Sources of intrinsic rotation in the low flow ordering

Transcription

1 PSFC/JA-10-5 Sources of intrinsic rotation in the low flow ordering Felix I. Parra,* Michael Barnes and Peter J. Catto *Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3NP, UK December 010 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 0139 USA This work was supported by the U.S. Department of Energy, Grant No. DE-FG0-91ER Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted. Submitted to Nuclear Fusion December 010

2 Sources of intrinsic rotation in the low flow ordering Felix I Parra and Michael Barnes Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3NP, UK f.parradiaz1@physics.ox.ac.uk Peter J Catto Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 0139, USA Abstract. Alowflow,δf gyrokinetic formulation to obtain the intrinsic rotation profiles is presented. The momentum conservation equation in the low flow ordering contains new terms, neglected in previous first principles formulations, that may explain the intrinsic rotation observed in tokamaks in the absence of external sources of momentum. The intrinsic rotation profile depends on the density and temperature profiles, the up-down symmetry and the type of heating. PACS numbers: 5.5.Fi, 5.30.Gz, 5.35.Ra Submitted to: Nuclear Fusion

3 Sources of intrinsic rotation in the low flow ordering 1. Introduction Experimental observations have shown that tokamak plasmas rotate spontaneously without momentum input [1]. This intrinsic rotation has been the object of recent work [1, ] because of its relevance for ITER [3], where the projected momentum input from neutral beams is small, and the rotation is expected to be mostly intrinsic. The origin of the intrinsic rotation is still unclear. There has been some theoretical work in turbulent transport of momentum using gyrokinetic simulations [4, 5, 6, 7, 8, 9, 10, 11, 1], and two main mechanisms have been proposed as candidates to explain intrinsic rotation. On the one hand, the momentum pinch due to the Coriolis drift [4] has been argued to transport momentum generated in the edge. On the other hand, it has also been argued that up-down asymmetry generates intrinsic rotation [7, 8]. However, neither of these explanations are able to account for all experimental observations. The up-down asymmetry is only large in the edge, generating rotation in that region that then needs to be transported inwards by the Coriolis pinch. Thus, intrinsic rotation in the core could only be explained by the pinch. The pinch of momentum is not sufficient because it does not allow the toroidal rotation to change sign in the core as is observed experimentally [13]. In this article we present a new model implementable in δf flux tube simulations [14, 15, 16, 17]. This model is based on the low flow ordering of [18], and self-consistently includes higher order contributions. As a result, new drive terms for the intrinsic rotation appear that depend on the gradients of the background profiles of density and temperature and on the heating mechanisms. We present two new effects, related to the ion-electron collisions and the heating, that were not treated in the original work [18]. In addition, we recast the results from [18] in a form similar to the equations in the high flow ordering [19, 0]. These are the equations that have been implemented in most gyrokinetic codes that are employed to study momentum transport. For this reason, the new form of the equations is useful to identify the differences with previous models. Finally, we discuss how the new contributions drive intrinsic rotation and we show that the intrinsic rotation resulting from these new processes depends on density and temperature gradients and on the heating mechanisms. In the remainder of this article we present the model, developed originally in [18], in a form more suitable for δf flux tube simulation. In Section we give the complete model, and in Section 3 we discuss its implications for intrinsic rotation. Appendix A contains the details of the transformation from the equations in [18] to the formulation in this article. In Appendix B we discuss the treatment of the ion-electron collision operator.

4 Sources of intrinsic rotation in the low flow ordering 3. Transport of toroidal angular momentum The derivation of the transport of toroidal angular momentum in the low flow regime, including both turbulence and neoclassical effects, is described in detail in [18]. To simplify the derivation, the extra expansion parameter B p /B 1wasemployed,with B the total magnetic field and B p its poloidal component. In this section, we review the results of reference [18], recast them in a more convenient form and add a collisional term and a term that depends on the heating mechanisms that were not treated previously. We assume that the turbulence is electrostatic and that the magnetic field is axisymmetric, i.e., B = I ζ + ζ, where is the poloidal magnetic flux, ζ is the toroidal angle, and we use a poloidal angle θ as our third spatial coordinate. With an axisymmetric magnetic field, in steady state and in the absence of momentum input, the equation that determines the rotation profile is Rˆζ P i T =0,where P i = d 3 v f i Mv v is the ion stress tensor, M is the ion mass, R is the major radius, ˆζ is the unit vector in the toroidal direction,... =V 1 dθ dζ.../b θ isthe flux surface average, V dv/d = dθ dζ B θ 1 is the derivative of the volume with respect to, and... T is the coarse grain or transport average over the time and length scales of the turbulence, much shorter than the transport time scale δ i a/v ti and the minor radius a. Here δ i = ρ i /a 1istheiongyroradiusρ i over the minor radius a, andv ti is the ion thermal speed. Note that we use the prime in v to indicate that the velocity is measured in the laboratory frame. Later we will find the equations in a convenient rotating frame where the velocity is v = v RΩ ζ ˆζ. In reference [18] we derived a method to calculate Rˆζ P i T to order B/B p δi 3p ir, withp i the ion pressure. We present the method again in different form to make it easier to compare with previous work in the high flow regime [19, 0]. In addition, instead of using the simplified ion Fokker-Planck equation of reference [18], f i t + v f i + Ze φ + 1 M c v B v f i = C ii {f i }, 1 where C ii is the ion-ion collision operator, φ is the electrostatic potential, Ze is the ion charge, and e and c are the electron charge magnitude and the speed of light, in this article we use the more complete equation f i t + v f i + Ze φ + 1 M c v B v f i = C ii {f i } + C ie {f i,f e } + S ht, where C ie {f i,f e } is the ion-electron collision operator and S ht δ i f iv ti /a is a source that models the different heating mechanisms. Applying the procedure in reference [18] to equation we find two additional terms in the expression for Rˆζ P i T that were not considered in [18]. In subsection.1 we explain how we split the distribution function and the electrostatic potential into different pieces, and we present the equations to selfconsistently obtain them. In subsection. we evaluate Rˆζ P i mploying the pieces of the distribution function and the potential obtained in subsection.1.

