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Notes on nonlinear gyrokinetic equationby Youjun Hu

Institute of Plasma Physics, Chinese Academy of SciencesEmail: [email protected]

Abstract

The nonlinear gyrokinetic equation given in Frieman-Chen's paper[3] is re-derived in this note, providingmore details on the derivation. The aspects that are relevant to numerical implementation of the gyro-kinetic model are also discussed, which are actually the emphasis of this note.

1 Introduction

Subtle things in a theory, which may be crucial for one to fully understand the theory, can be identi�ed andunderstood only when one re-derives the theory by oneself.

Presently, I do not understand the modern phase-space-Lagrangian Lie perturbation method of derivingthe gyrokinetic mode. So I am using the old style asymptotic expansion method in deriving the gyrokineticmodel, as was done in the original Frieman-Chen's paper[3]. The motivation of deriving the gyrokineticequation by myself is that I am developing a new kinetic model in the GEM code, which requires me tounderstand every details of the gyrokinetic theory.

2 Transform Vlasov equation from particle coordinates to guiding-center coordinates

To facilitate the derivation, we �rst need to choose some good variables to work with (why they are goodcan only be realized when we get the results, but some preliminary considerations can suggest that somevariables be good).

The Vlasov equation in terms of particle coordinates (x;v) is given by

@fp@t

+v � @fp@x

+qm(E+v�B) � @fp

@v=0; (1)

where fp= fp(x; v) is the particle distribution function, x and v are the location and velocity of particles.Next, we �rst de�ne the guiding-center transformation. Then we transform the Vlasov equation to theguiding-center coordinates, i.e., express the gradient operators @ /@x and @ /@v in terms of the guiding-centervariables.

2.1 Guiding-center transformation

The transformation from the particle variables (x;v) to the guiding-center variables X is de�ned by[1]

X(x;v)=x+v�ek(x)

(x); (2)

where x and v are particle position and velocity, ek=B0/B0, = qB0/m, B0=B0(x) is the equilibrium(macroscopic) magnetic �eld at the particle position. It is obvious that ��¡v� ek/ is the vector gyro-radius pointing from the the guiding-center to the particle position. This transformation involves bothposition and velocity of particles.

1

Given (x; v), it is straightforward to obtain X by using Eq. (2). On the other hand, the inverse trans-formation, i.e., given (X; v), to �nd x. This is in principle not easy because and ek depend on x, whichusually requires solving for the root of a nonlinear equation. Numerically, one can use

xn+1=X¡v�ek(xn)

(xn): (3)

as an iteration scheme to compute x. The initial guess can be chosen as x0=X. The equilibrium magnetic�eld we will consider has spatial scale length much larger than the thermal gyro radius �. In this case thedi�erence between the values of ek(x)/(x) and ek(X)/(X) is small and thus can be neglected. Then theinverse guiding-center transform can be written as

x�X¡v�ek(X)

(X); (4)

which can also be considered as we using the iterative scheme (3) to computer x and stopping at the �rstiteration. The di�erence between equilibrium �eld values evaluated at X and x is always neglected ingyrokinetic theory. Therefore it does not matter whether the above ek/ is evaluated at x or X. Whatmatters is where the perturbed �elds are evaluated: at x or at X. The values of perturbed �elds at x or at Xare di�erent and this is called the �nite Larmor radius (FLR) e�ect, which is almost all that the gyrokinetictheory is about.

2.2 Gyro-angle

The forward and backward guiding-center transformations (2) and (4) involve the velocity vector v. It isthe cross product between v and ek(x) or ek(X) that is actually used. Therefore, only two of the threecoordinates of v is actually needed in a local (at x or X) cylindrical coordinate system, namely v? and �,where � is the azimuthal angle of the velocity in the local cylindrical system. Here � will be called the gyro-phase or gyro-angle in the following.

