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Radial transport of toroidal angular momentum in tokamaks
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2015 Plasma Phys. Control. Fusion 57 075006
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1. Introduction
Intrinsic rotation is the toroidal rotation that tokamak plasmas can achieve without the need of external momentum input. There is a number of reasons why intrinsic rotation has become an active research field in the last years [1, 2]. First, fast toroidal rotation seems to improve confinement [3] and it quenches MHD instabilities [4]. Second, it is unlikely that rotation in the core of large and dense tokamaks can be pro-duced by means of neutral beam momentum injection [5, 6]. Therefore, in tokamaks like ITER or larger, the only rotation available will be intrinsic. Third, Parra and Catto showed that standard code implementations of the gyrokinetic equa-tions cannot determine the intrinsic rotation profile; they also pointed out that the correct equations had not been derived yet [7–10]. We explain the problem in more detail in what follows.
Gyrokinetic theory [11] is the appropriate framework to study microturbulence in fusion plasmas; that is, turbulence on the ion Larmor radius scale, ρ, with characteristic frequen-cies of the order of cs/L, where cs is the sound speed and L is the typical variation length of the magnetic field. The theory is formulated as an asymptotic expansion in ϵ ≔ ρ/L ≪ 1, in
which the gyromotion degree of freedom is eliminated order by order in ϵ, whereas non-zero Larmor radius effects are retained. The specific orderings assumed in electrostatic gyro-kinetics are explained in section 2. They lead to a consistent set of equations, as proven, in particular, by the fact that gyro-kinetic simulations [12–16] converge as long as the size of the simulation domain exceeds several gyroradii. Recently, it has explicitly been checked that the fluctuation spectrum reaches its maximum at wavelengths of the order of ρ [17]. Finally, dedicated experiments have confirmed that the scaling of the turbulence characteristics with the gyroradius are reproduced by gyrokinetic theory [18]. An introductory survey of gyroki-netics can be found in [19].
The so-called low-flow ordering, in which the plasma velocity is assumed to be of order ϵ cs, is probably the relevant one in the core of large and dense tokamaks. The reason is that sonic speeds are typically reached by neutral beam momentum injection, and we have already mentioned that its efficiency is reduced in large machines. In the low-flow regime, and in up-down symmetric tokamaks [20], Parra and Catto proved that finding the intrinsic rotation profile requires the gyroki-netic equations to O(ϵ2). Determining the toroidal rotation is equivalent to determining the long-wavelength radial electric
Plasma Physics and Controlled Fusion
Radial transport of toroidal angular momentum in tokamaks
Iván Calvo1 and Felix I Parra2,3
1 Laboratorio Nacional de Fusión, CIEMAT, 28040 Madrid, Spain2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK3 Culham Centre for Fusion Energy, Abingdon OX14 3DB, UK
E-mail: [email protected] and [email protected]
Received 2 December 2014, revised 20 April 2015Accepted for publication 27 April 2015Published 27 May 2015
AbstractThe radial flux of toroidal angular momentum is needed to determine tokamak intrinsic rotation profiles. Its computation requires knowledge of the gyrokinetic distribution functions and turbulent electrostatic potential to second-order in ϵ = ρ/L, where ρ is the ion Larmor radius and L is the variation length of the magnetic field. In this article, a complete set of equations to calculate the radial transport of toroidal angular momentum in any tokamak is presented. In particular, the O(ϵ2) equations for the turbulent components of the distribution functions and electrostatic potential are given for the first time without assuming that the poloidal magnetic field over the magnetic field strength is small.
Keywords: magnetic confinement fusion, plasma turbulence, tokamak, gyrokinetic theory, plasma rotation, angular momentum transport
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I Calvo and F I Parra
Radial transport of toroidal angular momentum in tokamaks
Printed in the UK
075006
PPCF
© 2015 EURATOM
2015
57
Plasma Phys. Control. Fusion
PLPHBZ
0741-3335
10.1088/0741-3335/57/7/075006
Paper
7
Plasma Physics and Controlled Fusion
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0741-3335/15/075006+19$33.00
doi:10.1088/0741-3335/57/7/075006Plasma Phys. Control. Fusion 57 (2015) 075006 (19pp)
I Calvo and F I Parra
2
field, and viceversa, because the neoclassical relation between the two quantities [21] still holds in the presence of microtur-bulence. Then, the difficulty of computing the toroidal rota-tion can be traced back to the intrinsic ambipolarity of the tokamak; that is, to the fact that to lowest order, and even in the presence of turbulence, the flux-surface-averaged ion and electron current densities cancel each other for any value of the long-wavelength radial electric field4. However, gyro-kinetic codes only have the equations implemented to such lowest order.
As advanced above, Parra and Catto [26] arrived at a for-mula for the radial flux of toroidal angular momentum that is written in terms of velocity integrals involving up to O(ϵ2) pieces of the ion distribution functions and electrostatic poten-tial. Hence, the Fokker–Planck and quasineutrality equa-tions have to be solved to O(ϵ2) to evaluate the radial flux of toroidal angular momentum. The set of equations appropriate for the limit B/Bp ≫ 1 has been given in [2, 26, 27]. Here, Bp is the magnitude of the poloidal magnetic field. However, the equations valid for a general tokamak had not been calculated.
In [28], by employing the results of [29], the first step to compute the toroidal angular momentum flux without using the approximation B/Bp ≫ 1 has been taken. In order to be more specific about what has been done and what remained to be done, we recall that in gyrokinetics the fields of the theory, i.e. the distribution functions and the electrostatic potential, are naturally decomposed as the sum of their long-wavelength and short-wavelength components. The short-wavelength components correspond to the turbulence scales and the long-wavelength ones to the macroscopic scales (see section 2 for more details on the decomposition). Both components of the distribution functions to O(ϵ2) enter the momentum transport equations. As for the electrostatic potential, the turbulent com-ponent to O(ϵ2) is needed, but the long-wavelength compo-nent is required only to O(ϵ). In [28], the long-wavelength gyrokinetic equations up to O(ϵ2) have been derived. In this paper, we compute the short-wavelength equations and then write explicitly all the equations that, when implemented in a global δ f code, give the toroidal angular momentum flux in any tokamak to the order needed to calculate intrinsic rotation.
In section 2, the orderings assumed in low-flow electrostatic gyrokinetics are explained. Section 3 summarizes the proce-dure employed by Parra and Catto to reach their expression for the radial flux of toroidal angular momentum. In section 4 we show that the evaluation of such an expression requires the long-wavelength and short-wavelength components of the dis-tribution functions to O(ϵ2), the long-wavelength component of the electrostatic potential to O(ϵ), and the short-wavelength component of the electrostatic potential to O(ϵ2). In section 5 the equations for the short-wavelength components are pro-vided. In section 5.1, we simply recall the well-known equa-tions for the O(ϵ) turbulent components. In sections 5.2 and 5.3, the equations for the O(ϵ2) turbulent components of the distribution functions and electrostatic potential are derived for the first time. Section 6 contains the results of [28] that are
needed to formulate a complete set of equations for toroidal angular momentum transport in any tokamak. These are the equations for the long-wavelength components of the distribu-tion functions up to O(ϵ2), the equation for the long-wavelength component of the electrostatic potential to O(ϵ), the transport equation for the density of each species, and the transport equa-tion for the total energy. In section 7 we summarize the results and list the equations of the main text that should be imple-mented in a code to determine the toroidal rotation profile.
2. Orderings and assumptions
The starting point in the derivation of the electrostatic gyro-kinetic equations consists of the Fokker–Planck equation for each species σ,
⎜ ⎟⎛⎝
⎞⎠
∑
φ∂ + ⋅ ∇ + −∇ + × ⋅ ∇
= [ ] +
σ σσ
σσ
σσσ σ σ σ
′′ ′
f fZ e
m cf
C f f S
v v B1
, ,
t r r v
(1)
and the quasineutrality equation,
∫∑ ( ) =σ
σ σZ e f t vr v, , d 0.3 (2)
The fields of the theory are the phase-space probability dis-tributions f σ(r, v, t) and the electrostatic potential φ(r, t). The magnetic field B(r) = ∇r × A(r) does not vary in time, c is the speed of light, e the charge of the proton, and Zσe and mσ are the charge and the mass of species σ. The Landau collision operator between species σ and σ′ reads
∫γ= ∇ ⋅
↔− ⋅ ∇
− ∇
′ ′
′ ′
σσ σ σ
σσ
σ σσ σ
σσ σ
′ ′
′′
′′ ′
⎛⎝⎜
⎞⎠⎟
C f f t
m mf t f t
mf t f t v
r v
W v v r v r v
r v r v
[ , ]( , , )
( )1
( , , ) ( , , )
1( , , ) ( , , ) d ,
v v
v3
(3)where
γ π≔ Λσσ σ σ′ ′Z Z e2 ln ,2 2 4 (4)
↔( ) ≔
↔−
W ww I ww
w,
2
3 (5)
lnΛ is the Coulomb logarithm, and ↔I is the identity matrix.
Finally, Sσ is a source term that represents heating and fuelling in the tokamak. Our equation does not explicitly include the transformer electric field because we assume that the electric field is purely electrostatic, but the effect of the transformer electric field can be recovered with suitable choices of Sσ.
Gyrokinetic theory relies on the existence of very different spatial and time scales when the plasma is strongly magnetized. The small spatial scale is given by the gyroradius of species σ, ρσ = vtσ/Ωσ, and the large spatial scale is given by L ∼ ∣∇r B/B ∣ −1, where B = ∣B∣ . Here, =σ σv T m/t 0 is the thermal
4 The intrinsic ambipolarity of the tokamak is a result of its axisymmetry. Therefore, the problem disappears for sufficiently rippled tokamaks or stel-larators far from quasisymmetry [22–25].
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velocity of species σ, where T0 is a typical value of the tem-perature, and Ωσ = Zσe B0/(mσc), the gyrofrequency, is defined in terms of a typical value of the magnetic field, B0. Strong magnetization implies that ϵσ ≔ ρσ/L ≪ 1; that is, the charged particle gyrates around a magnetic field line separating from it only a small distance of the order of the gyroradius. There is also a large separation in time scales. The fast time scale is the gyrofrequency Ωσ and the slow time scale is vtσ/L (rel-evant turbulence frequencies satisfy ω ∼ vtσ/L). In gyrokinetic theory, one eliminates the degree of freedom corresponding to the gyromotion by averaging out frequencies of the order of Ωσ, while non-zero Larmor radius effects are kept. Even if we often use ϵσ for convenience, we need a species-independent expansion parameter. We choose ϵs ≔ ρs/L, where ρs = cs/Ωi is the sound gyroradius and =c T m/s i0 is the sound speed. We assume that the dominant ion species, denoted by the sub-index i, is singly charged; i.e. Zi = 1. Gyrokinetics is then for-mulated as an asymptotic expansion in ϵs. Observe that the relation between ϵσ and ϵs is ϵs = Zστσϵσ, with
τ = =σσ
σ
v
c
m
m.t
s
i (6)
The presence of disparate time and space scales makes it useful to write the fields of the theory as the sum of two com-ponents, one associated to the microturbulence scales (the short-wavelength component) and the other to the macro-scopic scale (the long-wavelength component). To properly define these components, we need a set of flux coordinates {ψ, Θ, ζ}, where ψ is the poloidal magnetic flux, Θ is the poloidal angle, and ζ is the toroidal angle. The magnetic field of the tokamak is written as
ψ ζ ζ ψ= ( )∇ + ∇ × ∇IB .r r r (7)
The coordinate ζ is chosen to be the standard toroidal angle in the cylindrical coordinates whose axis is the tokamak sym-metry axis. In particular,∣∇rζ ∣ =R−1, where R is the distance to the symmetry axis.
The axisymmetric long-wavelength component of a func-tion g(ψ, Θ, ζ) is defined by
∫ ∫ ∫ ∫π ψψ ζ=
Δ Δ ΔΘΘ
ψ
π
Δ Δ ΔΘg
tt g
1
2d d d d ,
t
lw
0
2
(8)
where ϵs ≪ Δψ/ψ ≪ 1, ϵs ≪ ΔΘ ≪ 1, and L/cs ≪ Δt ≪ τE. Here,
τϵ
≔ L
cE
s s2 (9)
is the transport time scale (i.e. the time scale of variation of the profiles) and L/cs is the time scale of the turbulence. Then, the short-wavelength component is defined as
≔ −g g g .sw lw (10)
For any two functions g (r, t) and h (r, t) we have
[ ] =
[ ] =
[ ] = + [ ]
g g
g
gh g h g h
,
0,
.
lw lw lw
sw lw
lw lw lw sw sw lw
(11)
The decomposition into a short-wavelength and a long-wave-length component is not unambiguously defined because the integration limits of (8) are not specified. For the most part, this ambiguity is not a problem because the long- and short-wavelength pieces can be thought of as defined by the equations they satisfy. However, to implement some of the equations in this article, one needs to choose a time interval and a space volume over which the average […]lw is taken. For example, in equations (101), (110), (116) and (121), we need to take the average […]lw over nonlinear terms that con-tain products of two short wavelength pieces. Fortunately, the exact time interval and space volume over which the average […]lw is taken is irrelevant if the parameter ϵ is sufficiently small. This result can be deduced from the formal equiva-lence between the local and global approaches to gyrokinetics established in [30].
