J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Fluid moments in modied guiding-centre coordinates
Naoaki MIYATO, Bruce D. SCOTT1) and Shinji TOKUDA2)
Japan Atomic Energy Agency, Naka, Ibaraki 311-0193, Japan
Max-Planck-Institut für Plasmaphysik, D-85748 Garching, Germany
2)
Research Organization for Information Science and Technology, Tokai, Naka-gun, Ibaraki 319-1106, Japan
1)(Received: 27 October 2009 / Accepted: 12 February 2010)

A general uid moment in the particle coordinates is represented in terms of uid moments in modied
guiding-centre coordinates with strong E × B ow derived by the phase space Lagrangian Lie-transform method[N. Miyato et al., J. Phys. Soc. Jpn. 78, 104501 (2009)]. It is called the push-forward representation of theparticle uid moment associated with the guiding-centre transformation. The representation derived is similar tothat in terms of gyro-uid moments in the standard gyrokinetic theory in the long wavelength limit. Since theparticle coordinates are transformed to gyro-centre coordinates by the two-step transformation in the standard
gyrokinetic formulation, two exact push-forward representations are possible. Although the exact representation
usually used in the standard gyrokinetic theory has a diﬀerent form from that in the modied guiding-centre
case, the correspondence between the two cases is shown more clearly by considering the alternative form of thepush-forward representation for the standard gyrokinetic case.
Keywords: phase space transformation, push-forward representation, ow

1. Introduction

the standard gyrokinetic model formally [5]. The guidingcentre Hamiltonian also agrees with the standard gyrokinetic Hamiltonian in the long wavelength limit. Therefore
it is expected that the relation between the uid momentsin the modied guiding-centre coordinates and the particle uid moments be similar to the one obtained from thestandard gyrokinetic model in the long wavelength limit.
In this paper, we represent the particle uid moments interms of the uid moments in the modied guiding-centre
coordinates. The representation is compared with that obtained from the standard gyrokinetic model in the longwavelength limit.

It is well known that guiding-centre or gyro-centre
density is diﬀerent from particle density due to niteLarmor-radius (FLR) eﬀects [1–8]. Generally guidingcentre or gyro-centre uid moments are diﬀerent from corresponding particle uid moments. Any particle uid moment can be represented in terms of the guiding-centre or
gyro-centre uid moments. It is called the push-forward
representation of the particle uid moment associated withthe transformation from the particle phase space to theguiding-centre or gyro-centre phase space [5, 6]. The inverse of the representation was used to derive nonlinear reduced uid equations with FLR corrections from nonlinear
gyrouid equations [1]. Since the gyromotion of a charged
particle is removed at the kinetic level, this procedure allows one to bypass the issue of gyroviscous cancellations
and corrections in traditional derivations where an explicit
representation of the stress tensor is needed [9, 10]. An
expression for the gyroviscous force is obtained by comparing the reduced equations with the FLR corrections obtained from the gyrouid equations with the particle uid
momentum equation [3]. The correspondence between the
gyrouid and low-frequency uid equations is also shownby using the relation between the gyrouid moments andthe particle uid moments [4]. Recently we derived a modied guiding-centre fundamental 1-form with strong E × B
ow from which a guiding-centre Vlasov-Poisson system
was also constructed through the eld theory [11]. It is ofinterest to investigate a relation between uid moments inthe modied guiding-centre coordinates and the particleuid moments. In contrast to conventional formulations
with strong E × B ow [12–16], the symplectic part of theguiding-centre 1-form in our formulation does not include
the E × B drift velocity term and is the same as that in

2. Guiding-centre theory

We consider a transformation from particle coordinates (x, u, w, θ) to guiding-centre coordinates (X, U, μ, ξ)
given by [7, 11]
X = x − �ρ − �ρE + O(� 2 ),(1)

