Gyrokinetic theory and simulation of angular

PHYSICS OF PLASMAS 14, 122507 共2007兲

Gyrokinetic theory and simulation of angular momentum transport R. E. Waltz,a兲 G. M. Staebler, J. Candy, and F. L. Hintonb兲 General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA

共Received 20 August 2007; accepted 12 November 2007; published online 28 December 2007兲 A gyrokinetic theory of turbulent toroidal angular momentum transport as well as modifications to neoclassical poloidal rotation from turbulence is formulated starting from the fundamental six-dimensional kinetic equation. The gyro-Bohm scaled transport is evaluated from toroidal delta-f gyrokinetic simulations using the GYRO code 关Candy and Waltz, J. Comput. Phys. 186, 545 共2003兲兴. The simulations recover two pinch mechanisms in the radial transport of toroidal angular momentum: The slab geometry E ⫻ B shear pinch 关Dominguez and Staebler, Phys. Fluids B 5, 387 共1993兲兴 and the toroidal geometry “Coriolis” pinch 关Peeters, Angioni, and Strintzi, Phys. Rev. Lett. 98, 265003 共2007兲兴. The pinches allow the steady state null stress 共or angular momentum transport flow兲 condition required to understand intrinsic 共or spontaneous兲 toroidal rotation in heated tokamak without an internal source of torque 关Staebler, Kinsey, and Waltz, Bull. Am. Phys. Soc. 46, 221 共2001兲兴. A predicted turbulent shift in the neoclassical poloidal rotation 关Staebler, Phys. Plasmas 11, 1064 共2004兲兴 appears to be small at the finite relative gyroradius 共rho-star兲 of current experiments. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2824376兴 I. INTRODUCTION: INTRINSIC TOROIDAL ROTATION

The neoclassical transport of toroidal angular momentum is even smaller than for neoclassical energy transport and both are neglected here. Experimentally the turbulent plasma toroidal angular momentum and energy confinement times 共and diffusivities兲 are found to be nearly the same when there is strong central torque and toroidal rotation 关as in both co- and counter-directed neutral beam injection 共NBI兲兴.1,2 This is of course the normal expectation from the dominance of the E ⫻ B transport of the projected parallel momentum when compared to E ⫻ B transport of ion energy: the effective momentum and ion energy turbulent diffusivities are nearly the same. However, from the early work of Rice et al.,3 it now appears to be well established4 that tokamaks have a small but intrinsic (spontaneous) toroidal rotation even without auxiliary injected torque 共see Refs. 18–32 in Ref. 4兲. Such experimental results with small rotation have motivated a more careful treatment of momentum transport to answer the question: How can the momentum confinement time be significantly longer than that for energy? Since toroidal angular momentum is conserved, the core momentum in intrinsic toroidal rotation discharges must have been transported into the core from a source at the edge. Unlike energy or particle density, angular momentum is a directional quantity and we must refer to co-current and counter-current momentum. The edge is also a sink of both co- and counter-momentum via charge exchange from the neutrals tied to the wall or diverter at rest 共hence always a sink and never a source兲. There is always some plasma recycling at the edge and hence plasma transport outflow that necessarily provides some convective outflow of co- and counter- momentum at the edge 共depending on the local coor counter-toroidal rotation兲. Thus there must have been a net a兲

Electronic mail: [email protected]. Present address: University of California San Diego, La Jolla, California.

b兲

1070-664X/2007/14共12兲/122507/12/$23.00

inflow during the formation of the discharge. However at steady state, in the absence of auxiliary injected 共or otherwise identified兲 torque, there must exist some radius r = redge ⬍ a interior to which there is no source or sink of momentum and the 共flux surface average兲 radial transport momentum flux must be everywhere zero, i.e., the core must be in a null flow state, ⌫x␾关0 ⬍ r ⬍ redge兴 = 0. 关Here we assume there is no volumetric source or sink of co- or counter- momentum from nonlocal modes or low-n magnetic field perturbations tied to the wall. The phenomenon of ripple induced “magnetic breaking” 共sink兲 is well known and discounted here. Similarly rotating low-n magnetic field perturbations imposed at the wall can in principle be made a core source.兴 At present, we can only speculate on the momentum source mechanism at r ⬎ redge. The most likely co-current source is from the counter-current banana orbit loss of ions. Another 共likely weaker兲 co-current source could be the E ⫻ B flow from the diverter Debye sheath radial electric field. The intrinsic toroidal rotation is usually in the cocurrent direction for H-modes. Its peak is not necessarily on axis and some discharges have reversed 共i.e., countercurrent兲 rotation.4 Unless a counter-current edge source mechanism can be identified, we can only conclude that the net core toroidal angular momentum must always be in the co-current direction. For all practical purposes we can assume redge locates the H-mode pedestal top. Without a quantitative model of the edge source mechanism, it seems unlikely that the boundary condition on the toroidal rotation velocity u␾共redge兲 can be predicted with any more certainty that the pedestal temperature T共redge兲 共and likely not one without the other兲. For simplicity of discussion we will assume the plasma edge recycling does not penetrate significantly inside r ⬍ redge so we can ignore convection in the core. Here we focus on the transport rather than the source of

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toroidal angular momentum and the mechanisms allowing the core null flow steady state. When the intrinsic toroidal rotation is finite and peaked in the center −⳵u␾ / ⳵r ⬎ 0 共as sometimes observed兲, there must be some “pinching” mechanism to keep the momentum from flowing “down” the gradient in the rotation. The analogy with the well-known particle density pinch should not be missed. Plasma density profiles are typically peaked despite a negligible core particle source. The null flow of plasma down the density gradient is maintained mainly by an electron temperature gradient or thermal pinch5 at sufficiently low collisionality. It is useful to describe the pinched toroidal angular momentum flux 共⌫x␾兲 by breaking up the effective radial diffusivities 共or viscosity ␩兲 into parts, ⌫x␾/共Rnmi兲 − u␾⌫x = − ␩eff ⳵ u储/⳵ r 储 eff = − ␩eff 储z ⳵ u储/⳵ r − ␩储 y ⳵ u储/⳵ r

= − ␩储 ⳵ u储/⳵ r − 共␩␷/a兲u储 − ␩E ⳵ ␷E/⳵ r with u储 the parallel fluid velocity. We distinguish these definitions from the effective toroidal viscosity with u␾ the toroidal fluid −␩␾eff ⳵ u␾ / ⳵r ⬅ −␩eff 储 ⳵ u储 / ⳵r velocity. We shall ignore 共or discount兲 the convection u␾⌫x ⬃ 共Bt / B兲u储⌫x in the core 共i.e., assume ⌫x ⬃ 0兲. Momentum transport results from the three forces indicated: 共1兲 the parallel field 共z兲 forces; the parallel velocity shear ␥ P ⬵ −⳵u储 / ⳵r and 共2兲 the finite 共parallel velocity兲 Mach number M 储 = u储 / cs共cs = 冑Te / mi兲; and 共3兲 the perpendicular field 共y兲 force; the E ⫻ B shear ␥E ⬵ ⳵␷E / ⳵r. Parallel velocity shear provides the normal momentum flow down the gradient in parallel velocity at large ␥ P 共and strong toroidal rotation, i.e., much larger that diamagnetic level rotation兲. However at low ␥ P, the E ⫻ B shear ␥E can provide a significant pinch 共i.e., balancing flow up the gradient in parallel velocity兲 resulting in a null flow state. Dominguez and Staebler6 first discovered the E ⫻ B shear pinch in slab geometry models. Subsequently, Staebler et al.7 modeled “the toroidal rotation generated in heated plasmas even without external torque” in a transport code using the GLF23 共Ref. 8兲 transport model with an ad hoc E ⫻ B pinch. Transport code modeling of intrinsic toroidal rotation growing from the edge via the E ⫻ B pinch has been recently revisited in some detail by Gurcan et al.9 using a slab theory model. More recently still Peeters, Angioni, and Strintzi10 have noted that another pinch mechanism can arise from the “Coriolis” curvature drift at finite Mach number M 储 in toroidal geometry. In Sec. II, starting from the six-dimensional Vlasov electromagnetic kinetic equation 共Sec. II A兲 we formulate the radial continuity equation for toroidal angular momentum transport and the equation for poloidal rotation 共Sec. II B兲. The needed turbulent stresses and sources are evaluated from the “delta-f” gyrokinetic-Poison equations 共Sec. II C兲. The gyrokinetic equations for a parallel drifted Maxwellian background plasma including parallel velocity shear 共␥ P兲 as well as finite Mach number 共M 储兲, and E ⫻ B shear 共␥E兲 are derived in Appendix A. Both the E ⫻ B and Coriolis pinch effects are demonstrated and quantified with the GYRO gyroki-

netic code electrostatic cyclic flux tube simulations11 in Sec. III where we compute 共in effect兲 the pinch coefficients ␩E / ␩储 eff and ␩␷ / ␩储 or equivalently ␩eff relative to ␹Ei , the ion energy 储 transport, as a function of ␥E and M 储 at various ␥ P. To illustrate mechanisms determining intrinsic toroidal rotation it is useful to integrate the null flow ⌫x␾ = 0 considering only the E ⫻ B shear pinch for simplicity 共i.e., ␩␷ / ␩储 ⬃ 0兲. Null flow required a critical E ⫻ B shear for the given parallel velocity shear, ␥E = 共␩储 / ␩E兲␥ P. Although 共␩储 / ␩E兲 varies as a function of radius, treating it as a constant allows a simple radial integration of the null flow, u␾共r兲 = 兩兵− 共B p/Bt兲uneo * − 共 ␩ E/ ␩ 储 兲 r ⫻关u*共B/Bt兲 + uneo * 兴其兩r

edge

/兵1 − 共␩E/␩储兲共B p/Bt兲其共r兲

plus any contributions from the edge boundary condition . . . + 兵1 − 共␩E/␩储兲共B p/Bt兲其共redge兲u␾共redge兲/兵1 − 共␩E/␩储兲 ⫻共B p/Bt兲其共r兲, where the poloidal velocity was assumed to be purely neoclassical u␪ = uneo * ⬀ −⳵Ti / ⳵r 关u储 = 共Bt / B兲u␾ + 共B p / B兲u␪兴 and u* ⬀ 共−⳵ Pi / ⳵r兲 / n is the diamagnetic velocity resulting from the use of force balance 共u⬜ = ␷E + u*兲. It is immediately clear why the intrinsic toroidal rotation 共like neoclassical poloidal rotation itself兲 has its origin in “heating.”7Also unless significantly augmented by u␾共redge兲 at the edge, the intrinsic toroidal rotation will be at the diamagnetic level 关⬀cs␳* where ␳* = 共cs / ⍀i兲 / a is the relative gyroradius兴 and hence project to smaller levels at reactor scales. Also note that even without an explicit E ⫻ B pinch 关共␩E / ␩储兲 = 0兴 there is a small toroidal projection from the neoclassical poloidal rotation alone. Which brings us to our final point: intrinsic toroidal rotation cannot be accurately modeled 关given a u␾共redge兲兴 unless the poloidal rotation is accurately given by the neoclassical formulas. Recent experiments12 suggest otherwise. Staebler13 has argued that turbulence shifts the poloidal rotation away from the neoclassical values as we confirm in Sec. II C. In Sec. III we have evaluated turbulent shifts with GYRO simulations.

