Devitt, C.J.

Diamond, P.H.

Greenwald, M.J.

Gurcan, O.D.

Hahm, T.S.

Hubbard, A.E.

Hughes, J.W.

Kosuga, Y.

Marmar, E.S.

Podpaly, Y.A.

Reinke, M.L.

Rice, J.E.

Whyte, D.G.

DSpace@MIT

                       * University of California, San Diego ** Ecole Polytechnique, France *** Princeton Plasma Physics Laboratory  November 2010   Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge  MA  02139  USA            This work was supported by the U.S. Department of Energy, Grant No.  DE-FC02-99ER54512 .  Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.  PSFC/JA-10-39                   The Edge Temperature Gradient as Intrinsic Rotation Drive in Alcator C-Mod Tokamak Plasmas   Rice, J.E., Hughes, J.W., Diamond*, P.H., Kosuga*, Y., Podpaly, Y.A., Reinke, M.L., Greenwald, M.J., Gurcan**, O.D.,  Hahm***, T.S., Hubbard, A.E., Marmar, E.S., McDevitt*, C.J., Whyte, D.G.   The Edge Temperature Gradient as Intrinsic Rotation Drive in Alcator C-Mod Tokamak PlasmasJ.E. Rice†, J.W. Hughes†, P.H. Diamond‡, Y. Kosuga‡, Y.A. Podpaly†, M.L. Reinke†,M.J. Greenwald†, ¨O.D. Gu¨rcan, T.S. Hahm, A.E. Hubbard†, E.S. Marmar†, C.J. McDevitt‡ and D.G. Whyte††Plasma Science and Fusion Center, MIT, Cambridge, MA, USANFRI, Daejeon, Korea‡UCSD, San Diego, CA, USALPP Ecole Polytechnique, FrancePrinceton Plasma Physics Laboratory, Princeton, NJ, USAAbstractIntrinsic core toroidal rotation has been observed in I-mode plasmas from the Al-cator C-Mod tokamak, and is found to be very similar to the rotation in H-mode dis-charges, both in its edge origin, profile shape and in the scaling with global plasmapressure. Since I- and H-mode plasmas have similar pedestal temperature gradients,but completely different edge density profiles, it may be concluded that the drive ofthe intrinsic rotation is the edge temperature gradient rather than the pressure gradient.Evidence suggests that the connection between gradients and rotation is the residualstress, and a scaling for the intrinsic rotation from the conversion of thermodynamicfree energy to macroscopic flow is calculated.1Understanding the mechanism whereby turbulence drives a macroscopic shearedflow is a classical problem in the physics of self-organization. A notable exampleof this is the origin of solar differential rotation, i.e. how convection zone Rayleigh-Benard turbulence drives latitudinally and radially sheared flows [1] in the sun. Effortsto understand this self-acceleration phenomenon have motivated development of meanfield theory Reynolds stress closure models for convective and MHD turbulence [2].In tokamaks, sheared toroidal rotation appears spontaneously, seemingly as a result ofturbulence driven self acceleration.Rotation and velocity shear play important roles in the suppression of deleteriousMHD modes [3] and drift wave turbulence [4] in tokamak plasmas. Toroidal rotationis predominantly driven by neutral beam injection in present day tokamaks, but thismethod will be much less effective in future large scale devices and reactors. An alter-native approach is to take advantage of the self-driven flows (intrinsic rotation) whichhave been widely observed in plasmas without external momentum input [5]. A funda-mental understanding of this curious phenomenon is necessary in order to extrapolateconfidently to reactors. Since the transport and relaxation of toroidal momentum is dueto turbulence (with a Prandtl number χφ/χi ∼ 1), intrinsic rotation necessarily impliesthe existence of an agent or element of the momentum flux which can oppose turbulentviscosity. This off-diagonal flux, the residual stress Πres, is likely responsible for theself-acceleration of intrinsic flow [6]. The connection between observed intrinsic ro-tation and directly measured turbulent residual stress has been demonstrated