2.3 Distribution functions in terms of guiding-center variables

Denote the distribution function in terms of the guiding-center variables (X;v) by fg(X;v), then the relationbetween fg and fp is given by

fg(X;v)= fp(x;v); (5)

where X and x are related to each other via Eq. (2). Equation (5) along with Eq. (2) can be consideredas the de�nition of the guiding-center distribution function fg. Using Eq. (2), the above relation can beequivalently written as

fg(X;v) = fp(X+ �;v); (6)

or

fp(x;v)= fg(x¡ �;v): (7)

or fp(x;v)= fp(X+ �;v), or fg(X;v) = fg(x¡ �;v).As is conventionally adopted in multivariables calculus, both fp and fg are sometimes simply denoted

by f . Which one is actually assumed depends on the context, i.e., depends on which independent variablesare actually assumed: particle variables or guiding-center variables. This is one of the subtle (trivial?)things needed to be noted for gyrokinetic theory in particular and for multivariables calculus in general.This notation will cause confusions when, for example, fg(X;v) is evaluated at X=x, i.e., fg(x;v), which,if the subscript g is omitted and we rely on dependent variables to identify whether it is fp or fg, will bewrongly understood as fp. Therefore it seems better to use the accurate notation. One example where thisdistinguishing is important is encountered when we try to express the diamagnetic �ow in terms of fg, whichis discussed in Appendix A.

2 Section 2

In practice, fg is often called the guiding-center distribution function whereas fp is called the particledistribution function. They are the same distribution function expressed in di�erent variables.

In the above, we assume that X and x are always related to each other by the guiding-center transfor-mation (2) or (4) , i.e., x and X are not independent. For some cases, it may be convienent to treat x andX as independent variables and express the guiding-center transformation via an integral of the Dirac deltafunction. For example, expression (7) can be written as

fp(x;v) =

Zfg(X;v)�

3(X¡x+ �)dX; (8)

where x and X are considered as independent variables, �3(x¡X¡ �) is the three-dimensional Dirac deltafunction. [In terms of general coordinates (x1; x2; x3), the three-dimensional Dirac delta function is de�nedvia the 1D Dirac delta function as follows:

�3(x)=1

jJ j�(x1)�(x2)�(x3); (9)

where J is the the Jacobian of the general coordinate system. The Jacobian is included in order to make�3(x) satisfy the normalization condition

R�3(x)dx=

R�3(x)jJ jdx1dx2dx3=1.]

Expression (8) can be considered as a transformation that transforms an arbitrary function from theguiding-center coordinates to the particle coordinates. Similarly, equation (6) can be written as

fg(X;v)=

Zfp(x;v)�

3(x¡X¡ �)dx; (10)

which can be considered as a transformation that transforms an arbitrary function from the the particlecoordinates to the guiding-center coordinates.

2.4 Moments of distribution function expressed as integration over guiding-center variables

In terms of particle variables (x;v), it is straightforward to calculate the moment of the distribution function.For example, the number density n(x) is given by

n(x) =

Zfp(x;v)dv: (11)

However, it is a little di�cult to calculate n(x) at real space location x by using the guiding-center variables(X;v). This is because holding x constant and changing v, as required by the integration in Eq. (11), meansthe guiding-center variable X is changing, according to Eq. (2). Using Eq. (5), expression (11) is written as

n(x)=

Zfg(X(x;v);v)dv; (12)

As is mentioned above, the dv integration in Eq. (12) should be performed by holding x constant andchanging v, which means the guiding-center variable X=X(x; v) is changing. This means that, in (X; v)space, the above integration is a (generalized) line integral along the the line X(v) = x ¡ �(x; v) with xbeing constant. By using the Dirac delta function �, this line integral can be written as the following two-dimensional integration over the independent variables X and v:

n(x)=

ZZfg(X;v)�3(X¡x+ �)dvdX: (13)

In gyrokinetic particle in cell (PIC) simulations, the integral (13) is evaluated by using Monte-Carlo markers.Actually evaluated in the simulation is the integral of a special fg that is independent of the gyro-angle �.Using v? and vk as the two remaining velocity coordinates, then fg is written as fg= fg(X; v?; vk). In thiscase fg is a function of �ve variables (rather than six variables).