The distribution function and electrostatic potential are written using this decomposition,
φ φ φ
= +
= +σ σ σf f f ,
.
lw sw
lw sw (12)
The separation of time and space scales suggests the fol-lowing assumptions on the distribution functions and electro-static potential. First, we assume that the relative sizes of the short and long-wavelength components are
φ ϵ
φ
∼ ∼
∼ ∼
σ σ σ
σ σ
σ σ σ
σ σ
v f
n
Z e
m v
v f
n
Z e
m v
,
1,
t
e ts
t
e t
3 sw
0
sw
2
3 lw
0
lw
2
(13)
where ne0 is a characteristic value of the electron density. We also need an ordering scheme for the space and time deriva-tives. The natural assumptions are, for the long-wavelength components,
φ
φ ϵ
∇ ∇ ∼
∂ ∂ ∼σ
σ
f L
f c L
ln , ln 1/ ,
ln , ln / ,t t s s
r rlw lw
lw lw 2
(14)
and for the short-wavelength ones,
φ
φ ρ
φ
ˆ ⋅ ∇ ˆ ⋅ ∇ ∼
∇ ∇ ∼
∂ ∂ ∼
σ
σ
σ
⊥ ⊥
f L
f
f c L
b bln , ln 1/ ,
ln , ln 1/ ,
ln , ln / .
s
t t s
r r
r r
sw sw
sw sw
sw sw
(15)
Here,⊥ stands for perpendicular to B and ˆ ≔ −Bb B1 is the unit vector in the direction of B. Note the different sizes of the perpendicular and the parallel gradients of σf
sw and φsw. It is important to mention that the spatial dependence of σf
sw and φsw has two different characteristic scales in the direction per-pendicular to the magnetic field: the short length scale of the turbulent fluctuations, and the long length scale associated to the slow change of the turbulent average characteristics over the radius of the tokamak. These two different scales can be treated explicitly in the gyrokinetic equations [2, 30], but in this article we choose not to do it because we are deriving a global δ f gyrokinetic formulation.
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Finally, we assume that the source term in equation (1) only consists of the long-wavelength component, =σ σS S lw, and that it varies in the transport time scale τE. For consistency with the transport time scale τE, we assume
ϵ∼σσ
Sn c
Lv.s
e s
t
2 03 (16)
With the ordering detailed in this section, the gyromotion degree of freedom can be averaged out order by order in ϵs, and a consistent set of non-linear equations for plasma micro-turbulence is obtained [12–16].
3. Radial flux of toroidal angular momentum
In this section we proceed as in [2, 26] to obtain a convenient expression for the radial flux of toroidal angular momentum. In [26] a mass ratio expansion in ≪m m/ 1e i was performed in order to simplify the presentation, that consequently only needed to treat kinetically the ions (note that in [2, 27] the final equations for both kinetic ions and kinetic electrons are given). Here, we will not perform a mass ratio expansion.
We multiply the Fokker–Planck equation (1) by mσv and integrate in velocity space, obtaining
∫ ∫∑
φ∂ ( ) = −∇ ⋅↔
− ∇ + ×
+ [ ] +
σ σ σ σ σ σσ
σ σ
σσ
σσ σ σ σ σ′
′ ′
m N Z eNZ e
cN
m C f f v m S v
V P V B
v v, d d .
t r r
3 3
(17)
The density, velocity and stress tensor of species σ are defined by
∫( ) ≔ ( )σ σN t f t vr r v, , , d ,3 (18)
∫( ) ≔ ( )σσ
σtN
f t vV r v r v,1
, , d ,3 (19)
∫↔( ) ≔ ( )σ σ σt m f t vP r vv r v, , , d .3 (20)
Summing (17) over species,
∫∑ ∑ ∑∂ = −∇ ⋅↔
+ × +σ
σ σ σσ
σσ
σ σm Nc
m S vV P J B v1
d ,t r3
(21)where
∑( ) ≔σ
σ σ σt Z eNJ r V, (22)
is the electric current density. To write (21) we have employed the momentum conservation property of the collision operator and the quasineutrality equation (2), that can be expressed as ∑σZσeNσ = 0.
We define the flux-surface average of a function G(ψ, Θ, ζ) by
∫ ∫
∫ ∫
ψ ζ ζ
ζ⟨ ⟩ ≔
( Θ ) Θ
Θψ
π π
π πGg G
g
, , d d
d d,0
2
0
2
0
2
0
2 (23)
with
ψ ζ≔
∇ ⋅ (∇ Θ × ∇ )g
1
r r r (24)
the square root of the determinant of the metric tensor in coor-dinates {ψ, Θ, ζ}. It will also be useful to define the volume enclosed by the flux surface labeled by ψ,
∫ ∫ ∫ψ ψ ζ ψ( ) = Θ ( Θ)′ ′ψ π π
V gd d d , .0 0
2
0
2 (25)
In order to find the equation for toroidal angular momentum transport we take the scalar product of (21) with ζ ζˆ = ∇R R r
2 and flux-surface average. We get
∑ ζ∂ ˆ ⋅ = − ∂ ( Π) +′
′σ
σ σ σ
ψ
ψ ζm N RV
V TV1
,t (26)
where the prime stands for differentiation with respect to ψ, the external torque is
∫∑ ζ≔ ⋅ ˆζσ
σ σ
ψ
T R m S vv d ,3 (27)
and the radial flux of toroidal angular momentum is defined by
∑Π ≔ Πσ
σ, (28)
with
ζ ψΠ ≔ ˆ ⋅↔
⋅ ∇σ σψ
R P .r (29)
We have used that ζ∇ ( ˆ)Rr is antisymmetric and that
ζ ψˆ ⋅ ( × ) = ⋅ ∇ψ ψR J B J r . The latter has to be zero, as can be proven by noting that ∇r · J = 0, and that J · ∇rψ must vanish at ψ = 0.
In the low flow ordering, the magnitude of the ion flow is Vi ∼ ϵscs. Recalling the definition of the transport time scale given in section 2, τ ϵ= − L c/E s s
2 , we can estimate the size of the first term in (26). Namely,
∑ ζ ϵ∂ ˆ ⋅ ∼σ
σ σ σ
ψ
m N RRn T
LV .t s
e3 0 0 (30)
If the three terms in equation (26) are to be comparable, we deduce
ϵΠ ∼ n T RB Ls e3
0 0 0 (31)
and
ϵ∼ζTRn T
L.s
e3 0 0 (32)
Observe that we are assuming that the momentum injec-tion is smaller in ϵs than, for example, the particle input (see (16)). If Tζ were much larger than the estimate (32), it would dominate the toroidal rotation. Then, Vi would approach sonic values and the plasma would enter the so-called high-flow
Plasma Phys. Control. Fusion 57 (2015) 075006
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5
regime. As pointed out in the Introduction, it is unlikely that momentum input larger than the estimate (32) be avail-able in the core of large tokamaks, and therefore the only possible rotation will be of intrinsic nature. However, this is precisely the difficult case from the theoretical and com-putational perspective: in the low-flow regime, one needs to calculate Π to ϵ( )O s
3 .In [2, 26], a procedure was devised to write Π in such a
way that ‘only’ ϵ( )O s2 pieces of the distribution functions and
electrostatic potential are needed to evaluate it. We start by multiplying (1) by mσvv and integrating over velocities,
( )∫
∫
∫
∑
φ φ
Ω (↔
× ˆ − ˆ ×↔
)
= ∂↔
+ ∇ ⋅
+ (∇ + ∇ )
− [ ]
−
σ σ σ
σ σ σ
σ σ σ σ
σσ
σσ σ σ
σ σ
′′ ′
m f v
Z eN
m C f f v
m S v
P b b P
P vvv
V V
vv
vv
d
, d
d .
t r
r r
3
3
3
(33)
We are only interested in the long-wavelength component of this equation. We find the expression for Πσ
lw by taking the double-dot product of (33) with ζζˆ ˆR /22 , flux-surface aver-aging, and extracting the long-wavelength component,
∫
∫
∫
∑
ζ ζ
ζ
ζ
ζ
ζ
ψ
φ
Π = ∂ ˆ ⋅↔
⋅ ˆ
+ ∂ ( ⋅ ∇ ) ( ⋅ ˆ)
+ ∂ ( ⋅ ˆ)
− ( ⋅ ˆ)
− ( ⋅ ˆ)
′′
σσ
σσ
ψ
σ
σψ σ
ψ
ζ σ σ σ ψ
σ
σ σσσ
ψ
σ
σσ
ψ
′′
m c
Z eR
m c
Z e VV f R v
c RN m
m c
Z eC R v
m c
Z eS R v
P
v v
V
v
v
2
2
1d
2d
2d .
t
r
lw 2lw
2lw 2 2 3
lw
2lw 2 2 3
22 2 3
(34)
In [2, 26] it is shown that, up to terms that are small by a factor of ϵs, the right-hand side of (34) can be rewritten as
∫
∫
∫
∑
∑
ζ
ζ ζ
ζ
ζ
ζ
φ
φ
Π = ∂ ( ⋅ ˆ)
+ ∂ ∂ ( ˆ ⋅↔
⋅ ˆ) + ⟨ ⟩ ∂
− ( ⋅ ˆ)
− ∂ ( ⋅ ˆ)
− ( ⋅ ˆ)
′′
′′
σ ζ σ σ σ ψ
σ
σψ ζ σ
ψ
σ
σψ σ
σ
σ σσσ
ψ
σ
σψ
σσσ
ψ
σ
σσ
ψ
′′
′′
c RN m
m c
Z e VV R
m c
Z eR p
m c
Z eC R v
m c
Z e VV C R v
m c
Z eS R v
V
P
v
v
v
2
1
2
2d
6
1d
2d ,
t
lw lw
22
lw2
2lw 2 2 3
3 2
2 2lw 3 3 3
22 2 3
(35)
where pσ = nσT is the pressure of species σ, given by the O(ne0T0) term of the right side of (20). In section 4 we explic-itly show that the right-hand side of (35) can be recast in terms of the fields of the theory to ϵ( )O s
2 by rewriting it in such a way that the solutions of the gyrokinetic equations (derived in sec-tions 5 and 6) can be directly plugged into it.
4. Gyrokinetic expansion
The objective is to write the right side of (35) in terms of the pieces of the distribution functions and electrostatic potential that are obtained by solving the gyrokinetic equations. This is done in section 4.3. Before doing this, in sections 4.1 and 4.2 we briefly recall how the gyrokinetic expansion is defined and give the results of [28 and 29] that are needed in this article.
4.1. Dimensionless variables
The gyrokinetic expansion is more clearly understood by writing the equations in dimensionless variables. For time, space, magnetic field, electrostatic potential, particle density and temperature, we employ
φ φϵ
= = =
= = =σσ
σσ
t c t
L L B
e
Tn n
nT
T
T
r rB
B_ , _ , ,
_ , , .
s
s e
0
0 0 0
(36)
For velocities, distribution functions and sources, we use
ϵ= = =σ
σ
σσ
σσ σ
σσ
vf v
nf S
L
c
v
nSv v
, , .t
t
e s s
t
e
3
02
3
0 (37)
In dimensionless variables, the Fokker–Planck equation (1) becomes
∑
τ τ ϵ φϵ
τ ϵ
∂ + ⋅ ∇ + − ∇ + × ⋅ ∇
= +
σ σ σ σ σσ
σ
σσ
σσ σ σ σ′
′ ′
⎛⎝⎜
⎞⎠⎟
⎡⎣ ⎤⎦
f f Z f
C f f S
v v B
r v
_ _ _ _ _1
_ _
, (_, _ ) .
t s
s
r r v
2
(38)
The normalized collision operator reads, explicitly,
⎡⎣ ⎤⎦( )
( ( ) ( )( ) ( ))
∫γ τ τ
τ
τ
= ∇ ⋅↔
( − )
⋅ ∇
− ∇
′
′
′ ′
σσ σ σ
σσ σ σ
σ σ σ
σ σ σ
′ ′
′ ′
′
′ ′ ′
C f f
f t f t
f t f t v
r v
W v v
r v r v
r v r v
, _, _
_ _
_, , _ _ _, _, _
_, _, _ _, , _ d .
v
v
v3
(39)Here,
γ π≔ Λσσ
σ σ′
′Z Z n e L
T
2ln .
e2 2
04
02 (40)
is the collisionality (up to a factor of order unity with respect standard definitions). In this paper we assume γ ∼σσ′ 1.
Plasma Phys. Control. Fusion 57 (2015) 075006
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6
The quasineutrality equation (2) is recast as
∫∑ ( ) =σ
σ σZ f t vr v_, _, _ d 0.3 (41)
It is useful to list the ordering assumptions (13)–(15) in terms of dimensionless variables. The short-wavelength components of the electrostatic potential and the distribution functions satisfy
( ) ( )( ) ( )
( )
( )( )
( )φ
ϵ
φ
ϵ
φ
ϵ
∼
∼
ˆ ⋅ ∇ ∼
ˆ ⋅ ∇ ∼
∂ ∼
∂ ∼
σ
σ
σ
t
f t
t
f t
t
f t
r
r v
b r r
b r r v
r
r v
_ _, _ 1,
_, _, _ ,
_ _ _ _, _ 1,
_ _ _, _, _ ,
_ _ _, _ 1,
_ _, _, _ .
s
s
t
t s
r
r
sw
sw
sw
sw
sw
sw
(42)
The ordering for the long-wavelength components is
( )
( )
( )
( )
( )
( )
φ ϵ
φ ϵ
φ ϵ
ϵ
∼
∼
∇ ∼
∇ ∼
∂ ∼
∂ ∼
σ
σ
σ
t
f t
t
f t
t
f t
r
r v
r
r
r
r v
_ _, _ 1/ ,
_, _, _ 1,
_ _ _, _ 1/ ,
_ _, _ 1,
_ _ _, _ ,
_ _, _, _ .
s
s
t s
t s
r
r
lw
lw
lw
lw
lw
lw 2
(43)
Observe that the normalization of φ has an extra ϵ−s
1 that makes φ ϵ∼ −_ s
lw 1 and φ ∼_ 1sw , whereas ∼σf 1lw and ϵ∼σf ssw .
For the expansions ϵs ≪ 1 of both components of the normal-ized electrostatic potential, we write
φϵ
φ φ ϵ φ ϵ≔ + + + ( )t t t t Or r r r_ (_, _)1
(_, _ ) _ (_, _ ) _ (_, _ )s
s slw
0 1lw 2 lw 2
(44)
and
( ) ( ) ( )φ φ ϵ φ ϵ≔ + + ( )t t t Or r r_ _, _ _ _, _ _ _, _ .s ssw 1sw 2 sw 2 (45)
We do not underline variables from here on, but we assume that we are employing the non-dimensional ones.
4.2. Gyrokinetic coordinate transformation
The aim of gyrokinetic theory is to average equations (38) and (41) over time scales of the order of the ion gyrofrequency, assuming that the orderings of section 2 hold. The typical approaches to the problem try to find a coordinate transforma-tion in phase space such that in the new coordinates, called gyrokinetic coordinates, the fast degree of freedom is decou-pled. Iterative methods work directly on the equations of motion [7, 31–35], whereas Hamiltonian and Lagrangian methods employ techniques of analytical mechanics to obtain the coordinate transformation [29, 36–38].