5 Sources of intrinsic rotation in the low flow ordering 4 Table 1. Pieces of the potential. Potential Size Length scales Time scales φ 0, t /e ka 1 / t δi v ti/a φ nc 1, θ, t B/B p δ i /e ka 1 / t δi v ti/a φ nc, θ, t B/B p δi /e ka 1 / t δi v ti/a φ r,t φ 1 δ i /e k ρ i 1 / t v ti /a φ B/B pδi /e k a 1 Before presenting all the results, we emphasize that our results and order of magnitude estimates are valid for B p /B 1andforcollisionalityintherangeδ i qrν ii /v ti < 1 [18], where ν ii is the ion-ion collision frequency and q is the safety factor..1. Distribution function and electrostatic potential The electrostatic potential is composed to the order of interest by the pieces in Table 1 [18]. The axisymmetric long wavelength pieces φ 0, t, φ nc 1, θ, t andφ nc, θ, t are the zeroth, first and second order equilibrium pieces of the potential. The lowest order component φ 0 is a flux surface function. The corrections φ nc 1 and φ nc give the electric field parallel to the flux surface, established to force quasineutrality at long wavelengths the superscript nc refers to neoclassical because these are long wavelength contributions; however, we will show that turbulence can affect the final value of φ nc 1 and φnc. We need not calculate φ nc because it will not appear in the final expression for Rˆζ P i T. The piece φ r,tisturbulentandincludesbothaxisymmetriccomponentszonalflow and non-axisymmetric fluctuations. It is small in δ i but it has strong perpendicular gradients, i.e., k ρ i 1. Its parallel gradient is small, i.e., k a 1. The function φ is calculated to order B/B p δi /e, i.e.,φ = φ 1 + φ with φ 1 δ i /e and φ B/B p δi /e. It is convenient to keep both pieces together as φ as we do hereafter. To write the distribution function it will be useful to consider the reference frame that rotates with toroidal angular velocity Ω ζ = c φ 0 c/zen i p i,wheren i, t and p i, t arethelowestorderiondensityandpressure. Inthisnewreferenceframeit is easier to compare with previous formulations [19, 0]. To shorten the presentation, we perform the change of reference frame directly in the gyrokinetic variables. It is possible to do so easily because we are expanding in the parameter B/B p 1. We first present the gyrokinetic variables that we obtained for the laboratory frame and we argue later how they must be modified to give the gyrokinetic variables in the rotating frame. In [18] we used as gyrokinetic variables the gyrocenter position R = r + R 1 + R +...,the gyrokinetic kinetic energy E = E 0 +E 1 +E +...,themagneticmomentµ = µ 0 +µ and the gyrokinetic gyrophase ϕ = ϕ 0 +ϕ , where E 0 =v /istheparticlekinetic energy in the laboratory frame, µ 0 =v /B is the lowest order magnetic moment, ϕ 0 =arctanv ê /v ê 1 isthelowestordergyrophase,r 1 =Ω 1 i v ˆb δ i L is

6 Sources of intrinsic rotation in the low flow ordering 5 the first order correction to the gyrocenter position, E 1 = Zeφ φ/m δ i vti is the first order correction to the gyrokinetic kinetic energy, and the corrections R δi L, E δi v ti, µ 1 δ i vti /B and ϕ 1 δ i are defined in [1]. Here Ω i = ZeB/Mc is the ion gyrofrequency, ê 1 r andê r aretwoorthonormalvectorssuchthatê 1 ê = ˆb, and... =π 1 dϕ... R,E,µ,t is the gyroaverage holding R, E, µ and t fixed. When the ion distribution function is written as a function of these gyrokinetic variables, it does not depend on the gyrophase ϕ up to order B p /Bδi qrν ii/v ti f Mi [18, 1], where f Mi is the lowest order distribution function that is a Maxwellian. For the magnetic moment and the gyrophase, only the first order corrections µ 1 and ϕ 1 are needed because the lowest order distribution function f Mi does not depend on µ or ϕ. Moreover,since in [18] we expand on B/B p 1, the distribution function need only be known to order B/B p δi f Mi. Consequently, the piece of the distribution function that depends on the gyrophase, of order B p /Bδi qrν ii/v ti f Mi,isnegligible,andthegyrokinetic variables R, E, µ and ϕ only need to be obtained to order B/B p δi L,B/B pδi v ti, B/B p δ i vti /B and B/B pδ i,respectively,implyingthatthecorrectionsr, E, µ 1 and ϕ 1 are not needed for the final result. To change to the new reference frame, where the velocity is v = v RΩ ζ ˆζ, thedistributionfunctionthatisindependentofϕ has to be written as a function of the new gyrokinetic variables R, ε and µ. Notethatthe gyrocenter position and the magnetic moment are the same in both reference frames to the order of interest. The reason is that the toroidal rotation has two components, one parallel to the magnetic field, RΩ ζ ˆζ ˆb = IΩζ /B B/B p δ i v ti,andtheother perpendicular, RΩ ζ ˆζ ˆbˆb ˆζ = Ω ζ /B δ i v ti,andtheparallelvelocityislarger by B/B p 1. Since in [18] the gyrokinetic variables R and µ are to be obtained to order B/B p δi L and B/B p δ i vti/b, andinr and µ only the perpendicular velocity v = v + RΩ ζ ˆζ ˆbˆb ˆζ enters,wecansafelyneglectthecorrectionsduetothe change of reference frame because they are of order δi L and δ ivti /B, respectively. In contrast, the kinetic energy E as defined in [18] cannot be used in the rotating frame because it includes the parallel velocity v = v + IΩ ζ /B. Weuseanewkineticenergy variable ε that is related to the old kinetic energy variable by ε = E IΩ ζ u /B, where u = ± E µb isthegyrokineticparallelvelocityinthelaboratoryframe. Itis easy to check that u = ± [ε µb +I/B Ω ζ /] is equal to u = u IΩ ζ /B and it is the gyrokinetic parallel velocity in the rotating frame. With this relation, we find that another way to interpret the new energy variable ε = u + µb R Ω ζ 3 is realizing that it is the kinetic energy in the rotating frame plus the potential due to the centrifugal force. To write expression 3 we have used that I/B R for B p /B 1. In Appendix A we rewrite the results in [18] using the new gyrokinetic kinetic energy ε. The different pieces of the ion distribution function are given in Table [18]. In this table, m is the electron mass. The functions f Mi, Hi1 nc, Hnc i, H i, Hie i and Hht i are axisymmetric long wavelength contributions. The Maxwellian f Mi R,εisuniform in a flux surface. The first and second order corrections Hi1 nc and Hnc i are neoclassical