U = v� + O(�),mw2μ=+ O(�),
2B0
ξ = θ + O(�),

(2)
(3)
(4)

where x is the position of a particle with mass m and electric charge q, v� is the parallel velocity, w is the perpendicular velocity in the frame moving with D which is the E × B
drift velocity, θ is the gyrophase, ρ is the Larmor radiusevaluated by w, ρE = b̂ × D/Ω, Ω = qB0 /m and � ∼ ρ/Lis the small parameter with the background gradient scale
length L. Here it is assumed that the E × B drift velocity
is comparable to the ion thermal velocity vti . Therefore,ρE ∼ ρ for ions. The guiding-centre transformation for X
is diﬀerent from conventional ones in which the guiding
centre position X� is dened by X� ≡ x − �ρ [12–16]. X ismainly shifted by ρE from X� . The guiding-centre funda-

author’s e-mail: miyato.naoaki@jaea.go.jp546©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

N. Miyato et al., Fluid Moments in Modified Guiding-Centre Coordinates

mental 1-form is given by��2m
Γ = qA0 + �mU b̂ − � μW · dXqm
+� 2 μdξ − Hdt,
qParticle(5)BU = −� −1
· ∇H,
mB∗�

(8)

μ ≡ 0,

(9)

∗

q ∂H

ξ = � −2+ W · X,m ∂μZ

mkl (r) ≡vl� f δ3 (x − r)d3 xd3 v.

(10)

(11)

This particle uid moment can be written in terms of the
gyro-centre distribution function F¯ in the electrostatic limitas�
mkl (r) =d6 Z̄J(Z̄)
⎧�
⎡
⎤
�k ⎫
⎪
⎢⎢⎢ −1∗ ⎪⎪⎪⎥⎥⎨ mv2⊥l ⎬⎥⎢⎥⎥ (Z̄)v� ⎪× ⎢⎣T GC ⎪
⎪
⎪
⎩ 2B0
⎭⎦
¯ Z̄)δ3 (T −1
×[T ∗Gy F](
GC X̄ − r)���qφ̃ ∂F¯6k l ¯�d Z̄J(Z̄)μ̄ Ū F + �δB0 ∂μ̄
×δ3 (X̄ + ρ̄ − r)-1TGy

( X, U , P , [ )transformation associated with T GC , T Gy is the transformation from the guiding-centre phase space to the gyrocentre phase space, T ∗Gy is the pull-back transformation associated with T Gy , φ̃ = φ(X̄ + ρ̄) − �φ(X̄ + ρ̄)� is the gyrophase dependent part of the electrostatic potential, �·�denotes the gyrophase average, �δ is the small parameterfor the amplitude of φ and ρ̄ = ρ(Z̄). It is noted that¯ + O(� 2 ) withgenerally T ∗Gy F¯ is expressed as F¯ + �δ {S 1 , F}δ�¯ The phase space transformations in the
S 1 = (q/Ω) φ̃dξ.
standard gyrokinetic formulation [5, 17] are summarisedin Fig. 1. The particle phase space is transformed to the
gyro-centre phase space through two steps. First, the transformation from the particle phase space to the guidingcentre phase space is performed with � ∼ ρ/L to remove
gyrophase dependence from the single particle motion in
an equilibrium magnetic eld. After that, a small perturbation of electrostatic potential, φ, is introduced and thetransformation from the guiding-centre phase space to the
gyro-centre phase space is performed with �δ ∼ qφ/T to remove the gyrophase dependence reintroduced by φ. Thus
smallness of φ is necessary for the gyro-centre transformation. Push-forward of a scalar function is shown in Fig.
2 schematically. Here we consider a scalar function onz denoted by f and a transformation from the particle z
space to the guiding-centre Z space denoted by T GC . Then
we can represent f (z) in terms of Z through z = T −1
GC Z as−1−1∗
f (z) = f (T GC Z) = T GC f (Z). Thus we obtain a function−1∗
F ≡ T −1∗
GC f on Z. Since T GC “pushes forward” f on z to F
on Z, it is called the push-forward transformation associated with T GC . Note that the function of the transformation
is opposite to appearance of the symbol. Conversely, we
can obtain a function on z, T ∗GCG, from a function on Z
denoted by G. Then T ∗GC is called the pull-back transformation associated with T GC . Some explanation would be−1
needed for T −1
GC X̄. Originally [T GC x](Z) = X + ρ(Z) + · · ·Following Belova [3], we brief the relation between
the gyro-centre uid moments and the particle uid moments rst. A general particle uid moment is dened by
�k

qI
T

Fig. 1 Phase space transformations in the standard gyrokinetic
formulation.