It has been proven in [26] that the intrinsic ambipolarity of the tokamak implies that the gyrokinetic equations have to be computed to ϵ( )O s
2 if one wants to find the long-wavelength radial electric field. The same accuracy is needed to compute the intrinsic rotation profile in the low-flow ordering. The calculation of the gyrokinetic system of equations to second order is given in [29], in the phase-space Lagrangian for-malism, for general magnetic geometry. The correctness of the approach and the results in [29] has recently been confirmed by independent calculations of the most involved piece of the Lagrangian [39, 40]. In simplified geometries, the results of [29] reduce to those of seminal like [36].
However, reaching expressions sufficiently explicit to be implemented in a code still requires some work. In [28] this was done for tokamak geometry but only for the equa-tions that give the long-wavelength components of the fields. Here, we introduce some notation and refresh some results from [28 and 29] that will be needed in the following sections, where the equations determining the short-wavelength com-ponents of the distribution functions and electrostatic poten-tial are derived.
We denote by (R, u, μ, θ) the gyrokinetic coordinates, where R is the position of the gyrocenter, u is the parallel velocity, μ is the magnetic moment and θ is the gyrophase. The euclidean phase-space coordinates are denoted by X≡{r, v} and the transformation between both sets of coordinates by Tσ,
T μ θ( ) = ( )σ u tr v R, , , , , . (46)
In practice, the transformation is computed as a power series in ϵσ, where the lowest order terms of Tσ
−1 are given by
⎛⎝⎜
⎞⎠⎟
ϵ ϵ
ϵ
μ ϵ
θ ϵ
= −( )
ˆ ( ) × + ( )
= ⋅ ˆ ( ) + ( )
=( )
( − ⋅ ˆ ( ) ˆ ( )) + ( )
= ⋅ ˆ ( )⋅ ˆ ( )
+ ( )
σ σ
σ
σ
σ
BO
u O
BO
O
R rrb r v
v b r
rv v b r b r
v e rv e r
1,
,
1
2,
arctan .
2
2
2
1
(47)
Here, the unit vectors ˆ ( )e r1 and ˆ ( )e r2 are orthogonal to each other and to ˆ ( )b r , and satisfy ˆ ( ) × ˆ ( ) = ˆ ( )e r e r b r1 2 at every loca-tion r.
It will be useful to write some expressions in terms of σT *, the pull-back transformation induced by Tσ. Given a function g (X, t), σT g tZ* ( , ) is
T T* ( ) = ( ( ) )σ σg t g t tZ Z, , , . (48)
For the expansion of the transformation in powers of ϵσ we employ the notation
T T T T
T T T T
ϵ ϵ ϵ
ϵ ϵ ϵ
= + + + ( )
= + + + ( )
σ σ σ σ σ σ σ
σ σ σ σ σ σ σ− − − −
O
O
,
.
,0 ,12
,23
1,01
,11 2
,21 3
(49)
We will make an extensive use of the zeroth-order transforma-tion. The expression for Tμ θ( ) = ( )σ
−uR r v, , , ,,01 is
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⎛⎝⎜
⎞⎠⎟
μ
θ
=
= ⋅ ˆ ( )
=( )
( − ⋅ ˆ ( ) ˆ ( ))
= ⋅ ˆ ( )⋅ ˆ ( )
u
B
R r
v b r
rv v b r b r
v e rv e r
,
,
1
2,
arctan .
2
2
1
(50)
Higher-order terms of the change of coordinates will be com-puted or taken from [28] when needed. Given a function G (R, u, μ, θ), the following obvious identity will also be useful,
T∫ ∫ μ θ μ θ* ( ) = ( ) ( )σ− G v B G u ur v r r, d , , , d d d ,,0
1 3 (51)
where we have used that the Jacobian of Tσ,0 is B (r).Applying the gyrokinetic transformation to the the Fokker–
Planck equation (38), we get
T T T T∑τ τ τ θ
τ ϵ
∂ + ⋅ ∇ + ∂ + ∂
= * [ * * ]( ) + *σ σ σ σ σ σ θ σ
σσ
σ σσ σ σ σ σ σ σ− −
′′ ′ ′
F F u F F
C F F t S
R
Z
˙ ˙ ˙
, , ,
t u
s
R
1 1 2
(52)
where T≔ *σ σ σF f , T *σ−1 is the pull-back transformation that
corresponds to Tσ−1, i.e. T T* ( ) = ( ( ) )σ σ σ σ
− −F t F t tX X, , ,1 1 , and the particle equations of motion, R, u, and θ (note that μ = )˙ 0 are given in appendix A to the necessary order.
Since it will be also useful in this paper, we recall that in [29] the gyrokinetic transformation Tσ is written as the composition of two transformations, T T T=σ σ σPNP, , . First, we have the non-perturbative transfor-mation T T μ θ( ) = ( ) ≡ ( )σ σ ∥vr v Z R, , , ,g g g g gNP, NP, , defined by
ρ
ρ
ϵ μ θ
μ θ
= + ( )
= ˆ( ) + ( ) × ( )
σ
∥v
r R R
v b R R B R
, , ,
, , ,
g g g g
g g g g g g
(53)
with the gyroradius vector defined as
ρ μ θμ
θ θ( ) = −( )
[ ˆ ( ) − ˆ ( )]B
RR
e R e R, ,2
sin cos .g g gg
gg g g g1 2 (54)
Second, the perturbative transformation
Tμ θ μ θ( ) = ( )σ∥v u tR R, , , , , , , ,g g g g P, (55)
that is written as a power series in ϵσ,
ϵ ϵ
ϵ ϵ
μ μ ϵ μ ϵ
θ θ ϵ θ ϵ
= + + ( )
= + + ( )
= + + ( )
= + + ( )
σ σ σ
σ σ σ
σ σ σ
σ σ σ
∥
O
v u u O
O
O
R R R ,
, ,
,
.
g
g
g
g
2,2
3
,12
,12
,12
(56)
The corrections Rσ,2, uσ,1, μσ,1 and θσ,1 are given in appendix B.In gyrokinetic variables the quasineutrality equation (41)
reads
T∫∑ δ π∣ ( )∣ ( ( ( )) − ) =σ
σ σ σ σZ J F t ZZ rdet , d 0,r 6 (57)
where πr(r, v) ≔ r, and the Jacobian of Tσ to ϵ( )σO 2 is
∣ ( )∣= *σ σ∥J Bdet ,, (58)
with *σ∥B , defined in (A.7).Finally, we recall that the gyrokinetic equations are written
naturally in terms of a function ϕσ defined as
ρϕ μ θ φ ϵ μ θ( ) ≔ ( + ( ) )σ σt tR R R, , , , , , . (59)
It is useful to introduce its gyrophase-independent piece ⟨ϕσ⟩ and its gyrophase-dependent one ϕσ,
ϕ μ θ ϕ μ θ ϕ μ˜ ( ) ≔ ( ) − ⟨ ⟩( )σ σ σt t tR R R, , , , , , , , . (60)
The gyroaverage of a function G (R, u, μ, θ, t) is defined by
∫μπ
μ θ θ⟨ ⟩( ) ≔ ( )π
G u t G u tR R, , ,1
2, , , , d .
0
2 (61)
The ordering assumptions on φ, equations (42), imply that
ϕ ϕ ϵ ϕ ϵ
ϕ ϕ ϵ ϕ ϵ
⟨ ⟩ = ⟨ ⟩ + ⟨ ⟩ + ( )˜ = ˜ + ˜ + ( )
σ σ σ
σ σ σ
O
O
,
.
s s
s s
sw1
sw2
sw 2
sw1
sw2
sw 2 (62)
A Taylor expansion of ϕσlw around r = R is allowed. Employing
the convention :↔
= ⋅↔
⋅uv M v M u for an arbitrary matrix ↔M
and vectors u and v, we write
ϕ μϵ
φ φ
ϵ μτ
φ
φ ϵ
⟨ ⟩( ) = ( ) + ( )
+( )
(↔
− ( ) ( ))
∇ ∇ ( )
+ ( ) + ( )
σ
σ σ
⎛⎝⎜
⎞⎠⎟
t t t
Z B
t
t O
R R R
RI b R b R
R
R
, ,1
, ,
2ˆ ˆ
: ,
,
s
s
s
R R
lw0 1
lw
2 2
0
2lw 2
(63)
and
ρϕ μ θτ
μ θ φ ϵ˜ ( ) = ( ) ⋅ ∇ ( ) + ( )σσ σ
tZ
t OR R R, , ,1
, , , ,sRlw
0 (64)
giving ϕ = ( )σ O 1lw .
4.3. Gyrokinetic expansion of the radial flux of toroidal angular momentum
We start by writing the right-hand side of equation (35) in terms of dimensionless variables. We use the normalizations
ϵ
Π = Π
Π = Π
=↔
= ↔
σ σ
σ σ σ σ
σ σ
n T B L
n T B L
N n c N
n T
V V
P P .
e
e
s e s
e
0 0 02
0 0 02
0
0 0
(65)
Thus, in dimensionless variables (we do not underline them anymore),
∫τϵ
( ) ≔ ( )σ σσ
σN t f t vV r v r v, , , ds
3 (66)
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and
∫↔( ) ≔ ( )σ σt f t vP r vv r v, , , d .3 (67)
Also, we use the conventions
ϵ ϵ= ( ) + ( ) + ( )σ σ σ σ σ σ σ σN N N OV V V1 22 (68)
and
ϵ ϵ ϵ↔
=↔
+↔
+↔
+ ( )σ σ σ σ σ σ σOP P P P .0 12
23 (69)
In these variables, equation (35) reads
∫
∫
∫
∑
∑
ζ
ζ ζ
ζ
ζ
ζ
ϵτ
φ
ϵτ
φ
ϵτ
ϵτ
ϵτ
ϵτ
Π = ∂ ⋅
+ ∂ ∂ ⋅↔
⋅
+ ⟨ ⟩ ∂ − ⋅
− ∂ ⋅
− ⋅
′′
′′
σσ
ζ ϵ σ σ ψ
σ σψ ζ ϵ σ
ψ
σ σψ ϵ σ
σ σ σσσ
ψ
σ σψ
σσσ
ψ
σ σσ
ψ
′′
′′
RN
Z VV R
ZR p
ZC R v
Z VV C R v
ZS R v
V
P
v
v
v
( ˆ)
2
1( ˆ ˆ)
2 2( ˆ) d
6
1( ˆ) d
2( ˆ) d ,
s
s
st
s
s
s
lw2
2 /lw
2
2 /2
lw
3
22 lw 2 2 3
2
2 2lw 3 3 3
3
22 2 3
s
s
s2
(70)
where we have omitted the arguments of the collision opera-tors to ease the notation slightly.
We turn to write the right-hand side of (70) in terms of the solutions obtained from the gyrokinetic Fokker–Planck and quasineutrality equations that are given in sections 5.1–5.3 and 6. From the Fokker–Planck equation one obtains, order by order, Fσ0, Fσ1 and Fσ2, where our convention is
μ θ μ ϵ μ
ϵ μ θ ϵ
( ) = ( ) + ( )
+ ( ) + ( )
σ σ σ σ
σ σ σ
F u F u F u
F u O
R R R
R
, , , , , , ,
, , , .
0 1
22
3 (71)
In [28], it is shown that Fσ1 does not depend on the gyrophase and that Fσ0 is a Maxwellian with zero flow, whose density and temperature are flux functions,
⎛⎝⎜
⎞⎠⎟
ψπ ψ
μψ
= ( )( ( ))
− +( )σ
σFn t
T t
u
T t
,
2 ,exp
/2 B
,.0 3/2
2
(72)
Note the different notation employed for the density of the Maxwellian, nσ, and for the total density, Nσ, in (18). The temperatures of all species are equal because due to our ordering, τE is much larger than the collision time of all species, including electrons. In particular, we are not expanding in ≪m m/ 1e i . To see the effect that such an expansion would have on the result, we consider the equa-tions in [2], valid only for Bp/B ≪ 1. By exploiting that
≪m m/ 1e i , [2] finds momentum transport driven by the temperature difference between electrons and ions, Te −Ti. Our equations will have this effect, although only in the limit ∣ Te −Ti ∣ ≪Te ≃ Ti. The temperature difference Te −Ti is contained in the piece Fσ2.
The notation for the expansion of the electrostatic potential has been given in (44), (45), (62)–(64). As for the expansion of the collision operator, we define each coefficient by
ϵ ϵ ϵ= + + ( )σσ σ σσ σ σσ σ( ) ( )
′ ′ ′C C C O .1 2 2 3 (73)
Here, we have taken into account that the ϵ( )σO 0 term equals zero because the collision operator vanishes when acting on Maxwellians with the same temperatures.
Observe that Fσ(R, u, μ, θ) is obtained from the gyrokinetic Fokker–Planck equation, and is therefore expressed naturally in gyrokinetic coordinates, but the collision operator and other functions entering the integrals in (70) are written in coordi-nates (r, v). The simplest way to express (70) is given by trans-forming the relevant pieces of Fσ(R, u, μ, θ) to coordinates (r, v). In section 4.3.1 we write the necessary transformations. We employ the notation of section 4.2. In section 4.3.2, we manipulate the terms of (70) containing collision operators. In section 4.3.3, we give the final expression for the radial flux of toroidal angular momentum.