7 Sources of intrinsic rotation in the low flow ordering 6 Table. Pieces of the ion distribution function. Distribution function Size Length scales Time scales f Mi R,ε,t f Mi ka 1 / t δi v ti/a Hi1 nc R,θR,ε,µ,t B/B p δ i f Mi ka 1 / t δi v ti /a Hi nc R,θR,ε,µ,t B/B p δi f Mi ka 1 / t δi v ti /a Hi R,θR,ε,µ,t pv ti /qrν ii δi f Mi ka 1 / t δi v ti/a Hi ier,θr,ε,µ,t pδ i m/mfmi ka 1 / t δi v ti/a Hi htr,θr,ε,µ,t B/B pv ti /qrν ii S ht a/v ti ka 1 / t δi v ti/a fi R,ε,µ,t fi1 if Mi k ρ i 1 / t v ti /a fi pδi f Mi k a 1 Table 3. Pieces of the electron distribution function. Distribution function Size Length scales Time scales f Me R,ε,t f Me ka 1 / t δi v ti/a He1 nc R,θR,ε,µ,t B/B p δ i f Me ka 1 / t δi v ti/a fe R,ε,µ,t fe1 δ i f Me k ρ i 1 / t v ti /a fe B/B p δi f Me k a 1 corrections, and they are not the functions Fi1 nc and Fi nc in [18] because we are now working in the rotating frame. The function Hi is an axisymmetric piece of the distribution function that originates from collisions acting on the ions transported by turbulent fluctuations into a given flux surface [18]. The functions Hi ie and Hht i that were not included in [18] have their origin in the ion-electron collisions and in the heating mechanism. The function fi is the turbulent contribution. It will be determined self-consistently up to order B/B p δi f Mi, i.e.,fi = fi1 + f i with fi1 δ if Mi and fi B/B p δi f Mi.Itisconvenienttocombinebothpiecesoftheturbulentdistribution function into one function fi. The electron distribution function is very similar to the ion distribution function. It will have its own gyrokinetic variables that can be easily deduced from the ion counterparts. To the order of interest in this calculation, the electron distribution function is determined by the pieces in Table 3. The long wavelength, axisymmetric pieces f Me and He1 nc are the lowest order Maxwellian and the first order neoclassical correction. The second order long wavelength neoclassical correction is not needed for transport of momentum because of the small electron mass. The piece fe is the short wavelength, turbulent component that will be self-consistently calculated to order B/B p δi f Me. We now proceed to describe how to find the different pieces of the distribution function and the potential. We use the equations in [18] but we change to the new gyrokinetic kinetic energy ε. The details of this transformation are contained in Appendix A.

8 Sources of intrinsic rotation in the low flow ordering First order neoclassical distribution function and potential. The equation for Hi1 nc is [ uˆb R Hi1 nc + Zeφnc 1 Mε f Mi + 5 ] IufMi T i C l ii T i T i Ω i T i {Hnc i1 } =0, 4 where u = ± ε µb + R Ω ζ / ± ε µb isthegyrokineticparallelvelocity and C l ii is the linearized ion-ion collision operator. The correction Hi1 nc gives the parallel component of the velocity [, 3] Wi nc = ˆb d 3 vhi1 ncv = kcib/ze B T i, where k is a constant that depends on the collisionality and the magnetic geometry. Interestingly, the density perturbation due to Hi1 nc is small for qrν ii /v ti 1, i.e., d 3 vhi1 nc B/B p qrν ii /v ti δ i n i B/B p δ i n i [18]. This will be important when determining φ nc 1 below. The equation for He1 nc is similar to 4 and it is given by [, 3] { [ uˆb R H nc 1 1 p i f Me Zn i + 1 p e Mε p e + 5 ] } 1 IufMe e1 eφnc C l ee {Hnc e1 } Cl ei {Hnc e1 } = ef Me uˆb E A, 5 where Ω e = eb/mc is the electron gyrofrequency, E A is the electric field driven by the transformer, C ee l is the linearized electron-electron collision operator and C l ei is the linearized electron-ion collision operator. The lowest order solution for He1 nc is the Maxwell-Boltzmann response eφ nc 1 /f Me B/B p δ i f Me. The rest of the terms are small because they are of order B/B p δ e f Me B/B p m/mδ i f Mi B/B p δ i f Me, where δ e = ρ e /a is the ratio between the electron gyroradius ρ e and the minor radius a. Finally the poloidal variation of the potential is determined by quasineutrality, Z d 3 vhi1 nc + Z d 3 vhi = eφnc 1 n e, 6 Ω e giving eφ nc 1 / B/B p qrν ii /v ti δ i. We have included the density d 3 vhi B/B p v ti /qrν ii δi n i because it becomes important for qrν ii /v ti < fi /f Mi a/ρ i 1withfi /f Mi ρ i /a [18]..1.. Turbulent distribution function and potential. The turbulent piece of the ion distribution function is obtained using the gyrokinetic equation Dfi + uˆb + v M + v C + ve R fi C l { } ii h Dt i C l { } ie h i,h e [ 1 = ve n i Mε R n i T i T i T i + MIu ] Ω ζ f Mi ve R Hi1 nc BT i Zef Mi T i uˆb + v M + v C R φ + Ze M H nc i1 ε uˆb + v M R φ, 7 where D/Dt = t + RΩ ζ ˆζ R is the time derivative in the rotating frame, u = ± [ε µb + R Ω ζ /] ± ε µb istheparallelvelocityintherotatingframe, v M = µ/ω i ˆb R B +u /Ω i ˆb ˆb Rˆb are the B and curvature drifts,

