PSFC/JA-10-52
Sources of intrinsic rotation in the low flow ordering
Felix I. Parra,* Michael Barnes and Peter J. Catto
*Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Oxford, OX1 3NP, UK
December 2010
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge MA 02139 USA
This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER54109. Reproduction, translation, publication, use and disposal, in whole or in part, by or
for the United States government is permitted.
Submitted to Nuclear Fusion (December 2010)
Sources of intrinsic rotation in the low flow ordering
Felix I Parra and Michael Barnes
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1
3NP, UK
E-mail: f.parradiaz1@physics.ox.ac.uk
Peter J Catto
Plasma Science and Fusion Center, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
Abstract. A low flow, δf gyrokinetic formulation to obtain the intrinsic rotation
profiles is presented. The momentum conservation equation in the low flow ordering
contains new terms, neglected in previous first principles formulations, that may
explain the intrinsic rotation observed in tokamaks in the absence of external sources
of momentum. The intrinsic rotation profile depends on the density and temperature
profiles, the up-down symmetry and the type of heating.
PACS numbers: 52.25.Fi, 52.30.Gz, 52.35.Ra
Submitted to: Nuclear Fusion
Sources of intrinsic rotation in the low flow ordering
2
1. Introduction
Experimental observations have shown that tokamak plasmas rotate spontaneously
without momentum input [1]. This intrinsic rotation has been the object of recent
work [1, 2] because of its relevance for ITER [3], where the projected momentum input
from neutral beams is small, and the rotation is expected to be mostly intrinsic.
The origin of the intrinsic rotation is still unclear. There has been some
theoretical work in turbulent transport of momentum using gyrokinetic simulations
[4, 5, 6, 7, 8, 9, 10, 11, 12], and two main mechanisms have been proposed as candidates
to explain intrinsic rotation. On the one hand, the momentum pinch due to the Coriolis
drift [4] has been argued to transport momentum generated in the edge. On the other
hand, it has also been argued that up-down asymmetry generates intrinsic rotation
[7, 8]. However, neither of these explanations are able to account for all experimental
observations. The up-down asymmetry is only large in the edge, generating rotation in
that region that then needs to be transported inwards by the Coriolis pinch. Thus,
intrinsic rotation in the core could only be explained by the pinch. The pinch of
momentum is not sufficient because it does not allow the toroidal rotation to change
sign in the core as is observed experimentally [13].
In this article we present a new model implementable in δf flux tube simulations
[14, 15, 16, 17]. This model is based on the low flow ordering of [18], and self-consistently
includes higher order contributions. As a result, new drive terms for the intrinsic
rotation appear that depend on the gradients of the background profiles of density
and temperature and on the heating mechanisms.
We present two new effects, related to the ion-electron collisions and the heating,
that were not treated in the original work [18]. In addition, we recast the results from
[18] in a form similar to the equations in the high flow ordering [19, 20]. These are the
equations that have been implemented in most gyrokinetic codes that are employed
to study momentum transport. For this reason, the new form of the equations is
useful to identify the differences with previous models. Finally, we discuss how the new
contributions drive intrinsic rotation and we show that the intrinsic rotation resulting
from these new processes depends on density and temperature gradients and on the
heating mechanisms.
In the remainder of this article we present the model, developed originally in [18],
in a form more suitable for δf flux tube simulation. In Section 2 we give the complete
model, and in Section 3 we discuss its implications for intrinsic rotation. Appendix A
contains the details of the transformation from the equations in [18] to the formulation
in this article. In Appendix B we discuss the treatment of the ion-electron collision
operator.
Sources of intrinsic rotation in the low flow ordering
3
2. Transport of toroidal angular momentum
The derivation of the transport of toroidal angular momentum in the low flow regime,
including both turbulence and neoclassical effects, is described in detail in [18]. To
simplify the derivation, the extra expansion parameter Bp /B 1 was employed, with
B the total magnetic field and Bp its poloidal component. In this section, we review the
results of reference [18], recast them in a more convenient form and add a collisional term
and a term that depends on the heating mechanisms that were not treated previously.
We assume that the turbulence is electrostatic and that the magnetic field is
axisymmetric, i.e., B = I∇ζ + ∇ζ × ∇ψ, where ψ is the poloidal magnetic flux, ζ
is the toroidal angle, and we use a poloidal angle θ as our third spatial coordinate.
With an axisymmetric magnetic field, in steady state and in the absence of momentum
↔
input, the equation that determines the rotation profile is hhRζ̂· Pi ·∇ψiψ iT = 0, where
↔
R
Pi = d3 v 0 fi Mv0 v0 is the ion stress tensor, M is the ion mass, R is the major radius, ζ̂
R
is the unit vector in the toroidal direction, h. . .iψ = (V 0 )−1 dθ dζ (...)/(B · ∇θ) is the
R
flux surface average, V 0 ≡ dV /dψ = dθ dζ (B · ∇θ)−1 is the derivative of the volume
with respect to ψ, and h. . .iT is the coarse grain or “transport” average over the time
and length scales of the turbulence, much shorter than the transport time scale δi−2 a/vti
and the minor radius a. Here δi = ρi /a 1 is the ion gyroradius ρi over the minor
radius a, and vti is the ion thermal speed. Note that we use the prime in v0 to indicate
that the velocity is measured in the laboratory frame. Later we will find the equations
in a convenient rotating frame where the velocity is v = v0 − RΩζ ζ̂.
↔
In reference [18] we derived a method to calculate hhRζ̂· Pi ·∇ψiψ iT to order
(B/Bp )δi3 pi R|∇ψ|, with pi the ion pressure. We present the method again in different
form to make it easier to compare with previous work in the high flow regime [19, 20].
In addition, instead of using the simplified ion Fokker-Planck equation of reference [18],
1 0
Ze
∂fi
0
+ v · ∇fi +
−∇φ + v × B · ∇v0 fi = Cii {fi },
(1)
∂t
M
c
where Cii is the ion-ion collision operator, φ is the electrostatic potential, Ze is the ion
charge, and e and c are the electron charge magnitude and the speed of light, in this
article we use the more complete equation
1 0
Ze
∂fi
0
+ v · ∇fi +
−∇φ + v × B · ∇v0 fi = Cii {fi } + Cie {fi , fe } + S ht ,
(2)
∂t
M
c
where Cie {fi , fe } is the ion-electron collision operator and S ht ∼ δi2 fi vti /a is a source
that models the different heating mechanisms. Applying the procedure in reference [18]
↔
to equation (2) we find two additional terms in the expression for hhRζ̂· Pi ·∇ψiψ iT
that were not considered in [18].
In subsection 2.1 we explain how we split the distribution function and the
electrostatic potential into different pieces, and we present the equations to self↔
consistently obtain them. In subsection 2.2 we evaluate hhRζ̂· Pi ·∇ψiψ iT employing
the pieces of the distribution function and the potential obtained in subsection 2.1.
Sources of intrinsic rotation in the low flow ordering
4
Table 1. Pieces of the potential.