4.3.1. Some pieces of the transformation of Fσ to euclidean coordinates. We need the long-wavelength component of the transformation of the Maxwellian to first and second order. To O(ϵσ) the calculation is given in appendix C and the result is
T T
T
ϵ
φ
[ ] = + ⋅ + − ⋅
+ ⋅ ( × ∇ )
σ σ σ σσ
σσ
σσ σ
− −
−
⎡
⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤
⎦⎥
F FT
v
T
Z
BF
v V v V
v b
* *2
5
2
ˆ * ,
p T
r
,11
0lw
,01
0
2
0 ,01
0
(74)
where
≔ ˆ × ∇ ≔ ˆ × ∇σσ
σn Bp
BTV b V b
1,
1,p T
r r (75)
To ϵ( )σO 2 , the transformation is much more complicated. It was computed in [28],
φ
φ φ
φ
φ
τ ϕ τ φ
τ τ
φ
τ ϕ ϕ
= × × ∇ ∇ + ∇ ∇
− ∇ ∇ + ∇ ∇ + − ∇ ∇
− ∇ ∇ + × ⋅ ∇
+∇
+ − ∇
+ ⋅ ∇ +∇
+ − ∇
+ + × ⋅ ∇
+ Ψ + Ψ
+ Ψ +↔
− ∇ ∇
+ ∂ ⟨ ⟩
∼
∼
σ σ σσ
σ
σ σσ
σ
σσ σ
σ
σ
σσ σ
σ σσ σ σ σ
σ σ
σ σ σ ϕ σ σ σ σ ϕ σ
σ σσ
σ σσ σ μ σ σ σ
−
−
−
−
− −
− −
− ⊥
− −
⎡⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
⎡
⎣⎢
⎛⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟⎤
⎦⎥
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎡⎣
⎤⎦
⎡⎣⎢
⎡⎣ ⎤⎦⎤⎦⎥
T
T
T
T
T T
T T
T
T T
FB
nZ
T
Z
TT T
v
TT
v
TT T F
B
n
n
Z
T
v
T
T
TF
n
n
Z
T
v
T
T
TF
Z
TF
T
Z
B
Z Z
Z v
B
Z
BF
v b v b
v b
R
v b
I bb
[ * ]1
2( ˆ )( ˆ ) : ln
( )2
3
2ln
2*
1
2( ˆ )
2
3
2*
2
3
2*
2* ( ) *
1( ˆ )
* *
*4
( ˆ ˆ ) :
* * .
B
B
r r r r
r r r r r r
r rr
r r
r r r
r
r r
,21
0lw
2 0
2 0 0
2
2
3 ,01
0 2
02 2
,01
0
02lw 0
2
,01
0
4 2
2 ,01
1sw 2 lw
,01
0
2
1lw
4 2,01
,lw 2
,01
,lw
,01
,
2
2 0
4 2
,01
1sw
1sw lw
,01
0
(76)
Plasma Phys. Control. Fusion 57 (2015) 075006
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9
Here,
⎜ ⎟
⎜ ⎟
⎡⎣⎢⎛⎝
⎞⎠
⎛⎝
⎞⎠⎤⎦⎥
⎛
⎝⎜
⎞
⎠⎟
= + ×
+ × + ×∇
+ ⋅ ∇ + ⋅ ∇ ⋅
+ [ − ( × )( × )] ∇
+ ⋅ ∇ − ∇
∥ ⊥
∥ ⊥
∥⊥
∥⊥
⊥ ⊥ ⊥ ⊥
⊥ ⊥
⋅
⊥
Bv
vB
v
B
v
B
B
v
BB
v
BB
R b v v b
v b b vb
v b bb b v
bv v v b v b b
bb
1 1
4
1
4
8:
2 4,
r
r r
r
r r
02lw
2 2
2
2
3
2
3
(77)
where we have used the convention ×↔
= × ( ⋅↔
)ab M a b M.
,
μτ
φτ
φΨ = − ∇ ⋅ ∇ − (ˆ ⋅ ∇ ˆ) ⋅ ∇ϕ σσ σ σ σZ B
Bu
Z Bb b
3
2,B R R R R,
lw2 0
2
2 0
(78)
τφ ϕΨ = − ∇ − ∂ ⟨ ⟩∼
ϕ σσ σ
μ σ⎡⎣
⎤⎦Z B B
1
2
1
2( ) ,R,
lw2 2 2 0
21
sw 2 lw
(79)
and ΨB, σ is given in (A.10).We also need the long-wavelength component of the action
of T *σ−
,11 on σF 1
lw and σF 1sw. Employing repeatedly the results of
appendix C, it is straightforward to find that
T T
⎜
⎟
⎪
⎪⎧⎨⎩
⎛⎝
⎞⎠
⎛
⎝⎜⎜
⎞⎠⎟
⎫⎬⎭
ρ
ρ ρ ρ ρ ρ
ρ
ρ ρ ρ ρ
ρ ρ
μ μ
μ
φ
[ * ] = − * ⋅ ∇
+ ⋅ ∇ ⋅ + [ ( × ) + ( × ) ] ∇
− ⋅ ∇ × ∂ + − ⋅ ∇
− [ ( × ) + ( × ) ] ∇ + ⋅ ∇ ×
− ⋅ ∇ ⋅ − ⋅ ∇ ∂
σ σ σ
σμ σ
− −F
uB
BB
u u
B
u
B
Z
BF
b b b b b
b b
b b b b b
b b
4:
4:
u
R
R R
R R
R R
R R
,11
1lw lw
,01
2
0 1lw
(80)
and
⎡⎣⎢⎛⎝⎜
⎞
⎠⎟⎟
⎤
⎦⎥⎥
T Tτ
τ ϕ
[ * ] = − * (∇ Φ × ˆ) ⋅ ∇
− ∂
∼
σ σ σσ σ
ϵ σ ϵ
σ σ σμ σ
− −σ σ⊥ ⊥F
Z
B
Z
BF
b
.
R R,11
1sw lw
,01
2
2 / 1sw
/
21
sw
1sw
lw
(81)
The short-wavelength components of the action of T *σ−1 on Fσ
to first and second order, denoted respectively by T( * )σ σ− F1
1sw
and T( * )σ σ− F1
2sw, are also needed. We introduce an operator Tσ,0,
whose action on a phase-space function G (R, u, μ, θ) is given by
T⎛
⎝⎜⎜
⎛
⎝⎜
⎞
⎠⎟⎞
⎠⎟⎟
ϵ( ) ≔ −( )
( ) × ⋅ ( )
( )⋅ ( )⋅ ( )
σσ
⊥
G GB
v
B
r v rrb r v v b r
rv e r
v e r
, ^ , ^ ,
2, arctan
^
^.
,0
22
1
(82)
This operator is useful to write some expressions involving the short-wavelength pieces of the distribution functions and the potential, for which it is not possible to Taylor expand the dependence on ϵ− ( ) ˆ ( ) ×σ
−Br r b r v1 around r. Usually, G in (82) also depends slowly on R due to the quantities related to the magnetic field such as B (R) or ˆ ( )b R . In T ( )σ G r v,,0 this dependence is Taylor expanded around r.
With the help of (82) and appendix C we obtain
T TT Tτ ϕ( * ) = − ˜ *σ σ σ σ
σ σσ σ σ σ
− −F FZ
TF .1
1sw
,0 1sw
2
,0 1sw
,01
0 (83)
Finally, we have to compute T( * )σ σ− F1
2sw. For this, we need
again the expressions in appendix C and the calculation of the short-wavelength transformation of the Maxwellian to second order, explained in appendix D. The result is
T
T T T
T T T
T
μ μ
θ θ
μ
=
+ ⋅ ∇ − + ∂
− ∂ − + ∂
− ∂ + −
− × ⋅ ∇ + − ∇
+ ⋅ ∇ + − ∇
σ σ σ σ
σ ϵ σ σ σ σ σ μ
σ σ σ σ σ σ σ σ θ σ
σ σ σ μ σ σ σ σ σ
σ
σσ σ
σ
σσ σ
−
−
− −
− − −
−
−
σ
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
T
T
T T
T T T
T
T
F F
u F
FH
TF
H
TF
H
TB
n
n
v
T
T
TF
n
n
v
T
T
TF
R
v b
R
( * )
[( ( *ˆ ˆ )
*ˆ ( * ˆ ˆ ) ) ]
ˆ *[ ]
2* *
( ˆ )2
5
2*
2
3
2* ,
u
R
r r
r r
12sw
,0 2sw
02 ,0 / ,01
,1lw
,0 ,1sw
,0
,01
,1lw
,0 ,01
,1lw
,0 ,1sw
,0 1sw
sw
,0 ,1sw
,01
1lw 01
2 sw
2 ,01
002sw
,01
0
01sw 2
,01
0
02sw
2
,01
0
(84)
The corrections R02lw, R02
sw, H01lw, H01
sw and H02sw are provided in
(77), (D.5)–(D.7) and (D.9). Using the expressions for ˆσu ,1, μσ,1 and θσ,1 in (C.5), it is easy to obtain
μ
= − ⋅ ∇ ⋅ × − × + × ∇
− ⋅ ∇ ×
σ σ− ⊥ ⊥ ⊥ ⊥T u
u
B Bb b v b v b v v v b b
b b
*ˆ ˆ ˆ ˆ 1
4[( ˆ ) ( ˆ )] : ˆ
ˆ ˆ ,
r r
r
,01
,1lw
(85)
μ μ
μ φ
= × ⋅ ∇ + × + × ∇
+ ⋅ ∇ × + ⋅ ∇ ⋅ × + × ⋅ ∇
σ σ
σ
− ⊥ ⊥ ⊥ ⊥TB
Bu
B
u
B
u
B
Z
B
v b v b v v v b b
b b b b v b v b
*ˆ ( ˆ )4
(( ˆ ) ( ˆ )) : ˆ
ˆ ˆ ˆ ˆ ( ˆ ) ( ˆ ) ,
r r
r r r
,01
,1lw
2 2
2
2 2 0
(86)
T ⎛⎝⎜
⎞⎠⎟
θμ
μ
μφ
* = ⋅ ∇ + ⋅ ∇
− × ∇ ⋅ − (( × )( × ) − ) ∇
+ ⋅ ∇ + ⋅ ∇
σ σ
σ
− ⊥
⊥ ⊥ ⊥ ⊥
⊥
BB
u
u
u
BB
Z
v b b
b e e v b v b v v b
b v
1ln
2 B
8 B:
2 2 B,
r r
r r
r r
,01
,1lw 2
2 1 2
2 2 0
(87)
T Tμ τ ϕˆ ( ) = −( )
˜ ( )σ σσ σ
σ σtZ
Btr
rr, ,,0 ,1
sw2
,0 1sw
(88)
and
T Tθ τˆ ( ) =( )
∂ Φ ( )σ σσ σ
σ μ σtZ
Btr
rr, , .,0 ,1
sw2
,0 1sw
(89)
Plasma Phys. Control. Fusion 57 (2015) 075006
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10
4.3.2. Manipulations of terms in (70) that contain collision operators. We are ready to go back to (70) and write more explicitly the terms on the right side. The term before last in (70) is
∫
∫
∑
∑
ζ
ζ
ϵτ
ϵτ
− ∂ ( ⋅ ˆ)
= − ∂ ( ⋅ ˆ)
′′
′′
σ σψ
σσσ
ψ
σ σψ
σσσ
ψ
( )
′′
′′
Z VV C R v
Z VV C R v
v
v
6
1d
6
1d ,
s
s
2
2 2lw 3 3 3
3
3 31 lw 3 3 3
(90)
where
T
T T T
T T
(ττ
= ⋅ + − ⋅
+ + ⋅
+ − ⋅ +
σσ σσ σ σ σ σ
σ σ σ σσ σ
σ σσσ σ σ σ
σ σ σ σ σ
( ) −
− − −
− −
′ ′
′ ′′ ′
′ ′
′ ′ ′ ′ ′
⎡
⎣⎢⎢
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎤
⎦⎥⎥
C CT
v
TF
F FZ
ZC F
T
v
TF F
v V v V
v V
v V
1
2
5
2*
* , * * ,1
2
5
2* * .
p T
p
T
1 lw2
,01
0
,01
1lw
,01
0 ,01
0
2
,01
0 ,01
1lw
(91)
Here, we have used (74) and the Galilean invariance of the collision operator to drop the term containing φ0 in (74).
The fourth term on the right side of (70) has two contribu-tions of different orders,
∫
∫
∫
∑
∑
∑
ζ
ζ
ζ
ϵτ
ϵτ
ϵτ
− ( ⋅ ˆ)
= − ( ⋅ ˆ)
− ( ⋅ ˆ)
σ σ σσσ
ψ
σ σ σσσ
ψ
σ σ σσσ
ψ
( )
( )
′′
′′
′′
ZC R v
ZC R v
ZC R v
v
v
v
2d
2d
2d .
s
s
s
lw 2 2 3
2
2 21 lw 2 2 3
3
3 32 lw 2 2 3
(92)
The first-term on the right side can be computed by using (91). As for the second term, employing (74), (76), (80), (81), and (83) we immediately get
ττ
ττττ
τ ϕ
τϕ
= ++ +
+
+ + +
+ + +
+ −
−
∼
∼
σσ σσ σ σ σ σ
σ σ σ σ σ σ
σ σ
σ σσσ σ σ σ σ
σ σ σ σ σ σ
σ σ
σ σσσ σ σ σ σ σ σ σ σ
σ σ
σ σσσ σ σ
σ σσ σ σ σ σ σ
σ σσ σ σ σ
− −
− − −
− −
− − −
− − − −
−
−
′ ′
′ ′
′ ′′ ′ ′
′ ′ ′ ′ ′ ′
′ ′′ ′ ′ ′ ′
′ ′′ ′ ′
′ ′′ ′ ′ ′
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎡⎣⎢
⎤⎦⎥⎤⎦⎥
T T
T T T
T T
T T T
T T T T
T
T
C C F F
F F F
Z
ZC F F
F F FZ
ZC F F F F
Z
ZC F
Z
TF F
Z
TF
[ * [ * ]
[ * ] [ * ] , * ]
[ * , *
[ * ] [ * ] [ * ] ]
[ * [ * ] , * [ * ] ]
* ,
* .
(2)lw,01
2lw
,11
1lw lw
,11
1sw lw
,21
0lw
,01
0
2
,01
0 ,01
2lw
,11
1lw lw
,11
1sw lw
,21
0lw
,01
1lw
,11
0lw
,01
1lw
,11
0lw
,0 1sw
2
,0 1sw
,01
0 ,0 1sw
2
,0 1sw
,01
0
lw
T T T
T
(93)
4.3.3. Final expression for the radial flux of toroidal angular momentum. We turn to the first two terms on the right-hand side of (70). The second one is simple,
ζ ζ
ζ ζ
ϵτ
φ
ϵτ
φ
∂ ∂ ⋅↔
⋅
= ∂ ∂ ⋅↔
⋅
′′
′′
σ σψ ζ ϵ σ
ψ
σ σψ ζ ϵ σ
ψ
⎡⎣⎢
⎤⎦⎥
Z VV R
Z VV R
P
P
2
1( ˆ ˆ)
2
1( ˆ ˆ) ,
s
s
2
2 /2
lw
3
2 3 / 1sw 2
1
sw lw
s
s
(94)
because only the short-wavelength, O(ϵs) piece of the stress tensor contributes. Namely,
T∫↔( ) ≔ ( * ) ( )σ σ σ
−t F t vP r vv r v, , , d ,1
sw1
1sw 3 (95)
where T( * ) ( )σ σ− F tr v, ,1
1sw is defined in (83).
