Perturbative studies of toroidal momentum transport using neutral beaminjection modulation in the Joint European Torus: Experimentalresults, analysis methodology, and first principles modeling

P. Mantica,1 T. Tala,2 J. S. Ferreira,3 A. G. Peeters,4 A. Salmi,5 D. Strintzi,6 J. Weiland,7

M. Brix,8 C. Giroud,8 G. Corrigan,8 V. Naulin,9 G. Tardini,10 K.-D. Zastrow,8 andJET-EFDA Contributorsa�

JET-EFDA, Culham Science Centre, OX14 3DB Abingdon, United Kingdom1Istituto di Fisica del Plasma, EURATOM/ENEA-CNR Association, 20125 Milano, Italy2VTT, Association EURATOM-Tekes, P.O. Box 1000, FIN-02044 VTT, Finland3Centro de Fusão Nuclear, Associação EURATOM/IST, 1049-001 Lisbon, Portugal4Centre for Fusion Space and Astrophysics, University of Warwick, Coventry 7AL, United Kingdom5Association EURATOM-Tekes, Helsinki University of Technology, P.O. Box 2200, FIN-02150 TKK, Finland6Association Euratom-Hellenic Republic, National Technical University of Athens,GR-15773 Athens, Greece7Euratom-VR Association and Chalmers University of Technology, SE-412 96 Göteborg, Sweden8Euratom/CCFE Association, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom9Association Euratom-Risø, DTU, DK-4000 Roskilde, Denmark10Max-Planck-Institut für Plasmaphysik, EURATOM/MPI Association, D-85748 Garching, Germany

�Received 12 May 2010; accepted 20 July 2010; published online 28 September 2010�

Perturbative experiments have been carried out in the Joint European Torus �Fusion Sci. Technol.53�4� �2008�� in order to identify the diffusive and convective components of toroidal momentumtransport. The torque source was modulated either by modulating tangential neutral beam power orby modulating in antiphase tangential and normal beams to produce a torque perturbation in theabsence of a power perturbation. The resulting periodic perturbation in the toroidal rotation velocitywas modeled using time-dependent transport simulations in order to extract empirical profiles ofmomentum diffusivity and pinch. Details of the experimental technique, data analysis, and modelingare provided. The momentum diffusivity in the core region �0.2���0.8� was found to be close tothe ion heat diffusivity ��� /�i�0.7–1.7� and a significant inward momentum convection term, upto 20 m/s, was found, leading to an effective momentum diffusivity significantly lower than the ionheat diffusivity ���

eff /�ieff�0.4�. These results have significant implications on the prediction of

toroidal rotation velocities in future tokamaks and are qualitatively consistent with recentdevelopments in momentum transport theory. Detailed quantitative comparisons with the theoreticalpredictions of the linear gyrokinetic code GKW �A. G. Peeters et al., Comput. Phys. Commun. 180,2650 �2009�� and of the quasilinear fluid Weiland model �J. Weiland, Collective Modes inInhomogeneous Plasmas �IOP, Bristol, 2000�� are presented for two analyzed discharges.�doi:10.1063/1.3480640�

I. INTRODUCTION

The study of plasma rotation and momentum transport intokamaks is currently experiencing an intensive experimentaland theoretical research effort, in view of extrapolatingplasma performance from present devices to future machineslike ITER.1 The role of sheared rotation in the quenching ofturbulence with subsequent improvement in confinement2–4

is well known. In addition, the stabilizing effect of toroidalrotation on pressure limiting resistive wall modes has beenpointed out,5 although recent developments including kineticeffects have eased the situation on this front.6 The lowerrotation expected as a consequence of lower torque andlarger inertia may thus be detrimental for ITER performance.However, a reliable prediction of magnitude and profile of

toroidal rotation in a device like ITER is presently still notpossible, both because understanding of momentum transportis still incomplete, although good progress has been achievedrecently, and because we lack a precise knowledge of alltorque sources and plasma momentum self-generation pro-cesses. These reasons motivate further research on momen-tum transport, in particular experimental investigationsagainst which to validate theoretical models.

The momentum diffusivity �� and pinch velocity vpinch

�negative sign denotes inward� are related to the toroidal ve-locity v�, its gradient �v�, and the momentum flux ��, as-suming the absence of a significant particle flux, as follows:

�� � − n�� � v� + n vpinchv� = − n��eff � v�, �1�

where n is the ion density. It is always possible to combinethe diffusive and convective parts of the momentum flux intoan effective momentum diffusivity ��

eff, which can be cal-culated from steady-state rotation profiles once momentumsources are known. In the following the term “steady-state”

a�See Appendix of F. Romanelli et al., Fusion Energy Conference 2008�Proceedings of the 22nd International Conference, Geneva, 2008�, IAEA,Vienna �2008�.

PHYSICS OF PLASMAS 17, 092505 �2010�

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will be used to indicate a quantity taken at a given time �ortime-averaged over a given interval�, therefore not resolvingthe modulation time dependence, while the term “stationary”will be used to indicate conditions where plasma parametersand profiles do not exhibit significant slow trends.

On the theoretical side, substantial progress has beenachieved in the understanding of momentum transport drivenby ion temperature gradient �ITG� modes.7–12 Fluid modelsand gyrokinetic codes have been developed, which link in-herently momentum and ion heat transport, foreseeingsimilar diffusivities for both channels �Prandtl number,Pr=�� /�i�1�. In addition, the recently predicted existenceof an inward momentum pinch is a qualitatively new ingre-dient in assessing the achievable rotation peaking in ITER,where torque sources will be located mainly in the outerregions. As found for particle transport, the impact of thepinch existence on the plasma profiles would be higher formomentum than for heat, for which the presence of an in-ward pinch, if any, is always dominated by the high diffusivetransport component due to the core localization of heatsources.

On the experimental side, a significant improvement inthe characterization of the steady-state behavior of toroidalrotation has been achieved, both in plasmas with externaltorque source13–16 and in plasmas without external torquesources,17,18 which still exhibit nonzero toroidal rotation, theso-called “intrinsic rotation,” whose physical origin is stillunder investigation. A rather surprising recent observationfrom steady-state momentum studies is that the effective mo-mentum diffusivity is much lower than the ion heat diffusiv-ity �Pr

eff=��eff /�i

eff�0.4 using the definition of ��, as in

Eq. �1��,13,15,16 apparently contradicting the theoretical ex-pectations. It is clear that the existence of a momentum in-ward pinch would solve the apparent contradiction, allowingfor similar diagonal diffusivities at the same time as differenteffective values. However, a direct demonstration of the ex-istence of a momentum pinch is fairly difficult from steady-state data only. A clean way of identifying separately diffu-sive and convective transport components is by means ofperturbative experiments,19 i.e., exploiting the additional in-formation contained in the dynamic response of the plasmarotation to a time variation of the torque source. Modulationat a suitable frequency has the advantage of optimizing theS/N ratio by averaging over several cycles and has been usedsuccessfully to study electron and more recently ion heattransport, and also particle and impurity transport �see Ref.20 for a recent review�. Its application to momentum trans-port studies using neutral beam injection �NBI� modulationis a recent development, with a couple of studies reported inJT-60U,21,22 one experiment reported in the Joint EuropeanTorus �JET� �Refs. 23–25�, and one in DIII-D.26 The use ofmagnetic perturbations to brake the plasma was reported inDIII-D �Ref. 27� and in the National Spherical Torus Experi-ment �NSTX� �Refs. 26, 28, and 29� as another means toinduce a transient from which it is possible to infer diffusiv-ity and convection separately. In addition, DIII-D has theunique capability of being able to zero out applied torque atconstant power.

This paper describes momentum perturbative experi-ments using NBI modulation in the JET tokamak,50 present-ing both new experiments and more refined analysis andmodeling with respect to previous work23–25 and includingdetailed comparison of the results with recent gyrokineticand fluid theory predictions. Section II illustrates the experi-ments, Sec. III discusses the calculations of the time-dependent torque deposition, Sec. IV illustrates the ratherheavy experimental methodology, based on transport model-ing required to extract from the data the estimates of momen-tum diffusivity and pinch, Sec. V summarizes the linear gy-rokinetic predictions, Sec. VI describes first attempts toperform time-dependent modeling of the rotation dynamicresponse with 1D quasilinear fluid transport models. SectionVII summarizes the results and discusses future develop-ments.