Potential
Size
Length scales
Time scales
φ0 (ψ, t)
φnc
1 (ψ, θ, t)
φnc
2 (ψ, θ, t)
φtb (r, t)
Te /e
(B/Bp )δi Te /e
(B/Bp )2 δi2 Te /e
φtb
1 ∼ δi Te /e
2
φtb
2 ∼ (B/Bp )δi Te /e
ka ∼ 1
ka ∼ 1
ka ∼ 1
k⊥ ρi ∼ 1
k|| a ∼ 1
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ vti /a
Before presenting all the results, we emphasize that our results and order of magnitude
estimates are valid for Bp /B 1 and for collisionality in the range δi2 qRνii /vti <
∼1
[18], where νii is the ion-ion collision frequency and q is the safety factor.
2.1. Distribution function and electrostatic potential
The electrostatic potential is composed to the order of interest by the pieces in Table 1
nc
[18]. The axisymmetric long wavelength pieces φ0 (ψ, t), φnc
1 (ψ, θ, t) and φ2 (ψ, θ, t) are
the zeroth, first and second order equilibrium pieces of the potential. The lowest order
nc
component φ0 is a flux surface function. The corrections φnc
1 and φ2 give the electric
field parallel to the flux surface, established to force quasineutrality at long wavelengths
(the superscript nc refers to neoclassical because these are long wavelength contributions;
nc
however, we will show that turbulence can affect the final value of φnc
1 and φ2 ). We need
↔
not calculate φnc
2 because it will not appear in the final expression for hhRζ̂· Pi ·∇ψiψ iT .
tb
The piece φ (r, t) is turbulent and includes both axisymmetric components (zonal flow)
and non-axisymmetric fluctuations. It is small in δi but it has strong perpendicular
gradients, i.e., k⊥ ρi ∼ 1. Its parallel gradient is small, i.e., k|| a ∼ 1. The function
tb
tb
φtb is calculated to order (B/Bp )δi2 Te /e, i.e., φtb = φtb
1 + φ2 with φ1 ∼ δi Te /e and
2
tb
as we do
φtb
2 ∼ (B/Bp )δi Te /e. It is convenient to keep both pieces together as φ
hereafter.
To write the distribution function it will be useful to consider the reference frame
that rotates with toroidal angular velocity Ωζ = −c ∂ψ φ0 − (c/Zeni )∂ψ pi , where ni (ψ, t)
and pi (ψ, t) are the lowest order ion density and pressure. In this new reference frame it
is easier to compare with previous formulations [19, 20]. To shorten the presentation, we
perform the change of reference frame directly in the gyrokinetic variables. It is possible
to do so easily because we are expanding in the parameter B/Bp 1. We first present
the gyrokinetic variables that we obtained for the laboratory frame and we argue later
how they must be modified to give the gyrokinetic variables in the rotating frame. In
[18] we used as gyrokinetic variables the gyrocenter position R = r + R1 + R2 + . . ., the
gyrokinetic kinetic energy E = E0 +E1 +E2 +. . ., the magnetic moment µ = µ0 +µ1 +. . .
and the gyrokinetic gyrophase ϕ = ϕ0 +ϕ1 +. . ., where E0 = (v 0 )2 /2 is the particle kinetic
0 2
) /2B is the lowest order magnetic moment,
energy in the laboratory frame, µ0 = (v⊥
0
0
0
ϕ0 = arctan(v · ê2 /v · ê1 ) is the lowest order gyrophase, R1 = Ω−1
i v × b̂ ∼ δi L is
Sources of intrinsic rotation in the low flow ordering
5
the first order correction to the gyrocenter position, E1 = Ze(φ − hφi)/M ∼ δi vti2 is the
first order correction to the gyrokinetic kinetic energy, and the corrections R2 ∼ δi2 L,
E2 ∼ δi2 vti2 , µ1 ∼ δi vti2 /B and ϕ1 ∼ δi are defined in [21]. Here Ωi = ZeB/Mc is the ion
gyrofrequency, ê1 (r) and ê2 (r) are two orthonormal vectors such that ê1 × ê2 = b̂, and
H
h. . .i = (2π)−1 dϕ (. . .)|R,E,µ,t is the gyroaverage holding R, E, µ and t fixed. When
the ion distribution function is written as a function of these gyrokinetic variables, it
does not depend on the gyrophase ϕ up to order (Bp /B)δi2 (qRνii /vti )fM i [18, 21], where
fM i is the lowest order distribution function that is a Maxwellian. For the magnetic
moment and the gyrophase, only the first order corrections µ1 and ϕ1 are needed because
the lowest order distribution function fM i does not depend on µ or ϕ. Moreover, since
in [18] we expand on B/Bp 1, the distribution function need only be known to
order (B/Bp )δi2 fM i . Consequently, the piece of the distribution function that depends
on the gyrophase, of order (Bp /B)δi2 (qRνii /vti )fM i , is negligible, and the gyrokinetic
variables R, E, µ and ϕ only need to be obtained to order (B/Bp )δi2 L, (B/Bp )δi2 vti2 ,
(B/Bp )δi vti2 /B and (B/Bp )δi , respectively, implying that the corrections R2 , E2 , µ1 and
ϕ1 are not needed for the final result. To change to the new reference frame, where
the velocity is v = v0 − RΩζ ζ̂, the distribution function that is independent of ϕ has
to be written as a function of the new gyrokinetic variables R, ε and µ. Note that the
gyrocenter position and the magnetic moment are the same in both reference frames
to the order of interest. The reason is that the toroidal rotation has two components,
one parallel to the magnetic field, RΩζ ζ̂ · b̂ = IΩζ /B ∼ (B/Bp )δi vti , and the other
perpendicular, RΩζ |ζ̂ − b̂b̂ · ζ̂| = |∇ψ|Ωζ /B ∼ δi vti , and the parallel velocity is larger
by B/Bp 1. Since in [18] the gyrokinetic variables R and µ are to be obtained to
order (B/Bp )δi2 L and (B/Bp )δi vti2 /B, and in R and µ only the perpendicular velocity
0
= v⊥ + RΩζ (ζ̂ − b̂b̂ · ζ̂) enters, we can safely neglect the corrections due to the
v⊥
change of reference frame because they are of order δi2 L and δi vti2 /B, respectively. In
contrast, the kinetic energy E as defined in [18] cannot be used in the rotating frame
because it includes the parallel velocity v||0 = v|| + IΩζ /B. We use a new kinetic energy
variablepε that is related to the old kinetic energy variable by ε = E − IΩζ u0 /B, where
u0 = ± 2(E − µB) is the
q gyrokinetic parallel velocity in the laboratory frame. It is
easy to check that u = ± 2[ε − µB + (I/B)2 Ω2ζ /2] is equal to u = u0 − IΩζ /B and it is
the gyrokinetic parallel velocity in the rotating frame. With this relation, we find that
another way to interpret the new energy variable
R2 Ω2ζ
u2
ε=
+ µB −
(3)
2
2
is realizing that it is the kinetic energy in the rotating frame plus the potential due to
the centrifugal force. To write expression (3) we have used that I/B ' R for Bp /B 1.
In Appendix A we rewrite the results in [18] using the new gyrokinetic kinetic energy ε.
The different pieces of the ion distribution function are given in Table 2 [18]. In
nc
nc
tb
ie
ht
, Hi2
, Hi2
, Hi2
and Hi2
are
this table, m is the electron mass. The functions fM i , Hi1
axisymmetric long wavelength contributions. The Maxwellian fM i (ψ(R), ε) is uniform
nc
nc
and Hi2
are neoclassical
in a flux surface. The first and second order corrections Hi1
Sources of intrinsic rotation in the low flow ordering
6
Table 2. Pieces of the ion distribution function.