The first term on the right-hand side of (70) can be rewritten as
ζ
ζ
ζ
ζ
ϵτ
φ
ϵτ
φ
ϵτ
φ
ϵτ
φ
∂ ⋅
= ∂ ⋅
+ ∂ ⋅
+ ∂ ⋅
σζ ϵ σ σ ψ
σζ ϵ σ σ ψ
σ σζ ϵ σ σ ψ
σζ ϵ σ σ ψ
RN
R N
ZR N
R N
V
V
V
V
( ˆ)
( ) ˆ
( ) ˆ
( ) ˆ
s
s
s
s
2
2 /lw
2
2 / 1sw
1sw
lw
3
3 / 1sw
2sw
lw
3
2 / 2sw
1sw
lw
s
s
s
s
(96)
and therefore expressed in terms of
T∫τ( ) ( ) ≔ ( * ) ( )σ σ σ σ σ−N t F t vV r v r v, , , dj j
sw 1 sw 3 (97)
for j = 1, 2. The quantity T( * )σ σ− F1
1sw has been given in (83) and
T( * )σ σ− F1
2sw has been given in (84).
We have found that the radial flux of toroidal angular momentum is
ϵ ϵ ϵΠ = Π + Π + ( )O ,s s slw 2
2lw 3
3lw 4 (98)
where
∫∑ ∑
∑
ζ
ζ
τ
τφ
Π = − ⋅
+ ∂ ⋅
σ σ σ σσσ
ψ
σ σζ ϵ σ σ ψ
′′
⎡⎣⎢
⎤⎦⎥
ZC R v
R N
v
V
1
2( ˆ) d
1( ) ˆ
2lw
2 2(1)lw 2 2 3
2 / 1sw
1sw
lw
s
(99)
and
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
∫
∫
∫
∑ ∑
∑
ζ
ζ
ζ ζ
ζ
ζ
ζ
τ
τ τ
τφ
τφ
τφ
τ
Π = − ∂ ( ⋅ )
− ( ⋅ ) + ⟨ ⟩ ∂
+ ∂ ∂ ( ⋅↔
⋅ )
+ ∂ ( ) ⋅
+ ∂ ( ) ⋅
− ( ⋅ )
′′
′′
σ σ σψ
σσσ
ψ
σ σ σσσ
ψ σ σψ ϵ σ
σ σψ ζ ϵ σ
ψ
σ σζ ϵ σ σ
ψ
σζ ϵ σ σ
ψ
σ σσ
ψ
( )
( )
′′
′′
Z VV C R v
ZC R v
ZR p
Z VV R
ZR N
R N
ZS R v
v
v
P
V
V
v
1
6
1d
1
2d
1
2
1
2
1
1
1
1
2d .
t
3lw
3 31 lw 3 3 3
3 32 lw 2 2 3
22
2 3 / 1sw 2
1
sw lw
3 / 1sw
2sw
lw
2 / 2sw
1sw
lw
22 2 3
s
s
s
s
2
(100)
Plasma Phys. Control. Fusion 57 (2015) 075006
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11
The computation of the third term on the right-hand side of (100) indicates that the transport equations for nσ and T are needed. To calculate nσ, we use the density transport equation of each species. Since we are assuming that the temperatures of all species are equal, we can use the transport equation for the total energy to determine T. The required transport equations were calculated in [28] and can be found in section 6.3 of the present paper. Importantly, ⟨ ⟩σF 2
lw does not enter these transport equa-tions. The piece ⟨ ⟩σF 2
lw only enters the expression for Πlw through
σσ( )
′C 2 lw, and it is clear that adding a term τ φ( )σ σ σZ T F/3 22lw
0 to ⟨ ⟩σF 2lw
in those terms does not change the result. This is why the ϵ( )O s2
long-wavelength quasineutrality equation is not needed.Even though, in principle, Π2
lw dominates (98), this term is small in up-down symmetric tokamaks due to a symmetry of the gyrokinetic equation, and it is comparable to Π3
lw [2, 20, 41]. The feasibility of using external poloidal field coils to break up-down symmetry and generate intrinsic rotation has been explored in [42–44]. The largest second order momentum flux obtained from up-down asymmetry has been Π ∼Q/ 0.12
lw , where Q is the turbu-lent heat flux normalized by the gyroBohm heat flux. According to [2, 45], Π ∼Q B B/ / p3
lw . These estimates suggest that Π3lw domi-
nates over the up-down asymmetry contribution to Π2lw for (B/Bp)ϵ
≳ 0.1. For (B/Bp)ϵ ≲ 0.1, the importance of Π3lw depends on how
up-down asymmetric the tokamak is. Note that according to [44] not all possible up-down asymmetric shapes are equally effective.
The next sections are devoted to present the equations that give the short and long-wavelength components of the distri-bution function and electrostatic potential that are needed to evaluate the right sides of (99) and (100).
5. Short-wavelength equations
In this section the equations that determine the short-wavelength components of the distribution functions and electrostatic potential up to ϵ( )O s
2 are provided. The equations to O(ϵs) con-stitute the standard set of gyrokinetic equations. We give them for completeness in section 5.1. The equations for the ϵ( )O s
2 turbulent pieces are one of the main results of this paper and are derived in sections 5.2 and 5.3. Note that the ϵ( )O s
2 short-wavelength equations given in [2] are valid only if B/Bp ≫ 1.
5.1. Short-wavelength Fokker–Planck and quasineutrality equations to first order
The first-order, short-wavelength terms of the Fokker–Planck equation are
T∑
κ
τμ
τ ϕ
μ φ
τ ϕ
τ ϕ ϕ
∂ + ( ⋅ ∇ − ⋅ ∇ ∂ )
+ ( × ∇ ⟨ ⟩) ⋅ ∇
+ × + × ∇ + × ∇ ⋅ ∇
+ ( × ∇ ⟨ ⟩) ⋅ ∇
− ⋅ ∇ ⟨ ⟩ + × ( ⋅ ∇ ) ⋅ ∇ ⟨ ⟩ ∂
=
σσ σ
σ σϵ σ ϵ σ
σϵ σ
σ σϵ σ σ
σ σ σ ϵ σ σ
σσ σσ
( )
′′
σ σ
σ
σ
σ
⊥ ⊥
⊥
⊥
⊥⎜ ⎟
⎡⎣⎢
⎤⎦⎥
⎛⎝⎜
⎞⎠⎟
⎛⎝
⎞⎠
F u B F
Z
BF
u
B BB
Z
BF
Z
BF
Zu
BF
C
b b
b
b b b
b
b b b b
1 ˆ ˆ
ˆ
ˆ ˆ ˆ
ˆ
ˆ ˆ ˆ ˆ
* .
t u
u
R R
R R
R R R
R R
R R R
1sw
1sw
2
/ 1sw
/ 1sw
sw
2
0 / 1sw
2
/ 1sw
0
21
sw/ 1
sw0
NP,1 sw
(101)
Here we have used that Fσ1 does not depend on the gyrophase and
T T
T T
τ ϕ
ττ
τϕ
= −
+ −
σσ σσ σ σσ σ
σ σ σ σ σ σ
σ σ
σ σσσ σ σ σ σ
σ σσ σ σ σ
− −
− −
′ ′
′ ′′ ′ ′
′ ′′ ′ ′ ′
′ ′⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
T T
T T
C C FZ
TF F
Z
ZC F F
Z
TF
˜ * , *
* , ˜ * ,
(1)sw,0 1
sw2
,0 1sw
,01
0 ,01
0
,01
0 ,0 1sw
2
,0 1sw
,01
0
(102)where we have employed (83). The transformation
T μ θ( ) = ( )σ ur v R, , , ,NP, is defined in (53).As for the short-wavelength, first-order quasineutrality
equation, we have
⎡⎣⎢
⎤⎦⎥
∫∑ ρ
ρ
ϕ ϵ μ θ μ θ μ
τϵ μ θ μ μ θ
− ( − ( ) ) ( )( )
+ ( − ( ) ) =
∼
σσ σ σ
σ
σ
σσ σ
B Z tF u t
T t
F u t u
r rrr
r r
, , , , ,, , ,
,
1, , , , , d d d 0.
21
sw 0
1sw
(103)
These equations are well known and were derived in this form in [28]. They determine the lowest-order turbulent contribu-tions to the fields, σF 1
sw and φ1sw.
5.2. Short-wavelength Fokker–Planck equation to second order
In this section and in section 5.3, we compute the ϵ( )σO 2 terms of the short-wavelength Fokker–Planck and quasineutrality equations, respectively.
5.2.1. Gyrophase-dependent component of σF 2sw. The equa-
tion that determines − ⟨ ⟩σ σF F2sw
2sw comes from the gyrophase-
dependent part of the O(ϵσ) terms in the Fokker–Planck equation (52),
∑
∂ − ⟨ ⟩
= − −
θ σ σ
σσ σσ σ σσ
′′ ′T T( )
F F
BC C
( )
1* * .
2sw
2sw
NP,(1)sw
NP,(1)sw
(104)
5.2.2. Gyroaveraged component of σF 2sw. This subsection
is devoted to the manipulations leading to the final form of the second-order, short-wavelength, gyroaveraged Fokker–Planck equation. Recall equation (52) and recall that R and u are given in appendix A. We have to identify the pieces of these equations of motion that contribute to the gyroaveraged second-order Fokker–Planck equation at short wavelengths.
Equation (A.1) can be written as ϵ ϵ ϵ= + + + ( )σ σ σ σ σ σ σOR R R R˙ ˙ ˙ ˙,0 ,1
2,2
3 , where = ˆσ uR b˙ ,0 . The first-order terms are
κ μ
φ τ ϕ
= × + ∇
+ × ∇ + ∇ ⟨ ⟩
σ
σσ σ ϵ σσ⊥( )
Bu B
Z
BZ
R b
b
˙ 1 ˆ ( )
ˆ ,
R
R R
,12
0 / 1sw
(105)
where κ ≔ ˆ ⋅ ∇ ˆb bR . We remind the reader that we are using the conventions (44), (45) and (62), and also the formulae (63) and (64). The contribution of the second-order terms of (A.1) is
Plasma Phys. Control. Fusion 57 (2015) 075006
I Calvo and F I Parra
12
κμ
τ
τ τ
μ φ τ ϕ
τ φ τ ϕ
=
− ∇ × − × ∇ × ⋅
+ ∂ Ψ + ∂ Ψ
+ × ∇ Ψ + ∇ Ψ
− ∇ ×
⋅ × ∇ + ∇ + ∇ ⟨ ⟩
+ × ∇ + ∇ ⟨ ⟩
σ
σ σ ϕ σ σ
σ σ ϵ ϕ σ σ σ ϵ ϕ σ
σ σ σ ϵ σ
σ σ σ σ ϵ σ
⊥
σ σ
σ
σ
⊥ ⊥
⊥
⊥
( )
( )
u
B
u
B
Z
BZ Z
u
B
B Z Z
BZ Z
R
K b b b
b
b
b
bb
b
˙
( ) ( ˆ )( ˆ ) ˆ
( ) ˆ
1 ˆ ( )
( ˆ )
ˆ ˆ
1 ˆ ,
u B u B
B
R R
R R
R
R R R
R R
,2
3
2
2, ,
4 2/ ,
sw 2/ ,
sw
2
02
/ 1sw
21lw 3 2
/ 2sw
(106)
where K is defined in (A.6).The terms corresponding to the equation of motion
of the parallel velocity given in (A.2) can be written as ϵ ϵ ϵ= + + + ( )σ σ σ σ σ σ σu u u u O˙ ˙ ˙ ˙,0 ,1
2,2
3 , where
μ= − ˆ ⋅ ∇σu Bb˙ R,0 (107)
and
κ
τ ϕ φ
μ φ τ ϕ
= − ⋅ ∇ ⟨ ⟩ +
− × ⋅ ∇ + ∇ + ∇ ⟨ ⟩
σ σ σ σ
σ σ σ ϵ σσ⊥( )
u Z
u
BB Z Z
b
b
˙ ˆ ( )
( ˆ ) .
R
R R R
,12
1sw
1lw
02
/ 1sw
(108)As for the second-order terms, σu ,2, only the short-wavelength component contributes,
κ
κ
τ τ ϕ τ
τ τ ϕ τ
τ μ ϕ
=
− ⋅ ∇ ⟨ ⟩ + Ψ + Ψ
− × ⋅ ∇ ⟨ ⟩ + Ψ + Ψ
+ ∇ × ⋅ × + ∇ × ⋅ ∇ ⟨ ⟩
σ
σ σ σ σ σ σ σ ϕ σ ϕ σ
σ σ ϵ σ σ σ σ σ ϕ σ ϕ σ
σ σϵ σ⊥
σ
σ
⊥
⊥
⎛⎝⎜
⎞⎠⎟
u
Z Z Z
Zu
BZ Z
Z
B
u
B
b
b
b b b K
˙
ˆ ( )
( ˆ ) ( )
( ˆ ) ˆ ( ˆ ) ( ) .