II. FORMULATION A. Local Cartesian turbulent stress and force density from the Vlasov kinetic equation

Following Staebler,13 the “turbulent collision operator” 共D兲 method is used to write the six-dimensional 关xជ , ␷ជ 兴 Vlasov equation also keeping the collision operator 共C兲 and source 共S兲 in standard notation and Gaussian units

⳵f e ជ ជ 兴 · ⳵ f = C + D + S, + ␷ជ /c ⫻ B + ␷ជ · ⵜf + 关E ⳵t m ⳵ ␷ជ D=−

e m

再再

ជ + ␷ជ /c ⫻ ␦Bជ 兲 · ⳵ ␦ f 共␦E ⳵ ␷ជ

冎冎

,

共1兲

共2兲

ជ , ␦Bជ 兴 are the perturbed electric and magnetic field and ␦ f 关␦E

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Gyrokinetic theory and simulation…

is the perturbed distribution function, 兵兵 其其 denotes time or statistical average. 共Note statistical averages of linear perturbations 兵兵␦X其其 vanish, whereas bilinear 兵兵␦X␦Y其其 generally do not.兲 Taking the momentum moment mnuជ = m兰d3␷␷ជ f, results in ↔

ជ ·⌸+ⵜ ជ P − enEជ − enuជ ⫻ Bជ /c = Fជ , ⳵ 共mnuជ 兲/⳵ t + ⵜ

共3兲

where we have suppressed S and lumped the “collisional” C part and the “turbulent” D part together into a force density Fជ = m兰d3␷␷ជ 共C + D兲. The dynamic pressure and viscous stress ↔

↔

are given as P = 共1 / 3兲m兰d3␷␷2 f and ⌸ = m兰d3␷共␷ជ ␷ជ − I ␷2 / 3兲f. The species label is suppressed and otherwise taken to be an ion equation with charge e and mass m. The local B-field is in the z-direction and the normal to the flux surface is in the x-direction. We will assume that the flows in the flux surface are incompressible ⵜ · uជ = 0 with ux ⬃ 0 and first order is small compared to the thermal speed. The ordering is in the relative gyroradius ␳* = ␳s / a, where ␳s = cs / ⍀, cs = 冑Te / m, ⍀ = eB / mc, and a is the tokamak minor radius taken to be the unit of length. 共Strict ordering would require in-surface flows at the diamagnetic level, but in application larger parallel field flow will be allowed.兲 We assume leading order “force balance”

ជ P − enEជ − enuជ ⫻ Bជ /c = 0, ⵜ

共4兲

where P is a flux function so only ⵜx P ⫽ 0. We now proceed to write equations for the viscous stress taking m共␷i␷ j − ␦ij␷2 / 3兲 moments of Eq. 共1兲 共summing repeated indices兲

⳵ 共⌸ij兲/⳵ t + ⵜkQkij − en关Eiu j + E jui − 共2/3兲␦ijEkuk兴 − ⍀共Bl/B兲关␧ikl⌸kj + ␧ jkl⌸ki兴 = ⌬ij ,

冕

d3␷共␷i␷ j − ␷2␦ij/3兲共C + D兲.

共6兲

The first order Qkij = P关−共2 / 3兲uk␦ij + ui␦ jk + u j␦ik兴, then using first order force balance, plus incompressible flow in the surface ⵜiui = 0 共and ux = 0兲, we finally have 共Bl/B兲关␧ikl共⌸kj − mnuku j兲 + ␧ jkl共⌸ki − mnukui兲兴 = − Dij/⍀,

PWzz = ⌬zz .

共8c兲

In the momentum equation adding the electrons, the ជ / c term in Eq. 共3兲 electric field term vanished and the enuជ ⫻ B ជ ជ becomes j ⫻ B. Similarly, the D operator gives the turbulent ជ , and ion-electron collisions force density, Fជ becomes ␦ជj ⫻ ␦B conserve momentum. The 关⌬yz , ⌬yy , ⌬zz兴 are evaluated in Sec. II C from gyrokinetic simulations in Sec. III.

B. Flux surface average momentum radial continuity equations

For completeness we outline the standard treatment following Rosenbluth and Hinton14 共see also Ref. 15兲. The magnetic field coordinates and incompressible flow in the flux surface is given by the flow velocity

៝u = ␻共␺兲R␧ˆ ␾ + K共␺兲Bជ ,

共9a兲

␻共␺兲 = − c关⳵ ⌽/⳵ ␺ + 共1/ne兲 ⳵ P/⳵ ␺兴,

共9b兲

ជ , uជ p = K共␺兲B p

共9c兲

ជ = ␧ˆ I共␺兲/R + ␧ˆ /R ⫻ ⵜ ជ ␺ = B ␧ˆ + B ␧ˆ , B ␾ ␾ t ␾ p ␪

共9d兲

共5兲

where Qkij = m 兰 d3␷␷k关␷i␷ j − 共1 / 3兲␷2␦ij兴f and ⌬ij = m

As we will show in Sec. II B the projection of ⌸xz and ⌸xy in the toroidal direction determines the radial transport of toroidal angular momentum and the strain terms Wyz = ⵜyuz + ⵜzuy and Wyy = 2ⵜyuy will not contribute. The strain Wzz = 2ⵜzuz enters the magnetic pumping which constrains the poloidal rotation near its neoclassical value. From Eq. 共7兲 it is also clear that Dzz = 0 which implies

共7兲

where Dij = ⌬ij − PWij and Wij = ⵜiu j + ⵜ jui is the strain. Here we use the 关i , j , k兴 right-handed 关xˆ , yˆ , zˆ兴 coordinates with yˆ = zˆ ⫻ xˆ. From Bជ = 关0 , 0 , Bz兴, Eq. 共7兲 implies ⌸xz = 共Dyz + Dzy兲/2⍀ = 共⌬yz − Pⵜyuz − Pⵜzuy兲/⍀,

共8a兲

⌸xy = 共Dyy − Dxx兲/4⍀ = 共⌬yy − 2Pⵜyuy兲/⍀.

共8b兲

where ␻ is a toroidal rotation frequency, the poloidal rotation ជ ␺ 兩 / R. The velocity is found from K, and Bt = I / R and B p = 兩ⵜ usual 关␺ , ␪ , ␸兴 system is closely but not exactly aligned with the 关xˆ , yˆ , zˆ兴 triad, zˆ = Bជ / B ⬅ bˆ. It is useful to decompose Eq. 共9a兲 as uជ = uជ ⬜ + uជ 储 = ␻关R␧ˆ ␸ − 共I/B兲bˆ兴 + 关共I/B兲␻ + KB兴bជ ,

共10a兲

u⬜ = − ␻R共B p/B兲,

共10b兲

u储 = ␻R共Bt/B兲 + KB,

共10c兲

where Eqs. 共10b兲, 共9a兲, and 共9b兲 are consistent with radial force balance Eq. 共4兲. It is useful to note u␾ = ␻R + KB共Bt/B兲.

共10d兲

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With only 关␻ , K , I兴 constant on a flux surface it is easy to show14 the flux surface averages 具 典 are related by ¯ 2兲 ⳵ 具nB pu␪典/⳵ t = ⳵ 具nBu储典/⳵ t 共1 + 2q − 关BtR/具R2典兴 ⳵ 具nRu␾典/⳵ t, 共11a兲 where ¯q2 =

B2t R2 2具B2p典

关具R−2典 − 具R2典−1兴.

共11b兲

gence survives. V共r兲 is the volume enclosed by the midplane minor radius flux surface label r. 关More precisely, since the surface area is actually S共r兲 = V⬘共r兲具兩ⵜr 兩 典, the “surface average flux” is ⌫x␾ / 具兩ⵜr 兩 典.兴 From Eqs. 共8a兲, 共6兲, and 共2兲, ignoring the strain terms, turbulent stress

ជ · 关共␦Eជ + ␷ជ /c ⫻ ␦Bជ 兲␷ ␷ ␦ f兴共c/B兲 ⌸xz ⇒ ⌬Dyz/⍀ = m兰d3␷ⵜ ␷ y z =m

Thus we need a toroidal angular momentum equation for ⳵具nRu␾典 / ⳵t to get the toroidal rotation and an additional parallel momentum equation for ⳵具nBu储典 / ⳵t to get the poloidal rotation.