Distribution function
Size
Length scales
Time scales
fM i (ψ(R), ε, t)
nc
(ψ(R), θ(R), ε, µ, t)
Hi1
nc
Hi2 (ψ(R), θ(R), ε, µ, t)
tb
(ψ(R), θ(R), ε, µ, t)
Hi2
ie
Hi2 (ψ(R), θ(R), ε, µ, t)
ht
(ψ(R), θ(R), ε, µ, t)
Hi2
tb
fi (R, ε, µ, t)
fM i
(B/Bp )δi fM i
(B/Bp )2 δi2 fM i
2
(B/Bp )(vp
ti /qRνii )δi fM i
(B/Bp )δi m/MfM i
(B/Bp )(vti /qRνii )S ht a/vti
fi1tb ∼ δi fM i
fi2tb ∼ (B/Bp )δi2 fM i
ka ∼ 1
ka ∼ 1
ka ∼ 1
ka ∼ 1
ka ∼ 1
ka ∼ 1
k⊥ ρi ∼ 1
k|| a ∼ 1
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ vti /a
Table 3. Pieces of the electron distribution function.
Distribution function
Size
Length scales
Time scales
fM e (ψ(R), ε, t)
nc
(ψ(R), θ(R), ε, µ, t)
He1
tb
fe (R, ε, µ, t)
fM e
(B/Bp )δi fM e
tb
fe1
∼ δi fM e
tb
∼ (B/Bp )δi2 fM e
fe2
ka ∼ 1
ka ∼ 1
k⊥ ρi ∼ 1
k|| a ∼ 1
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ δi2 vti /a
∂/∂t ∼ vti /a
corrections, and they are not the functions Fi1nc and Fi2nc in [18] because we are now
tb
is an axisymmetric piece of the
working in the rotating frame. The function Hi2
distribution function that originates from collisions acting on the ions transported by
ie
ht
and Hi2
that were
turbulent fluctuations into a given flux surface [18]. The functions Hi2
not included in [18] have their origin in the ion-electron collisions and in the heating
mechanism. The function fitb is the turbulent contribution. It will be determined
self-consistently up to order (B/Bp )δi2 fM i , i.e., fitb = fi1tb + fi2tb with fi1tb ∼ δi fM i and
fi2tb ∼ (B/Bp )δi2 fM i . It is convenient to combine both pieces of the turbulent distribution
function into one function fitb .
The electron distribution function is very similar to the ion distribution function.
It will have its own gyrokinetic variables that can be easily deduced from the ion
counterparts. To the order of interest in this calculation, the electron distribution
function is determined by the pieces in Table 3. The long wavelength, axisymmetric
nc
are the lowest order Maxwellian and the first order neoclassical
pieces fM e and He1
correction. The second order long wavelength neoclassical correction is not needed
for transport of momentum because of the small electron mass. The piece fetb is the
short wavelength, turbulent component that will be self-consistently calculated to order
(B/Bp )δi2 fM e .
We now proceed to describe how to find the different pieces of the distribution
function and the potential. We use the equations in [18] but we change to the new
gyrokinetic kinetic energy ε. The details of this transformation are contained in
Appendix A.
Sources of intrinsic rotation in the low flow ordering
7
2.1.1. First order neoclassical distribution function and potential. The equation for
nc
Hi1
is
Zeφnc
Mε 5 IufM i ∂Ti
(`)
1
nc
nc
− Cii {Hi1
ub̂ · ∇R Hi1 +
fM i +
−
} = 0,
(4)
Ti
Ti
2 Ωi Ti ∂ψ
q
p
where u = ± 2(ε − µB + R2 Ω2ζ /2) ' ± 2(ε − µB) is the gyrokinetic parallel velocity
(`)
nc
gives the parallel
and Cii is the linearized ion-ion collision operator. The correction Hi1
R
nc
3
nc
component of the velocity [22, 23] Wi = b̂ d v Hi1 v|| = (kcIB/ZehB 2 iψ )∂ψ Ti ,
where k is a constant that depends on the collisionality and the magnetic geometry.
nc
is small for qRνii /vti 1, i.e.,
Interestingly, the density perturbation due to Hi1
R 3
nc
d v Hi1 ∼ (B/Bp )(qRνii /vti )δi ni (B/Bp )δi ni [18]. This will be important when
determining φnc
1 below.
nc
is similar to (4) and it is given by [22, 23]
The equation for He1
1 ∂pe
eφnc
Mε 5 1 ∂Te IufM e
1 ∂pi
1
nc
+
+
ub̂ · ∇R He1 −
fM e −
−
Te
Zni Te ∂ψ
pe ∂ψ
Te
2 Te ∂ψ
Ωe
efM e
(`)
(`)
nc
nc
−Cee
{He1
} − Cei {He1
}=−
ub̂ · EA ,
(5)
Te
where Ωe = eB/mc is the electron gyrofrequency, EA is the electric field driven by
(`)
(`)
the transformer, Cee is the linearized electron-electron collision operator and Cei is
nc
is the
the linearized electron-ion collision operator. The lowest order solution for He1
nc
Maxwell-Boltzmann response (eφ1 /Te )fM e ∼ (B/Bp )δi fp
M e . The rest of the terms are
small because they are of order (B/Bp )δe fM e ∼ (B/Bp ) m/Mδi fM i (B/Bp )δi fM e ,
where δe = ρe /a is the ratio between the electron gyroradius ρe and the minor radius a.
Finally the poloidal variation of the potential is determined by quasineutrality,
Z
Z
eφnc
3
nc
3
tb
(6)
Z d v Hi1 + Z d v Hi2 = 1 ne ,
Te
R 3
tb
giving eφnc
We have included the density dp
v Hi2
∼
1 /Te ∼ (B/Bp )(qRνii /vti )δi .
2
tb
(B/Bp )(vti /qRνii )δi ni because it becomes important for qRνii /vti <
∼ (fi /fM i ) a/ρi tb
1 with fi /fM i ∼ ρi /a [18].