Transform Vlasov equation from particle coordinates to guiding-center coordinates 3

In simulations, the position of a marker represents the guiding-center location (rather than the particlelocation), i.e., it is the guiding-center location that is directly sampled (the particle position is indirectlysampled, as is discussed below). For a marker with coordinators (X;v?; vk), we can calculate particle positionsby using the transformation (4). There are in�nite number of particle positions associated with the markersince the direction of v? (i.e., gyro-agnle) is not speci�ed. All these possible particle positions are on a circlearound the guiding-center position X. This circle is often called the gyro-ring. In particle simulations, thegyro-ring is descretized by several sampling points in calculating its integration contribution to grids. Wechoose N sampling points that are evenly distributed on the gyro-ring (N is usually 4 as a compromisebetween e�ciency and accuracy). Denote the Mote-Carlo weight of the j th marker by wj. Then the weightis evenly split by the N sub-markers on the gyro-ring since fg(X; v?; vk) is independent of the gyro-phase(put it another way, fg is uniformed distributed in the gyro-phase) Therefore each sub-marker have a Monte-Carlo weight wj/N . Then calculating the integration (13) at a grid corresponds to depositing all the N sub-markers associated with each guiding-center marker to the grid, as is illustrated in Fig. 1. A more accurateinterpretation of why the charge deposition can be done this way is discussed in Sec. 2.5.

A

B

C

Figure 1. The spatial grids in the plane perpendicular to the equilibrium magnetic �eld and one gyro-ring with 4sampling points (sub-markers) on it. For a guiding-center marker C with aMonte-Carlo weight of wj, the 4 sub-markersare calculated by using the transformation (4) (rebuilding the gyro-ring). TheMonte-Carlo weight of each sub-marker iswj/4. The value of integration (13) at a grid point is approximated by I /�V , where I is the Monte-Carlo integrationof all sub-markers associated with all guiding-center markers in the cell,�V is the cell volume. The cell associated witha grid-point (e.g., A) is indicated by the dashed rectangle (this is for 2D case, for 3D, it is a cube). If the Dirac deltafunction is used as the shape function of the sub-markers, then calculating the contribution of a sub-marker to a gridcorresponds to the nearest-point interpolation (for example, the 4 sub-markers will contribute noting to grid B sinceno-sub marker is located within the cell). In practice, the �at-top shape function with its support equal to the cell sizeis often used, then the depositing corresponds to linearly interpolating the weight of each sub-marker to the nearby grids.

The gyro-averaging of a perturbed �eld can be numerically calculated in a way similar to the above(discussed later).

2.5 Monte-Carlo sampling of 6D guiding-center phase-space

Suppose that the 6D guiding-center phase-space (X;v) is described by ( ; �; �; vk; v?; �) coordinates. TheJacobian of the coordinate system is given by J =J rv?, where J r is the Jacobian of ( ; �; �) coordinates.Suppose that we sample the 6D phase-space by using the following probability density function:

P ( ; �; �; vk; v?; �)=1

Vr

�m

2�T

�3/2

exp

"¡m (vk

2+ v?2 )

2T

#; (14)

4 Section 2

where Vr is the volume of the spatial simulation box, T is a constant temperature. (P given above is actuallyindependent of ; �; � and �.) It is ready to verify that P satis�es the following normalization condition:Z

Vr

ZPdvdX=

ZVr

Z¡1

+1Z0

1Z0

2�

Pv?d�dv?dvkJ rd d�d�=1: (15)

We use the rejection method to numerically generate Np markers that satisfy the above probability densityfunction. [The e�ective probability density function actually used in the rejection method is P 0, which isrelated to P by

P 0= jJ rJ vjP = jJ r jv?P : (16)

Note that P 0 does not depend on the gyro-angle �.] Then the phase-space volume occupied by a marker isgiven by

Vpj=1g=

1NpP

=Vr

Np

¡ m

2�T

�3/2exp

h¡m (vkj

2 + v?j2 )

2T

i; (17)

which is independent of j ; �j ; �j and �j. The weight of a marker is then given by wj=Vpj�fgj, where �fgis the perturbed distribution function evolved in simulations.