B
B
R
R
R R R
,2sw
22
sw 2,
sw,
sw
2/ 2
sw 2,
sw,
sw
2 2
/ 1sw
(109)
Then, the ϵ( )σO 2 , gyroaveraged terms of (52) at short wave-lengths are
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T T T∑
τ μ
τ τ
τ
τ τ
τ τ τ
τ
∂ ⟨ ⟩ + ( ˆ ⋅ ∇ − ˆ ⋅ ∇ ∂ )⟨ ⟩
+[ ⋅ ∇ ⟨ ⟩] + ∂ ⟨ ⟩
+ ⋅ ∇
+ ⋅ ∇ + [ ⋅ ∇ ]
+[ ∂ ] + ∂ + ∂
= * [ * * ]
σ σ σ
σ σ ϵ σ σ σ σ
σ σ σ
σ σ σ σ σ ϵ σ
σ σ σ σ σ σ σ σ σ
σσ
σ σσ σ σ σ σ− −
′′ ′ ′
σ
σ
⊥
⊥
F u B F
F u F
F
F F
u F u F u F
C F F
b b
R
R
R R
˙ ˙
˙
˙ ˙
˙ ˙ ˙
, ,
t u
u
u u u
R R
R
R
R R
2sw
2sw
,1 / 2sw sw
,0 2sw
,1sw
1lw
,2sw
0 ,2 / 1sw sw
,1 1sw sw
,1sw
1lw
,2sw
0
1 1
2
sw
(110)
where we have used that the right-hand sides of (A.1), (A.2), and (A.4) are gyrophase-independent, and so are Fσ1 and Fσ0. The gyrophase independence of the equations of motion is the reason why (110) does not contain contributions from the equation of motion for θ, (A.4).
Next, we deal with the collision terms in (110). First, we write them as
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T T T
T T T
T T T
T T T
T T T
T T T
∑
∑
∑
∑
∑
∑
τ
τ
τ
τ
τ
τ
* [ * * ]
= * [ * ( * ) ]
+ * [( * ) * ]
+ * [( * ) ( * ) ]
+ * [ * ( * ) ]
+ * [( * ) * ]
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
− −
− −
− −
− −
− −
− −
′′ ′ ′
′′ ′ ′
′′ ′ ′
′′ ′ ′
′′ ′ ′
′′ ′ ′
C F F
C F F
C F F
C F F
C F F
C F F
,
,
,
,
,
, .
1 1
2
sw
,0 ,01
01
2sw
,01
2sw
01
0
,01
11
1
sw
,1 ,01
01
1
sw
,11
1 ,01
0
sw
(111)
Let us start by writing the short-wavelength component of the action of T *σ,1 on any phase-space function f (r, v):
T T T
T T
T
T
μ θ
μ
θ
μ θ
[ * ] = ∂ ( * ) + ∂ ( * )
+[ ⋅ ∇ ( * )] + [ ∂ ( * )]
+[ ∂ ( * )] + ( ⋅ ∇
+ ∂ + ∂ + ∂ ) *
σ σ μ σ σ θ σ
σ ϵ σ σ μ σ
σ θ σ σ ϵ
σ σ μ σ θ σ
σ
σ
⊥
⊥
f f f
f f
f
u f
R
R
^ ^
,u
R
R
,1sw
,1sw
,0lw
,1sw
,0lw
,2sw
/ NP,sw sw
,1sw
NP,sw sw
,1sw
NP,sw sw
,2lw
/
,1lw
,1lw
,1lw
NP,sw
(112)
with ˆσu ,1, μσ,1, θσ,1 given in (C.5), and Rσ,2, uσ,1, μσ,1, θσ,1 given in (B.1)–(B.4). Note the difference between (Rσ,2, uσ,1, μσ,1, θσ,1) and ρ μ θ( ˆ ˆ ˆ )σ σ σu, , ,,1 ,1 ,1 . The former give the give the lowest order terms of the perturbative transformation defined in (55). The latter are the O(ϵσ) terms of the complete transformation Tσ from gyrokinetic coordinates to euclidean coordinates.
Equation (112) is useful to write the two last terms on the right side of (111),
∑
∑
∑
∑
∑
τ
τ
τ μ θ
τ μ θ
τ μ θ
+
= ∂ + ∂
+ ⋅ ∇ + ∂ + ∂
+ ⋅ ∇ + ∂ + ∂ + ∂
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ μ σ θ σ σσ
σσ
σ ϵ σ μ σ θ σ σσ
σσ
σ ϵ σ σ μ σ θ σ σσ
− −
− −
′′ ′ ′
′′ ′ ′
′′
′′
′′
σ
σ
⊥
⊥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T T T
T T T
T
T
T
C F F
C F F
C
C
u C
R
R
* [ * , ( * ) ]
* [( * ) , * ]
( ˆ ˆ ) *
( ) *
( ) * ,u
R
R
,1 ,01
01
1
sw
,11
1 ,01
0
sw
,1sw
,1sw
,0(1)lw
,2sw
/ ,1sw
,1sw
NP,(1)sw
sw
,2lw
/ ,1lw
,1lw
,1lw
NP,(1)sw
(113)
Plasma Phys. Control. Fusion 57 (2015) 075006
I Calvo and F I Parra
13
where σσ( )
′C 1 lw and σσ( )
′C 1 sw have been defined, respectively, in equations (91) and (102).
Define, for convenience,
T T T( * ) = * + [ * ]σ σ σ σ σ σ− − −F F F ,1
1lw
,01
1lw
,11
0lw (114)
where the last term is given in (74). Using (83) and (114) we can write the third term on the right side of (111) as the sum of three pieces:
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T T T
T T T
T T T
T T T
∑
∑
∑
∑
τ
τ
τ
τ
* [( * ) ( * ) ]
= * [( * ) ( * ) ]
+ * [( * ) ( * ) ]
+ * [( * ) ( * ) ]
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
σσ
σ σσ σ σ σ σ
− −
− −
− −
− −
′′ ′ ′
′′ ′ ′
′′ ′ ′
′′ ′ ′
C F F
C F F
C F F
C F F
,
,
,
, .
,01
11
1
sw
,01
1sw 1
1lw
,01
1lw 1
1sw
,01
1sw 1
1sw
sw
(115)
In order to reach explicit expressions for the first two terms on the right side of (111), one needs T( * )σ σ
− F12sw, which is given
in (84).
5.3. Short-wavelength quasineutrality equation to second order
The effort made in previous sections immediately gives the ϵ( )O s
2 short-wavelength contributions to (41) in terms of already calculated quantities. Namely,
T∫∑τ
( * ) =σ σ σ
σ σ−
ZF v
1d 0,
21
2sw 3
(116)
where T( * )σ σ− F1
2sw is given in (84).
6. Long-wavelength equations
The results of this section are taken from [28]. They are included in this paper because, as we have seen in section 4.3, the O(ϵs) and ϵ( )O s
2 long-wavelength pieces of the distribu-tion functions and the O(ϵs) long-wavelength pieces of the electrostatic potential are needed to compute the radial flux of toroidal angular momentum.
6.1. Long-wavelength Fokker–Planck equation up to second order
The equation for the first-order piece σFlw is conveniently written in terms of
⎜ ⎟⎧⎨⎩
⎛⎝
⎞⎠⎫⎬⎭
τ φ φ≔ + + ∂ + Υσ σσ σ σ
ψ σ σG FZ
T
Iu
B
Z
TF ,1
lw1
lw2
1lw
0 0 (117)
where
⎛⎝⎜
⎞⎠⎟
μΥ ≔ ∂ + + − ∂σ ψ σ ψnu
TTln
/2 B 3
2ln .
2
(118)
It reads
⎜ ⎟
⎜ ⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠⎤⎦⎥
T T T
T
T T
∑
∑
μ
ττ
( ˆ ⋅ ∇ − ˆ ⋅ ∇ ∂ )
= * * − Υ *
+ *
* * − Υ
σ
σσ σσ σ σ σ σ σ σ
σ
σ σ
σ σσ σσ
σ σ σ σ σ σ
− −
− −
′′ ′ ′
′ ′ ′′
′ ′ ′ ′
u B G
C GIu
BF F
Z
ZC
F GIu
BF
b b
,
, .
uR R 1lw
,0 ,01
1lw
0 ,01
0
,0
,01
0 ,01
1lw
0
(119)
In order to write the second-order, long-wavelength, gyroav-eraged Fokker–Planck equation, we define
T
T T T T
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛
⎝⎜
⎞
⎠⎟
⎛⎝
⎞⎠
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
⎡⎣
⎤⎦
⎛⎝
⎞⎠
∑ρ
τ φτ φ
μ φτ φ
τ φ φ
μ ψ φ
φ
μ
μ φ
τ ϕ
τ ϕ μ
= ⟨ ⟩ + + ∂ − ∂
− ( ∂ + ∂ )∂ −
− ∂ + Υ − ∂
− (( ) + ∇ ) − ∂ ∂
+ ∂ + Υ + ∂
+ + − ∂
− + (∂ ) + ∂
+⋅ ∇ Θ
( × ∇ ⟨ ⟩) ⋅ ∇ Θ
−⋅ ∇ Θ
⋅ ∇ Θ *
− ( ) + ⋅ ∇ × ∂ −∂
+ * [ * ] + * [ * ]
∼
σ σσ σ
σ ψ σσ σ
σ
ψ σ ψ σσ σ
σ
σ σσ
σψ σ ψ
σψ ψ
σψ σ ψ σ
ψ
ψσ
ψ σ
σ σσ ϵ σ
σσ σσ
σ σσ σ μ σ
σ σ σ σ σ σ
( )
− −
′′
σ⊥
G FZ
TF
Iu
BG
Z
uG
I
BB Z G
Z
TF
Z Iu
TBF
Z
T TT
BIu
Z
TT
Z
Tn
u
TT
u
TT
Z
TF
Z
uF
uC
Z
TF
u
BG
F F
Bb
b
b b
1
2
1
1
2B
2
ln
/2 B 3
2ln
/2 Bln
^
1^
2^ ^
,
u
u
u
R
RR R
R
R
R
2lw
2lw
3 2
2lw
0 1lw
21lw
1lw
0 1lw
21lw 2
0
2
1lw
0 0
22 2
2 0
0
22
22
22 2
0 0
2
1sw
/ 1sw
lw
,01 lw
4 2
2 1sw 2 lw
0 1lw
,0 ,11
1lw lw
,0 ,21
0lw
(120)
where the two last terms are computed in (E.2) and (E.3).Then, the equation is
Plasma Phys. Control. Fusion 57 (2015) 075006
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14
T
T
T
T
T
T T
T
⎜ ⎟
⎟
⎪
⎪
⎪
⎪
⎪
⎪
⎧⎨⎩
⎛⎝
⎞⎠
⎫⎬⎭
⎧⎨⎩
⎡⎣⎢
⎛⎝⎜
⎞⎠
⎤⎦⎥
⎫⎬⎭
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
∑
∑
∑
∑
∑
∑ ∑
ρ
ρ
ρ
ρ
μ τ
τ ϕ ψ
ψ
τ ϕ
μ ϕ
ϕ
μ φ
ψ μ φ
τ φμ
μ
μ
μ
τ
( ⋅ ∇ − ⋅ ∇ ∂ ) + ∂
− ⋅ ∇ Θ∂⋅ ∇ Θ
[ (∇ ⟨ ⟩ × ) ⋅ ∇ ]
+⋅ ∇ Θ
+ ⋅ ∇ *
− ∂ ⋅ ∇ ⟨ ⟩
+ ∂ ( × ∇ Θ) ⋅ ∇ ⟨ ⟩
+ [ × ( ⋅ ∇ )] ⋅ ∇ ⟨ ⟩
− ( ∂ + ∂ ) *
+ ∂ ⋅ ∇ ( ∂ + ∂ ) *
= − ∂ + ⋅ ∇ ×
− ⋅( ∂ ∇ Θ + ⋅ ∇ ) *
+ ∂ ⋅ ∇ ×
− ⋅( ∂ ∇ Θ + ⋅ ∇ ) *
+ [ * ] + *
+ *
σ σ σ ϵ σ
ψσ σ
σ ϵ σ
σσ σσ
σ σ σ σ
ϵ σ
ϵ σ
ψ σ ψσ
σ σσ
μ ψ σ ψσ
σ σσ
σ
σ σ
σ σσ
σμ
σ σσ
σσ σσ
σσ σσ
σ σ σ σ
( )
Θ
( )
( )
Θ( )
Θ( )
( ) ( )
′′
′′
′′
′
′
′
′
′′
′′
σ
σ
σ
⊥
⊥
⊥
u B G Z F
ZF
Iu
BC
Z F
uBB
u
B
I
BB Z C
BB Z C
Z
u
uB u C
u
B
BB u C
C C
Z S
b b
bB
b
b
b
b
b b b
b b
b b
b b
b b
1
1
1
1
.
u t
u
u
R R
RR
R R
RR
R
R R
R R
R
R
R R
R
R R
2lw 2
0
2
1sw
/ 1sw lw
,01 lw
21
sw1
sw
/ 1sw
/ 1sw
lw
0 ,01 lw
0 ,01 lw
21lw
2,0
1 lw
2,0
1 lw
,11 sw lw
,02 lw
2 2,0
s2
(121)
The term T *σ σσ( )
′C,02 lw is given in Appendix E, whereas the
gyroaverage of T[ * ]σ σσ( )
′C,11 sw lw was computed in [28] and the
result is
T
T T T
T T
T
T T
T
T
⎪
⎪
⎡
⎣⎢
⎧⎨⎩
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥⎫⎬⎭
⎤
⎦⎥
τ ϕ
τ ϕ
ττ
τϕ
[ * ] = −∂
* − * *
+ * *
− *
σ σσ μσ σ
σ
σ σσ σ σσ σ
σ σ σ σ σ σ
σ σ
σ σσ σσ σ σ σ σ
σ σσ σ σ σ
− −
−
−
′
′ ′ ′
′ ′′ ′ ′
′ ′′ ′ ′ ′
CZ
B
C FZ
TF F
Z
ZC F F
Z
TF
˜
˜ ,
,
˜ .
,1sw lw
2
1sw
NP, ,0 1sw
2
,0 1sw
,01
0 ,01
0
NP, ,01
0 ,0 1sw
2
,0 1sw
,01
0
lw
(122)
In equation (121) we have added the source term Tτ *σ σ σ σZ S2 2,0
that was not considered in [28].As explained in detail in section 5.1 of [28], the long-
wavelength Fokker–Planck equation, at any order in ϵσ, has
a non-zero kernel. In order to fix the component along the kernel, one has to impose some conditions. For instance, we can impose, for j = 1, 2,
∫
∫∑
μ θ σ
τμ μ θ
=
( + ) =
σψ
σ σ σσ
ψ
BG u
ZB u G u
d d d 0, for every , and
1/2 B d d d 0.
j
j j j
lw
2 lw
(123)
This is a natural way to fix the ambiguity but there are infi-nitely many different possibilities.