2.1.2. Turbulent distribution function and potential. The turbulent piece of the ion
distribution function is obtained using the gyrokinetic equation
D
E D (`) tb tb E
Dfitb
(`) tb
+ ub̂ + vM + vC + vE
· ∇R fitb − Cii htb
− Cie hi , he
i
Dt
Mε 3 1 ∂Ti MIu ∂Ωζ
1 ∂ni
tb
tb
nc
+
+
fM i − vE
= −vE · ∇R ψ
−
· ∇R Hi1
ni ∂ψ
Ti
2 Ti ∂ψ
BTi ∂ψ
nc Ze ∂Hi1
ZefM i ub̂ + vM · ∇R hφtb i,
ub̂ + vM + vC · ∇R hφtb i +
−
(7)
Ti
M ∂ε
where
q D/Dt = ∂t + RΩζ ζ̂ ·p∇R is the time derivative in the rotating frame, u =
± 2[ε − µB + R2 Ω2ζ /2] ' ± 2(ε − µB) is the parallel velocity in the rotating frame,
vM = (µ/Ωi )b̂ × ∇R B + (u2 /Ωi )b̂ × (b̂ · ∇R b̂) are the ∇B and curvature drifts,
Sources of intrinsic rotation in the low flow ordering
8
tb
vC = (2uΩζ /Ωi )b̂ × [(∇R × ζ̂) × b̂] is the Coriolis drift, vE
= −(c/B)∇R hφtb i × b̂ is the
(`)
(`)
tb
Cie {htb
turbulent E×B drift, Cii {htb
i } is the linearized ion-ion collision operator,
i , he }
H
is the linearized ion-electron collision operator, and h. . .i = (2π)−1 dϕ (. . .)|R,E,µ,t is
the gyroaverage holding R, E, µ and t fixed. The ion-electron collision operator can be
approximated by
tb
ne mνei (Te − Ti ) M v 2
eφ
(`)
tb
tb
Cie {hi , he } =
−3
fM i
pi M
Ti
Te
1
ne mνei
Te
tb
tb
∇v ·
∇v hi + vhi − (Ftb
+
− ne mνei Witb ) · vfM i ,
(8)
ni M
M
pi ei
√
3/2
where νei = (4 2π/3)Z 2 e4 ni ln Λ/m1/2 Te is the electron-ion collision frequency, ln Λ
R
is Coulomb’s logarithm, ni Witb = d3 v htb
i v is the turbulent ion flow, and
Z
2πZ 2 e4 ni ln Λ
tb
tb
d3 ve ∇ve ∇ve ve · ∇v htb
(9)
Fei = ne mνei Wi −
e1
m
is the friction force on the electrons due to collisions with ions. The functions that enter
in the collision operators are
nc
nc MfM i,0 ∂Hi1,0
Ze(φtb − hφtb i)
1 ∂Hi1,0
tb
tb
−
(10)
hi = fig +
+
+
M
Ti
∂ε0
B ∂µ0
and
1
mcfM e,0
tb
tb
htb
(v × b̂) · ∇fe0
+
(v × b̂) · ∇φtb .
(11)
e = fe0 −
Ωe
BTe
2
Here the subscript g in figtb = fitb (Rg , v 2 /2, v⊥
/2B, t) indicates that we have replaced
2
2
the variables R, ε and µ by Rg = r + Ω−1
i v × b̂, v /2 and v⊥ /2B; similarly,
nc
=
the subscript 0 in fM i,0 = fM i (ψ(r), v 2 /2, t), fM e,0 = fM e (ψ(r), v 2 /2, t), Hi1,0
nc
2
2
nc
nc
2
2
tb
Hi1 (ψ(r), θ(r), v /2, v⊥ /2B, t), He1,0 = He1 (ψ(r), θ(r), v /2, v⊥ /2B, t) and fe0 =
2
/2B, t) indicates that we have replaced the variables R, ε and µ by r,
fetb (r, v 2 /2, v⊥
2
2
v /2 and v⊥ /2B.
The equation for electrons is equivalent to the one for the ions, giving
(`) tb D (`) tb E
Dfetb 1 ∂ne
tb
tb
tb
+ ub̂ + vM + vE · ∇R fe − Cee he
− Cei he
= −vE · ∇R ψ
Dt
ne ∂ψ
efM e
Mε 3 1 ∂Te
fM e +
ub̂ + vM · ∇R hφtb i,
−
(12)
+
Te
2 Te ∂ψ
Te
(`)
(`)
where Cee is the linearized electron-electron collision operator and Cei is the linearized
electron-ion collision operator. If we were to neglect the effect of the trapped
electrons, the solution to this equation would simply be the adiabatic response fetb '
(ehφtb i/Te )fM e .
Finally, the electrostatic potential φtb is obtained from the quasineutrality equation
Z
nc
nc tb
tb
∂Hi1,0
MfM i,0
1 ∂Hi1,0
3 Ze(φ − hφ i)
−
Z dv
+
+
M
Ti
∂ε0
B ∂µ0
Z
Z
tb
+Z d3 v figtb = d3 v feg
.
(13)
Sources of intrinsic rotation in the low flow ordering
9
2.1.3. Second order, long wavelength distribution function. The long wavelength pieces
nc
tb
ie
ht
Hi2
, Hi2
, Hi2
and Hi2
are given by
Z
(`)
α
α
α
d3 v S α
ub̂ · ∇R Hi2 − Cii {Hi2 } = S −
Z
fM i
Mε 3
2Mε
3
α
+
−1
d vS
−
,
(14)
3Ti
Ti
2
ψ ni
where α = nc, tb, ie, ht, and
MIufM i ∂Ωζ
c
Mε 5 fM i ∂Ti
nc
vM · ∇R ψ − vC − ∇R φ1 × b̂ · ∇R ψ
S = −
−
BTi ∂ψ
B
Ti
2 Ti ∂ψ
I ∂pi
ZefM i
nc
nc
nc
b̂ · ∇R Hi1
− vM · ∇R Hi1
ub̂ · ∇R φnc
+
−
+
v
·
∇
φ
M
R 1
2
ni MΩi ∂ψ
T
i nc D
E
Ze
1 ∂pi
∂Hi1
(n`)
nc
nc
ub̂ · ∇R φnc
v
+
C
+
−
·
∇
ψ
{H
,
H
}
;
(15)
M
R
1
i1
i1
ii
M
Zeni ∂ψ
∂ε
D E Ze |u| ∂
|u|
B
B tb tb tb
tb
tb
f v
f
+
; (16)
ub̂ + vM · ∇R hφ i
S = − ∇R ·
B
|u| i E T
M B ∂ε |u| i
T
u
ne mνei
Te
ie
nc
nc
∇v ·
∇v Hi1 + vHi1
S =
− (Fnc
− ne mνei Winc ) · b̂fM i ,
(17)
ni M
M
pi ei
R
nc
where ni Winc = b̂ d3 v Hi1
v|| is the axisymmetric long wavelength ion flow and
Z
2πZ 2 e4 ni ln Λ
nc
nc
nc
d3 ve ∇ve ∇ve ve · ∇v He1
(18)
Fei = ne mνei Wi −
m
is the axisymmetric long wavelength friction force on the electrons due to collisions with
ions; and S ht is the source in the kinetic equation that mimics the heating mechanism.
For radiofrequency heating, S ht can be obtained from the quasilinear models that are
widely used. It is also possible to use model sources. For example, in [24, 25] simplified
sources were employed to study for the first time the effect of radiofrequency heating
on transport of momentum.
nc
2.2. Calculation of the momentum transport
↔
We obtain an equation for hhRζ̂· Pi ·∇ψiψ iT similar to equation (39) of [18] by
employing the same procedure that was used in that reference, but starting from the
more complete Fokker-Planck equation (2). The final result is as in equation (39) of
[18] plus the new terms
Z
Z
M 2c
3 0
2
0
2
3 0 ht
0
2
0
2
d v Cie {fi }R (v · ζ̂) + d v S (r, v )R (v · ζ̂)
.
(19)
−
2Ze
ψ T
Adding these new terms to expression (39) from [18] and using that for B/Bp 1,
Rv · ζ̂ ' Iv|| /B, we find
↔
tb
nc
nc
ie
ie
ht
hhRζ̂· Pi ·∇ψiψ iT = Πtb
−1 + Π0 + Π−1 + Π0 + Π−1 + Π0 + Π +
MchR2 iψ ∂pi
,
2Ze
∂t
(20)
Sources of intrinsic rotation in the low flow ordering
10
Table 4. Contributions to transport of momentum.