II. EXPERIMENTAL RESULTS

A. Outlook of experimental set-up

The toroidal momentum perturbative experiments havebeen carried out in low collisionality JET H-mode plasmas�BT=3 T, Ip=1.5 MA, ne0�4�1019 m−3, �n=0.4–0.5,�=0.01–0.015, P / PLH�2–3� with minimum level of mag-netohydrodynamic activity to prevent interference with theperturbation analysis, i.e., at high q95�7, to avoid large saw-teeth in the center, and with type III edge localized modes, toavoid large periodic edge crashes without changing the coretransport mechanisms. Total power levels were up to 13 MWfor NBI and 4 MW for ion cyclotron resonance heating in Hminority scheme. In these conditions, ITGs are the dominantinstability, making the coupling of momentum and ion heattransport, and thus the concept of Prandtl number, unambigu-ous. The NBI torque source is the best available tool on JETfor inducing a significant rotation perturbation, although it isnot ideal because its deposition profile is very broad, whichis a complicating factor in the analysis, unlike in the case oflocalized rf heat sources. On the other hand, the torquesources from NBI can be calculated more precisely thanradio-frequency power sources. Momentum transfer from theNBI fast ions to the thermal bulk plasma takes place via twomain mechanisms:30 �a� passing ions transfer toroidal angu-lar momentum to the bulk plasma by collisions, which is aslow process; �b� trapped ions transfer their momentum byJ�B forces, due to the �radial� displacement current thatsets in to ensure charge neutrality following the ion motionalong a banana orbit of finite width. This is practically aninstantaneous process �J denotes the displacement currentdensity and B the magnetic field�. The collisional torquedominates in the center while the J�B torque dominatesfrom midradius to the plasma edge. If the frequency of themodulation is fast enough compared to the fast ion slowingdown time, the J�B component will dominate over the col-lisional one in the perturbation source. In the analysis bothcollisional and J�B torque have been taken into account. Inthe JET experiments, the NBI power and torque were squarewave modulated with a duty cycle dc=50% or 33% at afrequency f =6.25 or 8.33 Hz, which is the highest techni-cally possible. Using the latter, J�B torque becomes the

092505-2 Mantica et al. Phys. Plasmas 17, 092505 �2010�

dominant source of torque perturbation. The modulation fre-quency is much slower than the 10 ms time resolution of thecharge exchange recombination spectroscopy �CXRS� diag-nostic used to measure the toroidal rotation profile � andion temperature Ti at 12 radial points, with an uncertainty of�5% on their absolute values and ��10% on theirgradients.31 Consequently, it is possible to observe changesin toroidal rotation in the beam on and beam off periods andperform the Fourier analysis of the modulated rotation. Themodulation took place in a stationary phase for several sec-onds following two basic schemes:

�1� Torque modulation with noncompensated power modu-lation, modulating �80 ms on, 80 ms off or 40 ms on,80 ms off� 3–4 tangential beams up to 6 MW of modu-lated power. This technique is the most straightforwardand allows a good S/N level in the toroidal rotationmodulation. As discussed above, the perturbed torquesource is the sum of the collisional and J�B torquedelivered by each tangential beam. Time traces of NBIpower, torque density at two radial locations as calcu-lated by TRANSP �Ref. 32� �see Sec. III�, toroidal an-gular rotation frequency �=v� /R �R is the major ra-dius�, and Ti and Te at midradius for nine of themodulation cycles are illustrated in Fig. 1. Steady-stateprofiles of ne, Te, Ti, �, and TRANSP calculated torquedensity are shown in Fig. 2. All spatial information inthe paper is provided using as radial coordinate the nor-malized toroidal radius �tor=�t /�t

edge, where�t= �� / B0, with � the toroidal magnetic flux, and�t

edge is the value of �t at the plasma boundary. One cansee in Fig. 1 the different time behavior of the J�B andcollisional torque, the first having the same square wave

modulation as power, due to instantaneous deposition,the second being integrated by the fast ion slowing downtime. A clear rotation modulation is visible in Fig. 1.Due to the concomitant power modulation, also the elec-tron and ion temperature are modulated, as well as thetotal energy content and plasma position, which arecomplications to be accounted for in the analysis, asdiscussed in Secs. II B and IV A 2.

�2� Torque modulation with compensated power modula-tion, modulating in antiphase three tangential and threenormal beams �50 ms on tangential, 110 ms on normal�.In this way there is no net power modulation, hence notemperature and plasma position modulation. The torquemodulation also partly cancels, the remaining torquemodulated component being due to the differences intorque deposition between tangential and normal beams.While beam particles absorbed at the high field side ofmagnetic axis become passing ions, those absorbed atthe low field side become either passing or trapped ionsdepending on the inverse aspect ratio of the flux surfacewhere they are created, �=r /R. For the parameters ofthe so-called normal neutral beams on JET, the fast ionsare passing, i.e., �=Rimp /R��t=�2� /1+� �where Rimp

is the impact radius�, for �tor�0.24, while for thetangential neutral beam bank they are passing for�tor�0.39. Therefore in a simplified picture, in the innerplasma region 0.24��tor�0.39 the perturbed sourcewill be dominated by the normal beams, while in anintermediate region outside �tor=0.39 there would becancellation of the J�B component of normal and tan-gential beams, leaving further out a J�B componentfrom tangential beams. The collisional torque compo-nent from tangential and normal beams should also tolarge extent cancel, decreasing its weight in comparisonwith the noncompensated case. Therefore this experi-

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FIG. 1. �Color online� Time traces of the modulated NBI power, �, Ti, andTe and the two different components of the torque density for JET shot66128 �noncompensated case�.

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FIG. 2. �Color online� Steady-state radial profiles at t=10 s of �, Ti, Te,ne, and torque density for JET shot 66128.

092505-3 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

ment should have resulted in a better spatial localizationof the perturbed torque source, although with somewhatpeculiar phasing, possibly allowing transport analysis ina source free region. However, as discussed more in de-tail in Sec. III on the basis of the Fourier analysis of theTRANSP calculated torque, the actual torque cancella-tion was not complete, and in addition also the S/N ofthe rotation perturbation in these experiments turned outmuch lower, as can be seen from the time traces in Fig.3 �Fourier analysis is the only way to detect the rotation

modulation�. Figure 4 shows the steady-state profiles forthis case. In conclusion this configuration did not deliverthe anticipated advantages in spite of the additionalcomplications of the experiment and has therefore beenabandoned in our later experiments on JET.

B. Fourier analysis of rotation and temperaturemodulation and corrections for oscillating plasmaposition

Standard fast Fourier transform has been applied to theexperimental time traces at various radial positions of �, Ti,and Te, to derive spatial profiles of amplitude �A� and phase��� of the perturbation at the first harmonic of the modula-tion frequency for dc=50% and also at the second harmonicfor dc=33%. The third harmonic is always below noiselevel. Using two harmonics improves the separate identifica-tion of the diffusive and convective components of the mo-mentum flux, since the impact of the momentum pinch onthe modulated data vanishes at high frequency.19 The profilesare shown in Fig. 5 for a noncompensated case at 6.25 Hz,50% dc, in Fig. 6 for a similar shot with noncompensatedmodulation at 8.33 Hz, 33% dc, and in Fig. 7 for a compen-sated case at 6.25 Hz, 33% dc. For the first noncompensatedcase also the temperature perturbations are shown, while forthe compensated case the temperature perturbation is null.Phase values are always calculated with respect to the phaseof the tangential beam power. The statistical error bars arecalculated for the amplitudes from the noise level outside thespectral peaks and for the phases by assuming that the noiselevel would add up to the signal with a 90° phase shift �worstcase�.

As mentioned in Sec. II A, in the noncompensated casethe plasma equilibrium changes periodically with the modu-lated NBI power. This is important as the experimentalCXRS data are measured at fixed positions in the laboratoryframe, resulting in a spurious measured �, Ti, and Te oscil-lation given by the product of their gradient and displace-ment �as a complex number�. This would falsify the transportanalysis if not properly taken into account. There is both arigid displacement of the whole plasma and a modulation ofthe Shafranov shift which affects only the core. They arecomparable in magnitude, each in the order of �5 mm forthe 5 MW NBI modulation in the low current plasmas usedfor these experiments. Rotation and temperature must there-fore be properly mapped into a plasma movement indepen-dent radial coordinate system before performing the transportanalysis and modeling. Here, this is done within TRANSPusing the time-dependent EFIT �Ref. 33� reconstructed equi-librium constrained by motional Stark effect measurementsto map all experimental data onto the same flux surface gridused for the torque calculation and the transport simulations.Both horizontal and vertical plasma oscillations are takeninto account. The outcome of the correction procedure isillustrated in detail in Figs. 5�a� and 5�b�, where the correctedA and � profiles are shown by dashed lines and compared touncorrected values in Fig. 5�a� for � and in Fig. 5�b� for Ti.