Although the distribution function �fg to be sampled in gyrokinetic simulations is independent of thegyro-angle �, we still need to sample the gyro-angle because we need to use the inverse guiding-centertransformation, which needs the gyro-angle. Each marker needs to have a speci�c gyro-angle value �j sothat we know how to transform its Xj to xj and then do the charge deposition in x space. To increase theresolution over the gyro-angle, we need to increase the number of markers. However, thanks to the fact thatboth sampling probability density function P and �fg are independent of �, the resolution over the gyro-angle can be increased in a simple way. Speci�cally, for a marker with (Xj ; vkj ; v?j), its gyro-angle value�j can be adjusted arbitrary without changing the value of its weight wj because both the phase-spacevolume Vpj and �fgj are independent of �j. In other words, we have the freedom of choosing the value of�j and the independence of Vpj and �fgj on �j guarantees that the weight of the marker is still equal to theoriginal value that this marker takes. Suppose that we choose 4 di�erent values of �j for the j th marker,denoted by �j1, �j2, �j3, and �j4. Then for each of the four gyro-angle values, we do the inverse guiding-center transformation and then do the charge deposition using their original weight wj for each marker andthen loop over all the markers. Denote a grid quantity (e.g. density) build by the deposition process byn1(x), n2(x), n3(x), and n4(x), corresponding to using the four gyro-angle values. Then the more accurateestimation of the grid quantity is given by

n(x) =n1(x)+n2(x)+n3(x) +n4(x)

4: (18)

This corresponds to sampling the 6D phase-space 4 separate times (each time with identical sampling pointsin (X; vk; v?) but di�erent sampling points in �) and then using the averaging of the 4 Monte-Carlo integralsto estimate the exact value. This estimation can also be (roughly) considered as a Monte-Carlo estimationusing 4 times larger number of markers as that is originally used (the Monte-Carlo estimation using truly 4time larger number of markers is more accurate than the result we obtained above because the former alsoincrease the resolution of (X; vk; v?) while the latter keeps the resolution of (X; vk; v?) unchanged.)

In numerical code, one may split the weight wj as wj / 4 in doing the deposition and then add thecontributions from all the 4 gyro-angles when doing the deposition and then one does not need to take theaveraging at the end. However, interpreting in this way is confusing to me because, with a single sampling ofthe phase-space, the phase-space volume or weight can not be easily split. I prefer the above interpretationthat the 6D phase space is sampled 4 separate times and thus we get 4 estimations and �nally we take theaveraging of these 4 estimations. It took me several days to �nally �nd this way of understanding.

In summary, the phase-space to be sampled in gyrokinetic simulations are still 6D rather than 5D. Inthis sense, the statement that gyrokinetic simulation works in a 5D phase space is misleading. We are stillworking in the 6D phase-space. The only subtle thing is that the sixth dimension, i.e., gyro-angle, can besampled in a easy way that is independent of the other 5 variables.


In numerical implementation, the gyro-angle may not be explicitly used. We just try to calculate 4arbitrary points on the gyro-ring that are easy to calculate. Some codes (e.g. ORB5) may introduce a randomvariable to rotate these 4 points for di�erent markers so that the gyro-angle is sampled more unbiased.

2.6 Choosing velocity coordinatesCoordinates of the velocity space can be chosen as (v; vk; �), or (v; v?; �), or (vk; v?; �), where vk and v? arethe parallel and perpendicular (with respect to B0(x)) velocity, respectively, � is the azimuthal angle in acylindrical/spherical velocity coordinate system. Here � is usually called the �gyro-angle�, which is the mostimportant variable we need because we need to directly perform averaging over this variable in deriving thegyrokinetic equation.