3. Push-forward representation of uid moments
3.1 Standard gyrokinetic case

mv2⊥
2B0

L

-1
TGC

Gyro-centre

where A∗ = A0 + (m/q)U b̂, B∗ = ∇ × A∗ and B∗� = b̂ · B∗ .
These are the same as those in the standard gyrokinetic
model formally.

� �

HG ~

TGy

The guiding-centre Hamilton equations, Z i = {Z i , H}, arewritten as(7)U( x, v )Guiding-centre
Z ( X, U , P , [ )where W = R + (b̂ · ∇ × b̂)b̂/2, b̂ ≡ B0 /B0 , R ≡ (∇e1 ) · e2 ,e1 and e2 are unit vectors spanning the plane perpendicular
to b̂, and the guiding-centre Hamiltonian is�mm �
U 2 + μB0 − D2
H = qφ + �
2
2
� �
�
�
2
m
mD
+� 2
b̂ · ∇ × D .
(6)
μ+
2q
2B0

∗
 = � −1 B ∂H + b̂ × ∇H,
X
mB∗� ∂U qB∗�

İ~

TGC

z

(12)

¯ are the gyro-centre coordinates,
where Z̄ = (X̄, Ū, μ̄, ξ)6¯d Z̄ = dX̄dŪdμ̄dξ, J = B∗� /m is the Jacobian, T −1
GC is theinverse of the transformation from the particle phase space
to the guiding-centre phase space, T −1∗
GC is the pull-backor
the push-forward
transformation associated with T −1
GC547

N. Miyato et al., Fluid Moments in Modified Guiding-Centre Coordinates

Particlezf

3.2 Modied guiding-centre case

Guiding-centre-1
TGC

In this subsection we consider a particle uid momentdened by

Z

7

-1*
GC

m̄kl (r) ≡

f {F

R

nV⊥ ≡

�

v⊥ f d3 v.

p⊥ = p�⊥ − mnD2 /2 + O(�).

(19)

(20)

(21)

Obviously, m̄kl → mkl as D → 0. The E×B drift velocity D
is sub-thermal in most cases of interest. Therefore, we assume D ∼ � 1/2 vti in the following. In terms of the modied
guiding-centre distribution function F, m̄kl can be writtenas�
d6 ZJ(Z)m̄kl (r) =
⎧�
⎫⎤
⎡
�
⎪
2 k
⎢⎢⎢ −1∗ ⎪
⎪
⎪
mw
⎨
⎬⎥⎥⎥⎥
l
(u) ⎪
× ⎢⎢⎣T GC ⎪
⎥ (Z)
⎪
⎪
⎩ 2B0
⎭⎦
×F(Z)δ3 (T −1
GC x − r)��
d6 ZJ(Z)(μ − Gμ1 )k U l F(Z)and we have assumed that (k⊥ ρ)2 ∼ �⊥ for a small O(�δ )perturbation, (k⊥ ρ) ∼ �⊥ for a O(1) moment and � ∼ �δ ∼
�⊥ . For k = l = 0, we have the well known push-forward
representation of the particle density n,

δ3 (X + ρ + ρE − r).

(22)

Here the μ component of the vector eld generating theguiding-centre transformation, Gμ1 , is kept because it has
the nonvanishing gyroaveraged part due to the E × B ow.
The delta function expanded in powers of ρ and ρE , theabove equation is rewritten as�m̄kl (r) �
d6 ZJ(μk − kμk−1Gμ1 )U l F(Z)�× δ3 (X − r) + ρE · ∇δ3 (X − r)�1
+ (ρ · ∇)2 δ3 (X − r)2��
d6 Zδ3 (X − r)μk U l FJ�
−k d6 Zδ3 (X − r)μk−1 �Gμ1 �U l FJ
�
+ d6 Zμk U l FJρE · ∇δ3 (X − r)�1
+ d6 Zμk U l FJ (ρ · ∇)2 δ3 (X − r).
2
(23)

(15)

where N and P⊥ are the gyro-centre density and the gyrocentre perpendicular pressure dened by�¯ Ūdμ̄dξ,
¯
N≡
FJd
(16)
¯ Ūdμ̄dξ,¯μ̄B0 FJd(18)