9 Sources of intrinsic rotation in the low flow ordering 8 v C =uω ζ /Ω i ˆb [ R ˆζ ˆb] isthecoriolisdrift,ve = c/b R φ ˆb is the turbulent E B drift, C l ii {h i } is the linearized ion-ion collision operator, Cl ie {h i,h e } is the linearized ion-electron collision operator, and... =π 1 dϕ... R,E,µ,t is the gyroaverage holding R, E, µ and t fixed. The ion-electron collision operator can be approximated by C l ie {h i,h e } = n emν ei T i Mv eφ 3 f Mi p i M T i + n emν ei n i M Te v M vh i + vh i 1 F ei p n emν ei Wi vf Mi, 8 i where ν ei =4 π/3z e 4 n i ln Λ/m 1/ T 3/ e is Coulomb s logarithm, n i W i = d 3 vh i F ei = n emν ei Wi πz e 4 n i ln Λ m is the electron-ion collision frequency, ln Λ v is the turbulent ion flow, and d 3 v e ve ve v e v h e1 9 is the friction force on the electrons due to collisions with ions. The functions that enter in the collision operators are h i = fig + Zeφ φ Mf Mi,0 M T + Hnc i1,0 + 1 H nc i1,0 10 i ε 0 B µ 0 and h e = f e0 1 Ω e v ˆb f e0 + mcf Me,0 B v ˆb φ. 11 Here the subscript g in fig the variables R, ε and µ by R g = f i R g,v /,v /B,t indicatesthatwehavereplaced = r +Ω 1 i v ˆb, v / and v /B; similarly, the subscript 0 in f Mi,0 = f Mi r,v /,t, f Me,0 = f Me r,v /,t, H nc i1,0 = e0 = Hi1 ncr,θr,v /,v /B,t, Hnc e1,0 = He1 ncr,θr,v /,v /B,t and f fe r,v /,v /B,t indicatesthatwehavereplacedthevariablesr, ε and µ by r, v /andv /B. The equation for electrons is equivalent to the one for the ions, giving Dfe + uˆb + v M + ve R fe { } C ee l h Dt e C l { } [ ei h 1 e = ve n e R n e Mε ] f Me + ef Me uˆb + v M R φ, 1 where C ee l is the linearized electron-electron collision operator and C l ei is the linearized electron-ion collision operator. If we were to neglect the effect of the trapped electrons, the solution to this equation would simply be the adiabatic response fe e φ / f Me. Finally, the electrostatic potential φ is obtained from the quasineutrality equation [ Z d 3 v Zeφ φ Mf Mi,0 H nc i1, H nc ] i1,0 M T i ε 0 B µ 0 +Z d 3 vfig = d 3 vfeg. 13

10 Sources of intrinsic rotation in the low flow ordering Second order, long wavelength distribution function. The long wavelength pieces Hi nc, H i, Hie i and Hht i are given by uˆb R Hi α Cl ii {Hα i } = Sα d 3 v S α Mε Mε + 1 d 3 v S α 3 f Mi, 14 3T i T i n i where α =nc,, ie,ht, and S nc = MIuf Mi BT i v M R H nc + Ze M S = u B B R u S ie = n emν ei n i M where n i W nc i Ω ζ v M R i1 + v C c B Rφ nc 1 ˆb Mε R 5 fmi T i T i T i uˆb R φ nc + v M R φ nc 1 T i I n i MΩ i p i ˆb R H nc i1 Zef Mi uˆb R φ nc 1 1 Zen i p i v M R v f i v E T + Ze M Te M vhi1 nc + vhnc i1 u B ε H nc i1 ε + C nl ii B fi uˆb + v M R φ u T {Hi1 nc,hnc i1 } ; 15 ;16 u p i F nc ei n emν ei W nc i ˆbf Mi, 17 = ˆb d 3 vhi1 nc v is the axisymmetric long wavelength ion flow and F nc ei = n emν ei Wi nc πz e 4 n i ln Λ d 3 v e ve ve v e v He1 nc 18 m is the axisymmetric long wavelength friction force on the electrons due to collisions with ions; and S ht is the source in the kinetic equation that mimics the heating mechanism. For radiofrequency heating, S ht can be obtained from the quasilinear models that are widely used. It is also possible to use model sources. For example, in [4, 5] simplified sources were employed to study for the first time the effect of radiofrequency heating on transport of momentum... Calculation of the momentum transport We obtain an equation for Rˆζ P i T similar to equation 39 of [18] by employing the same procedure that was used in that reference, but starting from the more complete Fokker-Planck equation. The final result is as in equation 39 of [18] plus the new terms M c d 3 v C ie {f i }R v Ze ˆζ + d 3 v S ht r, v R v ˆζ. 19 T Adding these new terms to expression 39 from [18] and using that for B/B p 1, Rv ˆζ Iv /B, wefind Rˆζ P i T =Π 1 +Π 0 +Π nc 1 +Π nc 0 +Π ie 1 +Π ie 0 +Π ht + Mc R p i Ze t, 0