(**incorrect**The parallel direction is fully determined by B0(x), but there are degrees of freedom inchoosing one of the two perpendicular basis vectors. It seems that, in order to make the azimuthal angle� fully de�ned, we need to choose a way to de�ne one of the two perpendicular directions. It turns outthat this is not necessary because the gyro-angle is de�ned only locally in space. At each spatial point, theperpendicular directions can be arbitrarily chosen. In other words, for a given v, the corresponding gyro-angles � at di�erent spatial locations are unrelated, i.e., @�/@xjv is not de�ned.** Update: we DO need tode�ne the perpendicular direction at each spatial location to make @�/@xjv de�ned, which is needed in theVlasov di�erential operators. However, it seems that terms related to @�/@xjv are �nally dropped due tothat they are of higher order, check*** )

The gyro-angle is a velocity variable that we will stick to. The other velocity variables, vk, v?, and v,can be replaced by other choices. In Frieman-Chen's paper, these variables are replaced by

"= "(v;x)� v2

2+q�0(x)m

; (19)

and

�= �(v?;x)�v?2

2B0(x); (20)

where �0(x) is the equilibrium (macroscopic) electrical potential, v? = jv?j, and v? = v ¡ v � ek is theperpendicular velocity. Note that ("; �; �) is not su�cient in uniquely determining a velocity vector. Anadditional parameter � = sign(vk) is needed to determine the sign of vk = v � ek. In the following, thedependence of the distribution function on � is often not explicitly shown in the variable list (i.e., � ishidden/suppressed), which, however, does not mean that the distribution function is independent of �.Another frequently used velocity coordinates are (�; vk; �). In the following, I will derive the gyrokineticequation in ("; �; �) coordinates. After that, I transform it to (�; vk; �) coordinates.

2.7 Spatial gradient operator in terms of guiding-center variablesUsing the chain-rule, the spatial gradient @f /@x is written

@fp@xjv=

@X@x

� @fg@X

+@"@x

@fg@"

+@�@x

@fg@�

+@�@x

@fg@�

: (21)

From the de�nition of X, Eq. (2), we obtain

@X@x

= I+v� @@x

� ek

�; (22)

where I is the unit dyad. From the de�nition of ", we obtain

@"@x

=¡ qmE0; (23)

where E0=¡@�0/@x. [From the de�nition of �, we obtain

@�@x

=¡ v?2

2B02

@B0@x

+12B0

@v?2

@x=¡ �

B0

@B0@x

+@v?@x

� v?B0

(24)

6 Section 2

Using@v?@x

=@[v¡ vkek]

@x=¡vk

@ek@x

¡@vk@x

ek; (25)

expression (24) is written as@�@x

=¡ �B0

@B0@x

¡ vk@ek@x

� v?B0; (26)

which agrees with Eq. (10) in Frieman-Chen's paper[3]. (Note that the partial derivative @/@x is taken byholding v constant. Since the direction of B0 is spatially dependent and thus v? and vk are also spatiallydependent when holding v constant.) To obtain an explicit formula for @�/@x, we need to choose a per-pendicular (to B0) direction. Then an explicit formula can be derived (not given here), which involves thegradient of B0.]

Using the above results, equation (21) is written as

@fp@xjv=

@fg@X

+v� @@x

� ek

�� @fg@X

¡ qmE0@fg@"

+@�@x

@fg@�

+@�@x

@fg@�

: (27)

For notation ease, de�ne

�B1=v�@@x

� ek

�� @@X

; (28)

and

�B2=@�@x

@@�

+@�@x

@@�; (29)

then the above expression is written as

@fp@xjv=

@fg@X

+ [�B1+�B2]fg¡qmE0@fg@"

: (30)

2.8 Velocity gradient operator in terms of guiding-center variablesNext, consider the form of the velocity gradient @f /@v in terms of the guiding-center variables. Using thechain rule, @f /@v is written

@fp@vjx=

@X@v

� @fg@X

+@"@v

@fg@"

+@�@v

@fg@�

+@�@v

@fg@�

: (31)

From the de�nition of X, we obtain

@X@v

=@@v

�v� ek

�=

@v

@v�ek

= I�ek: (32)

From the de�nition of ", we obtain@"@v

=v; (33)

From the de�nition of �, we obtain@�@v

=v?B0; (34)

From the de�nition of �, we obtain

@�@v

=1v?