Since V⊥ = D + O(�) in the strong E × B case, the relation
between p⊥ and p�⊥ is written as

where Mkl is the general gyrouid moment dened by�¯ Z̄)dŪdμ̄dξ,¯
μ̄k Ū l J(Z̄)F((14)
Mkl (X̄) ≡P⊥ ≡vl� f δ3 (x − r)d3 xd3 v.

wheredenotes the particle position in the guiding-centre phase−1
space. T −1
GC X̄ denotes [T GC x] whose argument is replacedby the gyro-centre coordinates Z̄, that is, T −1
GC X̄ � X̄ + ρ(Z̄)
and does not agree with the particle position in the gyro−1
centre phase space T −1
Gy T GC x. Expanding the delta functionand φ in powers of ρ̄ and integrating by parts lead to therepresentation of the particle uid moment mkl in terms ofgyrouid moments up to O(� 2 ) (see appendix A),
��
1
1
mkl (r) = Mkl (r) + ∇ ·∇⊥ Mk+1l (r)2qΩ��Mkl (r)∇⊥ φ(r) ,(13)+(k + 1)∇ ·
B0 Ω

�

�k

p⊥ = p�⊥ − mnV⊥ · D + mnD2 /2,

Fig. 2 Push-forward of a scalar function. F(Z) = T −1∗
GC f (Z) =f (T −1
GC Z) = f (z).

���∇ ⊥ P⊥
1
N∇⊥ φ + ∇ ·,n= N+∇·
ΩB0
2
qΩB0

mw2
2B0

Note that w = |v⊥ − D| is used for the denition of m̄kl ,while v⊥ is used for mkl . Therefore, m̄kl with k � 0 is
diﬀerent from mkl . For example, the relation between p⊥ ≡B0 m̄10 and p�⊥ ≡ B0 m10 is given by

Push-forward of f

�

� �

(17)

respectively. Although spatial derivatives are reduced tothe perpendicular Laplacian in ref. [3], we keep the divergence form to manifest the polarisation eﬀects here [5, 8].
It is easily shown by the volume integration of eq. (15)over the domain including the whole plasma that the number of particles included in the whole plasma agrees withthat of gyro-centres.

548

N. Miyato et al., Fluid Moments in Modified Guiding-Centre Coordinates

Since the main part of �Gμ1 � is given by [11]μ�Gμ1 � � − b̂ · ∇ × (D + U b̂)Ω �μ �� −∇ · D × b̂Ω ��μ∇⊥ φ ,= −∇ ·
ΩB0

where N and P⊥ are the guiding-centre density and theguiding-centre perpendicular pressure dened by�N≡
FJdUdμdξ,
(30)
P⊥ ≡

(24)

�

μB0 FJdUdμdξ,(31)respectively. Equation (29) is the same as the standard gyrokinetic case (15) formally.

the second term on the right hand side of eq. (23) becomes�−k d6 Zμk−1 �Gμ1 �U l FJδ3 (X − r)�
� k d6 Zδ3 (X − r)μk U l
�
�
� �
b̂
FJ
b̂ − D × · ∇FJ× ∇· D×ΩΩ� � k l��μU
63
� k d Zδ (X − r) ∇ ·FJ∇⊥ φΩB0�b̂ × ∇FJμk U l
−D ·
Ω
�
�
b̂ × ∇Mkl
Mkl
,
(25)
∇⊥ φ − kD ·= k∇ ·
ΩB0Ω

3.3 Variational derivation of the pushforward representation for the particle
density
The push-forward representation for the particle density is also derived from� t2 the functional derivative of theaction functional I = t Ldt with the Lagrangian for the1
Vlasov-Poisson system,��d6 Z� J s (Z� )F s (Z� , t� )L=swhere the guiding-centre uid moment Mkl is dened by�Mkl (X) ≡μk U l J(Z)F(Z)dUdμdξ.
(26)