in CSDXplasmas [7]. The residual stress has also been isolated in DIII-D plasmas by adjustingthe input torque from neutral beam injection to produce a null velocity profile with novelocity gradient [8]. Residual stress may be understood as a momentum flux driven bymacroscopic gradients which produces a directed bulk flow by converting radial inho-mogeneity to broken k‖ symmetry of the fluctuation spectrum. This process resemblesan engine, in that the turbulence converts thermodynamic free energy to macroscopicflow, just as a heat engine converts a temperature difference to useful work [9].Several questions arise, the first of which is concerned with the driving gradient.This is a subtle question, since the pressure gradient ∇P can drive instabilities, breakk‖ symmetry [10] by its contribution to the zonal flow shear 〈VE〉′, and also quenchturbulence [11] via 〈VE〉′. Other possibilities include ∇T and ∇n. Here, comparisonstudies of different confinement regimes (H-mode and I-mode plasmas) are used toisolate the driving gradient. The link between this gradient and the macroscopic flowis the residual stress. A simple theory of the intrinsic rotation engine, based on entropydynamics, is described. This theory yields a scaling for the intrinsic rotation velocity.The core intrinsic toroidal rotation has been found to be well correlated with theplasma stored energy (volume averaged pressure) in H-mode plasmas [12] in AlcatorC-Mod (R ∼ 0.67 m,a ∼ 0.21 m) and many other devices [5]. This effect is demon-strated in Fig.1, where H-mode (green dashed curves) was accessed using 2 MW ofICRF minority heating (zero momentum input) power in Alcator C-Mod. Followingthe L-mode to H-mode transition (∼0.62 s) there were increases in the electron densityand temperature, plasma stored energy and core toroidal rotation velocity (incrementin the co-current direction), the latter determined from the Doppler shifts of core x-raylines. Shown for comparison are the time histories for an I-mode plasma (solid redcurves). Both discharges had a magnetic field of 5.4 T and plasma current of 0.8 MA,20.4 0.6 0.8 1.0 1.2 1.4 1.6020406080100(kJ) WP0.4 0.6 0.8 1.0 1.2 1.4 1.60.00.51.01.52.02.53.0(1020/m3 )neI-modeEDA H-mode0.4 0.6 0.8 1.0 1.2 1.4 1.6-200204060(km/s)VTor(0)0.4 0.6 0.8 1.0 1.2 1.4 1.601234(keV) Te0.4 0.6 0.8 1.0 1.2 1.4 1.6t (s)0.00.51.01.52.02.5(MW)PICRFFigure 1: Time histories, from top to bottom, of the plasma stored energy, averageelectron density, central electron temperature, central rotation velocity (positive de-notes co-current) and ICRF power, for an EDA H-mode (green dashed) and an I-mode(red) plasma.3with similar target densities and ICRF hydrogen minority heating power. The maindifference was that the I-mode plasma was formed with the unfavorable ion B×∇Bdrift direction, in this case with an upper single null magnetic configuration. There wasvery little change in the electron density in the I-mode case, and a larger increase in theelectron temperature. The stored energy in these two discharges was very similar, aswas the magnitude of the core toroidal rotation velocity and velocity profile shape.I-mode is an operational regime [13] which exhibits H-mode energy confinementwith L-mode particle confinement. Shown in Fig.2 is a comparison of edge profilesfrom the EDA H-mode [14] and I-mode plasmas of Fig.1. These profiles are fromThomson scattering, and are averaged over the steady portion of the discharges. Both-25 -20 -15 -10 -5 0 5 100.00.51.01.52.02.53.0(1020/m3 )neEDA H-modeI-mode-25 -20 -15 -10 -5 0 5 100100200300400500600(eV)Te-25 -20 -15 -10 -5 0 5 10R - RLCFS (mm)02468101214(kPa)PeFigure 2: Edge profiles of the electron density (top frame), electron temperature (mid-dle frame) and electron pressure (bottom frame) from the EDA H-mode (green dots)and the I-mode (red asterisks) discharges of Fig.1. The curves are hyperbolic tangentfits.plasmas display an edge energy transport barrier, as manifested by the pedestals in theelectron temperature at 3 mm inside of the last closed flux surfaces. The temperaturegradient at this location is the same for both plasmas, about 70 keV/m. Only the EDAH-mode discharge has a pedestal in the electron density, however; in I-mode the edge4density profile is similar to that in L-mode. This demonstrates a favorable attribute of I-mode: good energy confinement with poor particle confinement. That heat and particletransport is decoupled in I-mode allows a unique opportunity for energy barriers tobe studied separately; in H-mode both transport channels are usually linked. Sincethere is no particle barrier in I-mode, the edge pressure gradient is considerably lower(∼0.7 MPa/m, about a factor of three in this comparison) compared to EDA H-mode.The pedestal radial electric field well depth is also shallower in I-mode compared toH-mode [15].For H-mode plasmas, it has been established that the intrinsic rotation originates atthe plasma edge, and propagates in to the core on a momentum confinement time scale[16]. Similar behavior is seen in I-mode plasmas, as shown in Fig.3. Following the0.6 0.7 0.8 0.9 1.0 1.1 1.2020406080100120(kJ)WP0.6 0.7 0.8 0.9 1.0 1.1 1.20.00.51.01.52.02.5ne (1020/m3 ) ICRFne0.6 0.7 0.8 0.9 1.0 1.1 1.20123(keV)Te0.6 0.7 0.8 0.9 1.0 1.1 1.2t (s)-200204060(km/s)VTorr/a=0.6 r/a=0.0Figure 3: Parameter time histories for an I-mode discharge. In the bottom frame is thenormalized chord-averaged toroidal rotation velocity at several radial locations, evenlyspaced from r/a=0.6 to r/a=0.0.transition to I-mode (0.7 s), the rotation velocity first appears at the plasma edge, thenpropagates in to the plasma center. The pedestal rotation velocity was not available forthis discharge, but in previous studies has been shown to develop immediately after theI-mode transition [15].5It is well documented that the change in the toroidal rotation velocity between L-and H-mode is proportional to the change in the plasma stored energy (normalized tothe plasma current) [12, 5], which is evident in Fig.1. Similar behavior is apparent inI-mode as well; the changes in rotation and stored energy are very similar in the twoplasmas of Fig.1. This scaling holds over a large range of plasma parameters (density,plasma current, magnetic field, ICRF power), as can be seen in Fig.4. The I-modepoints (red asterisks) overlay the H-mode points (green dots) from a large database,0 50 100 150'WP/IP (kJ/MA)020406080100120'V Tor (km/s)Figure 4: The change in the core toroidal rotation from L-mode to I-mode (red as-terisks) and to H-mode (green dots) as a function of the change in the stored energynormalized to the plasma current.suggesting a common phenomenology giving rise to the rotation.Given the evidence that the origin for the intrinsic rotation is in the pedestal region([16] and Fig.3), it would be natural to seek a local edge gradient rather than the globalstored energy as relevant to the rotation drive. Intrinsic rotation in JT-60U plasmas hasbeen linked to the ion pressure gradient [17]. Shown in Fig.5a is the change in therotation velocity as a function of the change in the pedestal electron pressure gradient.