11 Sources of intrinsic rotation in the low flow ordering 10 Table 4. Contributions to transport of momentum. Π Size [B/B p δ 3 i p ir ] Dependences Π 1 B p /B ud δ 1 i for ud > B/B p δ i Ω ζ, Ω ζ, ud, T i, n e,, T i 1for ud < B/B p δ i Π 0 1 T i, n e,, T i, n e, Π nc 1 Π nc Π ie 1 ud qrν ii /v ti δ 1 i for ud > B/B p δ i Ω ζ, ud, T i, n e, T i B/B p qrν ii /v ti for ud < B/B p δ i 0 B/B p qrν ii /v ti T i, n e, T i B p /BqRν ii /v ti δ i m/m Ti Π ie 0 qrν ii /v ti δ 1 i m/m T i, n e,, E A Π ht δ i S ht a/v ti f Mi Heating with c Π 1 = B φ ˆb 0 = M c 1 Ze V ci + B ˆb φ Π d 3 vf ig V c B φ ˆb d 3 vf ig IMv B Π nc 1 = c M d 3 vc l ii Ze {Hnc i1,0 + v Hnc i,0 }I B Π nc 0 = c v M d 3 vc nl ii {Hi1,0 nc Ze,Hnc i1,0 }I B M 3 c 6Z e 1 V IMv B + MRΩ ζ T I v d 3 vfig M c Ze V v 3 d 3 vc l ii {Hnc i1,0 }I3 B 3 B T T, 1 d 3 vc l ii {H i,0} I v B,, 3, 4 R 1+ eφnc 1 T i, 5 Π ie 1 = n emcν ei Ze Π ie 0 = M c d 3 vc l ii Ze {Hie i,0} I v B and Π ht = M c Ze d 3 vc l ii {Hht i,0} I v B + p emcν ei Zen i M c Ze I B d 3 vh nc i1 B Mv 1 6 d 3 v S ht I v. 7

12 Sources of intrinsic rotation in the low flow ordering 11 Recall that the subscript g indicates that R, ε and µ have been replaced by R g, v / and v /B, andthesubscript 0 that they have been replaced by r, v /andv /B. In Table 4 we summarize the size of all these contributions compared to the reference size B/B p δi 3p ir, andwewritewhattheydependon. Toobtainthesedependences, we use equations 4, 5, 6, 7, 1, 13 and 14. Most of the size estimates are taken from [18], except for Π ie 1, Π ie 0 and Π ht that are trivially found from the results here. We use ud to denote a measure of the flux surface up-down asymmetry. It ranges from zero for perfect up-down symmetry to one for extreme asymmetry. Notice that for extreme up-down asymmetry, Π 1 and Πnc 1 clearly dominate. The contribution Πie 1 is formally very large for qrν ii /v ti 1, but since the ion energy conservation equation requires that T i /T i B/B p v ti /qrν ii δi M/m, itwillalwaysbecomparable to B/B p δi 3 p i R. 3. Discussion We finish by showing how this new formalism gives a plausible model for intrinsic rotation. Until now, models have only considered the contribution Π 1,withfi and φ obtained by employing equations 7 and 13 without the terms that contain Hi1 nc. This is acceptable for RΩ ζ v ti or high up-down asymmetry ud 1. In this limit, Π 1 Ω ζ, Ω ζ ν Ω ζ Γ Ω ζ +Π ud. To obtain this last expression we have linearized around Ω ζ =0andΩ ζ =0forRΩ ζ /v ti 1. Here ν is the turbulent diffusivity, Γ is the turbulent pinch of momentum and Π ud udδi p ir is the value of Π 1 at Ω ζ =0and Ω ζ =0,andiszeroforperfectup-downasymmetrywhen equations 7 and 13 are solved without the terms that contain Hi1 nc [6]. Notice then that imposing Rˆζ P i T Π = ν Ω ζ Γ Ω ζ +Π ud =0givesintrinsic rotation only for up-down asymmetry or if momentum is pinched into the core from the edge. The complete model described in this article includes contributions that have not been considered before. On the one hand, the gyrokinetic equations 7 and 13 have new terms with Hi1 nc, giving Π 1 ν Ω ζ Γ Ω ζ +Π ud +Π 1,0, where Π 1,0 B/B pδi 3p ir is a new contribution due to the new terms in the gyrokinetic equation. On the other hand, there are the new terms Π nc 1,Πnc 0,Πie 1,Πie 0 and Π ht. As we did for Π 1,wecanlinearizeΠnc 1 Ω ζ around Ω ζ =0tofind Π nc 1 ν nc Ω ζ +Π nc ud +Πnc 1,0, where Π nc ud udb/b p qrν ii /v ti δi p i R and Π nc 1,0 B/B p qrν ii /v ti δi 3 p i R. Combining all these results and imposing that Rˆζ P i T =0,weobtain a Ω ζ = d Π int ν + ν nc exp d Γ = ν + ν nc = a +Ω ζ =a exp d Γ, 8 ν + ν nc =