6.2. Long-wavelength quasineutrality equation to first order
To order ϵs0, the quasineutrality equation simply gives a rela-
tion among the densities of Fσ0,
∑ ( ) =σ
σ σZ n tr, 0. (124)
In terms of the function σG 1lw defined in (117), the quasineu-
trality equation to first order gives
⎛⎝⎜
⎞⎠⎟
∫∑τ
μ μ θ
φ
( ) ( )
− ( ) ( ) =
σ σσ
σσ
B G u t u
Z
Tn t t
r r
r r
1, , , d d d
, , 0.
1lw
2
1lw
(125)
It is important to note that φ1lw can be determined from the
above equation only up to an additive function of ψ. This is connected to the ambiguity in the determination of σF 1
lw men-tioned at the end of section 6.1. In other words, only φˆ ⋅ ∇b R 1
lw can be found. Without loss of generality, we take
φ =ψ
0.1lw (126)
Again, a longer discussion on this is given in section 5.1 of [28].
6.3. Transport equations for density and energy
Momentum transport calculations require knowledge of the time evolution of the functions of ψ entering the Maxwellian, i.e. the density of each species and the temperature, which is the same for all of them. We take the relevant transport equa-tions from [28]. The equation for nσ is
T
∫
∫
∑ρ
ψψ
ψ μ θ
ϕ ψ
τψ
μ θ
∂ ( ) =( )
∂ ( )
[ (∇ ⟨ ⟩ × ) ⋅ ∇ ]
+ + ⋅ ∇
+
′′ϵ σ ψ
σ ϵ σ
σ σ σσσ
ψ
σ σψ
( )
′′
σ⊥
⎜ ⎟
⎪
⎪
⎪
⎪
⎧⎨⎩
⎛⎝
⎞⎠
⎫⎬⎭
n tV
V u
F
B
Z
Iu
BC
B S u
b
,1
d d d
ˆ
* d d d .
t
R R
R
1sw
/ 1sw lw
21 lw
,0
s2
(127)
Plasma Phys. Control. Fusion 57 (2015) 075006
I Calvo and F I Parra
15
As we have explained in section 4.3, with these equations and the transport equation for the total energy,
T
T
T
(
∫
∫
∫
∫
∑
∑
∑
∑
∑
∑
ρ
ψ ψ
ψψ μ
ϕ ψ
τψ μ θ
ϕ
μ ϕ
ϕ μ θ
τμ μ θ
μ μ θ
∂ ( ) ( )
=( )
∂ ( ) ( + )
[ (∇ ⟨ ⟩ × ) ⋅ ∇ ]
+ + ⋅ ∇
− ⋅ ∇ ⟨ ⟩
+ ( × ∇ ) ⋅ ∇ ⟨ ⟩
+ [ × ( ⋅ ∇ )] ⋅ ∇ ⟨ ⟩
+ ( + )
+ +
′′
ϵσ
σ
ψσ
σ ϵ σ
σ σ σσ σσ
ψ
σσ σ
ϵ σ
ϵ σψ
σ σ σ σσ σσ
ψ
σσ σ
ψ
( )
( )
′′
′′
σ
σ
σ
⊥
⊥
⊥
⎜ ⎟⎪
⎪
⎪
⎪
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎧⎨⎩
⎛⎝
⎞⎠
⎫⎬⎭
⎡
⎣⎢⎢
⎞⎠⎟
⎤
⎦⎥
⎡⎣ ⎤⎦
⎛⎝⎜
⎞⎠⎟
n t T t
VV u
F
B
Z
Iu
BC u
B F u
BB
u
Bu
ZB u C u
Bu
S u
b
b
b
b b b
3
2, ,
1/2 B
ˆ
* d d d
ˆ
ˆ
ˆ ˆ ˆ d d d
1/2 B * d d d
2B * d d d ,
t
R R
R
R
R R
R R
2
1sw
/ 1sw lw
2 ,01 lw
1sw
1sw
/ 1sw
2
/ 1sw
lw
,2
2,1
1 sw lw
2
,0
s2
(128)
the transport equation for the temperature can be calculated.Note that the calculation of density and energy transport
only requires gyrokinetic equations correct to O(ϵs). In other words, it is not necessary to know the long-wavelength radial electric field to solve the transport equations for nσ and T. Higher order gyrokinetic equations are needed to determine the transport of toroidal angular momentum. This is in con-trast to what happens in the high-flow ordering [46–50], where the plasma velocity is assumed to be O(cs). In that set-ting, density, energy and toroidal angular momentum trans-port can be calculated from the solution of the gyrokinetic equations to O(ϵs).
7. Conclusions
Recently, it has been proven that the computation of radial transport of toroidal angular momentum in a tokamak requires, in the low flow regime, high-order gyrokinetics [1]. This issue has received much attention because the problem is equivalent to determining the tokamak intrinsic rotation profile. In [2, 26, 27], the equations needed to calculate intrinsic rotation have been given under the assumption B/Bp ≫ 1, where Bp is the magnitude of the poloidal magnetic field. However, a set of equations valid for tokamaks with large Bp was missing, and it has been derived in this paper. In this section, we restrict ourselves to point out the equations in the text that have to be solved by a code intended to calculate radial transport of toroidal angular momentum in a tokamak. Denote by ϵs ∼ ρi/L the gyrokinetic expansion parameter, where ρi is the ion Larmor radius and L is the variation length of the magnetic field. One needs
The Fokker–Planck equation at short-wavelengths up to ϵ( )O s
2 , equations (101), (104) and (110); The quasineutrality equation at short-wavelengths up to
ϵ( )O s2 , equations (103) and (116);
The Fokker–Planck equation at long-wavelengths up to ϵ( )O s
2 , equations (119) and (121); The quasineutrality equation at long-wavelengths to
O(ϵs), equation (125); The transport equation for the density of each species σ,
(127), and for the total energy, (128); The formula for the radial flux of toroidal angular
momentum, equations (98)–(100).
Acknowledgments
IC acknowledges the hospitality of Merton College and of the Rudolf Peierls Centre for Theoretical Physics, University of Oxford, where part of this work was completed. This work has been carried out within the framework of the EUROfu-sion Consortium and has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opin-ions expressed herein do not necessarily reflect those of the European Commission. This research was supported in part by grant ENE2012-30832, Ministerio de Economía y Com-petitividad, Spain, by the RCUK Energy Programme (grant number EP/I501045) and by US DoE grant DE-SC008435.
Appendix A. Gyrokinetic equations of motion
The gyrokinetic equations of motion derived in [29] are repro-duced here,
τ ϵ ϵ
ϵ μ τ ϵ ϕ
τ ϵ τ ϵ
τ ϵ τ ϵ ϵ
=
( + ∂ Ψ + ∂ Ψ )**
+*
ˆ × ( ∇ + ∇ ⟨ ⟩
+ ∇ Ψ + ∇ Ψ
+ ∇ Ψ + ∇ Ψ + ∇ Ψ )
σ σ σ ϕ σ σ ϕσ
σ
σσ σ σ σ ϵ σ
σ σ σ ϵ ϕ σ σ σ σ ϵ ϕ σ
σ σ σ ϕ σ σ σ σ ϕ σ σ σ
∥
∥σ
σ σ
⊥
⊥ ⊥
u ZB
BB Z
Z Z
Z Z
R
B
b
˙
1
,
u B u B
B
B B
R R
R R
R R R
2 2,
2,
,
,
2/
4 2 2/ ,
2 2/ ,
4 2 3,
2 3,
3,
(A.1)
μ τ ϵ ϕ
τ ϵ τ ϵ
ϵ
ϵ μ τ ϵ ϕ
τ ϵ τ ϵτ ϵ τ ϵ ϵ
=−
** ⋅ ∇ − ˆ ⋅ ∇ ⟨ ⟩
− ˆ ⋅ ∇ Ψ − ˆ ⋅ ∇ Ψ
− ˆ ⋅ ∇ Ψ −*
[ ˆ × (ˆ ⋅ ∇ ˆ)
− (∇ × ) ] ⋅ ( ∇ ⟨ ⟩
+ ∇ Ψ + ∇ Ψ+ ∇ Ψ + ∇ Ψ + ∇ Ψ )
σσ σ σ σ σ
σ σ σ ϕ σ σ σ σ ϕ
σ σσ
σ σ σ σ ϵ σ
σ σ σ ϵ ϕ σ σ σ σ ϵ ϕ
σ σ σ ϕ σ σ σ σ ϕ σ σ σ
∥
∥
⊥
σ
σ
σ σ σ
⊥
⊥ ⊥
u
BB Z
Z Z
Bu
Z
Z Z
Z Z
B b
b b
b b b b
K
˙
1
,
B
B
B
B B
R R
R R
R R
R R
R R
R R R
,
2
4 2 2,
2 2,
2,
,
2/
4 2 2/ ,
2 2/ ,
4 2 3,
2 3,
3,
(A.2)
μ =˙ 0, (A.3)
θϵ
= − + ( )σ
B O˙ 11 . (A.4)
Plasma Phys. Control. Fusion 57 (2015) 075006
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16
Here,
μ ϵ ϵ μ*( ) ≔ ( ) + ∇ × ˆ( ) − ∇ × ( )σ σ σu uB R B R b R K R, , R R2 (A.5)
and
( ) ≔ ˆ( ) ˆ ( ) ⋅ ∇ × ˆ( ) − ∇ ˆ ( ) ⋅ ˆ ( )K R b R b R b R e R e R1
2.R R 2 1 (A.6)
The last term of (A.6) depends on the definition of the gyr-ophase, i.e. on the choice of ˆ ( )e R1 and ˆ ( )e R2 . It is common to say that this term is ‘gyrogauge dependent’. However, only ∇R × K (R) enters the gyrokinetic equations, which was shown to be gyrogauge independent in [51].
The parallel component of *σB ,
μ μ
ϵ
ϵ μ
≔ ⋅
= + ⋅ ∇ ×
− ⋅ ∇ ×
σ σ
σ
σ
∥B u u
B u
R B R b R
R b R b R
b R K R
* ( , , ) *( , , ) ˆ ( )
( ) ˆ ( ) ˆ ( )
ˆ ( ) ( ),
R
R
,
2
(A.7)
is the Jacobian of the gyrokinetic transformation Tσ to ϵ( )σO 2 . Finally,
ϕ
ϕ
Ψ = ∇ Φ ⋅ × ∇
− ∂ ⟨ ⟩
∼
∼
ϕ σ ϵ σ ϵ σ
μ σ
σ σ⊥ ⊥( )B
B
b1
2ˆ
1
2,
R R, 2 ( / ) ( / )
2
(A.8)
ρ
ρ
ρρ ρ ρ
ρ ρ
ρ ρ ρ ρ
ρ ρ ρ ρ
ϕ
μ ϕ ϕ
ϕ
ϕμ
ϕ
ϕ
μϕ
Ψ = − ∇ × ⋅ ∇ ⋅
− ∇ ⋅ ∇ ⟨ ⟩ − ∇ ⋅ ⟨ ⟩
− ∇ ⋅ − × × ⋅ ∇
− ⋅ ∇ ⋅ ∂ − ⋅ ∇ ⋅ ⟨ ⟩
+ ∇ ∂ × + ×
+ ∇ × + ×
∼
∼
∼
∼ ∼
∼
∼
ϕ σ ϵ σ
ϵ σ σ
ϵ σ
μ σ σ
μ σ
σ
σ
σ
σ
⊥
⊥
⊥
( )u
B
BB
BB
BB
u
B
u
u
u
b b
b b
b b b b
b b b
b b b
ˆ ˆ
2
1
1
4[ ( ˆ )( ˆ )]
ˆ ˆ2 B
ˆ ˆ
4ˆ : [ ( ˆ ) ( ˆ ) ]
4ˆ : [ ( ˆ ) ( ˆ ) ]
B R R
R R R
R R
R R
R
R
, ( / )
2 ( / )
( / )
2 2
(A.9)
and
μ
μ
μ μ
μ μ
μ μ
μ
μ μ
Ψ = − ⋅ ∇ ⋅ ∇
+↔
− ∇ ∇ ⋅
− ∣∇ ∣ + ∇ ∇
+ − ∇ ∇
− + ∇ ⋅
+ − ∣ ⋅ ∇ ∣
+ − ⋅ ∇ ×
σ
⊥
⊥ ⊥
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
u
BB
B
BB
u
B
u
B
u
B
u
B
u
B
u
B
b b
I bb B b
b b
b b
b
b b
b b
3
2ˆ ˆ
4( ˆ ˆ ) : ˆ
3
4 2ˆ : ˆ
8 4ˆ : ( ˆ )
3
8 16( ˆ )
3
2 2ˆ ˆ
8 16(ˆ ˆ ) .
B R R
R R
R R R
R R
R
R
R
,
2
2
2
2
22
2
2 2T
2 22
2 4
2
2
2 22
(A.10)
Here ↔M
T is the transpose of an arbitrary matrix matrix
↔M and
∫μ θ ϕ μ θ θΦ ( ) ≔ ( ) ∼ ′ ′σθ
σt tR R, , , , , , d , (A.11)
where the lower limit of the integral is chosen such that ⟨Φ ⟩ = σ 0.
Appendix B. Lowest-order terms of the perturbative transformation, T σP,
The expressions for the corrections Rσ,2, uσ,1, μσ,1, and θσ,1 found in [29], and valid for arbitrary magnetic geometry, are
ρ ρ
ρρ ρ ρ
ρρ τ
= − ⋅ ∇ ⋅ × − × ∇ ⋅
− − × × ∇
− ⋅ ∇ − × ∇ Φ
σ
σ σϵ σσ⊥
u
B
u
B
BB
Z
B
R bb b b b b
b b b b
b
2 ˆ ˆ ˆ ( ˆ ) ˆ ˆ
1
8ˆ [ ( ˆ )( ˆ )] : ˆ
1
2ˆ ,
R R
R
R R
,2
2
2 ( / )
(B.1)
ρ
ρ ρ ρ ρ
= ⋅ ∇ ⋅
− × + × ∇
σu u
B
b b
b b b
ˆ ˆ
4[ ( ˆ ) ( ˆ ) ] : ˆ ,
R
R
,1
(B.2)
ρ
ρ ρ ρ ρ
μτ ϕ
= − − ˆ ⋅ ∇ ˆ ⋅
+ [ ( × ˆ ) + ( × ˆ) ] : ∇ ˆ
∼σ
σ σ σZ
B
u
Bu
b b
b b b4
,
R
R
,1
2 2
(B.3)
ρ
ρρ ρ ρ
ρ
θ τμ
μ
= ∂ Φ + ˆ ⋅ ∇ ˆ ⋅ ( × ˆ )
+ [ − ( × ˆ)( × ˆ )] : ∇ ˆ
+ ( × ˆ) ⋅ ∇
σσ σ
μ σZ
B
u
u
BB
b b b
b b b
b
2 B
8
1.