(In C-Mod plasmas, the edge ion and electron pressure profiles are the same [15].) Forthe H-mode points, there is a very good correlation between the core rotation velocity60 1 2 3 4 5Change in Edge Grad P (MPa/m)020406080100'V Tor (km/s)(a)0 50 100 150 200Change in Edge Grad T (keV/m)020406080100(b)Figure 5: The change in the core rotation velocity as a function of the change in thepedestal electron pressure gradient (a) and pedestal electron temperature gradient (b)for H-mode (green dots) and I-mode (red asterisks) plasmas.7and edge pressure gradient. However, the I-mode points do not coincide, because therotation is the same while the pressure gradients are lower than in H-mode (as seen inFig.2). The two sets of points overlay very well for the pedestal temperature gradient,as demonstrated in Fig.5b, suggesting that it is ∇T which is key in driving intrinsicrotation. The association of intrinsic rotation with the ion temperature gradient hasbeen demonstrated in LHD ITB plasmas [18].Ongoing work is focused on a study of the relation ΔVT (0) ∼ ∇T/Bθ|edge asan alternative scaling expressed in terms of local quantities, and on the correlation ofedge electric field shear with ΔVT (0). The high density regime of C-Mod makes itdifficult to separate Te from Ti. Thus, the driving gradient could be either ∇Te or ∇Ti,depending on the nature of the turbulence (i.e. CTEM or ITG). To this end, it’s usefulto recall that gyrokinetic particle simulations [19] have shown that intrinsic rotationin ITG turbulence correlates with ∇Ti while in CTEM turbulence, it correlates with∇Te. Either is possible in C-Mod, and the ultimate resolution of the question requiresdetailed edge fluctuation measurements.The approach for addressing the origin of intrinsic flow may be cast in terms offluctuation entropy and describes intrinsic rotation as a thermodynamic engine. In theframework of residual stress, the generation process of flows can be understood as aconversion of thermal energy, which is injected into a system by heating, into kineticenergy of macroscopic flow by drift wave turbulence excited by ∇T , ∇n, etc. Usingthe physical picture of flow generation as an energy conversion, an explicit expressionfor the efficiency of the conversion process may be formulated by comparing rates ofentropy production/destruction due to thermal relaxation/flow generation [20].For a simple model with drift kinetic ions and adiabatic electrons, the time evolu-tion of entropy or δf2 is given by∂t∫dΓ〈δf2〉〈f〉=∫d3x{− nTiLTQiturb −nv2thi〈V⊥〉′〈V˜rV˜⊥〉 − nv2thi〈V‖〉′〈V˜rV˜‖〉+ 1Ti〈J˜ i‖E˜‖〉}(1)apart from collisional dissipation and boundary terms. Here the notation is standard.The right hand side can be further simplified for a stationary state. Using a simplemodel for turbulent flux and neglecting the pinch, the entropy production rate, P , be-comesP ≡∫d3x{nχi(∇TT)2− nK( 〈VE〉′vthi)2+nχφ( 〈V‖〉′vthi)2− nΠresr‖2v2thiχφ}(2)where K ≡∑k c2sτZF (ρ2sk2θ)/(1 + k2⊥ρ2s)2{−kr∂ηk/∂kr}, ηk ≡ (1 + k2⊥ρ2s)2|φˆk)|2.The first term in the right hand side is the entropy production rate due to thermal re-laxation. The second term is the entropy destruction rate due to zonal flow generation.Note that this term destroys entropy only when zonal flow grows, i.e. γZF ∝ K > 0.The third term is the entropy production rate due to the relaxation of the velocity gra-dient. The last term is the entropy destruction rate due to the generation of intrinsictoroidal rotation.8The production rate P contains terms with a definite order. The last two terms aresmaller than the first two terms by the order of O(k‖/k⊥) where k is a representativeof the mode number of drift waves. Hence a stationary state is achieved by balancingthe production and destruction rates order by order. To the lowest order, the balance isbetween the production rate from thermal relaxation and the destruction rate from zonalflow growth. The balance yields 〈VE〉′2 = (χi/K)(v2thi/L2T ) which relates zonal flowstrength to the temperature gradient directly. To the next order, the third and the fourthterms in Eq. 2 cancel since the total parallel momentum vanishes for a stationary stateas 〈V˜rV˜‖〉 = −χφ〈V‖〉′ + Πresr‖ = 0.To calculate 〈V‖〉, a model of the residual stress is required. Here, a case