13 Sources of intrinsic rotation in the low flow ordering 1 where a is the poloidal flux at the edge, Ω ζ =a is the rotation velocity in the edge and Π int =Π ud +Π 1,0 +Π 0 +Πnc ud +Πnc 1,0 +Πnc 0 +Πie 1 +Πie 0 +Πht. Notice that this equation gives a rotation profile that depends on Π int that in turn depends on the gradient of temperature and density, the geometry and the heating mechanism. The typical size of the rotation is Ω ζ B/B p δ i v ti /R for ud < B/B p δ i and Ω ζ ud v ti /R for ud > B/B p δ i. This new model for intrinsic rotation has been constructed such that the pinch and the up-down symmetry drive, discovered in the high flow ordering, are naturally included. By transforming to the frame rotating with Ω ζ we have made this property explicit. Acknowledgments Work supported in part by the post-doctoral fellowship programme of the UK EPSRC, by the Junior Research Fellowship programme of Christ Church at University of Oxford, by the U.S. Department of Energy Grant No. DE-FG0-91ER at the Plasma Science and Fusion Center of the Massachusetts Institute of Technology, by the Center for Multiscale Plasma Dynamics of University of Maryland and by the Leverhulme network for Magnetised Turbulence in Astrophysical and Fusion Plasmas. Appendix A. Equation for the distribution function in the rotating frame In this Appendix we derive equations 4, 7 and 14 for the different pieces of the ion distribution function, equations 5 and 1 for the different pieces of the electron distribution function, and equations 6 and 13 for the different pieces of the potential. These equations are valid in the frame rotating with angular velocity Ω ζ,andwededuce them from the results in [18], obtained in the laboratory frame. In reference [18] we showed that in the limit B p /B 1, and neglecting the ionelectron collisions and the effect of the heating, the ion distribution function is given by f i R,E,µ,t = f Mi R,E,t + Fi1 nc nc R,θR,E,µ,t + Fi R,θR,E,µ,t + fi R,E,µ,t, wherethesizeofthesedifferentpiecesisfi1 nc B/B p δ i f Mi, Fi nc B/B p δi f Mi and fi = fi1 + fi,withfi1 δ i f Mi and fi B/B p δi f Mi. The equations for the different pieces were obtained from the gyrokinetic equation f i t + Ṙ Rf i + Ė f i E = C ii{f i }, where the time derivative Ṙ is Ṙ = u ˆbR+v M c B R φ ˆb and the time derivative Ė is Ė = Ze M [u ˆbR+v M ] R φ. A.1 A. A.3

14 Sources of intrinsic rotation in the low flow ordering 13 Here, u = ± E µb isthegyrokineticparallelvelocityinthelaboratoryframe, and v M = µ Ω i ˆb R B + u Ω i ˆb ˆb Rˆb A.4 are the B and curvature drifts in the laboratory frame. Equations 19 and 0 of [18] for Fi1 nc and equation 4 of [18] for Fi nc are obtained from the long wavelength axisymmetric contributions to A.1 of order δ i f Mi v ti /a and B/B p δi f Mi v ti /a, respectively. Equation 5 of [18] for Fi is also a long wavelength axisymmetric component of A.1. In particular, it is the contribution of order δi f Miv ti /a that does not become of order B/B p δi f Miν ii when the equation is orbit averaged. Equation 55 of [18] for fi is the sum of the short wavelength components of A.1 of order δ i f Mi v ti /a and B/B p δi f Mi v ti /a. In this article, we extend equation A.1 to account for ion-electron collisions and the effect of the different heating mechanisms. For this reason, we use f i t + Ṙ Rf i + Ė f i E = C ii{f i } + C ie {f i } + S ht, A.5 where C ie {f i } m/mν ii f i is the ion-electron collision operator, treated in detail in Appendix B. Moreover, we want to write the equation in the rotating frame, that is, we need to use the new gyrokinetic variable ε = E IΩ ζ u /B. Thus,thenewgyrokinetic equation is f i t + Ṙ Rf i + ε f i ε = C ii{f i } + C ie {f i } + S ht. A.6 The time derivative of the new gyrokinetic variable ε is ε = Ṙ Rε + Ė ε E. A.7 In Ṙ, usingu = u + IΩ ζ /B, withu = ± ε µb + R Ω ζ /, leads to Ṙ = uˆb + IΩ ζ B ˆb + v M + v C c B B R φ ˆb + O δ 3 Bp i v ti, A.8 with v M = µ Ω i ˆb R B + u Ω i ˆb ˆb Rˆb A.9 the B and curvature drifts in the rotating frame, and v C =uiω ζ /BΩ i ˆb ˆb Rˆb the Coriolis drift. To obtain this expression for Ṙ we have used u = u +IΩ ζ u/b + O[B/B p δi vti] towritev M = v M + v C + O[B /Bpδ i 3 v ti ]. The usual result for the Coriolis drift v C =uω ζ /Ω i ˆb [ R R ˆζ ˆb] canberecoveredbyrealizingthat for B p /B 1, ˆb ˆζ, ˆb Rˆb R R/R and I/B R, giving v C = IuΩ ζ BΩ i ˆb ˆb Rˆb uω ζ Ω i ˆb [ R R ˆζ ˆb]. A.10