×L s [Z s (Z� , t� ; t), Z s (Z� , t� ; t), t]�1
F : F,− d3 x4μ0(32)

where L s is the single particle Lagrangian of species s obtained from the 1-form eq. (5), s + m s μ s ξ s − H s ,L s = q s A∗s · X(33)qsIntegrating by parts the third term on the right hand side ofeq. (23), we have�
d6 Zμk U l FJρE · ∇δ3 (X − r)���b̂ × D63k l= − d Zδ (X − r)∇ · μ U FJ
Ω
�
�
�
∇⊥ φ
6
3
k l
=
d Zδ (X − r)∇ · μ U FJ
ΩB0
�
�
Mkl
∇⊥ φ .
(27)=∇·
ΩB0
The rst and last terms on the right hand side of eq. (23)are the same as those in the standard gyrokinetic model.
As a consequence, the push-forward representation of m̄kl
in terms of the modied guiding-centre uid moments isgiven by�
�
1
1∇⊥ Mk+1l (r)m̄kl (r) = Mkl (r) + ∇ ·2qΩ��Mkl (r)∇⊥ φ(r)+(k + 1)∇ ·B0 Ω

Z� ≡ (X� , U � , μ� , ξ� ) and Z s (Z� , t� ; t) denotes the guidingcentre coordinates of the particle at t with the initial condition, Z s (Z� , t� ; t� ) = Z� . The guiding-centre HamiltonianH s is given by eq. (6). The last part of L is the Lagrangian
for the electromagnetic elds in which μ0 is permeability of vacuum, the electromagnetic eld tensor F is dened by F μν ≡ ∂μ Aν − ∂ν Aμ and F : F ≡ F μν F μν . Whenημν = ημν = diag(−1, +1, +1, +1) is taken as a metric tensor
of Minkowski spacetime, the covariant four vector potential and the four gradient operator are Aμ = (−φ/c, A0 ) and∂μ = ((1/c)∂t, ∇), respectively [8,18]. From δI/δφ = 0, we
obtatin the guiding-centre Poisson equation [11],
�
�
��
∇⊥ φ
�0 ∇2 φ(x, t) = −qs Ns + qs ∇ · Ns
B0 Ω s
s
�
��
��
b̂
b̂
×
−∇ · ∇ × P⊥s
B0
2Ω s
�
qs ns ,
(34)
= −b̂ × ∇Mkl (r).
(28)−kD(r) ·ΩThis representation is the same as eq. (13) except the lastterm formally. The last term goes to the higher order thanO(� 2 ) in the same ordering used in deriving eq. (13). Theneq. (28) agrees with eq. (13) up to O(� 2 ). The last term has
no eﬀect for k = 0. For k = l = 0, therefore, we have����∇ ⊥ P⊥
N1
n= N+∇·∇⊥ φ + ∇ ·
,
(29)
ΩB0
2
qΩB0

where �0 is permittivity of vacuum. It is noted that wehave assumed D ∼ � 1/2 vti and (m2 D2 /4qB0 )b̂ · ∇ × D in the
Hamiltonian (6) has been neglected. The right hand side of
the above equation is the sum of charge density of species
s. Noting
�
�
�
�
b̂
b̂
b̂
b̂
� ∇P⊥ ×∇ × P⊥
×
×
B0
2Ω
B0
2Ω
∇⊥ P⊥
,
(35)
� −
2B0 Ω

s

549

N. Miyato et al., Fluid Moments in Modified Guiding-Centre Coordinates

we have the same relation between the particle density n s
and the guiding-centre uid moments as eq. (29).

Using the above result and expanding the delta functionand φ, we have���qd6 Z̄J μ̄k − kμ̄k−1 φ̃(X̄ + ρ̄)mkl (r) �B0l ¯ 3
×Ū Fδ (X̄ + ρ̄ + ρ̄1 − r)���
d6 Z̄J F¯μ̄k Ū l δ3 (X̄ − r)
�
1
3
2 3
+�ρ̄1 � · ∇δ (X̄ − r) + (ρ̄ · ∇) δ (X̄ − r)2�q−k d6 Z̄J F¯μ̄k−1 Ū l φ(X̄ + ρ̄)B0×ρ̄ · ∇δ3 (X̄ − r)����∇⊥ Mk+1l
Mkl
1∇⊥ φ + ∇ ·� Mkl + ∇ ·
ΩB0
2
qΩ
�
�
Mkl
∇⊥ φ ,
(40)
+k∇ ·
ΩB0