冕

d3␷关␷z共c␦Ey/B + ␷z␦Bx/B − ␷x␦Bz/B兲

+ ␷y共c␦Ez/B + ␷x␦By/B − ␷y␦Bx/B兲兴␦ f . 共16兲 Similarly from Eq. 共8b兲 D 兲/4⍀ ⌸xy ⇒ 共⌬Dyy − ⌬zz

1. Toroidal angular momentum transport

Dotting the combined momentum equation 关Eq. 共3兲兴 with R␧ˆ ␾,

=m

↔

ជ · ⌸ 兲 · R␧ˆ 典 + 具jជ · ⵜ␺典/c + 具R␧ˆ · Fជ 典. ⳵ 具mnRu␾典/⳵ t = − 具共ⵜ ␾ ␾ i 共12兲 共We will ignore the electron stress.兲 The momentum source from radial current can also be written as 具jជ · ⵜ␺典 / c = 具RjxB p典 / c. Note that from Maxwell’s equation ជ ␺典 / c + 具⳵Eជ / ⳵t · ⵜ ជ ␺典 = 具ⵜ ជ ⫻ Bជ · ⵜ ជ ␺典 = 具ⵜ ជ · 共Bជ ⫻ ⵜ ជ ␺兲典 = 0. 4␲具jជ · ⵜ Thus ignoring the stress and angular momentum source, the total angular momentum of the plasma and the field ជ ␺ · Eជ 典 / 4␲c is conserved. The turbulent source is 具mnRu␾典 + 具ⵜ

=m

冕 冕

ជ · 关共␦Eជ + ␷ជ /c ⫻ ␦Bជ 兲共␷ ␷ − ␷ ␷ 兲␦ f/4兴共c/B兲 d 3␷ ⵜ ␷ y y x x d3␷ 关␷y共c␦Ey/B + ␷z␦Bx/B − ␷x␦Bz/B兲␦ f兴.

In Eq. 共17兲 we have used the fact that on gyrokinetic evaluation 共see Sec. II C兲

m

冕

d3␷ 关␷y共␦Ey + ␷z/c␦Bx − ␷x/c␦Bz兲␦ f兴共c/B兲

=−m

ជ 典 = 具R␦ j ␦B − R␦ j ␦B 典/c 具R␧ˆ ␾ · F x p p x

冕

d3␷ 关␷x共␦Ex − ␷z/c␦By + ␷y/c␦Bz兲␦ f兴共c/B兲. 共18兲

= 具R关共Bt/B兲共␦ jx␦By − ␦ j y␦Bx兲 − 共B p/B兲共␦ jz␦Bx − ␦ jx␦Bz兲兴典/c,

共13兲

which we will come to shortly. The turbulent viscous stress is ↔

ជ · ⌸ 兲 · R␧ˆ 典 = 具ⵜ ជ · 关␧ˆ ⌸ 共− B /B兲R + ␧ˆ ⌸ 共B /B兲R兴典 具共ⵜ ␾ x xy p x xz t i = V⬘共r兲−1⳵r兵V⬘共r兲具兩ⵜr兩关⌸xz共Bt/B兲R − ⌸xy共− B p/B兲R兴典其.

2ជ

The strain terms in Eq. 共8a兲 and in Eq. 共8b兲 ⵜyuz = 共Bt/B兲共兩ⵜ␪兩⳵␪兲关␻R共Bt/B兲 + KB兴, ⵜzuy = − 共B p/B兲共兩ⵜ␪兩兲⳵␪关␻R共B p/B兲兴, ⵜyuy = − 共Bt/B兲共兩ⵜ␪兩⳵␪兲关␻R共B p/B兲兴,

共14兲

The average flux of the toroidal angular momentum is ␾ + ⌫␾xy . ⌫x␾ = 具兩ⵜr兩关⌸xz共Bt/B兲R − ⌸xy共B p/B兲R兴典 = ⌫xz

共17兲

共15兲

We have used the fact that 共R␧ˆ ␾兲 = R ⵜ␾ is constant on a flux surface 关see Ref. 16, Eq. 共2.91兲兴 and only the radial diver-

have been ignored since they vanish on flux surface average by up-down symmetry. 2. Poloidal rotation equation

Dotting the combined momentum equation Eq. 共3兲 with ជB and using Eq. 共11兲 with the toroidal rotation equation Eq. 共12兲, the poloidal rotation is given by

↔

ជ · 共ⵜ ជ · ⌸兲典 − 关B R/具R2典兴关具ⵜ ជ · ␧ˆ ⌸ 共B /B兲R典兴 ¯ 2兲 ⳵ 具B pu␪典/⳵ t = − 具B mn共1 + 2q t x xy p ជ · ␧ˆ ⌸ 共B /B兲R兲兵1 − 关共B/B 兲2具R2典/R2兴其 + 具B共␦ j ␦B − ␦ j ␦B 兲典/c + 关BtR/具R2典兴具ⵜ x xz t t x y y x − 关BtR/具R2典兴关具R关共Bt/B兲共␦ jx␦By − ␦ j y␦Bx兲 − 共B p/B兲共␦ jz␦Bx − ␦ jx␦Bz兲兴典 − 具jxB p典兴/c.

共19兲

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122507-5

Phys. Plasmas 14, 122507 共2007兲

Gyrokinetic theory and simulation…

The parallel stress term ⌸xz nearly cancel since 兵1 − 关共B / Bt兲2具R2典 / R2兴其 ⬃ 0 and the dominant turbulent perpendicular stress ⌸xy 共or essentially the “poloidal viscosជ · 关␧ˆ ⌸ 共B / B兲R典兴 ity”兲 is contained in −关BtR / 具R2典兴关具ⵜ x xy p ␾ 2 −1 = 关BtR / 具R 典兴V⬘共r兲 ⳵r兵V⬘共r兲⌫xy其 关see Eq. 共15兲兴 must be compared with the likely controlling neoclassical magnetic ↔ ជ · 共ⵜ ជ · ⌸兲典. Previous experience pumping drag contained in −具B

with the GLF23 transport model8 suggests that the turbulent perpendicular tress is subdominant in current tokamaks in the plateau neoclassical regime; u␪ will be dragged to the neoclassical poloidal rotation uneo * . Following Ref. 13, we now proceed to argue that the turbulence will only “shift” the u␪ from uneo * by 共what turns out to be兲 a small amount. It can be shown 共see Ref. 15, p. 222兲 that since the diagonal terms of the stress tensor are ↔

↔

dominant with ⌸ = 共P储 − P⬜兲共zˆzˆ − I / 3兲 关and the off-diagonal terms are at most O共␳*兲兴, then ↔

ជ · 共ⵜ ជ · ⌸兲典 = 具共P − P储兲zˆ · ⵜB典 = − 具共3/2兲共zˆ · ⵜB兲⌸ 典. 具B ⬜ zz 共20兲 In the neoclassical collisional regime the parallel stress ⌸zz = −␩Wzz and for incompressible flow in the flux surface Wzz = 2ⵜzuz ⇒ 2共u␪ − uneo * 兲共zˆ · ⵜB兲 / B p 共see Ref. 15, p. 228兲. The strong stress is relieved only when the strain vanishes, c = PWzz i.e., u␪ ⇒ u␪neo. From Eq. 共8c兲 the collisional part ⌬zz 13 D − ⌬zz, Staebler infers there must be a turbulent shift in the neoclassical rotation, ↔

ជ · 共ⵜ ជ · ⌸兲典 = ␩具共3共zˆ · ⵜB兲2共u − uneo兲/B 典 具B ␪ p * ⇒ ␩具3关共zˆ · ⵜB兲2共u␪ − uneo * 兲/B p − 共zˆ ·

D ⵜB兲⌬zz /2P兴典.

共21兲

The inference by analogy to neoclassical theory is based on the additivity of the collisional part C and turbulent part D in Eq. 共8c兲 combined with the fact that the parallel classical viscosity ␩ increases with collisionality 共at least in the banana regime兲. Thus the turbulent shift in the neoclassical rotation is D /2P典/具共zˆ · ⵜB兲2典, ⌬u␪D = B p具共zˆ · ⵜB兲⌬zz

共22a兲

with D ជ ␦ f兴. ⌬zz = e兰d3␷关共4/3兲␷z␦Ez␦ f − 共2/3兲␷ជ ⬜ · ␦E ⬜

共22b兲

Appendix B provides a formal derivation 共valid in banana, plateau, and collisional regimes兲 of Eq. 共22兲 共and its limitations兲 missing from Ref 13. There we argue that there may be additional and likely smaller shifts on top of Eq. 共22兲. While the rotation equation Eq. 共19兲 may be evolved with any 共external兲 momentum sources and losses 共or gains兲 associated with the turbulent perpendicular stress ⌸xy 共and ⌸xz兲 included, the parallel field magnetic pumping is usually taken to be so strong that the shifted neoclassical poloidal D rotation u␪ ⬃ uneo * + ⌬u␪ should be a good first estimate. However it should be noted that uneo * / cs 共and the neoclassical drag term itself兲 is O共␳*兲, whereas the shift ⌬u␪D / cs is O共␳2*兲

hence likely small. Since the cross field Reynolds stress term ជ · 关␧ˆ ⌸ 共B / B兲R典兴 is also O共␳2 兲 in Eq. 共19兲, −关BtR / 具R2典兴关具ⵜ x xy p * ជ · ␧ˆ ¯ 典 ⬃ O共␳ 兲兴 a 关assuming the usual scale separation 具ⵜ x * further small shift cannot be precluded. We focused on ⌬u␪D / uneo * since it is easiest to quantify without actually solving a transport equation with momentum sources or solving for the neoclassical equilibrium uneo * .