Π
Size [(B/Bp )δi3 pi R|∇ψ|]
Dependences
Πtb
−1
(Bp /B)∆ud δi−1 for ∆ud >
∼ (B/Bp )δi
<
1 for ∆ud ∼ (B/Bp )δi
1
∆ud (qRνii /vti )δi−1 for ∆ud >
∼ (B/Bp )δi
(B/Bp )(qRνii /vti ) for ∆ud <
∼ (B/Bp )δi
(B/Bp )(qRνii /vti ) p
−2
(Bp /B)(qRνii /v
m/M
ti )δi
p
−1
(qRνii /vti )δi
m/M
−2
ht
δi (S a/vti fM i )
∂ψ Ωζ , Ωζ , ∆ud , ∂ψ Ti , ∂ψ ne , ∂ψ Te , ∂ψ2 Ti
Πtb
0
Πnc
−1
Πnc
0
Πie
−1
Πie
0
Πht
with
Πtb
−1 = −
*
c
(∇φtb × b̂) · ∇ψ
B
Z
d3 v figtb
∂ψ Ti , ∂ψ ne , ∂ψ Te , ∂ψ2 Ti , ∂ψ2 ne , ∂ψ2 Te
∂ψ Ωζ , ∆ud , ∂ψ Ti , ∂ψ ne , ∂ψ2 Ti
∂ψ Ti , ∂ψ ne , ∂ψ2 Ti
Ti − Te
∂ψ Ti , ∂ψ ne , ∂ψ Te , EA
Heating
IMv||
+ MRΩζ
B
+
ψ
,
(21)
T
**
+ +
Z
2 2
2
I
v
M
c
∂
1
c
||
V0
(∇φtb × b̂) · ∇ψ d3 v figtb 2
Πtb
0 = −
2Ze V 0 ∂ψ
B
B
T ψ
*Z
+
*
+
Z
2 2
2
v
I
IM
v
M
c
cI
||
||
(`)
tb
b̂ · ∇φtb d3 v figtb
d3 v Cii {Hi2,0
+
−
} 2
, (22)
B
B
2Ze
B
ψ
T
Πnc
−1
Πnc
0
M 2c
=−
2Ze
M 2c
=−
2Ze
*Z
*Z
3
dv
(`)
nc
Cii {Hi1,0
+
nc
Hi2,0
}
(n`)
nc
nc
d3 v Cii {Hi1,0
, Hi1,0
}
M 3 c2 1 ∂ 0
V
− 2 2 0
6Z e V ∂ψ
*Z
ψ
+
2
I 2 v||
B2
I 2 v||2
B2
,
ψ
+
ψ
(`)
nc
Cii {Hi1,0
}
d3 v
(23)
I 3 v||3
B3
+
,
(24)
ψ
eφnc
ne mcνei
1
ie
2
R 1+
(Ti − Te ),
Π−1 =
Ze
Te
ψ
*Z
+
*
Z
2 2
2
v
I
M
p
c
mcν
I2
||
e
ei
(`)
ie
3
ie
nc
d v Cii {Hi2,0 } 2
d3 v Hi1
+
Π0 = −
2Ze
B
Zeni
B2
(25)
Mv||2
Te
ψ
and
M 2c
Πht = −
2Ze
*Z
d3 v
(`)
ht
Cii {Hi2,0
}
I 2 v||2
B2
+
ψ
M 2c
−
2Ze
*Z
d3 v S
I 2 v||2
ht
B2
+
.
ψ
−1
!+
(26)
ψ
(27)
Sources of intrinsic rotation in the low flow ordering
11
Recall that the subscript g indicates that R, ε and µ have been replaced by Rg , v 2 /2
2
2
/2B, and the subscript 0 that they have been replaced by r, v 2 /2 and v⊥
/2B. In
and v⊥
Table 4 we summarize the size of all these contributions compared to the reference size
(B/Bp )δi3 pi R|∇ψ|, and we write what they depend on. To obtain these dependences,
we use equations (4), (5), (6), (7), (12), (13) and (14). Most of the size estimates are
ie
ht
that are trivially found from the results
taken from [18], except for Πie
−1 , Π0 and Π
here. We use ∆ud to denote a measure of the flux surface up-down asymmetry. It ranges
from zero for perfect up-down symmetry to one for extreme asymmetry. Notice that
nc
ie
for extreme up-down asymmetry, Πtb
−1 and Π−1 clearly dominate. The contribution Π−1
is formally very large for qRνii /vti ∼ 1, but sincep
the ion energy conservation equation
2
requires that (Ti − Te )/Ti ∼ (B/Bp )(vti /qRνii )δi M/m, it will always be comparable
to (B/Bp )δi3 pi R|∇ψ|.
3. Discussion
We finish by showing how this new formalism gives a plausible model for intrinsic
tb
rotation. Until now, models have only considered the contribution Πtb
−1 , with fi and
nc
.
φtb obtained by employing equations (7) and (13) without the terms that contain Hi1
This is acceptable for RΩζ ∼ vti or high up-down asymmetry ∆ud ∼ 1. In this limit,
tb
tb
tb
Πtb
−1 (∂ψ Ωζ , Ωζ ) ' −ν ∂ψ Ωζ − Γ Ωζ + Πud . To obtain this last expression we have
linearized around ∂ψ Ωζ = 0 and Ωζ = 0 for RΩζ /vti 1. Here ν tb is the turbulent
2
diffusivity, Γtb is the turbulent pinch of momentum and Πtb
ud ∼ ∆ud δi pi R|∇ψ| is the
value of Πtb
−1 at Ωζ = 0 and ∂ψ Ωζ = 0, and is zero for perfect up-down asymmetry when
nc
[26]. Notice then
equations (7) and (13) are solved without the terms that contain Hi1
↔
that imposing hhRζ̂· Pi ·∇ψiψ iT ' Πtb = −ν tb ∂ψ Ωζ − Γtb Ωζ + Πtb
ud = 0 gives intrinsic
rotation only for up-down asymmetry or if momentum is pinched into the core from the
edge.
The complete model described in this article includes contributions that have
not been considered before. On the one hand, the gyrokinetic equations (7) and
nc
tb
tb
tb
tb
, giving Πtb
(13) have new terms with Hi1
−1 ' −ν ∂ψ Ωζ − Γ Ωζ + Πud + Π−1,0 ,
3
where Πtb
−1,0 ∼ (B/Bp )δi pi R|∇ψ| is a new contribution due to the new terms in the
nc
ie
ie
gyrokinetic equation. On the other hand, there are the new terms Πnc
−1 , Π0 , Π−1 , Π0
nc
and Πht . As we did for Πtb
−1 , we can linearize Π−1 (∂ψ Ωζ ) around ∂ψ Ωζ = 0 to find
nc
nc
nc
nc
2
Πnc
−1 ' −ν ∂ψ Ωζ + Πud + Π−1,0 , where Πud ∼ ∆ud (B/Bp )(qRνii /vti )δi pi R|∇ψ| and
2
3
Πnc
−1,0 ∼ (B/Bp ) (qRνii /vti )δi pi R|∇ψ|. Combining all these results and imposing that
↔
hhRζ̂· Pi ·∇ψiψ iT = 0, we obtain
!