10864444240

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FIG. 3. �Color online� Time traces of the total NBI power, �, Ti, and Te at�=0.3 and the total torque density at �=0.2 and �=0.5 for JET shot 73700�compensated case�.

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Torque

FIG. 4. �Color online� Steady-state radial profiles at t=6.1 s of �, Ti, Te,ne, and torque density for JET shot 73700.

092505-4 Mantica et al. Phys. Plasmas 17, 092505 �2010�

The details of Te modulation are not important for the fol-lowing analysis and modeling, so only the uncorrected dataare shown for the sake of information. The effect of oscillat-ing displacement is found rather important on core Ti oscil-

lations but minor on � oscillations. This is due to the higherpeaking of the Ti profile compared to the � profile. The datain Fig. 6 are already corrected. This correction is obviouslynot needed in the compensated case.

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FIG. 5. �Color online� Radial profiles of A and � at 6.25 Hz in shot 66128for �a� �, �b� Te, and �c� Ti. In �a� and �c� the data corrected for oscillatingplasma position are shown.

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092505-5 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

It can be noted in Fig. 5�a� that the rotation perturbationA and � profiles are highly influenced by the nonlocalizedsource profiles, made by two different components with atime shift, rather than by transport processes only, at variancewith most perturbative heat transport experiments. In fact theamplitude is maximum in the center, corresponding to thecollisional torque source which is dominant there at 6.25 Hz,but the phase is minimum around ��0.6, which is the regionwhere the collisional torque has vanished and only the in-stantaneous J�B torque is acting. The increase in � towardthe center is not only due to propagation of the J�B rotationwave, but also to the high phase values associated to the fastion slowing down that provides the central collisional torquesource. This is a clear indication that, due to the nonlocaliza-tion of the modulated torque source, a simple determinationof the momentum diffusivity and pinch directly from thespatial derivatives of the amplitude and phase of the modu-lated � is not viable. Therefore, time-dependent transportmodeling of � is required to extract the transport propertiesfrom the plasma dynamic response, assuming that torquesources are known with reasonable accuracy from numericalcalculations.

Figure 7 illustrates the measured A and � for the com-pensated experiment. The results are in line with the expec-tation of a much narrower torque source in the center, sinceat midradius �0.3��tor�0.7�, there is a zone of partial can-cellation between normal and tangential beams. However,a priori we would have expected a much clearer separationof normal and tangential peaks in the amplitude profiles �seediscussion on torque source in Sec. III A�. The transitionfrom normal to tangential torque component is clearly visiblein the large increase of phase between �tor=0.2 �where nor-mal beams dominate� and �tor=0.7 �where tangential beamsdominate, although with much smaller amplitudes�, due tothe 180° phase jump in the torque of the two sources. Thevery external region is again dominated by normal beams, ascan be seen from the phase values in the region �tor�0.8.This could be due to a larger deposition of trapped particlesin the external region by the normal beams. This region��tor�0.8� is outside the scope of this paper which focuseson core momentum transport. Also TRANSP torque calcula-tions become less reliable in this region where the plasmaprofiles are not known with similar accuracy as in the core.

III. NUMERICAL COMPUTATIONOF THE TIME-DEPENDENT TORQUE SOURCE

A. Power and torque calculations using TRANSPand ASCOT

Due to the absence of a source free region where thetransport analysis could be done independently of the detailsof the source, the calculation of the time-dependent torqueprofile is an essential step for the derivation of the momen-tum transport coefficients from the data. The transport analy-sis will in fact rely on full transport simulations in which thesource is input as known, with the obvious consequence thatany error in the source will determine an error on the derivedtransport coefficients. Great care has then been dedicated tosuch torque calculations and associated uncertainties. In the

following we will consider the NBI as the only torquesource, since residual stress or other intrinsic sources aredefinitely negligible in the presence of 10–15 MW NBIpower, as discussed in Ref. 25.

The TRANSP code32 has been used to calculate NBIpower and torque profiles as a function of time, given thetime-dependent experimental profiles of all plasma param-eters, but neglecting the small temporal oscillations ofplasma parameters due to the NBI modulation itself, to avoidpolluting with noise the main torque oscillation coming fromthe NBI oscillating waveform. Ti has been taken fromCXRS, Te from electron cyclotron emission radiometer andlight detection and ranging �LIDAR�, ne from interferometerand LIDAR, Zeff from CXRS and spectroscopic measure-ments �Zeff�2 in these discharges, a radially constant profilehas been assumed�. Error bars are typically �5% for Ti andTe, �10% for ne, �20% for Zeff. The internal TRANSP equi-librium solver with EFIT data as boundary condition wasused to calculate equilibrium consistently with the slow timevariations of profiles. The fast equilibrium variations due tomodulation have been neglected because as discussed in Sec.II B their effect has been removed from the data before theyare compared with simulations. TRANSP contains the NU-BEAM Monte Carlo solver,34 which has always been usedwith a high number �160 000� of particles. A test of the sen-sitivity of torque amplitudes and phases to the number ofMonte Carlo particles has shown that in excess of 80 000particles are needed to reach convergence in the results. Thetorque calculations are reliable up to �tor=0.8, due to uncer-tainties in the plasma parameters in the external region.

Figure 8 shows A and � of the NBI total torque densityand of the ion and electron power densities calculated forshot 66128. The steady-state torque density for this shot wasalready shown in Fig. 2. The central torque density perturba-tion is typically in the order of 10% while the central powerdensity perturbation is �3%. One can notice the zero-phase,instantaneous torque deposition for �tor�0.4 due to theJ�B torque, while in the central region the main torquecomponent is the delayed collisional one. The ion and elec-tron power phases also indicate collisional deposition fromfast NBI ions to thermal population, with longer collisionaltimes to ions than to electrons.

Figure 9 shows A and � of the calculated torque densityfor shot 73701, at two harmonics of the modulation fre-quency, indicating separately the J�B and collisional com-ponents. In this shot due to higher modulation frequency�8.33 Hz rather than 6.25 Hz of shot 66128�, the J�B torquedominates at all radii but the very central region.

In order to gain confidence in the correctness of theTRANSP NUBEAM calculations, the modulated torque for73701 has been calculated also with the orbit followingASCOT code.35 ASCOT is similar to NUBEAM modulewithin TRANSP. Both codes are guiding center followingMonte Carlo codes with internal NBI birth profile generator.In ASCOT calculations the small temporal variations in equi-librium due to the NBI modulation are neglected and con-stant EFIT reconstructed equilibria with motional Stark ef-fect �MSE� constraint are used together with time-averagedplasma profiles. Time varying NBI waveforms provide the

092505-6 Mantica et al. Phys. Plasmas 17, 092505 �2010�

dynamics in torque profiles. The results for A and � areshown in Fig. 10, again indicating the J�B and collisionalcomponents. Figure 11 compares the steady-state torque den-sity calculated with TRANSP and ASCOT. The agreementbetween the two very different and independent codes is re-

markable and constitutes the most powerful confirmation thatwe can rely on the torque calculations for our transportanalysis.

Finally, the A and � of modulated torque density for thecompensated case, shot 73700, are shown in Fig. 12 calcu-

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FIG. 8. �Color online� Radial profiles of A and � at 6.25 Hz in shot 66128 �noncompensated� for TRANSP calculated: �a� total torque density �black full line:A, red dashed line: ��; �b� ion power density �black full line: A, red dashed line: �� and electron power density �gray dashed-dotted line: A, green dashed line:��.

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FIG. 9. �Color online� Radial profiles of A and � of TRANSP calculated torque density at 8.33 and 16.66 Hz in shot 73701 �noncompensated�, distinguishingcollisional and J�B components.