C. Gyrokinetic evaluation of turbulent stresses and sources D We must evaluate the turbulent stresses 关⌸xz , ⌸D xy 兴 in D Eqs. 共16兲 and 共17兲 as well as ⌬zz in Eq. 共22兲. We will show the “magnetic flutter” sources, 共␦ jx␦By − ␦ j y␦Bx兲 and 共␦ jz␦Bx − ␦ jx␦Bz兲, vanish to leading order or are very small D and ⌸D on local radial average. The leading terms in ⌸xz xy have radial E ⫻ B motion, ␦␷Ex = c␦Ey / B. To evaluate these turbulence terms with the time average correlations of ␦␷Ex with ␦ f obtained from gyrokinetic simulations, we follow 关see Ref. 17, Eq. 共120兲兴 to write in leading order

␦ f共xជ , ␷ជ 兲 = ␦ f共xជ , ␷z, ␮, ␣兲 ⇒ ␦ f GK共xជ − ␳ជ , ␷z, ␮兲 = ␦ f adia共xជ , ␷z, ␮兲 + ␦g共xជ − ␳ជ , ␷z, ␮兲,

共23兲

2 where ␮ = 共1 / 2兲m␷⬜ / B and ␣ is the gyrophase angle as it appears in ␷ជ ⬜ = ␷⬜关cos共␣兲eˆx + sin共␣兲eˆy兴. Only the nonadiabatic part ␦g will be in phase with ␦␷Ex = c␦Ey / B so the adiabatic part does not contribute to transport. It is useful to expand in the local Fourier modes of the turbulence and note the following relations:

␦g共xជ − ␳ជ 兲 = 兺 k exp关ik⬜␳⬜ sin共␣ − ␤兲兴␦gk共0兲exp共ikជ · xជ 兲, 共24a兲

␦Eជ ⬜共xជ 兲 = 兺 k − ikជ ⬜␦␾k共0兲exp共ikជ · xជ 兲,

共24b兲

␦Bជ ⬜共xជ 兲 = 兺k共ikជ ⬜ ⫻ zˆ兲␦Azk共0兲exp共ikជ · xជ 兲,

共24c兲

where ␳ជ = ␳⬜bជ ⫻ sជ = −␳⬜关sin共␣兲eជ x − cos共␣兲eជ y兴, sជ ⬅ ␷ជ ⬜ / ␷⬜, kជ = k⬜共cos ␤␧ជ x + sin ␤␧ជ y兲, so kជ · ␳ជ = kជ · bជ ⫻ sជ = −k⬜␳⬜ sin共␣ − ␤兲. We will use 养d␣ / 2␲ exp关ik⬜␳⬜ sin共␣ − ␤兲 − in␣兴 J−n共k⬜␳⬜兲 = 共−1兲nJn共k⬜␳⬜兲, and = exp共−i␤n兲Jn共k⬜␳⬜兲, Here xជ 2nJn共k⬜␳⬜兲 = 共k⬜␳⬜兲关Jn−1共k⬜␳⬜兲 + Jn+1共k⬜␳⬜兲兴. means the local displacement, so for ␦gk共0兲, the “0” is the local point 共which we suppress兲. The velocity space integrands 兰d3␷ ⇒ 2␲兰Bd␮d␷z / m after averaging over the phase 兰d␣ / 2␲ will be denoted by 兵其. Thus we have 兵␷z␦ f k其 = 兵␷zJ0共k⬜␳⬜兲␦gk其,

共25a兲

兵␷y␦ f k其 = 兵␷⬜共ikx/k⬜兲J1共k⬜␳⬜兲␦gk其,

共25b兲

兵␷x␦ f k其 = − 兵␷⬜共iky/k⬜兲J1共k⬜␳⬜兲␦gk其.

共25c兲

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122507-6

Phys. Plasmas 14, 122507 共2007兲

Waltz et al.

We can now write the average flux of toroidal angular momentum Eq. 共15兲 as

⌫x␾共conv兲 =

D ␾ ␾ 共Bt/B兲R − ⌸D ⌫x␾ = 具兩ⵜr兩关⌸xz xy 共B p/B兲R兴典 = ⌫xz + ⌫xy , 共26a兲

D = ⌸xz

⌸D xy ⬇

冕

冕

d3␷兺共␦␷Ex兲−km␷z关J0共k⬜␳⬜兲␦gk兲, k

共26b兲

2 /2兲共c/ZeB兲关J0共k⬜␳⬜兲␦gk兴, d3␷ 关 兺 共␦␷Ex兲−kmⵜx共m␷⬜ k

共26c兲

␷⬜共ikx / k⬜兲J1共k⬜␳⬜兲␦gk where we approximated 2 = ⵜx共m␷⬜ / 2兲共c / ZeB兲关共J0共k⬜␳⬜兲␦gk + J2共k⬜␳⬜兲␦gk兴 by dropping the small J2共k⬜␳⬜兲 term. Equation 共26b兲 is clearly the E ⫻ B transport of m␷z zˆ-momentum and Eq. 共26c兲 of a “diamagnetic” m共␷*兲y or yˆ -momentum as expected. Since the gyrokinetic simulation refers to ␦gk driven by a parallel drifted Maxwellian 共see Appendix A兲, we must then explic␾ + ⌫␾xy + ⌫x␾共conv兲 where itly add the convective flow, ⌫x␾ ⇒ ⌫xz

⌬u␪D = B p共4/3兲具共zˆ · ⵜB兲e ⬃ 共4/3兲具共2R sin ␪兲e

冕 冕

冓

兩ⵜr兩mR共Bt/B兲uz

⫻

冕

d3␷ 兺 共␦␷Ex兲−k关J0共k⬜␳⬜兲␦gk兲兴 k

冔

共26d兲

while deleting it from Eq. 共26b兲. This means that ␷z = ␷z⬘ + uz is replaced by ␷z⬘ 共the deviation from the bulk parallel drift ␾ 兲. In practice ⌫x␾共conv兲 uz兲 in Eq. 共26b兲 共and in ⌫xz ⬃ 具mR共Bt / B兲uz典⌫x is often replaced with ⬃具mRu␾典⌫x where ⌫x is the average radial ion particle flux. Formulation of the gyrokinetic equation used to evaluate the correlation of ␦gk with 共␦␷E兲−k is given in Appendix A for the GYRO code.11 The formal derivation is for the special case of purely toroidal rotation velocity projected into perpendicular E ⫻ B and parallel velocity, but the application is to the general case. From Eqs. 共25b兲 and 共25c兲 it is clear that on flux surface ជ · ␦ជj 典关 average 具 典 and local radial average 兴 关 that 兴具␦E ⬜ ⬜ =0. This proves Eq. 共18兲. It also means that the second term D can be dropped. The turbulent shift in the in Eq. 共22b兲 for ⌬zz neoclassical rotation Eq. 共22兲 becomes

d3␷兺k␷z共␦Ez兲−k关J0共k⬜␳⬜兲␦gk兴/2P典/具共zˆ · ⵜB兲2典 d3␷ 兺 k ␷z共␦Ez兲−k关J0共k⬜␳⬜兲␦gk兴/2P典,

with the latter in the infinite aspect ratio circular sˆ − ␣ limit where B ⬃ B0 / 关1 + 共r / R兲cos ␪兴. Similarly, the flux surface and local radial average momentum source terms vanish,

兴具R共Bt/B兲共␦ jx␦By − ␦ j y␦Bx兲/c典关 =兴具R共Bt/B兲共␦ jxⵜx␦Az + ␦ j yⵜy␦Az兲/c典关=0.

共28兲

Neglecting ␦Bz the remaining momentum source in Eq. 共13兲 ␦ jz␦Bx / c = JxBz / c is proportional to the nonambipolar radial current carried by the magnetic flutter. However for local microturbulence, where we are permitted a local radial average in addition to the flux surface average, the nonambipolar radial current is known to vanish.18 We hasten to add the neglect momentum sources from perturbed magnetic fields applies only to local homogenous microturbulence and larger scale nonlocal magnetic modes are capable of “magnetic breaking” 共or “dragging on”兲 rotation. Thus some magnetic turbulence momentum “source” 具␦ jz␦Bx / c典 may appear in a global gyrokinetic simulation. In a global code there is no local radial averaging 共i.e., 兺ky,kx ⇒ 兺ky兲 at each local radius as in a local cyclic flux-tube simulation. D and ⌸D While the leading terms in ⌸xz xy have radial

共27兲

E ⫻ B motion, substitution of the combination ␦␷Ex = c␦Ey / B ⇒ c␦Ey / B + ␷z␦Bx / B provides the additional small magnetic flutter transport. It is represented by c共−ⵜy关␦␾ − ␷z / c␦Az兴兲 / B using the generalized potential19 ␦␾ ⇒ 关␦␾ − ␷z / c␦Az兴. We ignore the ␦Bz terms with the provision that grad-B drift is replaced by curvature drift 共as in the MHD D approximation.兲 The other terms in ⌸xz with m␷y共c␦Ez / B + ␷x␦By / B − ␷y␦Bx / B兲␦ f are O共␳*兲 smaller. 关More accurately m␷y共c␦Ez / B兲␦ fis O共␳*兲 smaller but m␷y共␷x␦By / B − ␷y␦Bx / B兲␦ f is O共k⬜␳⬜ / 2兲 smaller than the m␷z共␷z␦Bx / B兲␦ f term included in the magnetic flutter.兴

TABLE I. Simulations of GA-standard case at nonzero ␥ P, ␥E, and M 储. Diffusivities are in gyro-Bohm units: 关cs / a兴␳s2 = csa␳2*. The turbulent poloidal rotation shift is in units of cs␳2*.

␥P

␥E M 储

1.0 0.2 0.2 0.2

0.0 0.0 0.2 0.0

␩eff 储

␩储zeff

eff ␩储y

␹eff Ei

0.0 11.5 11.1 0.42 13.3 0.0 9.61 9.47 0.14 13.0 0.0 −1.29 0.15 −1.44 4.14 0.4 −11.1 −10.2 −0.85 14.8