Z ψa
Z ψ0
int
tb
Π
Γ
dψ 0 tb
exp
dψ 00 tb
Ωζ = −
nc nc ν
+
ν
ν
+
ν
0
ψ
ψ
ψ=ψ
ψ=ψ00
!
Z ψa
Γtb 0
,
dψ tb
+Ωζ |ψ=ψa exp
ν + ν nc ψ=ψ0
ψ
(28)
Sources of intrinsic rotation in the low flow ordering
12
where ψa is the poloidal flux at the edge, Ωζ |ψ=ψa is the rotation velocity in the edge
tb
tb
nc
nc
nc
ie
ie
ht
and Πint = Πtb
ud + Π−1,0 + Π0 + Πud + Π−1,0 + Π0 + Π−1 + Π0 + Π . Notice that
this equation gives a rotation profile that depends on Πint that in turn depends on the
gradient of temperature and density, the geometry and the heating mechanism. The
typical size of the rotation is Ωζ ∼ (B/Bp )δi vti /R for ∆ud <
∼ (B/Bp )δi and Ωζ ∼ ∆ud vti /R
for ∆ud >
∼ (B/Bp )δi .
This new model for intrinsic rotation has been constructed such that the pinch
and the up-down symmetry drive, discovered in the high flow ordering, are naturally
included. By transforming to the frame rotating with Ωζ we have made this property
explicit.
Acknowledgments
Work supported in part by the post-doctoral fellowship programme of the UK EPSRC,
by the Junior Research Fellowship programme of Christ Church at University of Oxford,
by the U.S. Department of Energy Grant No. DE-FG02-91ER-54109 at the Plasma
Science and Fusion Center of the Massachusetts Institute of Technology, by the Center
for Multiscale Plasma Dynamics of University of Maryland and by the Leverhulme
network for Magnetised Turbulence in Astrophysical and Fusion Plasmas.
Appendix A. Equation for the distribution function in the rotating frame
In this Appendix we derive equations (4), (7) and (14) for the different pieces of the
ion distribution function, equations (5) and (12) for the different pieces of the electron
distribution function, and equations (6) and (13) for the different pieces of the potential.
These equations are valid in the frame rotating with angular velocity Ωζ , and we deduce
them from the results in [18], obtained in the laboratory frame.
In reference [18] we showed that in the limit Bp /B 1, and neglecting the ionelectron collisions and the effect of the heating, the ion distribution function is given by
fi (R, E, µ, t) = fM i (ψ(R), E, t) + Fi1nc (ψ(R), θ(R), E, µ, t) + Fi2nc (ψ(R), θ(R), E, µ, t) +
fitb (R, E, µ, t), where the size of these different pieces is Fi1nc ∼ (B/Bp )δi fM i , Fi2nc ∼
(B/Bp )2 δi2 fM i and fitb = fi1tb + fi2tb , with fi1tb ∼ δi fM i and fi2tb ∼ (B/Bp )δi2 fM i . The
equations for the different pieces were obtained from the gyrokinetic equation
∂fi
∂fi
+ Ṙ · ∇R fi + Ė
= hCii {fi }i,
(A.1)
∂t
∂E
where the time derivative Ṙ is
0
Ṙ = u0 b̂(R) + vM
−
c
∇R hφi × b̂
B
and the time derivative Ė is
Ze
0
Ė = − [u0 b̂(R) + vM
] · ∇R hφi.
M
(A.2)
(A.3)
Sources of intrinsic rotation in the low flow ordering
13
p
Here, u0 = ± 2(E − µB) is the gyrokinetic parallel velocity in the laboratory frame,
and
µ
(u0 )2
0
b̂ × (b̂ · ∇R b̂)
(A.4)
= b̂ × ∇R B +
vM
Ωi
Ωi
are the ∇B and curvature drifts in the laboratory frame. Equations (19) and (20) of
[18] for Fi1nc and equation (24) of [18] for Fi2nc are obtained from the long wavelength
axisymmetric contributions to (A.1) of order δi fM i vti /a and (B/Bp )δi2 fM i vti /a,
respectively. Equation (25) of [18] for Fi2tb is also a long wavelength axisymmetric
component of (A.1). In particular, it is the contribution of order δi2 fM i vti /a that does
not become of order (B/Bp )δi2 fM i νii when the equation is orbit averaged. Equation (55)
of [18] for fitb is the sum of the short wavelength components of (A.1) of order δi fM i vti /a
and (B/Bp )δi2 fM i vti /a.
In this article, we extend equation (A.1) to account for ion-electron collisions and
the effect of the different heating mechanisms. For this reason, we use
∂fi
∂fi
+ Ṙ · ∇R fi + Ė
= hCii {fi }i + hCie {fi }i + S ht ,
(A.5)
∂t
∂E
p
where Cie {fi } ∼ m/Mνii fi is the ion-electron collision operator, treated in detail in
Appendix B. Moreover, we want to write the equation in the rotating frame, that is, we
need to use the new gyrokinetic variable ε = E − IΩζ u0 /B. Thus, the new gyrokinetic
equation is
∂fi
∂fi
+ Ṙ · ∇R fi + ε̇
= hCii {fi }i + hCie {fi }i + S ht .
(A.6)
∂t
∂ε
The time derivative of the new gyrokinetic variable ε is
∂ε
.
(A.7)
ε̇ = Ṙ · ∇R ε + Ė
∂E
q
In Ṙ, using u0 = u + IΩζ /B, with u = ± 2(ε − µB + R2 Ω2ζ /2), leads to
2
IΩζ
c
B 3
b̂ + vM + vC − ∇R hφi × b̂ + O
(A.8)
Ṙ = ub̂ +
δ vti ,
B
B
Bp2 i
with
vM =
µ
u2
b̂ × ∇R B + b̂ × (b̂ · ∇R b̂)
Ωi
Ωi
(A.9)
the ∇B and curvature drifts in the rotating frame, and vC = (2uIΩζ /BΩi )b̂ × (b̂ ·∇R b̂)
the Coriolis drift. To obtain this expression for Ṙ we have used (u0 )2 = u2 + 2IΩζ u/B +
0
= vM + vC + O[(B 2 /Bp2 )δi3 vti ]. The usual result for the
O[(B/Bp )2 δi2 vti2 ] to write vM
Coriolis drift vC = (2uΩζ /Ωi )b̂ × [(∇R R × ζ̂) × b̂] can be recovered by realizing that
for Bp /B 1, b̂ ' ζ̂, b̂ · ∇R b̂ ' −∇R R/R and I/B ' R, giving
vC =
2uΩζ
2IuΩζ
b̂ × (b̂ · ∇R b̂) '
b̂ × [(∇R R × ζ̂) × b̂].
BΩi
Ωi
(A.10)
Sources of intrinsic rotation in the low flow ordering
14
nc
tb
In addition, using I b̂/B = Rζ̂ + b̂×∇ψ/B, φ = φ0 +φnc
1 +φ2 +φ , hφ0 i = φ0 (ψ(R), t)+
nc
3
nc
2
2 2
O(δi2 Te /e), hφnc
1 i = φ1 (ψ(R), θ(R), t) + O[(B/Bp )δi Te /e] and hφ2 i = O[(B /Bp )δi vti ],
we can simplify equation (A.8) to
1 ∂pi
c
b̂ × ∇ψ + vC − ∇R φnc
1 × b̂
ni MΩi ∂ψ
B
2
c
B 3
tb
− ∇R hφ i × b̂ + O
δ vti .