15 Sources of intrinsic rotation in the low flow ordering 14 In addition, using I ˆb/B = Rˆζ+ˆb /B, φ = φ 0 +φ nc 1 +φnc +φ, φ 0 = φ 0 R,t+ Oδi /e, φ nc 1 = φnc 1 R,θR,t+O[B/B pδi 3/e] and φ nc = O[B /Bp δ i v ti], we can simplify equation A.8 to Ṙ = uˆb + RΩ ζ ˆζ + vm 1 p i n i MΩ i ˆb + v C c B Rφ nc 1 ˆb c B B R φ ˆb + O δ 3 Bp i v ti. A.11 The time derivative ε in A.7 can be written as ε = Ė Iu B Ω ζ Ṙ R Ω ζ Ṙ R Iu B IΩ ζ Bu Ė. A.1 To simplify this equation we use φ = φ 0 + φ nc 1 + φ nc + φ, φ 0 = φ 0 R,t+ Oδi /e, φ nc 1 = φnc 1 R,θR,t+O[B/B pδi 3/e], φ nc = φnc R,θR,t+ O[B /Bp δ4 i /e], u = u + O[B/B p δ i v ti ]and Iu Ṙ R = u ˆb Iu Iu R cb Iu R φ ˆb R B B = Ze Mc v M R + With these results, we obtain ε = Ze M [uˆbr+v M + v C ] Iu B Ω ζ + v M R B ZeI MBu v M R φ + O 1 Zen i p i R + R φ nc 1 + Rφ nc v M c B R φ ˆb R + O B Bp B δ i v ti. A.13 δ i v ti a + R φ. A.14 To obtain the result in A.13, we have employed v M R = u ˆb R Iu /Ω i ; c Iu B R φ ˆb R = ZeI [ ] µ ˆb R B u I ˆb R ln B MBu R φ Ω i Ω i B = ZeI [ ] µ ˆb R B + u Bp ˆb ˆb Rˆb MBu R φ + O Ω i B δ ivti = ZeI MBu v M R φ + O Ω i Bp B δ iv ti, A.15 where we have used I/B = R+O[Bp /B R] andˆb Rˆb = R R/R+O[B p /BR 1 ]; and Iu v M [ R = u R u ˆb u ˆbˆb R B Ω ˆb Iu B p ] R = O i B B δ iv ti,a.16 where we have used ˆb R ˆb B p /Ba 1, ˆb R Iu /B Rv ti /qr B p /BR/av ti and R u ˆb R Iu /B = R [u ˆb R Iu /B] = R [Iu /B R ζ R Iu /B] + R [u /B ζ R Iu /B] = R { R ζ R [I u /B ]} ζ [u /R B R Iu /B] = 0.

16 Sources of intrinsic rotation in the low flow ordering 15 With equations A.6, A.11 and A.14, we can now easily obtain equations 4, 7 and 14 for Hi1 nc, f i, Hi nc, H i, Hie i and Hht i. To obtain 4, we take the long wavelength axisymmetric contribution to A.6 to order δ i f Mi v ti /a, giving uˆb R Hi1 nc + v M R f Mi Ze f Mi uˆb R φ nc 1 1 p i M ε Zen i v M R = C l ii {Hnc i1 }. A.17 This equation differs from equations 19 and 0 of [18], and gives a function Hi1 nc different from the function Fi1 nc defined in [18]. The reason is that f Mi R,ε+Hi1 nc + Hi nc must be equal to the function f Mi R,E+Fi1 nc + Fi nc defined in [18] to the order of interest, but how the terms of first and second order in δ i are assigned to one or the other piece differs depending on the frame. For this reason, we have changed the name of the functions. The final result in 4 is obtained from A.17 by using v M R = uˆb R Iu/Ω i foru = ± ε µb + R Ω ζ / ± ε µb. Equation 7 is the sum of the short wavelength contributions to A.6 of order δ i f Mi v ti /a and B/B p δi v ti/a. The equation is almost straightforward if we apply the same methodology as in [18]. Only two terms require some care. On the one hand, the ion-electron collision operator that was not treated in [18] is now considered in detail in Appendix B. On the other hand, in Ṙ there is a drift n imω i 1 p i ˆb that is not included in 7. To study the effect of the perpendicular drift n i MΩ i 1 p i ˆb, it is better to consider the local approximation. In the local approximation, the length scale of the turbulence is so small that the background quantities can be represented by their local value and their local derivative, i.e., the drift n i MΩ i 1 p i ˆb isgivenbyitsvalue n i MΩ i 1 p i ˆb =0 at the point = 0 around which we want to calculate the turbulent fluctuations and the linear dependence 0 [n i MΩ i 1 p i ˆb ] =0. The characteristic scale of the turbulence is the ion gyroradius, giving 0 ρ i RB p and 0 [n i MΩ i 1 p i ˆb ] =0 δi v ti. Since we only need to keep terms up to order B/B p δi f Miv ti /a, 0 [n i MΩ i 1 p i ˆb ] =0 R fi δi f Miv ti /a is negligible. Only the constant drift n i MΩ i 1 p i ˆb =0 remains. Aconstantdriftdoesnotchangethecharacteroftheshortwavelengthstructuresand can be safely ignored. Equation 14 is found from the long wavelength axisymmetric components of A.6 to order δi f Miv ti /a. Note that to this order we have the time derivative t f Mi [18]. Using t f Mi =[n 1 i t n i +Mε/T i 3/T 1 i t T i ]f Mi and realizing that t n i = d 3 v S α and 3/ t n i T i = d 3 v S α Mε,wherethesummations are over α = nc,, ie, ht, we find the final form in 14. The equations for Hi nc and Hi are obtained in the same way as equations 4 and 5 in [18], i.e., the equation for Hi nc is the axisymmetric long wavelength component of A.6 of order B/B p δi f Miv ti /a, andtheequationforhi is the axisymmetric long wavelength component of order δi f Miv ti /a that when it is orbit averaged does not reduce to a piece of order B/B p δi ν iif Mi. The equations for Hi ie and Hht i where not considered in [18].