eff ␹Ee

Deff

3.57 3.38 1.53 3.75

−1.97 −2.10 −0.87 −2.35

eff ␩eff ⌬uD␪ 储 / ␹ Ei

0.86 0.74 −0.31 −0.74

+43 +53 +19 +48

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122507-7

Gyrokinetic theory and simulation…

Phys. Plasmas 14, 122507 共2007兲

III. SIMULATIONS WITH TOROIDAL ANGULAR MOMENTUM PINCHES AND A TURBULENT POLOIDAL ROTATION SHIFT

To illustrate the pinch mechanisms in the radial flux of toroidal angular momentum we have made GYRO local fluxtube simulations about the GA-standard case: R / a = 3, r / a = 0.5, q = 2, sˆ = 1, a / LT = 3, a / Ln = 1, Te / Ti = 1, ␯ei / 关cs / a兴 = 0, and ␤ = 0. These kinetic electron electrostatic simulations are in the infinite aspect ratio sˆ − ␣ circular geometry limit with the MHD ␣ = 0. The GYRO website20 has a downloadable transport database of 300+ kinetic electron electrostatic simulations centered around the GA-standard case and a record of many published simulations with parameter scans about this case. Table I gives details of one high- and three low-rotation cases. A number of observation are clear from Table I: From high to low parallel velocity shear 共␥ P ⬅ −R ⳵ 共具u储典 / R兲 / ⳵r ⬃ −⳵u储 / ⳵r兲, without the effect of E ⫻ B shear 兵␥E ⬅ 共r / q兲 ⳵ 关␷E / 共B pR兲兴 / ⳵r ⬃ ⳵␷E / ⳵r其 or finite Mach number 共M 储 = u储 / cs兲, the effective momentum diffusivity is about the same as the total ion energy diffusivity as expected form the simplest theoretical arguments as well as eff experimental observation.2 The same ratio 共␩eff 储 / ␹ Ei ⬃ 0.8兲 共for strong ␥ P and weak ␥E兲 reported in earlier gyrokinetic simulations21,22 and quasilinear calculations.23 In the absence of E ⫻ B shear, the parallel component of momentum diffueff sivity dominates over the perpendicular 共␩储z  ␩eff 储y , where eff eff eff ␩储 ⬅ ␩储z + ␩储y 兲. See the first equations in Sec. I and discussion of Eq. 共26兲 removing convection. However at low parallel velocity shear, it is the perpendicular component that eff contains the E ⫻ B shear pinch 共␩eff 储 ⬃ ␩ 储y ⬍ 0兲. The finite M 储 eff pinch clearly operates on the parallel component 共␩eff 储 ⬃ ␩ 储z ⬍ 0兲. As illustrated in Fig. 1共a兲, the E ⫻ B shear pinching6 is only effective at low values of parallel velocity shear ␥ P. For M 储 = 0 the null flow point at ␥E ⬃ 0.069 共0.034兲 for ␥ P ⬃ 0.2 共0.1兲 and there is no null point for ␥ P ⬎ 0.3 in this sˆ = 1 case. 共For sˆ = 0.5 we find the null flow point at ␥E ⬃ 0.036 for ␥ P ⬃ 0.2 which conforms to the null E ⫻ B shear rate ␥E ⬀ sˆ␥ P as expected from the ballooning mode representation where the symmetry breaking E ⫻ B shear enters in the combination ␥E / sˆ against the symmetry breaking from ␥ P.兲 As seem in Fig. 1共b兲, the sign of ␥E has no effect on the E ⫻ B shear stabilization with the low-k quench point at ␥E = ± 0.5,21,24 however the E ⫻ B momentum pinch effect requires ␥E aligned with ␥ P. The GA-standard case has a significant particle pinch 共Deff ⬍ 0 in Table I兲 driven by the electron temperature gradient.5 Convection has been discounted from effective viscosity 共␩eff 储 兲 whereas the energy eff eff , ␹Ee兲 count convection. None of the diffudiffusivities 共␹Ei eff eff , ␹Ee, and Deff sivities are strongly effected by ␥ P and the ␹Ei all decrease in proportion to increasing ␥E. As illustrated in Fig. 2, finite parallel velocity itself can contribute a pinch even in the absence of E ⫻ B shear.10 Figure 2共a兲 is plotted so the characteristic Coriolis “pinch velocity” 共in the absence of E ⫻ B shear兲 can be obtained from Vpinch = ␩储共L␷ = 0兲 / L␷共␩储 = 0兲. From Table I ␩储共L␷ = 0兲 ⬃ 9.6csa␳2* and from Fig. 2共a兲 L␷共␩储 = 0兲 ⬃ 1.5a so that

FIG. 1. 共Color online兲 Ratio of effective parallel viscosity to ion energy diffusivity in 共a兲 and ion energy diffusivity 共in gyro-Bohm units兲 in 共b兲 vs E ⫻ B shear rate for the GA-standard case of Table I with M 储 = 0.

Vpinch ⬃ 6cs␳2* for the GA-standard case at ␥E = 0. Figure 3 shows that the E ⫻ B 共Ref. 6兲 and Coriolis10 pinches at ␥ P = 0.5 and ␥E = 0.2 are additive as expected. Furthermore, it is clear that null flow can be obtained for ␥ P ⬎ 0.3 when finite parallel velocity is taken into account. Going back to the last column in Table I, it is apparent that for the GA-standard case at hand, the shift in the neoclassical poloidal velocity13 is about ⌬u␪D ⬃ 50cs␳2* 共with ␥E = 0兲. For a / LTi = 3, the neoclassical rotation is uneo * ⬃ 共−3.5⇒ 5.1兲cs␳* from the banana to the collisional regime. Since ⌬u␪D / uneo * ⬀ ␳*, and core values of ␳* ⬃ 0.2% −0.6% are typical of current experiments, ⌬u␪D ⬃ 共0.1− 0.3兲cs␳* appears to be less than about 10% of uneo * . Note that the shift contains an ion component and an electron component, but the electron component is strongly oscillating about zero and should be ignored. In the formulation for determining the poloidal

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122507-8

Waltz et al.

Phys. Plasmas 14, 122507 共2007兲

FIG. 3. Ratio of effective parallel viscosity to ion energy diffusivity vs finite parallel velocity M 储 GA-standard case of Table I with ␥E = 0 and ␥E = 0.2 at ␥ P = 0.5. The convective transport of momentum has been discounted.

FIG. 2. 共Color online兲 Ratio of effective parallel viscosity to ion energy diffusivity vs finite parallel velocity M 储 in 共a兲 and 共same data兲 vs the parallel velocity gradient length LV / a in 共b兲. GA-standard case of Table I with ␥E = 0. The convective transport of momentum has been discounted.

rotation Sec. II B 2 and Eq. 共19兲, we noted that the parallel field viscose stress term 共or neoclassical drag兲 is likely to dominate holding the poloidal rotation close to a slightly shifted neoclassical rotation. However at vanishing collisionality, the neoclassical drag vanishes and the cross field Reynolds stress 共or actually its divergence兲 −关BtR / 具R2典兴 ជ · 关␧ˆ ⌸ 共B / B兲R典兴 = 关B R / 具R2典兴V⬘共r兲−1⳵ 兵V⬘共r兲⌫␾ 其 must ⫻关具ⵜ x xy p t r xy be considered. It is easy to see from the definitions that the Reynolds stress ⌸xy ⬇ −共B / B p兲关nimi兴␩储y␥ p can be obtained from the perpendicular viscosity ␩储y given in Table I. IV. SUMMARY AND CONCLUSIONS

Gyro-Bohm scaled delta-f gyrokinetic equations for the turbulent radial transport of toroidal angular momentum and turbulent shift from the neoclassical poloidal rotation13 have

been formulated from the more fundamental six-dimensional Vlasov kinetic equations. The 共slab geometry兲 E ⫻ B shear pinch6 关Fig. 1共a兲兴 and 共toroidal geometry兲 finite parallel velocity “Coriolis” pinch10 关Fig. 2共a兲兴 mechanisms required to sustain the steady-state null toroidal angular momentum transport in discharges with intrinsic toroidal rotation 共in the absence of injected auxiliary torque兲 were demonstrated and quantified with GYRO 共Ref. 11兲 simulations. From Table I, in discharges with strong parallel velocity shear 共␥ P兲 but sufficiently weak E ⫻ B shear 共␥E兲, the effective momentum transport diffusivity 共␩eff 储 兲 is comparable to eff 兲 and parallel prothe ion energy transport diffusivity 共␹Ei eff 兲 dominates over jected toroidal momentum diffusivity 共␩储z the perpendicular projection. However at low parallel velocity shear and sufficiently strong E ⫻ B shear, the perpendicular projection of toroidal momentum diffusivity 共␩eff 储y 兲 domieff 兲, and pinched flow up nated over the parallel projection 共␩储z the parallel velocity shear gradient results. The toroidal Coriolis pinch with finite parallel velocity or Mach number 共M 储 = u储 / cs兲 acts on the parallel projected toroidal momentum eff diffusivity 共␩储z 兲 and adds to the E ⫻ B shear pinch 共Fig. 3兲. Since toroidal angular momentum transport is driven by a combination of parallel forces 共␥ P , M 储兲 and perpendicular force 共␥E兲, small diamagnetic level intrinsic toroidal rotation cannot be separated from the diamagnetic level poloidal rotation. The turbulent shift from the neoclassical poloidal rotation13 appears to be less than about 10% at the finite-␳* values of current experiments and even smaller on reactor scales since the shift as a fraction of the neoclassical poloidal rotation scales as ␳*. The turbulent shift as calculated here seems unlikely to account for the experimentally observed12 deviations from neoclassical poloidal rotation.

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122507-9

Phys. Plasmas 14, 122507 共2007兲

Gyrokinetic theory and simulation…

␦ f = 兵关e␦␾ − e␦A储␷储/c兴 − e具␦␹典eiL其关⳵ F0/⳵ ␧

ACKNOWLEDGMENTS

R.E.W. thanks Dr. Clemente Angioni 共IPP-Garching兲 for carefully reading the manuscript and catching a significant factor 10 error. This work was supported by the U.S. Department of Energy under Cooperative Agreement No. DE-FC0204ER54698. APPENDIX A: GYROKINETIC FORMULATION

For completeness we give an abbreviated formulation of the ␦ f gyrokinetic and Poisson–Ampere equations for the 11 GYRO code with finite parallel Mach number M 储 = u储 / cs of the ions isolating the parallel velocity shear ␥ p = −Rd共u储 / R兲 / dr and E ⫻ B shear ␥E = 共r / q兲d共Ex / RB p兲 / dr. Following Antonsen and Lane 共AL兲,25 and adding the nonជ ␦h, the gyrokinetic equation is linear term ␷ជ ␹ · ⵜ

共A4兲

We now assume a parallel drifted Maxwellian model F0共xជ ,␧, ␮兲 = n0共m/2␲T0兲3/2 2 ⫻exp关− m/2T0共␷⬜ + 共␷储 − u储兲2兲兴,

共A5兲

2 where ␷⬜ = 2␮B / m and ␷储 = ␴关2 / m共␧ − ␮B − e⌽兲兴1/2 with ␴ = ± 1. The derivatives of the Maxwellian needed and ⳵F0 / ⳵␧ are 关⳵F0 / ⳵␧ + 1 / B ⳵ F0 / ⳵␮兴 = −共m / T0兲F0 = −共m / T0兲F0共1 − u储 / ␷储兲. The gradient of the Maxwellian at constant ␧ is

ជ ln F = ⵜ ជ ln n + 关共m/2T 兲共␷2 + 共␷储 − u储兲2兲 ⵜ 0 0 0 ⬜ ជ ln T − m/T 关− 共e/m兲共1 − u储/␷储兲ⵜ ជ⌽ − 3/2兴ⵜ 0 0

ជ ␦h + C␦h ⳵ ␦h/⳵ t + 共␷储zˆ + ␷ជ E + ␷ជ d + ␷ជ ␹兲 · ⵜ

ជ B − 共␷储 − u储兲ⵜ ជ u储兴. + 共u储/␷储兲␮ⵜ

ជ F + 共␷储zˆ + ␷ជ + ␷ជ 兲 · ⵜ ជ 具␦␹典关e ⳵ F /⳵ ␧兴, = − ␷ជ ␹ · ⵜ 0 E d 0 共A1兲 where

␦ f = 关e␦␾兴关⳵ F0/⳵ ␧ + 1/B ⳵ F0/⳵ ␮兴 − 关e␦A储␷储/c兴1/B ⳵ F0/⳵ ␮ + gALeiL .