B
Bp2 i
Ṙ = ub̂ + RΩζ ζ̂ + vM −
The time derivative ε̇ in (A.7) can be written as
Iu0 ∂Ωζ
Ṙ · ∇R ψ − Ωζ Ṙ · ∇R
ε̇ = Ė −
B ∂ψ
Iu0
B
−
IΩζ
Ė.
Bu0
(A.11)
(A.12)
nc
tb
To simplify this equation we use φ = φ0 + φnc
1 + φ2 + φ , hφ0 i = φ0 (ψ(R), t) +
nc
3
nc
nc
O(δi2 Te /e), hφnc
1 i = φ1 (ψ(R), θ(R), t) + O[(B/Bp )δi Te /e], hφ2 i = φ2 (ψ(R), θ(R), t) +
O[(B 2 /Bp2 )δi4 Te /e], u0 = u + O[(B/Bp )δi vti ] and
0
0
0
0
c
Iu
Iu
Iu
Iu
0
0
= u b̂ · ∇R
+ vM · ∇R
− (∇R hφi × b̂) · ∇R
Ṙ · ∇R
B
B
B
B
B
Ze 0
ZeI 0
Bp 2 2
=
vM · ∇R ψ +
δ v .
(A.13)
v
·
∇
hφi
+
O
R
M
Mc
MBu0
B i ti
With these results, we obtain
Ze
1 ∂pi
nc
nc
tb
ε̇ = − [ub̂(R) + vM + vC ] · −
∇R ψ + ∇R φ1 + ∇R φ2 + ∇R hφ i
M
Zeni ∂ψ
2 2
c
Iu ∂Ωζ δ v
tb
vM − ∇R hφ i × b̂ · ∇R ψ + O i ti .
(A.14)
−
B ∂ψ
B
a
0
To obtain the result in (A.13), we have employed vM
· ∇R ψ = u0 b̂ · ∇R (Iu0 /Ωi );
0
ZeI
(u0 )2
c
Iu
µ
I
=
· ∇R hφi
B
−
ln
b̂
×
∇
b̂
×
∇
− (∇R hφi × b̂) · ∇R
R
R
B
B
MBu0 Ωi
Ωi
B
ZeI
(u0 )2
µ
Bp 2
δi vti
=
b̂ × ∇R B +
b̂ × (b̂ · ∇R b̂) · ∇R hφi + O
MBu0 Ωi
Ωi
B
ZeI 0
Bp 2
=
δi vti ,
(A.15)
v · ∇R hφi + O
MBu0 M
B
where we have used I/B = R+O[(Bp2 /B 2 )R] and b̂·∇R b̂ = −∇R R/R+O[(Bp /B)R−1 ];
and
2
0
0
i
Bp 2
u0 h
Iu
Iu
0
0
0
=
=O
∇R × (u b̂) − u b̂b̂ · ∇R × b̂ ·∇R
δi v ,(A.16)
vM ·∇R
B
Ωi
B
B 2 ti
where we have used b̂ · ∇R × b̂ ∼ (Bp /B)a−1 , b̂ · ∇R (Iu0 /B) ∼ Rvti /qR ∼
(Bp /B)(R/a)vti and ∇R × (u0 b̂) · ∇R (Iu0 /B) = ∇R · [u0 b̂ × ∇R (Iu0 /B)] = ∇R ·
[(Iu0 /B)∇R ζ × ∇R (Iu0 /B)] + ∇R · [(u0 /B)(∇ζ × ∇ψ) × ∇R (Iu0 /B)] = ∇R · {∇R ζ ×
∇R [I 2 (u0 )2 /2B 2 ]} − ∂ζ [(u0 /R2 B)∇ψ · ∇R (Iu0 /B)] = 0.
Sources of intrinsic rotation in the low flow ordering
15
With equations (A.6), (A.11) and (A.14), we can now easily obtain equations (4),
nc
nc
tb
ie
ht
(7) and (14) for Hi1
, fitb , Hi2
, Hi2
, Hi2
and Hi2
. To obtain (4), we take the long
wavelength axisymmetric contribution to (A.6) to order δi fM i vti /a, giving
Ze ∂fM i
1 ∂pi
nc
nc
ub̂ · ∇R φ1 −
vM · ∇R ψ
ub̂ · ∇R Hi1 + vM · ∇R fM i −
M ∂ε
Zeni ∂ψ
(`)
nc
}.
= Cii {Hi1
(A.17)
nc
This equation differs from equations (19) and (20) of [18], and gives a function Hi1
nc
different from the function Fi1nc defined in [18]. The reason is that fM i (ψ(R), ε) + Hi1
+
nc
nc
nc
Hi2 must be equal to the function fM i (ψ(R), E) + Fi1 + Fi2 defined in [18] to the
order of interest, but how the terms of first and second order in δi are assigned to one
or the other piece differs depending on the frame. For this reason, we have changed
the name of the functions. The final q
result in (4) is obtained from (A.17) by using
p
vM · ∇R ψ = ub̂ · ∇R (Iu/Ωi ) for u = ± 2(ε − µB + R2 Ω2ζ /2) ' ± 2(ε − µB).
Equation (7) is the sum of the short wavelength contributions to (A.6) of order
δi fM i vti /a and (B/Bp )δi2 vti /a. The equation is almost straightforward if we apply the
same methodology as in [18]. Only two terms require some care. On the one hand,
the ion-electron collision operator that was not treated in [18] is now considered in
detail in Appendix B. On the other hand, in Ṙ there is a drift −(ni MΩi )−1 ∂ψ pi (b̂ ×
∇ψ) that is not included in (7). To study the effect of the perpendicular drift
−(ni MΩi )−1 ∂ψ pi (b̂ × ∇ψ), it is better to consider the local approximation. In the
local approximation, the length scale of the turbulence is so small that the background
quantities can be represented by their local value and their local derivative, i.e., the
drift −(ni MΩi )−1 ∂ψ pi (b̂ × ∇ψ) is given by its value −(ni MΩi )−1 ∂ψ pi (b̂ × ∇ψ)|ψ=ψ0 at
the point ψ = ψ0 around which we want to calculate the turbulent fluctuations and
the linear dependence −(ψ − ψ0 )∂ψ [(ni MΩi )−1 ∂ψ pi (b̂ × ∇ψ)]|ψ=ψ0 . The characteristic
scale of the turbulence is the ion gyroradius, giving ψ − ψ0 ∼ ρi RBp and −(ψ −
ψ0 )∂ψ [(ni M Ωi )−1 ∂ψ pi (b̂ × ∇ψ)]|ψ=ψ0 ∼ δi2 vti . Since we only need to keep terms up
to order (B/Bp )δi2 fM i vti /a, −(ψ − ψ0 )∂ψ [(ni MΩi )−1 ∂ψ pi (b̂ × ∇ψ)]|ψ=ψ0 · ∇R fitb ∼
δi2 fM i vti /a is negligible. Only the constant drift −(ni MΩi )−1 ∂ψ pi (b̂×∇ψ)|ψ=ψ0 remains.
A constant drift does not change the character of the short wavelength structures and
can be safely ignored.