092505-7 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

lated with TRANSP and in Fig. 13 calculated with ASCOT,and indicating separately the J�B and collisional compo-nents. To help visualization of the complex torque dynamics,in Fig. 14 we also show the A and � of the torque densitycalculated with ASCOT indicating separately the normal andtangential beams. The phase is always taken with respect tothe tangential beams. One can see that the peak in A of theperpendicular beams is shifted slightly inside with respect tothe one of the tangential beams, and the two are in phaseopposition, so that there is a partial cancellation leading to amore narrow total deposition �with respect to the noncom-pensated case� dominated by the perpendicular beams in thecentral region. However at the frequency used �6.25 Hz�there is still a significant central collisional component thatcomes mainly from the tangential beams. Moreover, the factthat the cancellation in the external region is not perfectleads to the impossibility of considering such region assource free for the analysis, which in practice nullifies theusefulness of this approach, as will be discussed in detail inSec. IV B.

B. Analysis of sensitivity to plasma parameters

Having assessed by comparing two different codes thereliability of the torque calculations for a given set of plasmaparameters and profiles, in this section we deal with the as-sessment of uncertainties in torque following from uncertain-

ties in plasma parameters. We have evaluated the impact ofthe most important players, i.e., ne, Te, Zeff. The J�B torqueis essentially instantaneous and depends only on the initialdeposition �governed by ne and Zeff to lesser extent�. Te andZeff affect the slowing down rate and ion to electron heatingratio and thus the torque deposition dynamics. Ti is not avery sensitive player as it has practically no role in the initialdeposition and only affects the ion slowing down at lowenergies, but has no effect on electron slowing down andthus its overall effect on torque dynamics is usually small. Ofcourse, additionally, NBI acceleration voltage and beam frac-tions play a role but these can be assumed to be known.

Figure 15 shows the variation of A and � of the torquedensity when �a� ne is varied between 0.8 and 1.2 of theexperimental value, �b� Te is varied between 0.8 and 1.2 ofthe experimental value, �c� Zeff is decreased from the experi-mental value of 1.9 to 1. Once can see that the impact is onlyon amplitudes, not on phases. As we will discuss inSec. IV A 1, this implies that the momentum diffusivity, orPrandtl number, is a robust quantity, insensitive to errors inplasma parameters. The pinch value, instead, depends on thetorque amplitudes and therefore its determination will be af-fected by larger uncertainties, reflecting those in the mea-sured plasma parameters. Among these, the main impact isgiven by density, with a variation of about �15% when ne isvaried in an interval of �20% of the measured value. Te and

0.2 0.4 0.6 0.80

A(torque)(N/m

2 )A(torque)(N/m

2 )

ϕ(torque)(deg)

ϕ(torque)(deg)

ρtor0.2 0.4 0.6 0.8 1.01.0

ρtor

f = 8.33HzTotalJ × BCollisional

f = 16.66HzTotalJ × BCollisional

0

0.04

0.04

0.02

0.02

100

50

0

100

50

0

FIG. 10. �Color online� Radial profiles of A and � of ASCOT calculated torque density at 8.33 and 16.66 Hz in shot 73701 �noncompensated�, distinguishingcollisional and J�B components.

092505-8 Mantica et al. Phys. Plasmas 17, 092505 �2010�

Zeff have much smaller impact. This is due to the fact that inthis discharge the dominant torque is the J�B, which ismainly affected by ne through a change in ionization profile,while Te and Zeff affect the slowing down time of the ions,i.e., only the collisional part of the torque. Particular care hasthen to be paid in assessing the best ne profile. For theLIDAR measurements used in this paper the uncertainty onne is �10% and this has been further improved following therecent introduction of ne measurements using high resolutionThomson scattering ��ne� �5%�. Therefore, we do not ex-pect uncertainties larger than �4%–8% in torque profilesassociated to uncertainties in plasma parameters. Taking intoaccount the uncertainties on Ti, rotation, ne, torque, and ion

heat fluxes discussed above, we can estimate that �� and �i

are separately determined with ��18% accuracy and theirratio with ��25% accuracy.

IV. TRANSPORT ANALYSIS OF TIME-DEPENDENTTOROIDAL ROTATION USING JETTO

A. The noncompensated modulation case

1. Empirical transport model with diffusivity constantin time

The power and torque sources calculated by TRANSPand described in Sec. III have been used in the transportanalysis of toroidal rotation using the 1.5D code JETTO.36

The transport equation for � is solved while q-profile, Ti,Te, and ne are frozen to their experimental values. Theboundary conditions for steady-state �, amplitudes A���and phases ���� of the modulated � are chosen to fit theexperimental data at �=0.8 as the edge plasma transport isbeyond the scope of interest in this study. The transportsimulations for shot 66128 are carried out over the ninemodulation cycles shown in Fig. 1.

The transport model for toroidal momentum is an em-pirical one, constant in time, featuring a diagonal momentumdiffusivity given by ��= Pr ·�i

eff �where Pr is the Prandtlnumber and �i

eff is the effective ion heat diffusivity, which isa good estimator of the diagonal �i, because there is no ex-perimental evidence of an ion heat pinch in JET from exist-ing Ti modulation experiments37,38� and a pinch velocityvpinch. The values of Pr and vpinch are chosen in such a waythat they fit the available data with sufficient accuracy ac-cording to visual inspection �in principle their profiles arecompletely free with no outside constraints�.

The simulations are run with three logical steps: �1� runa fully interpretative simulation to calculate �i

eff; �2� run pre-dictive simulations for � varying the Pr value and profileuntil a reasonable reproduction of the experimental phaseprofile of the modulated � is obtained, as the phase is ratherinsensitive to vpinch; and �3� vary vpinch value and profile toreproduce also the amplitude of the modulated � and itssteady-state profile. Step 1 is straightforward as �i

eff can becalculated from the measured Ti data and calculated powerdeposition profiles. �i

eff is shown in Fig. 16. Step 2 leads toa rather precise identification of the acceptable range of Pr

values, since Pr is the only unknown �the sources are takenfrom the NUBEAM calculations�. This resolves the indeter-minacy associated with the analysis of only the steady-stateprofile, as the latter can be reproduced by an unlimited num-ber of possible combinations for �� and vpinch yielding thesame ��

eff. Once Pr is identified, step 3 allows us to identifyalso the convective component that enables us to reproducethe steady-state � and amplitude with the chosen Pr. As aninitial choice, Pr was taken uniform along radius, and as arefinement, Pr has been chosen to have a radial profile, assuggested from the gyrokinetic simulations described in Sec.V, but most of all because it provides a better fit to most ofthe shots analyzed.

In Fig. 17 we show the simulations of the modulateddata of shot 66128 corresponding to the two most obviousoptions for �� �i.e., Pr� and vpinch: in Fig. 17�a� we fix

0

0.2

0.4

0.6

0.2 0.4 0.6 0.80 1

torque

(N/m2)

Total

JxBColl

ρtor

(a)

0

0.2

0.4

0.6

0.2 0.4 0.6 0.80 1

torque

(N/m2)

Total

JxBColl

ρtor

(b)

FIG. 11. �Color online� Radial profiles of steady-state torque density in shot73701 �noncompensated�, distinguishing collisional and J�B components,calculated by �a� TRANSP and �b� ASCOT.

092505-9 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

Pr=0.5 to yield �����eff and vpinch=0 while in Fig. 17�c�

we follow the above described best-fit procedure, whichleads to Pr�2 as the best value to fit the phase profiles, andto the vpinch profile shown in Fig. 16, with values up to 20m/s, to fit amplitudes as well as steady-state. It can be seen inFig. 17�b� that both simulations predict the steady-state �

within 10% accuracy in the region of interest, i.e., 0.2��tor�0.8. Inside �tor�0.2, neoclassical transport starts todominate ion heat transport, and the predictions are worse asthe use of the ITG based Pr for calculating � is not appro-priate. In Figs. 17�a� and 17�c� one can see that the twooptions differ in reproducing the A��� and ���� profiles.The case with Pr=0.5 and vpinch=0 clearly disagrees with theexperiment. The simulated phase is too large inside and thisis an indication of too low ��, i.e., too low Pr used in thesimulation. On the other hand, the simulated amplitude is toolow toward the plasma center, which could only be cured bylowering �� further. This shows that a model with vpinch=0 isnot compatible with the experimental data. In the secondcase, the agreement between the simulated and experimentalamplitudes and phases improves dramatically. Using Pr=2the �� is now large enough to yield perfect match of thephase profile, and the pinch is required to provide the ob-served amplitude peaking. This is a direct indication of theexistence of a convective term in the toroidal momentumtransport. The same simulation using, instead of uniform Pr,

a radially varying Pr profile, as shown in Fig. 16, and withthe same vpinch gives in this case as good agreement withexperimental A and � as the Pr=2 simulation �Fig. 17�d��,but a better steady-state rotation profile. A sensitivity analy-sis using the experimental uncertainties on A and � and onplasma parameters and torque source shows that the range ofPr and vpinch values compatible with such uncertainties is inthe range of 20%–30%, outside which the simulated phase,amplitude, and steady-state deviate unacceptably from theexperimental values.