4. Summary and discussion

In the previous section it was shown that the pushforward representation of the particle uid moment interms of the modied guiding-centre uid moments is very
similar to that in terms of the standard gyrouid momentsin the appropriate limit. However, there is an apparent difference between the exact representations (12) and (22).
Equation (22) has the standard form of the push-forward
representation associated with the guiding-centre trnasformation, while eq. (12) is the mixed representation of thepush-forward transformation associated with T GC and thepull-back transformation assciated with T Gy . The representation (12) is derived by considering Z in the integral
on the right hand side of eq. (22) a dummy variable, replacing Z by Z̄ and using F = T ∗Gy F¯ [17]. Since the perturbation of the electrostatic potential is introduced after theguiding-centre transformation in the standard gyrokinetic
formulation, its eﬀect is not included in T GC , but in T Gy .
Therefore the eﬀect of φ is included only in T ∗Gy F¯ in eq.
(12). On the other hand, it is included in T GC in the modied guiding-centre case. The correspondence between thestandard gyrokinetic case and the modied guiding-centrecase becomes more transparent by considering the alternative form similar to eq. (22) given by [19]�mkl (r) =d6 Z̄J(Z̄)
⎧�
⎤
⎡
�k ⎫
⎪
⎢⎢⎢ −1∗ −1∗ ⎪
⎪
⎪⎥⎥
mv2⊥
⎨
l ⎬⎥
⎥⎥
v
× ⎢⎢⎣T Gy T GC ⎪
⎪
⎪
�⎪
⎩ 2B0
⎭⎦
−1
¯ Z̄)δ3 (T −1
×F(
Gy T GC x − r).

where the contribution from Gμ1 has been dropped because
in the standard guiding-centre theory �Gμ1 � = −μU b̂ · ∇ ×b̂/Ω and variation of the magnetic eld has been assumed
to be very mild, and the gyroiud moment is dened byeq. (14). Thus the alternative form of the push-forward
representation yields the same representation as eq. (13)and gives the direct correspondence to the representationin the modied guiding-centre case.

Acknowledgment

This work is partly supported by the Grant-in-Aid for
Scientic Research (C) from the Japan Society for the Promotion of Science (19560832).

A. Derivation of the push-forward representation in the standard gyrokinetic model(36)

This is more straightforward representation than eq. (12).
The particle position in the gyro-centre phase space−1
T −1
Gy T GC x is given by−1
T −1
Gy T GC x = X̄ + ρ̄ + ρ̄1 + · · · ,The rst term on the right hand side of eq. (12) yields
�¯ Z̄)δ3 (X̄ + ρ̄ − r)d6 Z̄μ̄k Ū l F(��¯ Z̄) δ3 (X̄ − r)=d6 Z̄μ̄k Ū l F(
�
1
2 3
¯
+ (ρ̄ · ∇) δ (X̄ − r) + · · ·
2
�
�
1
1
∇⊥ Mk+1l (r) .
� Mkl (r) + ∇ ·
(41)
2
qΩ

(37)

where ρ̄1 = −{S 1 , X̄ + ρ̄} is the gyro-centre displacement vector [5, 6]. The gyroangle average of ρ̄1 corresponds to ρE in the small amplitude limit. Since the standard guiding-centre magnetic moment μ is related with the
gyro-centre coordinates asμ� μ̄ − {S 1 , μ̄}∂S 1 ¯� μ̄ −{ξ, μ̄}∂ξ¯qq
� μ̄ − φ̃(X̄ + ρ̄) ,Ωm

The second term of eq. (12) becomes�qφ̃ ∂F¯ 3
d6 Z̄μ̄k Ū l
δ (X̄ + ρ̄ − r)
B0 ∂μ̄�¯q¯ 3 (X̄ − r)¯ X̄) ∂F ρ̄ · ∇δ�
d6 Z̄μ̄k Ū l ρ̄ · ∇φ(
B0
∂μ̄
�
=
d6 Z̄δ3 (X̄ − r)��¯
k l F ¯¯∇⊥ φ(X̄)×∇ · (k + 1)μ̄ Ū
B0 Ω��Mkl (r)∇⊥ φ(r) .
= (k + 1)∇ ·
(42)
B0 Ω

(38)

we have−1∗
T −1∗
Gy T GC��
mv2⊥
� μ(Z̄) − Gμ1 (Z̄)
2B0q� μ̄ − φ̃(X̄ + ρ̄) − Gμ1 (Z̄).
B0

(39)

550

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