共A2兲

The g used by AL is gAL = ␦h − e关⳵F0 / ⳵␧ + 1 / B ⳵ F0 / ⳵␮兴具␦␹典 and the h is hAL = h − e ⳵ F0 / ⳵␧具␹典. L is the gyrophase. ជ ⌽, ␷ជ = 共c / B兲zˆ ⫻ ⵜ ជ 具␦␹典, and ␷ជ = 共c / B兲zˆ ␷E = 共c / B兲zˆ ⫻ ⵜ ␹ d 2 2 2 ជ ⫻ 共m␷储 zˆ · ⵜzˆ + ␮ⵜB兲 with ␧ = mv / 2 + e⌽ and ␮ = m␷⬜ / 2B. The gyroaveraged perturbed generalized potential is 具␦␹典 = J0共␥兲关␦␾ − ␷储␦A储兴 +

+ 1/B ⳵ F0/⳵ ␮兴 + 关e␦A储␷储/c兴 ⳵ F0/⳵ ␧ + ␦heiL .

2 共m␷⬜ /e兲共J1共␥兲/␥兲␦B储/B,

共A3兲

ជ 具␦␹典 ⬅ 0. When we evaluate the with ␥ = k⬜␷⬜ / ⍀. Note ␷ជ ␹ · ⵜ Poisson–Ampere relations, we take velocity space integrals over

共A6兲

At this point we make an approximation going to the limit of large and purely toroidal flow 共which ignores ជ = cⵜ ជ ⌽ and diamagnetic level flows兲, uជ = eˆ␾u␾ with uជ ⫻ B u储 / c = −共I / B兲⌽⬘共␺兲. Taking I = RBT⬵ constant, and u⬘储 / c = −共I / B兲⌽⬙共␺兲 − 共I / B兲⬘⌽⬘共␺兲, we have

ជ ln F = 兵ⵜ ជ ln n + 关共m/2T 兲共␷2 + 共␷储 − u储兲2兲 ⵜ 0 0 0 ⬜ ជ ln T − m/T 关− 共e/m兲共1 − u储/␷储兲⌽⬘ − 3/2兴ⵜ 0 0 ជ␺ + 共␷储 − u储兲cI/B⌽⬙兴ⵜ ជ B/B兴. − 共m/T0兲u储关␮B/␷储 + 共␷储 − u储兲兴ⵜ After much algebra the right-hand side 共RHS兲 of Eq. 共A1兲 can be evaluated with much cancellation 关without the low-␤ approximation zˆ ⫻ ⵜB / B ⇒ zˆ ⫻ 共zˆ · ⵜzˆ兲兴

ជ ␦h + C␦h = − 兵n⬘/n + 关共m/2T 兲共␷2 + ␷⬘储 2兲 − 3/2兴T⬘/T − 共m/T 兲␷⬘储 cI/B⌽⬙其ⵜ ជ ␺ · zˆ ⫻ ⵜ ជ 具␹典F ⳵ ␦h/⳵ t + 共␷储zˆ + ␷ជ E + ␷ជ ␹ + ␷ជ d兲 · ⵜ 0 0 0 0 0 ⬜ 0 0 ជ B/B共␮B + u储␷⬘储 + u2储 兲 − 共e/T0兲共c/B兲关zˆ ⫻ ⵜ ជ zˆ兴 · ⵜ ជ 具␦␹典F − 共e/T 兲␷⬘储 zˆ · ⵜ ជ 具␦␹典F . + m共␷⬘储 2 + u储␷⬘储 − u2储 兲zˆ ⫻ zˆ · ⵜ 0 0 0

The phase space can be written in terms of ␷⬘储 = ␷储 − u储 and 2 兲. Note in particular that vE the Doppler ␧⬘ = 共m / 2兲共␷⬘储 2 + ␷⬜ shift occurs only on the LHS where it can only result in a frequency shift of the normal modes. In fact the LHS terms can be rewritten as ␷储zˆ + ␷ជ E = ␷储zˆ + 共c / B兲zˆ ˆ 兴共c / B兲⌽⬘共␺兲 = zˆ␷⬘ − Re ˆ c⌽⬘共␺兲 ⫻ ⵜ␺⌽⬘共␺兲 = ␷储zˆ + 关Izˆ − BRe 储 ␾ ␾ ជ ␺ ⫻ eˆ / R. The bounce points are at ˆ +ⵜ using Bជ = I / Re ␾ ␾

共A7兲

zˆ · eˆ␪共␷储 − u储兲 = 0 or 共␷储 − u储兲 = ␷⬘储 = 0. Furthermore it is easy to ˆ c⌽⬘共␺兲 · ⵜ␦h see that the Doppler rotation term is −Re ␾ → i共nq / r兲关c / Bunit共⳵⌽ / ⳵r兲兴␦hn, where following Waltz and ¯ / 共r ⳵ r兲 with Miller26 we define Bunit = B0␳ ⳵ ␳ / 共r ⳵ r兲 = ⳵␹ ¯ ¯ q = ⳵␹ / ⳵␺. ␹ is the toroidal flux and n is the toroidal mode number. The E ⫻ B shear rate can be written ␥E = −共r / q兲d关c共q / r兲 / Bunitd⌽ / dr兴 / dr. It is useful to note that

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122507-10

Phys. Plasmas 14, 122507 共2007兲

Waltz et al.

the n = 0 potential in the Doppler rotation term 共⌽兲 is additive to the n = 0 perturbed potential 共␦␾0兲 in the correspondជ ␦h in the small ␳ aping part of the nonlinear term ␷ជ ␹ · ⵜ * proximation 关dropping the slow 共1 / r兲⳵␪ derivative兴: i共nq / r兲 ⫻关c / Bunit共⳵具␦␹典0 / ⳵r兲兴␦hn. Using c共I / B兲⌽⬙ 兩 ⵜ␺ 兩 = 共R / R0兲␥ p 兩 ⵜr兩 the ␻*-term 关first term on RHS of Eq. 共A7兲兴 becomes − 兵共⳵ n0/⳵ r兲/n0 + 关␧⬘ − 3/2兴共⳵ T0/⳵ r兲/T0 − 2共m/2T0兲␷⬘储 共R/R0兲␥ p其 ⵜ r · bˆ ⫻ ⵜ具␹典F0 .

共A8兲

−R−1 0 ⳵ 共具u 储典 / R0兲 / ⳵r

Note it is shear in the angular velocity which drives the Kelvin-Helmholtz instability. In this formulation, n0, T0, and ⌽ as well as u储 / R are flux-functions although u储 is not. F0共xជ , ␧ , ␮兲 = n0共m / 2␲T0兲3/2 exp关−␧⬘ / T0兴 is flux function only for constant ␧⬘, i.e., if used as a grid. ˆ ⬘, as a grid, so that ⳵ / ⳵r GYRO actually uses ␧⬘ / T0 = ␧ → ⳵ / ⳵r + 共⳵␧ˆ ⬘ / ⳵r兲 ⳵ / ⳵␧ˆ ⬘ when operating on the perturbations, but these energy derivatives are order ␳* small and may be dropped. The structure of the GYRO code is not changed 共␧ˆ ⬘ and ␷⬘储 / ␷th replace ␧ˆ and ␷储 / ␷th兲 but now there are finite-u储 terms added to the drifts. Note that the LHS drift term has ␷2储 = 共␷⬘储 + u储兲2 = ␷⬘储 2 + 2␷⬘储 u储 + 共u2储 兲. The RHS drift term 关setting ជ zˆ兲 as proper when dropping ␦B储兴 has bˆ ⫻ ⵜB / B to zˆ ⫻ 共zˆ · ⵜ 2 2 2 共␷⬘储 + u储兲 − u储 = ␷⬘储 + 2␷⬘储 u储. Thus to keep GYRO coding simple, we will drop the u2储 in the LHS drift and assume ␦B储 = 0 so grad-B is set to curvature. Thus both RHS drift term 关second term on the RHS of Eq. 共A7兲兴 and the LHS drift term have the combinations ␷2储 ⇒ ␷⬘储 2 + 2␷⬘储 u储. This is in agreement with a similar approximation made in Ref. 10 that dropped u2储 terms. The parallel gradient terms in Eq. 共A7兲 共the LHS ␦h and

2 ⵜ⬜ ␦A储 = − 4␲/c 兺 s

冕

d 3␷ e s

冋

RHS 具␦␹典兲 become in the GYRO11 ballooning mode eikonal representation

ជ = ␷⬘储 共zˆ · ⵜ␪兲⳵ ⇒ 共␷⬘储 /Rq兲⳵ , ␷⬘储 zˆ · ⵜ ␪ ␪

in the sˆ − ␣ large aspect circular geometry limit 共see Ref. 22兲. ␷⬘储 = ␴冑2m共␧⬘ − ␮B兲. The parallel momentum transport 关Eq. 共26b兲兴 is determined by the broken symmetry of ␦g 共hence of ␦h兲 under ␷⬘储 ↔ −␷⬘储 or equivalently from Eq. 共A9a兲 under ␪ ↔ −␪. The later is the same as up-down symmetry or symmetry of poloidal harmonics about singular surfaces. The perpendicular momentum transport 关Eq. 共26c兲兴 from depends on symmetry under ⵜx ↔ −ⵜx. From Ref. 23, ⵜx = i共nq/r兲关关共kx/ky兴兴 + 兩ⵜr兩⳵r ⇒ i共nq/r兲关sˆ共␪ − ␪0兲 − ␣ sin ␪兴,