Another example is the simulation of discharge 73701�two harmonics available� shown in Fig. 18. This has beensimulated using the same procedure and the profiles of ��

and vpinch derived from the best-fit procedure are shown inFig. 19. Again, a high Pr number and a significant pinch arerequired to fit the experimental data.

2. Effect of fluctuating �i due to Ti oscillations

A significant complicating factor in the noncompensatedexperiment is the Ti modulation induced by the NBI powermodulation. In fact, being the ITG the supposed drive oftoroidal momentum transport, a time varying Ti and/or R /LTi

implies a time varying ITG driven transport, both for ionheat and momentum. The induced oscillation in �� producesa component in the rotation modulation, adding an extra con-

0

0

0.2 0.4 0.6 0.80

A(torque)(N/m

2 )A(torque)(N/m

2 )

ϕ(torque)(deg)

ϕ(torque)(deg)

ρtor0.2 0.4 0.6 0.8 1.01.0

ρtor

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.05f = 6.25HzTotalJ × BCollisional

200

150

100

50

0

200

150

100

50

0

f = 12.5HzTotalJ × BCollisional

FIG. 12. �Color online� Radial profiles of A and � of TRANSP calculated torque density at 6.25 and 12.5 Hz in shot 73700 �compensated�, distinguishingcollisional and J�B components.

092505-10 Mantica et al. Phys. Plasmas 17, 092505 �2010�

tribution to the A��� and ���� profiles, which, if not prop-erly taken into account, may lead to an erroneous evaluationof the Pr and vpinch values. Therefore it is important to assessquantitatively the relevance of this mechanism on the resultsof Sec. IV A 1.

As seen in Figs. 5�b� and 5�c� for shot 66128, the ion andelectron temperatures are modulated with peak amplitudesaround 70 eV, i.e., a perturbation of �1%, while the modu-lation amplitude in � is around 4%. To estimate the impactof such Ti modulation on the determined Pr and vpinch, atime-dependent �i has been calculated for shot 66128 usingthe critical gradient model �CGM� model based on the criti-cal gradient length concept.39 This model has no implicationson momentum transport and here it has been used only toevaluate the expected modulation in �i �and through Pr in��� because it is a model commonly used in JET to investi-gate ion heat transport from Ti modulation experiments.37,38

Typical values for ion threshold and stiffness found in JET inshots similar to the ones discussed here have been used tomodel the modulated Ti and the associated time variation of�i and ��. The reason why we did not use directly the time-dependent experimental �i

eff instead of the one modeled byCGM is the very high level of noise that is fed into to theexperimental �i

eff by the experimental determinationof the Ti gradient. The same noise problems hold for the

determination of a time-dependent �� directly from the mo-mentum flux and rotation gradient, with in addition the com-plication of having a non-negligible and unknown convectivepart. Figure 20 shows the time traces of �i at different radii,Fig. 21�a� the simulated Ti modulation, and Figs. 21�b� and21�c� the � modulation, with the same two assumptions forPr and vpinch as in Figs. 17�b� and 17�d�. One can see that thereproduction of Ti modulation in Fig. 21�a� is satisfactory,given the large uncertainties associated with the Ti modula-tion data after subtraction of the oscillating displacement ef-fect, which for Ti is significant, as was shown in Fig. 5�b�.This allows us to assume that the level of oscillation of �i inFig. 20 is representative of the amount of effect to be ex-pected, even if we cannot have a precise determination ofthreshold and stiffness level for this shot. One can also seethat the simulations of � with oscillating �i �and conse-quently ��� in Figs. 21�b� and 21�c� do not differ signifi-cantly from their counterparts with constant �i in Figs.17�b�–17�d�. One can conclude that, owing to the small am-plitude of the Ti modulation and of the induced �i and ��

modulation, the effect on the values determined for Pr andvpinch is insignificant. The same conclusion is actually de-rived from the first principles simulations using the quasilin-ear fluid Weiland model40 discussed in Sec. VI.

0.2 0.4 0.6 0.80

A(torque)(N/m

2 )A(torque)(N/m

2 )

ϕ(torque)(deg)

ϕ(torque)(deg)

ρtor0.2 0.4 0.6 0.8 1.01.0

ρtor

f = 6.25HzTotalJ × BCollisional

f = 12.5HzTotalJ × BCollisional

200

150

100

50

0

200

150

100

50

0

0

0

0.04

0.04

0.02

0.02

FIG. 13. �Color online� Radial profiles of A and � of ASCOT calculated torque at 6.25 and 12.5 Hz in shot 73700 �compensated�, distinguishing J�B andcollisional beam components.

092505-11 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

B. The compensated modulation case

The same procedure described in Sec. IV A 1 has beenapplied to the compensated modulation case, shot 73700,using the TRANSP time-dependent torque described in Sec.III. In this case the logic behind the best-fit is however com-pletely different, because it is no longer true that phases areuniquely determined by diffusivities. In fact, best-fitting thedata has turned out very cumbersome, because the phaseprofile is the result of two propagating waves �normal andtangential beams� of modulated rotation, in antiphase witheach other, so that the amplitude of each component plays adominant role in determining the resultant phase, and there-fore momentum pinch enters in the phase dynamics as well.The core being dominated by the normal beams, while theouter region by the tangential, the rapidity of the increase by180° of the phase between the two components, dependscrucially on the Prandtl number: low Pr number yields a veryfast transition, while high Pr number a smooth one, with thepinch value determining the precise phase profile. In addi-tion, a low Pr number would yield two very well definedpeaks of amplitudes, while a high Pr number would tend tosmooth them down. It is clear from the data shown in Fig. 7that experimentally the transition is very smooth, and there isno pronounced peak of the tangential component, which are

nontrivial confirmations of the presence of a high Pr number�implying then a pinch in order to match the steady-state�.The best simulation achieved for the compensated shot73700 is compared against the data in Fig. 22 and shows afair agreement in profile and magnitude for modulated andsteady-state data. The Pr numbers and pinch profiles used inthe simulation are shown in Fig. 23. We conclude that thecompensated modulation supports the previous conclusionsabout the existence of a pinch. In general, however, this pro-cedure has turned out too cumbersome to be used for a sys-tematic study of several shots such as when performing para-metric scans. In addition, we note that the S/N level in suchcompensated modulation is much lower than in the noncom-pensated case, as amplitudes are reduced due to compensa-tion. In particular outside �=0.5, which is the region wherethe pinch is expected to be largest, the signal is very near tonoise level, so the sensitivity of this technique to the pinchvalue is reduced. Since on the other hand the problem ofmodulated power is not found to seriously affect the data ofthe noncompensated case, as discussed in Sec. IV A, thecompensated technique has been abandoned at JET and thenoncompensated one has been used for further studies ofparametric dependencies of momentum transport, which willbe the object of a future paper.

0.2 0.4 0.6 0.80

A(torque)(N/m

2 )A(torque)(N/m

2 )

ϕ(torque)(deg)

ϕ(torque)(deg)

ρtor0.2 0.4 0.6 0.8 1.01.0

ρtor

f = 6.25HzTotalPerpTang

f = 12.5HzTotalPerpTang

200

150

100

50

0

200

150

100

50

0

0

0

0.04

0.04

0.02

0.02

FIG. 14. �Color online� Radial profiles of A and � of ASCOT calculated torque at 6.25 and 12.5 Hz in shot 73700 �compensated�, distinguishing tangentialand perpendicular beam components.

092505-12 Mantica et al. Phys. Plasmas 17, 092505 �2010�

V. THEORETICAL PREDICTIONS OF TOROIDALMOMENTUM TRANSPORT FROM LINEARGYROKINETIC SIMULATIONS

Effective transport of momentum needs, like all turbu-lent transport, a phase shift between the turbulent velocityfluctuations and the momentum fluctuations. Such a phaseshift is established by a finite average rotation but also gen-erally through symmetry breaking, in which case residualstresses or asymmetry momentum fluxes arise.7,9 For negli-gible residual stress terms, the resulting equation for thetransport of momentum can be directly written as the sum ofa diffusive and pinch contribution in the local limit. Off-diagonal terms, i.e., a momentum flux directly driven by thedensity and temperature gradients, have been investigated inglobal simulations in Ref. 41 and were found to be one ordersmaller in normalized Larmor radius and therefore negligiblefor JET parameters.