is con-sidered where toroidal rotation is driven by a two step process: first, a stationary stateis achieved by balancing the entropy production rate due to thermal relaxation by theentropy destruction due to zonal flow growth. Secondly, the zonal flow E × B shearset by the dominant balance gives rise to symmetry breaking and momentum flux viak space scattering. In such a process, the residual stress is calculated as [10, 6]Πresr‖ = −ρ∗Ls2csK〈VE〉′2 = −ρ∗ Ls2cs χi(∇TT)2v2thi. (3)It is worthwhile to comment that the strong scaling of the residual stress with ∇T/T isentirely consistent with the observed ΔVT (0) vs ∇T |ped correlation discussed earlier.The condition 〈V˜rV˜‖〉 = 0 implies a strongly nonlinear relation between ∇V‖ and∇T/T , i.e.〈V‖〉′ =Πresr‖χφ= −12ρ∗χiχφLscs(∇TT)2v2thi. (4)Simple integration then establishes the relation between edge∇T and the flow velocity,〈V‖〉vthi∼= 12ρ∗χiχφLsLT√TiTe(5)where (T′/T)′ = -(T′/T)2 + T′′/T ∼= -(T′/T)2 has been used. The sign conventions hereare such that the rotation is predicted to be co-current. For typical H- and I-modedischarges in C-Mod, ρ∗ ∼ 0.006, Te ∼ Ti, Prandtl number ∼ 1, Ls ∼ 0.6 m and LT∼ 0.01 m, Eq.5 yields a thermal Mach number of 0.18, which is also typical [5]. Eq.5captures the ∇T scaling of Fig.5 (and in LHD [18]) from 1 / LT and the 1 / Ip scalingof Fig.4 from Ls ∝ q/sˆ ∝ 1 / Ip.In this paper, progress toward a physics based phenomenology of intrinsic rotationin H-mode and I-mode has been described. The principal results of this paper are asfollows.1. studies in Alcator C-Mod indicate a close correlation between change in centralvelocity ΔVT (0) and ∇T |edge, both in H-mode and I-mode. This, in turn, sug-gests that the macroscopic scaling ΔVT (0) ∼ ΔWp/Ip may be replaced by arelation expressed in terms of local parameters i.e. ΔVT (0) ∼ ∇T/Bθ. Thecentral velocity does not correlate with edge ∇P in I-mode.92. a theory of the entropy balance for a turbulent plasma is presented, which yieldsa scaling for the residual stress and the toroidal rotation velocity. The theorypredicts that V‖ scales with ∇T and q.The authors thank the Alcator C-Mod operations and ICRF groups for expert run-ning of the tokamak. Work supported by DoE Contract No. DE-FC02-99ER54512 atMIT, DE-AC02-09CH11466 at PPPL and DE-FG02-04ER54738 at UCSD, the W.C.I.program of MEST, Korea and A.N.R. contract ANR-06-PLAN-0084 in France.References[1] G.Rudiger, “Differential Rotation and Stellar Convection”, Gordon-Breach (NewYork) (1989)[2] U.Frisch et al., Physics D 28 (1987) 382.[3] E.J.Strait et al., Phys. Rev. Lett. 74 (1994) 2483.[4] T.S.Hahm, Phys. Plasmas 1 (1994) 2940.[5] J.E.Rice et al., Nucl. Fusion 47 (2007) 1618.[6] P.H.Diamond et al., Nucl. Fusion 49 (2009) 045002.[7] Z.Yan et al., Phys. Rev. Lett. 104 (2010) 065002.[8] W.M.Solomon et al., Phys. Plasmas 17 (2010) 056108.[9] ¨O.D.Gu¨rcan et al., Phys. Plasmas 17 (2010) 032509.[10] ¨O.D.Gu¨rcan et al., Phys. Plasmas 14 (2007) 042306.[11] H.Biglari et al., Phys. Fluids B 2 (1990) 1.[12] J.E.Rice et al., Nucl. Fusion 39 (1999) 1175.[13] D.G.Whyte et al., Nucl. Fusion 50 (2010) 105005.[14] M.Greenwald et al., Fusion Sci. Technol. 51 (2007) 266.[15] R.M.McDermott et al., Phys. Plasmas 16 (2009) 056103.[16] W.D.Lee et al., Phys. Rev. Lett. 91 (2003) 205003.[17] M.Yoshida et al., Phys. Rev. Lett. 100 (2008) 105002.[18] K.Ida et al., Nucl. Fusion 50 (2010) 064007.[19] W.X.Wang et al., Phys. Plasmas 17 (2010) 072511.[20] Y.Kosuga et al., Phys. Plasmas 17 (2010) 102313.10

The Edge Temperature Gradient as Intrinsic Rotation Drive in Alcator C-Mod Tokamak Plasmas

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The Edge Temperature Gradient as Intrinsic Rotation Drive in Alcator C-Mod Tokamak Plasmas

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