2 ␦␾ = 4␲ 兺 ⵜ⬜ s

d3␷es关Jo␦h

− F0共es/T兲兵共1 − J20兲共␦␾ − ␦A储u储兲 2 − 共J0J1/␥兲共m␷⬜ /es兲␦B储/B其兴,

For completeness we give here a formal derivation of Eq. 共22a兲 共and its limitations兲 missing from Ref 13. Following the Hirshman review27 where possible, the neoclassical of poloidal rotation involves solving four equations. The first and second equations are for parallel momentum and viscous heating,

ជ 兲 · Bជ 典 = 具ⵜ ជ · ⌸ · Bជ 典 = − 具⌸a zˆ · ⵜB典, 具共Fជ a + eanaE a zz

共B1兲

共A10a兲

where the LHS can be set to zero ignoring the DeBye charge separation,

␦B储/B = − 4␲/B2 兺 s

冕

2 d3␷ 关共m␷⬜ J 1/ ␥ 兲 ␦ h

2 /esJ1/␥␦B储/B其兴, + F0共es/T兲兵J0共␦␾ − u储␦A储兲 + m␷⬜

共A10b兲

册

共A10c兲

.

↔

ជ · Bជ 典 = 具ⵜ ជ · R · Bជ 典 = − 具Ra zˆ · ⵜB典 . . . , 具共⌰ a a zz ↔ ↔

↔

冕

2 + u储关共1 − J20兲共␦␾ − u储␦A储兲 − 共m␷⬜ /esJ1J0/␥␦B储/B兲兴其

APPENDIX B: TURBULENT SHIFT IN POLOIDAL ROTATION

共A9b兲

which again is related to symmetry breaking under ␪ ↔ −␪. For completeness we write the Poisson–Ampere system following AL,

J0␦h共␷⬘储 + u储兲 − F0共es/T0兲兵␷⬘储 2J20␦A储

Note finite u储 couples Poisson and Ampere for electromagnetic simulations, but since ion parallel current is very small, we can ignore finite u储 effects in the field equations.

共A9a兲

共B2兲

↔

ជ兴 where 关⌸ , R 兴 = m 兰 d3␷关1 , m␷2 / 2T兴共␷ជ ␷ជ − I ␷2 / 3兲f and 关Fជ , ⌰ = m 兰 d3␷关1 , m␷2 / 2T兴␷ជ C with 具 典 the flux surface average and a 共and b兲 species label and C is the “collision” source. The “turbulent” source D 关Eq. 共2兲兴 has been momentarily suppressed. The parallel electric field can have some small effect, but we ignore it here. Expanding the equilibrium distribution function f in first few Laguerre polynomials of order 3/2 the LHS can be written as

ជ · Bជ 典 = 具Bជ · ⌺ 关ᐉab · uជ − 2/5ᐉab · qជ + ¯ 典, 具共F b a b 11 b 12

共B3兲

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122507-11

Phys. Plasmas 14, 122507 共2007兲

Gyrokinetic theory and simulation…

ជ · Bជ 典 = 具Bជ · ⌺ 关ᐉab · uជ + 2/5ᐉab · qជ − ¯ 典, 具共⌰ a b 21 b b 22

共B4兲

where n关uជ , qជ 兴 = 兰d3␷关1 , m␷2 / 2T兴␷ជ f. These are equivalent to Eqs. 共4.2兲 and 共4.3兲 of Ref. 27. Notice that from the incompressible plasma flow conditions Eq. 共10兲 and the analogous incompressible heat flow condition Bជ · uជ = B关␻R共Bt/B兲 + u␪共B/B p兲兴,

共B5兲

ជ · qជ = B关共5/2兲共− c/e ⳵ T/⳵ ␺兲R共B /B兲 + q 共B/B 兲兴. B t ␪ p

共B6兲

Since we want to find the poloidal rotation component of u储 共namely u␪兲 we can set the toroidal component ␻ 关Eq. 共9b兲兴 to zero without loss of generality. The 共ion兲 temperature gradient 关first term in Eq. 共B6兲兴 is the driver of the neoclassical poloidal flow velocities 关u␪ , q␪兴 for plasma and heat. Equations 共B5兲 and 共B6兲 are equivalent to Eqs. 共3.47兲 and 共3.48兲 of Ref. 27. To get the RHS of the first 关Eq. 共B1兲兴 and second 关Eq. 共B2兲兴 equations, the third and fourth equations are

ជ B兲/B = ⌬a /P , 2关u␪a + 共2/5兲q␪a兴共zˆ · ⵜ p zz a

共B7兲

ជ B兲/B = ⌺a /P , 7关u␪a + 共9/2兲q␪a兴共zˆ · ⵜ p zz a

共B8兲 ↔

where 关⌬zz , ⌺zz兴 = zˆ · m兰兰d3␷关1 , m␷2 / 2T兴共␷ជ ␷ជ − I ␷2 / 3兲 · zˆC. These equations are usually not given explicitly. These fola / Pa 关Eq. 共8c兲兴 and low from a generalization of Wzz = ⌬zz a 共7 / 2兲Wzz = ⌺zz / Pa 关respectively, Eq. 共142兲 and a corrected Eq. 共161兲 of Ref. 13兴 combined with the rate of stain ជ B兲 / B 共again see Ref. 15, p. 228兲 for incomWzz = 2u␪共zˆ · ⵜ p pressible flow in the surface, to include heat flow. 关There is ជ B兲 / B which an analogous “rate of strain” Y zz / P = 2q␪共zˆ · ⵜ p follows from the continuity of energy flow in the surface. Essentially, the energy flux tensor Qkij following Eq. 共6兲 共and the next ␷2 moment兲 must be generalized beyond convection to include heat.兴 Again by expanding the equilibrium distribution function f in the first few Laguerre polynomials to evaluate the collision operator, we obtain a a a = ␭11⌸zz + ␭12Rzz , ⌬zz

共B9兲

a a a . = ␭21⌸zz + ␭22Rzz ⌺zz

共B10兲

Substituting these into the third and fourth Eqs. 共B7兲 and ជ B兲 followed by flux surface 共B8兲 and multiplying by 共zˆ · ⵜ average gives the familiar relations 关Eqs. 共4.18兲 and 共4.19兲 or Ref. 27兴 for the RHS of the first and second Eqs. 共B1兲 and 共B2兲, a ជ B典 = − 3具共z · ⵜ ជ B兲2典/B 关␮a ua + ␮a qa兴, zˆ · ⵜ 具⌸zz p 11 ␪ 12 ␪

共B11兲

a ជ B典 = − 3具共z · ⵜ ជ B兲2典/B 关␮a ua + ␮a qa兴. zˆ · ⵜ 具Rzz p 32 ␪ 21 ␪

共B12兲

↔

↔ ↔

↔

Note the 2 ⫻ 2 matrix ␮ = ␭−1M where M is the 2 ⫻ 2 matrix of coefficients on the LHS of Eqs. 共B7兲 and 共B8兲. Equating the RHS Eq. 共B11兲 关Eq. 共B12兲兴 to the LHS Eq. 共B3兲 关Eq. 共B4兲兴 and using the driver Eqs. 共B5兲 and 共B6兲 共with the ion temperature gradient兲 we can solve the 2 ⫻ 2 matrices for the

neoclassical poloidal flow velocities 2-vector 关u␪ , q␪兴 neo ⬅ 关uneo * , q* 兴. Because we have explicitly written the intermediate Eqs. 共B7兲 and 共B8兲, we can add the turbulent part D to the colliC D C D + ⌬xx , ⌺xx + ⌺xx 兴. sional part D of the RHS 关⌬zz , ⌺zz兴 ⇒ 关⌬xx D D We can then absorb 关⌬xx , ⌺xx兴 on the RHS of the modified Eqs. 共B7兲 and 共B8兲 into “shifted” velocities on the LHS,

ជ B兲2典兴 − 共2/5兲⌬qDa , ជ B⌬Da典/关2Pa具共zˆ · ⵜ ⌬u␪Da = B p具zˆ · ⵜ zz ␪ 共B13兲

ជ B共⌺Da − 共7/2兲⌬Da典/关共7/5兲2Pa具共zˆ · ⵜ ជ B兲2典兴, ⌬q␪Da = B p具zˆ · ⵜ zz zz 共B14兲 with 关u* , q* The leading term in Eq. 共B13兲 is the Eq. 共22兲 we seek. We have dropped the smaller higher order moment term given by Eq. 共B14兲. However there is a further approximation unnoticed in Ref. 13. The turbulence also adds a source to the RHS of the first 关Eq. 共B1兲兴 and second 关Eq. 共B2兲兴 equations 关or Eqs. 共B3兲 ជ · B典兴 ⇒ 关具Fជ C · B典 + 具Fជ D · B典 , 具⌰ ជ C · B典 and 共B4兲兴, 关具Fជ a · B典 , 具⌰ a a a a ជ D · B典兴. It clear that the C forces are O共␳ 兲 smaller than + 具⌰ * a the D source forces, however these further shifts are of the same ␳* order as 关⌬u␪D , ⌬q␪D兴. Unfortunately to derive these additional shifts, we would have to put the turbulent source 3 2 ជ ជD ជ ជ ជ Da典 into forces 关具Fជ D a · B典 , 具⌰a · B典兴 = 具B · m兰d ␷关1 , m␷ / 2Ta兴␷ the standard neoclassical matrix problem and solve numerically 共as normally done兲 to get a modified 共further shifted兲 uneo * . In the neoclassical solution procedure, there is actually ជ D · Bជ 典 cancels a sum over species. Thus from Eq. 共2兲, 具F i ជ ជD ជ 具Fជ D e · B典 and only the higher moment forces 具⌰a · B典 come in. Hence we speculate that these likely produce even smaller shifts than ⌬u␪D given by Eq. 共22兲. The possibility that the ជ D · Bជ 典 could modify uneo has parallel turbulent heat flow 具⌰ a * 28,29 long been recognized. neo