Gyrokinetic studies of the diffusion coefficient �� arepresented in Refs. 8 and 11, whereas the pinch term vpinch

was derived in Ref. 9 using the comoving reference frame�for more details on the derivation see Ref. 42�. In the co-moving frame the toroidal rotation enters the equations onlythrough the Coriolis drift velocity. Consequently, any toroi-dal momentum pinch is directly related to the Coriolis drift.Since this drift, unlike the drifts due to the magnetic fieldinhomogeneity, is linear in the parallel velocity, it generates acoupling between density/temperature perturbations and theperturbations in the parallel velocity. Over this coupling thetemperature perturbations of the ITG generate parallel veloc-ity perturbations that are then transported by the E�B ve-locity leading to a finite radial flux of parallel momentum.Furthermore, the drift in the electrostatic potential perturba-tions accelerates the particles and also leads to the generationof parallel velocity fluctuations and consequently to a mo-mentum flux. Although no complete expression for the fluxhas been derived for the laboratory frame, the latter effect isrelated to the turbulent equipartition �TEP� �Ref. 43� theoryand has recently been worked out in Ref. 10. Obviously, thecalculated radial fluxes should be frame independent and itcan be shown that the TEP contribution is incorporated in thedescription that uses the Coriolis drift �see Ref. 44�.

The gyrokinetic flux tube code GKW �Ref. 45� is usedhere to calculate the values of the momentum transport co-efficients under experimental conditions. All parameters aretaken directly from the experiment, but the following ap-proximations have been made. �a� The geometry is approxi-mated using the s-� model, with the circular radius chosensuch that the number of trapped particles is roughly equal tothe experimental case. This approximation is motivated byhaving found negligible impact of full with respect to circu-lar geometry on the ITG simulations made for very similardischarges within the ion heat transport studies described in

JXB

JXB

Coll

CollA(troque)(N-M/M

3 )

ϕ(torque)(deg)

0

0.04

0.03

0.02

0.01

150

-100

0

(a)

JXB

JXB

Coll

Coll

0

-100

A(troque)(N-M/M

3 )

ϕ(torque)(deg)

0

0.04

0.03

0.02

0.01

150(b)

JXB

JXB

Coll

CollA(troque)(N-M/M

3 )

ϕ (torque)(de g)

0

0.04

0.03

0.02

0.01

0

150

-100

(c)

ρtor0 0.40.2 0.6 0.8

FIG. 15. �Color online� Radial profiles of A and � of TRANSP calculatedtorque at 8.33 Hz in shot 73701 with �a� ne varying between 0.8 and 1.2 ofexperimental value; �b� Te varying between 0.8 and 1.2 of experimentalvalue; and �c� Zeff varying between 1.9 and 1. Red full lines are A ofJ�B torque, blue dashed lines are A of collisional torque, violet dashed-dotted lines are � of J�B torque, and green dotted lines are � of collisionaltorque. The arrows indicate the relevant changes due to parameter variation.

20

15

10

5

3

2

1

00

25

0.2 0.4 0.6 0.80

-vpinch(m/s),

χ i(m

2 /s)

-vpinch

P r

Pr

ρtor

χi

FIG. 16. �Color online� Radial profiles of �i,eff, vpinch, and Pr used in the

simulations of shot 66128 discussed in the text.

092505-13 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

Ref. 37. �b� A pure deuterium plasma is considered andCoulomb collisions are neglected. The latter approximationis motivated by the small influence of the collisions.46 Thesimulations presented here have been performed for the mostunstable mode k��s=0.3.

The Pr number ��� /�i� and the pinch number�Rvpinch /��� profiles calculated by GKW are compared withthe experimental ones in Figs. 24 and 25 for the JET pulses66128 and 73701, respectively. The determination of bothPrandtl number and pinch requires two simulations per radialposition: one with zero rotation �v�=0� and one with zerogradient �grad �=0�. In the latter case the diagonal contri-bution is zero, and hence the momentum flux, which is ob-tained from the code directly, gives the pinch contribution�vpinch=�� /nv��, whereas in the former case the pinch con-tribution is zero and the momentum flux gives informationon the diffusion coefficient ���=−�� /n grad v��. Of course,this procedure assumes that the fluxes are linear in both v�

and grad �, an assumption that is well satisfied as shown inRef. 9.

One can see that for shot 73701 �low density� the pre-dicted and experimental Prandtl and pinch numbers are ingood agreement with experiment, although only a statisticalstudy outside the scope of this work would be able to assesswhether such agreement is real or fortuitous. However inshot 66128 �higher density� experimentally the reduction in�i due to increased density is larger than the one in ��, thus

2000

1500

1000

500

200

150

100

50

00

2500

0.2 0.4 0.6 0.80

A(ωφ)(rad/s)

ϕ(ω

φ)(deg)

ρtor

1

2

3

ExperimentPr (r) with pinchPr = 0.5 no pinchPr = 2 with pinch

4

5

0

6

ωφ(rad/s)(104)

2000

1500

1000

500

200

150

100

50

00

2500

0.2 0.4 0.6 0.80ρtor

A(ωφ)(rad/s)

ϕ(ω

φ)(deg)

2000

1500

1000

500

200

150

100

50

00

2500

A(ωφ)(rad/s)

ϕ(ω

φ)(deg)

(a) (b)

(c) (d)

FIG. 17. �Color online� Radial profiles of experimental �dots� and simulated �lines� A �black squares� and � �red circles� of � in shot 66128 using�a� Pr=0.5 and vpinch=0 m /s; �c� Pr=2 and vpinch as shown in Fig. 16; �d� a Pr profile increasing with radius �Fig. 16� and vpinch as shown in Fig. 16. In�b� the radial profiles of experimental and simulated angular rotation using the three different transport options are shown.

1500

200

150

100

150

50

0

0

100

50

1000

500

0

500

1000

1500

0

60

40

20

0

2000f = 8.33Hz

f = 16.66Hz

Steady-state

0.25 0.500 0.75

ωφ

(krad/s)

A ωφ

(rad/s)

ϕ ωφ

(deg)

ρ

FIG. 18. �Color online� Radial profiles of experimental �dots� and simulated�lines� A �black squares� and � �red circles� of � at two harmonics of themodulation frequency and steady-state � in shot 73701. The simulationderives �by best-fitting the data� the Pr and vpinch profiles, as shown inFig. 19.

092505-14 Mantica et al. Phys. Plasmas 17, 092505 �2010�

giving origin to higher experimental Pr and pinch numbers,while the theory prediction remains basically unchanged.Therefore a discrepancy by a factor of 1.5–2 is present in thislatter shot between GKW estimates and experiment. Thisseems a trend with density which however will be furtherinvestigated with more shots in a future work. On the otherhand, the theoretical description of momentum transport isstill evolving and at present it may already be quite satisfac-tory to find an agreement with experiment within a factor of2. Indeed more than one reason can be put forward. Nonlin-

ear effects, for instance, might lead to different transport co-efficients. Furthermore, several effects such as the E�Bshearing,47 the current symmetry breaking,48 and the residualstress49 are not kept in the current description. It is clear thenthat further theory development and also more extensivevalidation against data are required, in view of improvingour predictive capabilities. It is also fair, however, to recog-nize that theory has made substantial steps forward in a fieldlittle explored until recently, and for which the exercise ofvalidation against data has just started. The agreement withina factor of 2 found within this first and challenging exerciseof validation against perturbative momentum transport ex-periments is therefore encouraging.

Finally we remark that with linear simulations only nor-malized quantities can be compared with experiment, but notabsolute values of transport coefficients. This will have towait for nonlinear gyrokinetic simulations of momentumtransport. This observation justifies the attempt made in Sec.VI to compare with experiment the predictions of the quasi-linear fluid Weiland model, which can be used for a fulltransport simulation of the time evolution of rotation, therebyallowing a complete comparison with experiment.

20

15

10

5

4

1

2

3

00

25

0.2 0.4 0.6 0.80

-vpinch(m/s),χ φ,χ

i(m2 /s)

-vpinch

P r

Pr

ρtor

χi

χφ

FIG. 19. �Color online� Radial profiles of ��, �i, −vpinch, and Pr from theempirical simulation of shot 73701.