17 Sources of intrinsic rotation in the low flow ordering 16 The equation for Hi ie is the axisymmetric long wavelength contribution that includes the ion-electron collision operator that is treated in detail in Appendix B. The equation for Hi ht is the axisymmetric long wavelength contribution that has the source Sht. The equations 5 and 1 for the electron distribution function in the rotating frame are derived in the same way as the equations for the ion distribution function. The only differences are that the Coriolis drift v C and the term in A.14 that is proportional to Ω ζ are small by m/m and hence negligible, and that we include the electric field E A driven by the transformer, leading to a modified time derivative for the energy ε = e m uˆb E A + e m [uˆbr+v M ] 1 p i Zen i R + R φ nc 1 + R φ. A.18 Finally, the equations for the different pieces of the electrostatic potential 6 and 13 are easily deduced from the results in [18] by realizing that moving to a rotating reference frame does not modify the quasineutrality equation. Appendix B. Ion-electron collisions In this Appendix we discuss how we treat the ion-electron collision operator, given by [ C ie {f i,f e } = γ ie v d 3 v e g g g f e v f i Mm ] f i ve f e, B.1 where γ ie =πz e 4 ln Λ/M, g = v v e and g g =g I gg/g 3. Both the ion velocity, v, andtheelectronvelocity,v e,aremeasuredintherotatingframe. We must write the ion-electron collision operator up to order m/mν ii δ i f Mi. For the wavelengths of interest, between the minor radius a and the ion gyroradius, f i R,ε,µ,t = f Mi R,ε,t+Hi1 nc R,θR,ε,µ,t + fi R,ε,µ,t +... and f e R,ε,µ,t = f Me R,ε,t + He1 nc R,θR,ε,µ,t + fe R,ε,µ,t +...,where He1 nc =eφnc 1 /f Me +O[B/B p δ e f Me ]andfe =eφ / f Me +Oδ e f Me. The electron distribution function is then a Maxwell-Boltzmann response eφ/ f Me δ i f Me plus acorrectionoforderδ e f Me m/mδ i f Me that is smaller by m/m. Expanding the ion distribution function around R g = r +Ω 1 i v ˆb, v / and v /B, we obtain f i = f Mi,0 + Hi1,0 nc + h i,whereh i = fig [Zeφ φ /T i ]f Mi,0. For the electrons, since we are considering wavelengths larger than the electron gyroradius, it is possible to expand around r, v /andv /B, givingf e = f Me,0 + He1,0 nc + h e,where h e = fe0 Ω 1 e v ˆb fe0 +[mcv ˆb φ /B ]f Me,0. Here, the subscript 0 in f Mi,0, f Me,0, Hi1,0 nc, Hnc e1,0 and fe0 indicates that the variables R, ε and µ have been replaced by r, v /andv /B. The subscript g in f ig indicates that R, ε and µ are replaced by R g = r +Ω 1 i v ˆb, v /andv /. Using these expressions for f i and f e, and the relation g g g = ve ve v e v ve ve ve v e + O m M 1 v te, B.

18 Sources of intrinsic rotation in the low flow ordering 17 we find that C ie {f i,f e } = Mγ ie v T i +γ ie v Mγ ie m v Mγ ie v Mγ [ ie v +O [1+ eφnc 1 [ v Hi1 nc + vh [f Mi [1+ eφnc 1 + eφ f Mi v i d 3 v e f Me ve ve v e ] d 3 v e f Me ve ve v e ] d 3 v e ve ve v e ve H nc e1 + v e h + eφ f Mi v e ] d 3 v e f Me ve ve ve v e v e ] ] Hi1 nc + h i v d 3 v e f Me ve ve ve v e v e m M ν iiδ i f Mi. B.3 Here we have used that ve f Me = mv e / f Me and ve ve v e v e = 0. Using ve ve ve v e v e = ve ve v e and d 3 v e f Me ve ve v e =/3 I d 3 v e f Me /v e = /3 πn e m/te I,wefind C ie {f i,f e } = n emν ei n i M 1+ eφnc 1 + n emν ei n i M v + eφ Te Mv 1 3 f Mi T i T i [ ] Te M vhi1 nc + vh i +vhnc i1 + h i 1 [F nc ei p + F ei n emν ei Wi nc i +O + W i ] vf Mi m M ν iiδ i f Mi. B.4 Here ν ei =4 π/3z e 4 n i ln Λ/m 1/ Te 3/ is the electron-ion collision frequency, F nc ei is the long wavelength axisymmetric friction force on electrons due to collisions with ions, given in 18, F ei is the short wavelength turbulent friction force, given in 9, Wi nc = n 1 i ˆb d 3 vhi1 ncv is the long wavelength axisymmetric ion average velocity in the rotating frame, and Wi = n 1 i d 3 vh i v is the short wavelength turbulent ion average velocity. References [1] Rice J E et al 007 Nucl. Fusion [] Nave M F F et al 010 Phys. Rev. Lett [3] Ikeda K et al 007 Nucl. Fusion 47 E01. [4] Peeters A G, Angioni C and Strinzi D 007 Phys. Rev. Lett [5] Waltz R E, Staebler G M, Candy J and Hinton F L 007 Phys. Plasmas Waltz R E, Staebler G M, Candy J and Hinton F L [6] Roach C M et al 009 Plasma Phys. Control. Fusion [7] Camenen Y, Peeters A G, Angioni C, Casson F J, Hornsby W A, Snodin A P and Strintzi D 009 Phys. Rev. Lett [8] Camenen Y, Peeters A G, Angioni C, Casson F J, Hornsby W A, Snodin A P and Strintzi D 009 Phys. Plasmas

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