兴 = 关u␪ − ⌬u␪D , q␪ − ⌬q␪D兴.

neo

1

S. D. Scott, P. H. Diamond, R. J. Fonck, R. J. Goldston, R. B. Howell et al., Phys. Rev. Lett. 64, 531 共1990兲. 2 J. S. deGrassie, D. R. Baker, K. H. Burrell, P. Gohil, C. M. Greenfield, R. J. Groebner, and D. M. Thomas, Nucl. Fusion 43, 142 共2003兲. 3 J. E. Rice, M. Greenwald, I. H. Hutchinson, E. S. Marmar, Y. Takase, S. M. Wolf, and F. Bomarda, Nucl. Fusion 38, 75 共1998兲. 4 J. E. deGrassie, J. E. Rice, K. H. Burrell, R. J. Groebner, and W. M. Solomon, Phys. Plasmas 14, 056115 共2007兲. 5 C. Estrada-Mila, J. Candy, and R. E. Waltz, Phys. Plasmas 12, 022305 共2005兲; 12, 049902共E兲 共2005兲; 13, 074505 共2006兲. 6 R. R. Dominguez and G. M. Staebler, Phys. Fluids B 5, 387 共1993兲. 7 G. M. Staebler, J. E. Kinsey, and R. E. Waltz, Bull. Am. Phys. Soc. 46, 221 共2001兲. 8 R. E. Waltz, G. M. Staebler, W. Dorland, G. W. Hammett, M. Kotschenreuther, and J. A. Konings, Phys. Plasmas 4, 2483 共1997兲. 9 O. D. Gurcan, P. H. Diamond, T. S. Hahm, and R. Singh, Phys. Plasmas 14, 042306 共2007兲. 10 A. G. Peeters, C. Angioni, and D. Strintzi, Phys. Rev. Lett. 98, 265003 共2007兲. 11 J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 共2003兲; ibid. 91, 045001 共2003兲. 12 W. M. Solomon, K. H. Burrell, R. Andre, L. R. Baylor, R. Budny, P. Gohil, R. J. Groebner, C. T. Holcomb, W. A. Houlberg, and M. R. Wade, Phys. Plasmas 13, 056116 共2006兲. 13 G. M. Staebler, Phys. Plasmas 11, 1064 共2004兲. 14 M. N. Rosenbluth and F. L. Hinton, Nucl. Fusion 36, 55 共1996兲. 15 P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plas-

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122507-12

mas 共Cambridge University Press, Cambridge, 2002兲. F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲. 17 F. L. Hinton and R. E. Waltz, Phys. Plasmas 13, 102301 共2006兲. 18 R. E. Waltz, Phys. Fluids 25, 1269 共1982兲. 19 E. A. Freeman and Liu Chen, Phys. Fluids 25, 502 共1982兲. 20 GYRO website, http://fusion.gat.com/theory/Gyro 21 J. E. Kinsey, R. E. Waltz, and J. Candy, Phys. Plasmas 12, 062302 共2005兲. 22 R. E. Waltz, J. Candy, F. L. Hinton, C. Estrada-Mila, and J. E. Kinsey, 16

Phys. Plasmas 14, 122507 共2007兲

Waltz et al.

Nucl. Fusion 45, 741 共2005兲. A. G. Peeters and C. Angioni, Phys. Plasmas 12, 072515 共2005兲. 24 R. E. Waltz, R. L. Dewar, and X. Garbet, Phys. Plasmas 5, 1784 共1998兲. 25 T. M. Antonsen and B. Lane, Phys. Fluids 23, 1205 共1980兲. 26 R. E. Waltz and R. L. Miller, Phys. Plasmas 6, 4265 共1999兲. 27 S. P. Hirshman and D. S. Sigmar, Nucl. Fusion 21, 1079 共1981兲. 28 K. C. Shing, Phys. Fluids 31, 2249 共1988兲. 29 H. Sugamma and W. Horton, Phys. Plasmas 2, 2989 共1995兲. 23

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PHYSICS OF PLASMAS 16, 079902 共2009兲

Erratum: “Gyrokinetic theory and simulation of angular momentum transport” †Phys. Plasmas 14, 122507 „2007…‡ R. E. Waltz,a兲 G. M. Staebler, J. Candy, and F. L. Hintonb兲 General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA

共Received 11 June 2009; accepted 12 June 2009; published online 7 July 2009兲 关DOI: 10.1063/1.3167808兴 A programming error in GYRO was brought to our attention. The B p / B factor in Eq. 共14兲 projecting the perpendicular stress ⌸xy into the toroidal stress ⌸x␾ was misevaluated for the simple infinite aspect ratio circular “sˆ-␣” geometry used in the illustrated GYRO simulations. For the GAstandard parameters 共used in every case兲 this primarily resulted in a factor of 4 too large of a value for perpendicular viscosity ␩eff 储 y defined in the first unnumbered equation of section. This results in a somewhat weaker E ⫻ B shear momentum pinch 共Ref. 6 in Ref. 1兲. After repeating all the simulations in the paper, a revised Table I and Fig. 1共a兲 are given below. Figure 1共b兲 is unchanged. Subsequent Figs. 2 and 3 illustrating the Coriolis pinch 共Ref. 10 in Ref. 1兲 are negligibly corrected since it resides mostly in parallel viscosity ␩eff 储z . Comparing the revised Table I to the published Table I, the turbulent shift in the neoclassical poloidal rotation ⌬u␪D was found to be two to five times larger in the repeated simulations. From Eq. 共27兲, there is no reason to connect the revised ⌬u␪D with the projection error discussed above. In fact we have not identified the source of this discrepancy, except to note that ⌬u␪D has a highly intermittent time trace 共e.g., 221⫾ 83兲. The repeated simulations were run much longer and the time average sampling errors were thereby reduced. The larger ⌬u␪D values at first appear to call into question our earlier conclusion that the turbulent shift 共⌬u␪D / uneo ⴱ ⬀ ␳ⴱ兲 is negligible at ␳ⴱ values for DIII-D. Based on the GA-standard case ␥E = 0 with values ␹i ⬃ 15csa␳2ⴱ and values of ⌬u␪D = 50cs␳2ⴱ, our estimated ⌬u␪D / uneo ⴱ ⬍ 10% would be significantly revised upward. However ⌬u␪D is proportional to ␹i. The core of DIII-D operates closer to threshold 共with significant E ⫻ B shear兲 than suggested by the

GA-standard case, so the experimental ␹i values are typically an order of magnitude smaller than ␹i ⬃ 15csa␳2ⴱ The identity used in Eq. 共14兲 was mis-stated as “R␧ˆ ␾ ជ ␾ is constant on a flux and only the radial divergence = R 2ⵜ survives.” This resulted in no error but we should have said “effectively constant.” The actual identity used 关from Ref. 16 in Ref. 1, Eq. 共2.91兲兴 is

J 典 = 具ⵜ · ⌸ J · R 2ⵜ ជ␾ · ⵜ · ⌸ ជ ␾典 具R2ⵜ i i = V⬘共r兲−1⳵r兵V⬘共r兲具兩ⵜr兩⌸x␾典其. In formulating the gyrokinetic equations for a strongly toroidally rotating 共u␾兲 plasma in Appendix A, we adopted a projected parallel velocity 共u储兲 shifted Maxwellian to model the background distribution 共while staying in the laboratory frame gyrokinetics兲. The neglect of the projected and smaller perpendicular velocity u⬜ ⬃ 共r / Rq兲u储 misestimates the Coriolis force by about 10%. Further neglecting the centrifugal forces 关⬀共u储 / cs兲2兴, our approximate formulation is in agreement with derivations by Artun and Tang,2 Brizard,3 as wells as Sugama and Horton.4 We were unaware of the earlier formal and more exact derivations at the time of publication. An updated GYRO now includes a path to the more exact formulation while retaining the approximate formulation for quantitative comparison. This work was supported by the U.S. Department of Energy under Cooperative Agreement No. DE-FC0204ER54698. One of us 共R.E.W.兲 thanks Dr. Cemente Angioni 共IPP-Garching兲 for bringing the GYRO code error to our attention.

a兲

Electronic mail: [email protected]. Present address: University of California, San Diego.

b兲

1070-664X/2009/16共7兲/079902/2/$25.00

16, 079902-1

© 2009 American Institute of Physics

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079902-2

Phys. Plasmas 16, 079902 共2009兲

Waltz et al.

TABLE I. Simulations of GA-standard case at nonzero ␥ P, ␥E, and M 储. Diffusivities are in gyro-Bohm units: 关cs / a兴␳s2 = csa␳2ⴱ. The turbulent poloidal rotation shift is in units of cs␳2ⴱ.

␥P

␥E

M储

␩eff 储

␩eff 储z

␩eff 储y 0.10

␹eff Ei

eff ␹Ee

Deff

eff ␩eff 储 / ␹ Ei

⌬uD␪

1.0

0.0

0.0

11.3

11.2

14.1

3.55

⫺2.17

0.80

+221

0.2 0.2

0.0 0.2

0.0 0.0

14.3 ⫺0.43

14.18 ⫺0.05

0.16 ⫺0.39

14.9 4.32

3.77 1.56

⫺2.25 ⫺0.89

0.96 ⫺0.10

+223 +35

0.2

0.0

0.4

⫺7.56

⫺7.52

⫺0.13

14.9

3.68

⫺2.26

⫺0.51

+177

FIG. 1. 共Color online兲 共a兲 Ratio of effective parallel viscosity to ion energy diffusivity.

R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 共2007兲. M. Artun and W. M. Tang, Phys. Plasmas 1, 2682 共1994兲. 3 A. Brizard, Phys. Plasmas 2, 459 共1995兲. 4 H. Sugama and W. Horton, Phys. Plasmas 5, 2560 共1998兲. 1 2

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