0.10.20.30.40.50.60.70.80.9

8

1

9

9.8 10.0 10.2 10.4 10.6 10.8

6

7

5

3

2

4

11.0

χ i(m

2 /s)

Time (s)

FIG. 20. �Color online� Time traces of �i at different radii as calculated forshot 66128 using the CGM model with typical values of ion threshold andstiffness derived from previous ion heat transport studies.

80(a)

(b)

(c)

60

40

20

200

150

100

50

00

0

100

A(Ti)(eV)

ϕ(Ti)(deg)

2000

1500

1000

500

150

100

50

0

ϕ(ω

φ)(deg)

2000

1500

1000

500

150

100

50

00 0.2 0.4 0.6 0.80

A(ω

φ)(rad/s)

A(ω

φ)(rad/s)

ϕ(ω

φ)(deg)

ρtor

FIG. 21. �Color online� Radial profiles in shot 66128 of experimental �dots�and simulated �lines� A and � of �a� Ti using �i from the CGM model; �b�� using Pr=0.5 and vpinch=0 m /s and �i from the CGM model; and �c� �

using the Pr profile increasing with radius �Fig. 16� and vpinch as shown inFig. 16 and �i from the CGM model.

092505-15 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

VI. TIME-DEPENDENT FLUID MODELING

This section describes first attempts to do time-dependent simulations of the JET NBI modulation experi-ments using the quasilinear toroidal drift wave fluid modeldescribed in Ref. 40, which due to the inclusion of a spacedependent nonlinear frequency shift �giving zonal flows� isno longer quasilinear, in which a new treatment of momen-tum transport has been recently implemented,12 making itunique in the class of quasilinear momentum transport mod-

800

600

400

200

0

800

600

400

200

0

300

0

200

100

0

300

400

200

100

6

4

2

0

1000f = 6.25 Hz

f = 12.5 Hz

Steady-State

0.2 0.4 0.60 0.8

ωφ(rad/s10

4 )A(

ωt)(rad/s)

ϕ(ω

φ)(deg)

ϕ(ω

φ)(deg)

A(ω

φ)(rad/s)

ρtor

FIG. 22. �Color online� Radial profiles of experimental �dots� and simulated�lines� A �black squares� and � �red circles� of � at two harmonics of themodulation frequency and steady-state � in shot 73700 �compensatedcase�. The simulation derives �by best-fitting the data� the Pr and vpinch

profiles shown in Fig. 23.

20

15

10

5

4

3

2

1

00

25

30

0.2 0.4 0.6 0.80

-vpinch(m/s),χ

φ,χ i(m

2 /s)

-vpinch

P r

Pr

ρtor

χi

χφ

FIG. 23. �Color online� Radial profiles of ��, �i, −vpinch, and Pr from theempirical simulation of shot 73700.

4

2

2.0

1.0

1.5

0.5

00

6

0.2 0.4 0.6

GKWExpt

0.80

-Rv pinch/χ

φ

P r

Pr

Pr

ρtor

-Rvpinch / χφ

-Rvpinch / χφ

FIG. 24. �Color online� Comparison between experiment and linear gyroki-netic simulations using GKW of the radial profile of Pr and −Rvpinch /�� forshot 66128.

3

1

2

2.0

1.0

1.5

0.5

00

4

0.2 0.4 0.6 0.80

-Rv pinch/χ

φ

-Rvpinch / χφ

P r

Pr

ρtor

GKWExpt

FIG. 25. �Color online� Comparison between experiment and linear gyroki-netic simulations using GKW of the radial profile of Pr and −Rvpinch /�� forshot 73701.

092505-16 Mantica et al. Phys. Plasmas 17, 092505 �2010�

els available for time-dependent simulations of rotation. Newaspects are the inclusion of stress tensor effects for the tor-oidal momentum.9–12 As it turns out, toroidal effects from thestress tensor9 are important for the parallel ion motion whichwe here use as an approximation for the toroidal motion. Thetoroidal curvature effects enter both for the diagonal andconvective parts of the momentum transport:

m �

�t+ �vE + 2vD� · �v�

= e�E� + �v�xB� ��� −1

ne�� · �P −

1

n� · �off. �2�

Here the diagonal contributions from the stress tensor wereincluded explicitly but the off-diagonal contributions are stillkept in the remaining part �off of the stress tensor. The di-agonal element of the toroidal momentum transportbecomes9,12

�� =�3/k2

�r − 2Di�2 + �2 . �3�

The convective fluxes are here taken from the kinetic formu-lation in Ref. 10. The same fluxes have, however, recentlybeen derived from a fluid formulation in Ref. 12. They aredivided into TEP �Refs. 10 and 44�

�TEP = Re� k��cs2 − V�0�De

− 2�Di�̂vEr

�� �4�

and thermodiffusion

�term = Re� k��cs2 − V�0�De

− 2�Di

�p

�PvEr

�� . �5�

The magnetic drift frequency has here been averaged overthe mode profile as indicated by the bar in �Di. Moreover k�� has been averaged over the radial mode profile. It isnonzero due to symmetry breaking effects of flowshear andmagnetic curvature. The expression is quite complicated andgiven in Ref. 10. The curvature parts of Eqs. �4� and �5�usually dominate except close to marginal stability.

Since the diagonal element for the ion thermal conduc-tivity is obtained from Eq. �3� by replacing 2 in the denomi-nator by 5/3, we realize that the Prandtl number of the diag-onal components has to be very close to 1. Of the newconvective parts, the thermodiffusion is usually the dominantpart and is also usually directed inward. As compared to amodel without the toroidal stress tensor effects, both diago-nal and convective fluxes are considerably larger in thepresent model. Although the steady-state sometimes does notchange very much with the new model, the transient trans-port does.

This model has been used for 1.5D transport simulationsof the JET discharges 66128 and 73701 �noncompensatedNBI modulation cases� described previously. Time-dependent simulations were carried out by predicting �,, Ti,Te and keeping ne fixed to the experimental time-averagedsteady-state values. Figure 26 shows the rotation steady-state�a� and modulation A and � �b� for shot 66128, comparingthe experiment with the simulation. Figure 26�c� shows for

the same simulation the profiles of Ti and Te at a given time.Figure 27 shows the radial profiles of turbulent �i �not in-cluding the neoclassical component�, diagonal and effective�the simulation in fact foresees the existence of an ion heatpinch�, of momentum diffusivity ��, diagonal and effective,and of momentum convective velocity vpinch predicted by themodel. For comparison also the experimental �i

eff and ��

diagonal and vpinch are plotted. We note that in the model thediagonal �i and also �� �Eq. �3�� depend on the eigenfre-quency. Through that they depend also on other gradientsand thus their definition as diagonal elements is not reallyappropriate. This may be particularly important for thismodel where diagonal and effective ion thermal conductivi-ties differ significantly. In Fig. 28 the Pr and pinch numberfrom the simulation �both defined using the diagonal diffu-sivities for �� and �i� and from experiment �where we usethe diagonal �� but the effective �i since so far we have noexperimental evidence of an ion heat pinch� are compared.Figures 29–31 show the same quantities �excluded the Ti andTe profiles� for shot 73701. One can see that for both shotsthe model predicts the existence of a momentum pinch, how-ever, quantitatively in absolute magnitude significantlysmaller than found in experiment �vpinch�3–7 m /s in simu-lations against vpinch�10–20 m /s in experiment�. Also the

ωφ(rad/s10

4 )

200

150

100

50

T(keV)

(c)

(b)

(a)

ϕ(ω

φ)(deg)

0.2 0.4 0.6 0.80

A(ω

φ)(rad/s)

5

6

4

3

1

2

0

0

2000

1500

1000

500

6

5

4

3

2

1

ρtor

Ti exptTi WeilandTe exptTeWeiland

FIG. 26. �Color online� Radial profiles in shot 66128 of experimental �dotswith line� and simulated �lines only� �a� time-averaged steady-state �; �b�A and � of �; and �c� time-averaged steady-state Ti and Te. The simulationsare carried out using the Weiland model with Ti, Te, and � predicted and ne

fixed to the experimental profile.