Equation (14) is found from the long wavelength axisymmetric components of
(A.6) to order δi2 fM i vti /a. Note that to this order we have the time derivative
− 3/2)T −1 ∂ T ]f and realizing that
∂t fM i [18]. Using ∂t fM i = [n−1
i ∂t ni + (Mε/T
PR 3 α
Pi R 3 α i t i Mi
h d v S iψ and (3/2)∂t (ni Ti ) =
h d v S Mεiψ , where the summations
∂ t ni =
are over α = nc, tb, ie, ht, we find the final form in (14). The equations for
nc
tb
and Hi2
are obtained in the same way as equations (24) and (25) in [18], i.e.,
Hi2
nc
is the axisymmetric long wavelength component of (A.6) of
the equation for Hi2
2
tb
is the axisymmetric long wavelength
order (B/Bp )δi fM i vti /a, and the equation for Hi2
2
component of order δi fM i vti /a that when it is orbit averaged does not reduce to a piece
ie
ht
and Hi2
where not considered in [18].
of order (B/Bp )δi2 νii fM i . The equations for Hi2
Sources of intrinsic rotation in the low flow ordering
16
ie
The equation for Hi2
is the axisymmetric long wavelength contribution that includes
the ion-electron collision operator that is treated in detail in Appendix B. The equation
ht
is the axisymmetric long wavelength contribution that has the source S ht .
for Hi2
The equations (5) and (12) for the electron distribution function in the rotating
frame are derived in the same way as the equations for the ion distribution function. The
only differences are that
p the Coriolis drift vC and the term in (A.14) that is proportional
to ∂ψ Ωζ are small by m/M and hence negligible, and that we include the electric field
EA driven by the transformer, leading to a modified time derivative for the energy
e
e
1 ∂pi
A
nc
tb
∇R ψ + ∇R φ1 + ∇R hφ i .
ε̇ = ub̂ · E + [ub̂(R) + vM ] · −
(A.18)
m
m
Zeni ∂ψ
Finally, the equations for the different pieces of the electrostatic potential (6) and
(13) are easily deduced from the results in [18] by realizing that moving to a rotating
reference frame does not modify the quasineutrality equation.
Appendix B. Ion-electron collisions
In this Appendix we discuss how we treat the ion-electron collision operator, given by
Z
M
3
(B.1)
d ve ∇g ∇g g · fe ∇v fi − fi ∇ve fe ,
Cie {fi , fe } = γie ∇v ·
m
↔
where γie = 2πZ 2 e4 ln Λ/M 2 , g = v − ve and ∇g ∇g = (g 2 I −gg)/g 3 . Both the ion
velocity, v, and the electron velocity, ve , are measured in the rotating frame.
p
m/Mνii δi fM i .
We must write the ion-electron collision operator up to order
For the wavelengths of interest, between the minor radius a and the ion gyroradius,
nc
(ψ(R), θ(R), ε, µ, t) + fitb (R, ε, µ, t) + . . . and
fi (R, ε, µ, t) = fM i (ψ(R), ε, t) + Hi1
nc
(ψ(R), θ(R), ε, µ, t) + fetb (R, ε, µ, t) + . . ., where
fe (R, ε, µ, t) = fM e (ψ(R), ε, t) + He1
nc
tb
tb
= (eφnc
He1
1 /Te )fM e +O[(B/Bp )δe fM e ] and fe = (eφ /Te )fM e +O(δe fM e ). The electron
distribution function is then a Maxwell-Boltzmann
response (eφ/T
p
pe )fM e ∼ δi fM e plus
a correction of order δe fM e ∼ m/Mδi fM e that is smaller by m/M . Expanding
2
2
the ion distribution function around Rg = r + Ω−1
i v × b̂, v /2 and v⊥ /2B, we
nc
tb
tb
tb
tb
+ htb
obtain fi = fM i,0 + Hi1,0
i , where hi = fig − [Ze(φ − hφ i)/Ti ]fM i,0 . For the
electrons, since we are considering wavelengths larger than the electron gyroradius, it
2
nc
/2B, giving fe = fM e,0 + He1,0
+ htb
is possible to expand around r, v 2 /2 and v⊥
e , where
tb
tb
−1
tb
tb
he = fe0 − Ωe (v × b̂) · ∇fe0 + [mc(v × b̂) · ∇φ /BTe ]fM e,0 . Here, the subscript 0
nc
nc
tb
in fM i,0 , fM e,0 , Hi1,0
, He1,0
and fe0
indicates that the variables R, ε and µ have been
2
2
replaced by r, v /2 and v⊥ /2B. The subscript g in figtb indicates that R, ε and µ are
2
2
replaced by Rg = r + Ω−1
i v × b̂, v /2 and v⊥ /2. Using these expressions for fi and fe ,
and the relation
m 1
,
(B.2)
∇g ∇g g = ∇ve ∇ve ve − v · ∇ve ∇ve ∇ve ve + O
M vte
Sources of intrinsic rotation in the low flow ordering
17
we find that
Z
Mγie
eφnc
eφtb
1
3
Cie {fi , fe } = −
fM i v · d ve fM e ∇ve ∇ve ve
∇v · 1 +
+
Ti
Te
Te
Z
nc
tb
3
+γie ∇v · (∇v Hi1 + ∇v hi ) · d ve fM e ∇ve ∇ve ve
Z
Mγie
3
nc
tb
−
∇v · fM i d ve ∇ve ∇ve ve · (∇ve He1 + ∇ve he )
m
Z
eφnc
eφtb
Mγie
1
3
fM i v · d ve fM e ∇ve ∇ve ∇ve ve · ve
∇v · 1 +
+
−
Te
Te
Te
Z
Mγie
nc
tb
3
−
∇v · (Hi1 + hi )v · d ve fM e ∇ve ∇ve ∇ve ve · ve
Te
m
νii δi fM i .
+O
(B.3)
M
Here we have used that ∇ve fM e = −(mve /Te )fM e and ∇ve ∇ve ve · ve = 0. Using
↔ R
R
∇ve ∇ve ∇ve ve · ve = −∇ve ∇ve ve and d3 ve fM e ∇ve ∇ve ve = (2/3) I d3 ve fM e /ve =
↔
p
√ √
(2 2/3 π)ne m/Te I , we find
eφnc
ne mνei
eφtb
Te
Mv 2
1
Cie {fi , fe } =
1+
+
−1
− 3 fM i
ni M
T
Te
Ti
Ti
e
ne mνei
Te
nc
tb
nc
tb
∇v ·
(∇v Hi1 + ∇v hi ) + v(Hi1 + hi )
+
ni M
M
1
tb
nc
tb
− [Fnc
ei + Fei − ne mνei (Wi + Wi )] · vfM i
pi
m
νii δi fM i .
(B.4)
+O
M
√
3/2
Here νei = (4 2π/3)Z 2 e4 ni ln Λ/m1/2 Te is the electron-ion collision frequency, Fnc
ei
is the long wavelength axisymmetric friction force on electrons due to collisions with
ions, given in (18), Ftb
ei is the short wavelength turbulent friction force, given in (9),
R 3
−1
nc
nc
Wi = ni b̂ d v Hi1 v|| is the long wavelength axisymmetric ion average velocity in
R 3 tb
d v hi v is the short wavelength turbulent ion
the rotating frame, and Witb = n−1
i
average velocity.
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