, comment by Peeters PoP 09, related response Hahm PoP 09, and Diamond NF 09 Simple fluid model in Peeters PRL 07 consistently derived from GK equation ( see Peeters to be publ. PoP 09, for a very detailed derivation ) It is easy to show (as we just did) that this model is equivalent to that derived in Hahm PoP 07 and, e.g, applied in Diamond NF 09. With Te = Ti, and adiabatic electron response, Peeters PRL 07 derives the final result from the fluid Are TEP (Hahm PoP 08) and Thermoelectric pinch (Diamond NF 09) included in this expression ? Why this expression includes a dependence on R/Ln and not one on R/LT ? or

23. Turbulent Equipartition ( ExB compression term ) TEP (Hahm PoP 08) In Peeters’ work, the toroidal velocity is considered In Hahm’s work, the total angular momentum is considered By simple algebra, pinch to conductivity ratio are transformed one in the other The TEP result is equal to the ExB compression part in Peeters fluid (PRL 07, Comment PoP 09) This is NOT the total fluid result

24. R/LT dependence pointed out in thermoelectric pinch Recently, Diamond et al, NF 2009 decompose explicitly the thermoelectric pinch in the expression for the flux In this expression, however, the eigenfrequency is not determined through the dispersion relation where

25. Consistent calculation through the dispersion relation The effect of the thermoelectric pinch can be consistently computed making use of the dispersion relation. For ITG modes, the limit of adiabatic electrons is considered (Peeters et al, Phys Plasmas submitted 2008) Decompose momentum flux in diagonal diffusion, ExB compression (TEP) and thermoelectric pinch where

26. Use the dispersion relation to compute  Use the dispersion relation in expression of the thermo- electric pinch to compute the dependence on the eigenfrequency (iterative procedure, neglect u ) Dependence on R/LT removed by the inclusion of the dispersion relation 2

27. Final expression clarifies the controversy Final expression agrees with that previously derived, with  = 1 (Peeters PRL 2007): no term was omitted there ! It does not involve explicitly the eigenfrequency, since consistently computed through the dispersion relation Diagonal diffusion ExB compression (TEP) Thermoelectric pinch All non-diagonal terms arise from the presence of the Coriolis drift in the initial gyrokinetic equation No explicit dependence on R/LT in the fluid model with adiabatic electrons, which explains the weak dependence found with GK codes in the strong ITG limit

28. Dependence on R/LTi in ITG and TEM domains Sizable dependence on the logarithmic ion temperature gradient is obtained in TEM and in ITG close to the ITG TEM transition, where the kinetic electron response is strong and where mode structure varies (change in k parallel square) The dependence on R/LTi reverses sign The dependence becomes weaker in the strong ITG limit TEM ITG N. Kluy, C. Angioni, Y. Camenen, A.G. Peeters et al, in preparation R/LTi [ R/LTe = 9 ] RV  /  

29. Gyrokinetic results required to compare with experiment For quantitative comparisons with experiments, gyrokinetic results have to be considered Simple fluid models overestimate the pinch, since do not include properly the compensation effect by the self-consistent mode structure through parallel dynamics Effect of particle flux is small (negligible in experimental conditions,  close to the null) GKW

30. Conclusions The Coriolis drift generates a pinch in the momentum flux Due to the ‘compensation effect’, trapped particles have a remarkable influence and the pinch scales with the root of the inverse aspect ratio Collision, however, do not have a strong impact on the magnitude of the pinch (unless extremely high collisionality is reached) The contribution of the thermoelectric pinch can be decomposed and computed analytically, and the weak dependence on R/LT obtained with codes in the strong ITG limit can be explained When the kinetic electron response becomes more important, the dependence of the momentum pinch on R/LTi can become sizable, and exhibits a sign reversal (similar to particle transport) A new effect related with the up-down asymmetry of the equilibrium is identified. This effect can be important and requires the exact geometry to be computed consistently in the calculations

31. THE END

32. Nonlinear simulations

33. Nonlinear gyro-kinetic simulations Waltz standard case (q = 2, s = 1,  = 0.16, R/L T = 9, R/L N = 3) Based on GKW Value is obtained (in gyro-Bohm units)  i = 11.3 ± 0.9 (GYRO 11)   = 8.2 ± 0.7 (GYRO ~8.8) Prandtl number Pr = 0.73 ± 0.07 (GYRO ~0.8)

34. Diagonal part In linear theory the Prandtl number strongly increases with the poloidal wave vector Same trend in nonlinear theory, but far less strong Reasonable difference in the Prandtl number is obtained at poloidal wave number for which the growth rate is maximum 1  0.7 Reasonable agreement for the value for which the nonlinear transport is maximum

35. Diagonal part Prandtl number shows little parameter variation Is larger only close to the threshold of the ITG

36. Nonlinear simulations First non-linear simulations of this effect obtained by R.E. Waltz using the GYRO code [6] Here confirmed by GKW, in red RV  /   (diffusion separate run) Waltz standard case (q = 2, s = 1,  = 0.16, R/L T = 9, R/L N = 3) [6] R.E. Waltz, et al., Phys. Plasmas (2007) GKW

37. Nonlinear simulations Magnitude similar to the linear theory (though somewhat smaller) Dependence on plasma parameters follows roughly (but not exactly) linear theory Dependence on temperature gradient is stronger, but mainly through a change in the diagonal term Main dependence appears to be the density gradient