092505-17 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

absolute value of the diagonal �� is significantly smaller inthe simulations, which gives origin to steeper simulatedphase profiles in Figs. 26 and 29. So in spite of a goodagreement with experiment of steady-state rotation profilesand amplitudes, the disagreement on the phases clearly indi-cates that both momentum diffusivity and pinch are too low.Interestingly, the Pr and pinch number are not far off withrespect to experiment, because �i, ��, and vpinch are all toolow, and their ratios remain close to the experimental ones

0

6

0

5

10

15

2

4

0.2 0.40 0.6

χφWeilandχφeff Weilandχieff Weilandχi Weilandχieff exptχφexpt

0.8

χ(m

2 /s)

-vpinch(m/s)

ρtor

vpinch Weilandvpinch expt

FIG. 27. �Color online� Radial profiles of turbulent �i �diagonal and effec-tive�, �� �diagonal and effective�, and momentum vpinch for the Weilandmodel simulation of shot 66128 shown in Fig. 25, compared with profiles ofeffective �i �turbulent+neoclassical�, diagonal ��, and vpinch obtained bybest-fitting experimental data.

6

4

2

2.0

1.0

1.5

0.5

00

10

8

0.2 0.4 0.6

Pr WeilandPr exptPinch number WeilandPinch number expt

0.80

-Rv pinch/χ

φ

P r

ρtor

FIG. 28. �Color online� Comparison of radial profiles of Prandtl and pinchnumbers from Weiland simulation and experiment for shot 66128.

1500

1000

500

1500

1000

500

0

0

300

0

200

100

0

300

400

200

100

6

4

2

0

2000 (a)

(b)

(c)

0.2 0.4 0.60 0.8 1.0

ωφ(rad/s10

4 )A(ωφ)(rad/s)

ϕ(ω

φ)(deg)

ϕ(ω

φ)(deg)

A(ωφ)(rad/s)

ρtor

FIG. 29. �Color online� Radial profiles of experimental �dots� and simulated�lines� time-averaged � �c� and its A �black squares� and � �red circles� atfirst �a� and second �b� harmonics in shot 73701 using the Weiland modelwith Ti, Te, and � predicted and ne fixed to the experimental profile.

0

20

0

5

10

15

20

10

5

15

0.2 0.40 0.6 0.8

χ(m

2 /s)

-vpinch(m/s)

ρtor

vpinch Weilandvpinch expt

χφWeilandχφeff Weilandχieff Weilandχi Weilandχieff exptχφexpt

FIG. 30. �Color online� Radial profiles of turbulent �i �diagonal and effec-tive�, �� �diagonal and effective�, and momentum vpinch for the Weilandmodel simulation of shot 73701 shown in Fig. 29, compared with profiles ofeffective �i �turbulent+neoclassical�, diagonal ��, and vpinch obtained bybest-fitting experimental data.

092505-18 Mantica et al. Phys. Plasmas 17, 092505 �2010�

�and to the GKW ones�. This shows clearly that it is notsufficient to compare only dimensionless ratios to validate amodel, but we need also the absolute values of the diffusivi-ties. This is thus a question of overall normalization of thetransport, a question which has not been addressed by theGKW results. We remark that also for the Weiland model asfor GKW the agreement is better at low density, where the Pr

number matches the experimental one at least in the innerpart of the plasma �with a different, rather flat profile of Pr

number, at variance with GKW�, while at high density theexperimental Pr number is well above the simulated one.

VII. CONCLUSIONS

This paper describes in detail the technique set-up at JETto perform perturbative studies of momentum transport usingNBI modulation. Experimentally the simplest method ofsquare wave modulation of �5 MW of NBI power hasturned out the most robust to be used in future studies. Itgives a good S/N for the rotation modulation, and complicat-ing effects associated to power modulation �such as modu-lated plasma displacement and modulated ion heat transport�have been shown small and can in any case be accounted forin the analysis. The idea of compensating the power modu-lation by modulating in antiphase normal and tangentialbeams yields a much poorer S/N, requires a rather cumber-some fitting procedure of the data, and was shown to be lesssensitive to the pinch value in the region of interest.

The method set-up for extracting the toroidal momentumdiffusivity and pinch values is a full time-dependent simula-tion of a number of NBI modulation cycles using the JETTOcode, with empirically adjusted profiles of Prandtl numberand pinch in order to fit both the steady-state rotation profilesand amplitudes and phases at two harmonics of the rotationmodulation. The key for the uniqueness of the result is thatphases are dominantly sensitive to Pr number, while ampli-

tudes and steady-state also to the pinch. The main source ofuncertainty in the analysis is represented by the need of as-suming the torque deposition profile from numerical compu-tations. Therefore a lot of effort has been dedicated to sensi-tivity studies of the time-dependent torque calculations usingTRANSP-NUBEAM Monte Carlo code. A comparison withthe time-dependent torque calculated by a totally indepen-dent tool, the orbit following Monte Carlo code ASCOT, hasgiven further confidence on the reliability of the calculatedtorque source. An overall uncertainty of �20% on Prandtlnumber and �30% on pinch value has been estimated.

The physics results yielded by the analysis of NBImodulation data have provided valuable insight into momen-tum transport, allowing to solve the apparent discrepancybetween the low effective momentum diffusivity �Pr

eff

�0.4� measured in steady-state experiments, and the ITGbased theory prediction of a Pr number close to 1. In fact, itis clear from the phase response to the modulation that thediagonal momentum diffusivity is fairly large �Pr�1–2� andthat the low effective diffusivity is due to the presence of asignificant momentum pinch �vpinch up to 20 m/s in the outerpart of the plasma�. This finding, on one hand, is in verygood qualitative agreement with the recent theory develop-ments described in Secs. V and VI; on the other hand itopens interesting perspectives for the possibility of havingpeaked rotation profiles also in machines with low torqueinput, allowing to exploit the beneficial effects of flow shear.

For this reason the experimental results for two shotshave been compared with linear gyrokinetic predictions of Pr

number and pinch number using the code GKW. In addition,the full time evolution of the rotation during several modu-lation cycles has been computed with the recently updatedquasilinear Weiland model for momentum transport andcompared with steady-state, amplitude, and phase profiles.The GKW theoretical predictions are in good agreement withdata both in value and profile shape for the lower densityshot, while the higher density shot has a larger experimentalPr number than predicted by the theory. The reason for thehigh experimental Pr number is that the increasing densityreduces �i more than ��, in experiment, thereby inducing asignificant variation of Pr number. The theory predicted Pr

number is instead rather insensitive to variations in mainplasma parameters. The reason for such discrepancy remainsto be investigated with further experiments at varying plasmaparameters such as density, collisionality, and safety factor.Future work will also address the dependence of the momen-tum pinch on plasma parameters in both experiment andtheory. We note also that linear simulations do not yield ab-solute values of diffusivity and pinch, so it is not possible tocompare quantitative predictions of transport with data untilnonlinear simulations are available. This is the reason whyfull simulations of the data using the Weiland model havebeen attempted. This is presently the only fluid model thathas incorporated recent theoretical advances in momentumtransport. The main result is that although the dimensionlessPr and pinch number are rather similar to the GKW valuesand to experiment, the absolute values of �i, ��, and pinchare all too low compared to experiment, thereby yieldingsteeper profiles of amplitude and phases. Another observa-

6

4

2

2.0

1.0

1.5

0.5

00

10

8

0.2 0.4 0.6 0.80

-Rv pinch/χ

φ

P rρtor

Pr WeilandPr exptPinch number WeilandPinch number expt

FIG. 31. �Color online� Comparison of radial profiles of Prandtl and pinchnumbers from Weiland simulation and experiment for shot 73701.

092505-19 Perturbative studies of toroidal momentum transport… Phys. Plasmas 17, 092505 �2010�

tion is that the Pr profile is rather flat in the Weiland model,while the experimental and GKW ones are increasing withradius. These modeling results indicate that further work isneeded both in terms of theory development and validationagainst experiment toward the availability of a physics basedmodel for momentum transport to be used for extrapolationto future devices. On the other hand, this first exercise ofdetailed validation of recent theory developments againstchallenging data as are those of perturbative experiments is afirst encouraging step forward, since it indicates the qualita-tive agreement of the new theory predictions with experi-ment as far as the existence of a sizeable momentum pinch isconcerned, and also a basic quantitative agreement.

ACKNOWLEDGMENTS

This work, supported by the European Communities un-der the contract of Association between EURATOM/ENEA-CNR, was carried out within the framework of the EuropeanFusion Development Agreement. The views and opinions ex-pressed herein do not necessarily reflect those of the Euro-pean Commission.

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