R
R
R
,1
2 2
(B.4)
Appendix C. Gyrokinetic transformation, Tσ, to first order
In this appendix we provide explicit expressions for the gyro-kinetic transformation T μ θ( ) = ( )σ u tr v R, , , , , to order ϵσ. Define
≔ ⋅ ˆ( )∥v v b r , (C.1)
μ ≔( − ˆ( ))
( )∥v
B
v b r
r2,0
2
(C.2)
⎛⎝⎜
⎞⎠⎟θ ≔ ⋅ ˆ ( )
⋅ ˆ ( )v e rv e r
arctan .02
1 (C.3)
The result is
Plasma Phys. Control. Fusion 57 (2015) 075006
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17
ρϵ ϵ
ϵ ϵ
μ μ ϵ μ ϵ
θ θ ϵ θ ϵ
= + + ( )
= + ˆ + ( )
= + ˆ + ( )
= + ˆ + ( )
σ σ
σ σ σ
σ σ σ
σ σ σ
∥
O
v u u O
O
O
r R ,
,
,
,
2
,12
0 ,12
0 ,12
(C.4)
where
ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ
ρ
ρ
ρρ ρ ρ
μ
μ μ
μ τ ϕ
θμ
μ
τ
= ⋅ ∇ ⋅ + × + × ∇
− ⋅ ∇ ×
= − ⋅ ∇ − × + × ∇
+ ⋅ ∇ × − ⋅ ∇ ⋅ −
= × ⋅ ∇ + ⋅ ∇ − × ∇ ⋅
− − × × ∇
+ ⋅ ∇ + ∂ Φ
σ
σ
σ σσ
σ
σ σμ σ
⎛⎝⎜
⎞⎠⎟
u uB
BB
u
u
B
u
B
Z
B
Bu
u
u
BB
Z
B
b b b b b
b b
b b b
b b b b
b b b b e e
b b b
b
ˆ ˆ ˆ4
[ ( ˆ ) ( ˆ ) ] : ˆ
ˆ ˆ ,
ˆ4
( ( ˆ ) ( ˆ ) ) : ˆ
ˆ ˆ ˆ ˆ ˜ ,
ˆ ( ˆ ) ln2 B
ˆ ˆ ˆ ˆ ˆ
8( ( ˆ )( ˆ )) : ˆ
2ˆ ˜ .
R R
R
R R
R R
R R R
R
R
,1
,1
2 2
1
,1
2
2 1
2
2
1
(C.5)
Note the difference between (Rσ,2, uσ,1, μσ,1, θσ,1), defined in appendix B, and ρ μ θ( ˆ ˆ ˆ )σ σ σu, , ,,1 ,1 ,1 . The former give the give the lowest order terms of the perturbative transformation defined in (55). The latter are the O(ϵσ) terms of the complete transformation Tσ from gyrokinetic coordinates to euclidean coordinates.
It is useful to have the long-wavelength limit of the pre-vious expressions at hand. Employing (63) and (64), we get
ρ ρ ρ ρ ρ
ρ ρ
ρ
ρρ ρ ρ
ρ
μ μ
μ φ
θμ
μ
μφ
=
= − ⋅ ∇ − × + × ∇
+ ⋅ ∇ × − ⋅ ∇ ⋅ − ⋅ ∇
= × ⋅ ∇ + ⋅ ∇ − × ∇ ⋅
− − × × ∇
+ ⋅ ∇ + × ⋅ ∇
σ σ
σ
σ
σ
σ
⎛⎝⎜
⎞⎠⎟
u u
BB
u
u
B
u
B
Z
B
Bu
u
u
BB
Z
b b b
b b b b
b b b b e e
b b b
b b
ˆ ˆ
ˆ4
( ( ˆ ) ( ˆ ) ) : ˆ
ˆ ˆ ˆ ˆ ,
ˆ ( ˆ ) ln2 B
ˆ ˆ ˆ ˆ ˆ
8( ( ˆ )( ˆ )) : ˆ
2ˆ
2 B( ˆ ) .
R R
R R R
R R R
R
R R
,1lw
,1
,1lw
2
0
,1lw
2
2 1
2 0
(C.6)
To write (74) in section 4.3, we need to calculate the long-wavelength limit of T *σ σ
− F10 to first order in ϵσ. Inverting
(C.4) to first order, and recalling (C.6) and the relations ∂uFσ0 = −(u/T) Fσ0, ∂μFσ0 = −(B/T) Fσ0, one finds (74).
The results of this appendix are also used in equations (80) and (81).
Appendix D. Second-order transformation of the Maxwellian at short-wavelengths
In this section we derive T[ * ]σ σ− F,21
0sw. We proceed in a way
analogous to that employed in Appendix G of [28]. Since Fσ0 is a Maxwellian that depends only on R and u2/2 + μB(R), we deduce that
= × × ∇ ∇ + − ∇ ∇
− ∇ ∇ + ∇ + − ∇ ∇
+ − ∇
+ ⋅ ∇ + − ∇
− × ⋅ ∇ + − ∇
+ −
σ σ
σ
σ
σ
σ
σ
σ σ
σ
σσ σ
σ
σ
σ σ
σ σ σ σ
−
−
−
−
− −
⎡
⎣⎢
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟⎛⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟⎤
⎦⎥
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
T
T
T
T
T T
F
Bn
v
TT
v
TT T
n
n
v
T
T
T
n
n
v
T
T
TF
n
n
v
T
T
TF
BH
n
n
v
T
T
T
F
T
HF
TH
F
T
v b v b
R
v b
*
1
2( ˆ )( ˆ ) : ln
2
3
2ln
2 2
3
2
2
3
2*
2
3
2*
1( ˆ )
2
5
2
*
1
2
* *,
r r r r
r rr r r
r
r r
r r
,21
0
2
2
2
3
2
2
,01
0
02
2
,01
0
01
2,01
0
012 ,0
10
2 02,01
0
(D.1)
where the functions R02, H01 and H02 are given by
ϵ ϵ ϵ= + × ˆ + + ( )σσ σ
BOR r v b R2
023
(D.2)
and
⎛⎝⎜
⎞⎠⎟T μ ϵ ϵ ϵ* + ( ) = + + + ( )σ σ σ σ
− u vH H OR
2B
2.1
2 2
012
023 (D.3)
Unlike in [28], we need the short-wavelength component of (D.1),
T
T
T
T T
[ ]
= ⋅ ∇ + − ∇
− ( × ) ⋅ ∇ + − ∇
+ [ ] −
σ σ
σ
σσ σ
σ
σ
σ σ
σ σ σ σ
−
−
−
− −
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
F
n
n
v
T
T
TF
BH
n
n
v
T
T
T
F
T
HF
TH
F
T
R
v b
*
2
3
2*
1 ˆ2
5
2
*
1
2
* *.
r r
r r
,21
0sw
02sw
2
,01
0
01sw
2,01
0
012 sw ,0
10
2 02sw ,0
10
(D.4)
The coefficients R02sw and H01 are obtained following appendix
G of [28], easily arriving at
Tτ= ˆ × ∇ Φσ σσ ϵ σσ⊥
Z
BR b ,R02
sw2
2 ,0 / 1sw (D.5)
φ= − ( × ˆ) ⋅ ∇σHZ
Bv b ,r01
lw0 (D.6)
Tτ ϕ= ˜σ σ σ σH Z .01sw 2
,0 1sw
(D.7)
Plasma Phys. Control. Fusion 57 (2015) 075006
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18
In order to find H02sw we recall (G.9) in [28],
⎛⎝⎜
⎞⎠⎟T τ ϵ φ
μ τ ϵ ϕ μ
τ ϵ τ ϵ ϵ
τ ϵ ϵ
* + ( )
= + ( ) + ⟨ ⟩( )
+ Ψ + Ψ + Ψ
− ∂ Φ + ( )
σ σ σ σ
σ σ σ σ
σ σ σ ϕ σ σ σ σ ϕ σ σ σ
σ σ σσ σ
vZ t
uZ t
Z Z
Z
BO
r
R R
2,
2B , ,
.
B B
t
22
22
4 2 2,
2 2,
2,
2 23
(D.8)
It is almost immediate to see that
T
T T T
T T T
T
T
τ φ τ ϕ
τ μ ϕ τ μ ϕ
τ ϕ τ τ
τ
= ( ) − [ ⋅ ∇ ⟨ ⟩]
+ [ ˆ ∂ ⟨ ⟩] + *ˆ ∂ ⟨ ⟩
− ⟨ ⟩ − Ψ − Ψ
+ ∂ Φ
σ σ σ σ σ ϵ σ
σ σ σ σ σ μ σ σ σ σ σ σ μ σ
σ σ σ σ σ σ σ ϕ σ σ σ σ ϕ σ
σ σσ σ
−
σ⊥H Z t Z
Z Z
Z Z Z
Z
B
r R,
,
B
t
R02sw 3 2
2sw 2
02 ,0 / 1sw sw
2,0 1
sw,0 1
sw sw 2,01
1lw
,0 1sw
3 2,0 2
sw 4 2,0 ,
2,0 ,
2
,0sw
(D.9)
where T μ*ˆσ σ−
,01
1lw and T μσ σ,0 1
sw can be found in (86) and (88).Equation (D.4), together with (D.5)–(D.7) and (D.9), give
an explicit expression for T[ * ]σ σ− F,21
0sw.
Appendix E. Computation of the gyroaverage of σσ
( )′C 2 lw
In this appendix we calculate the gyroaverage of (93). First, we can write
T T T
T
ττ
ττ
ττ
τ ϕ
τϕ
= +
+ +
+
+
+
+ +
+ −
−
∼
∼
σ σσ
σ σσ σ σ σ σ σ σ
σ σ σ σ σ σσ σ
σ σσ
σσ σ σ
σ σ σ σ σ σ
σ σ σ σ
σ σ
σ σ
σ σσ σ σ σ σ σ σ σ σ
σ σ
σ σσ σσ σ σ
σ σσ σ σ σ σ σ
σ σσ σ σ σ
− − −
− − −
−
− − −
− −
− − − −
−
−
′
′
′ ′′ ′
′
′ ′ ′ ′ ′ ′
′ ′ ′ ′
′ ′
′ ′ ′ ′ ′
′ ′′ ′ ′
′ ′′ ′ ′ ′
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎡⎣⎢
⎤⎦⎥⎤⎦⎥
T
T T T T T
T T T T T
T
T T T T
T T T
T T T T T
T T
T
C
C F F
F FZ
Z
C F
F F
F
Z
Z
C F F F F
Z
ZC F
Z
TF F
Z
TF
*
* [ * * * [ * ]
* * [ * ] , * ] *
[ * ,
* * * [ * ]
* * [ * ] ]
* [ * [ * ] , * [ * ] ]
* * ,
* ,
,0(2)lw
,0 ,01
2lw
,01
,0 ,11
1lw lw
,01
,0 ,21
0lw
,01
0
2
,0
,01
0
,01
2lw
,01
,0 ,11
1lw lw
,01
,0 ,21
0lw
,0 ,01
1lw
,11
0lw
,01
1lw
,11
0lw
,0 ,0 1sw
2
,0 1sw
,01
0 ,0 1sw
2
,0 1sw
,01
0
lw
(E.1)
where we have used that T T* [ * ] =σ σ σ− F 0,0 ,1
11
sw lw . Here,
⎜ ⎟⎛⎝
⎞⎠T T μ* [ * ] = ˆ ⋅ ∇ × ˆ ∂ − ∂σ σ σ μ σ
− Fu
BFb b ,uR,0 ,1
11
lw lw1
lw (E.2)
and
T T
μ
μ
μ φ μ μ
μ φ μ
μ φ μ
τ ϕ φ
τ ϕ μ φ
φ
μ φ
* [ * ]
= (↔ − )
∇ ∇ + + − ∇ ∇
− ∇ ⋅ ∇ − + ∣∇ ∣
+ ∇ +∇
+ + − ∇
− ∇ ⋅ ∇ +∇
+ + − ∇
+ ( ) + − ∣∇ ∣
− ∂ ( ) − ∇ ⋅ ∇
− ⋅ ∇ ⋅ ∇ + Ψ
+ (↔ − ) ∇ ∇
∼
∼
σ σ σ
σ σ
σσ σ
σ
σ
σσ
σ
σ
σσ
σ σσ σ
σ
σ σμ σ
σ
σσ
σσ
−
⊥
⊥
⎡
⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤
⎦⎥
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞
⎠⎟
⎡⎣
⎤⎦
⎡⎣⎢
⎡⎣
⎤⎦
⎤⎦⎥
F
B
nu
TT F
B
Z
TTF
B
u
TT F
B
n
n
Z
T
u
T
T
TF
BB
n
n
Z
T
u
T
T
TF
Z
TF
T
Z
B
Z
B
Z
BB
Z u
B
Z
BF
I bb
b b
I bb
2ˆ ˆ
: ln/2 B 3
2ln
2
/2 B
2
/2 B 3
2
2
/2 B 3
2
2
1
2
2
3
2
ˆ ˆ
ˆ ˆ : .
B
R R R R
R R R
R R R
RR R R
R
R R
R R
R R
,0 ,21
0lw
2
0
2 0 0
2
32
0
02 2
0
20
2
0
4 2
2 1sw 2 lw
0
2
2 02
4 2
1sw 2 lw
2 0
2
2 0 ,
0 0
(E.3)
This last result has been obtained by gyroaveraging (76).
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