�ek�

v?v?

�=e�v?; (35)

where e� is de�ned by

e�= ek�v?v?: (36)

Using the above results, expression (31) is written

@fp@vjx=

I� ek

�@fg@X

+v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

: (37)


2.9 Time derivatives in terms of guiding-center variablesIn the guiding-center variables, the time partial derivative @f /@t appearing in Vlasov equation is written as

@fp@tjx;v=

@fg@tjX;V+

@X@t� @fg@X

+@V@t� @fg@V

; (38)

where V = ("; �; �). Here @X/@t and @V/@t are not necessarily zero because the equilibrium quantitiesinvolved in the de�nition of the guiding-center transformation are in general time dependent. This timedependence is assumed to be evolving slow in the gyrokinetic ordering discussed later.

2.10 Final form of Vlasov equation in guiding-center coordinatesUsing the above results, the Vlasov equation in guiding-center variables is written

@fg@t

+@X@t� @fg@X

+@V@t� @fg@V

+ v ��@fg@X

+ [�B1+�B2]fg¡q

mE0@fg@"

�+

qm(E+v�B) �

�I� ek

�@fg@X

+ v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

�= 0; (39)

Using tensor identity a � I �b= a�b, equation (39) is written as

@fg@t

+@X@t� @fg@X

+@V@t� @fg@V

+ v ��@fg@X

+ [�B1+�B2]fg¡qmE0@fg@"

�+

q

m(E+v�B)�

� ek

�� @fg@X

+q

m(v�B) �

�e�v?

@fg@�

�+

qmE ��v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

�= 0; (40)

This is the Vlasov equation in terms of guiding-center variables.

3 Perturbed Vlasov equation in guiding-center variablesSince the de�nition of the guiding-center variables (X; "; �; �) involves the macroscopic (equilibrium) �eldsB0 and E0, to further simplify Eq. (40), we need to separate electromagnetic �eld into equilibrium andperturbation parts. Writing the electromagnetic �eld as

E=E0+ �E (41)and

B=B0+ �B; (42)

then substituting these expressions into equation (40) and moving all terms involving the perturbed �eldsto the right-hand side, we obtain

@fg@t

+@X@t� @fg@X

+@V@t� @fg@V

+ v � @fg@X

+v ��[�B1+�B2]fg¡

qmE0@fg@"

�+

qm(E0+v�B0)�

� ek

�� @fg@X

+qm(v�B0) �

�e�v?

@fg@�

�+

qmE0 �

�v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

�= �Rfg; (43)

8 Section 3

where �R is de�ned by

�R = ¡ qm(�E+v� �B)�

� ek

�� @@X

¡ qm(v� �B) �

�e�v?

@@�

�¡ qm�E �

�v@@"

+v?B0

@@�

+e�v?

@@�

�: (44)

Next, let us simplify the left-hand side of Eq. (43). Note that

qmE0�

� ek

�� @fg@X

= c

�E0� ekB0

�� @fg@X

=vE0 �@fg@X

; (45)

where vE0 is de�ned by vE0= cE0�ek/B0, which is the macroscopic (equilibrium) �ow due to E0�B0 drift.Further note that

qmv�B0

c�� ek

�� @fg@X

= [(v� ek)�ek] �@fg@X

= [vkek¡v] �@fg@X

; (46)

which can be combined with v � @fg/@X term, yielding vkek � @fg/@X. Finally note that

q

m(v�B0) �

�e�v?

@fg@�

�=q

m(v�B0) �

ek�v?v?2

@fg@�

=¡@fg@�

: (47)

Using Eqs. (45), (46), and (47), the left-hand side of equation (43) is written as

@fg@t

+@X@t� @fg@X

+@V@t� @fg@V

+ (vkek+VE0) �@fg@X

+v � [�B1+�B2]fg¡@fg@�

+qmE0 �

�v?B0

@fg@�

+e�v?

@fg@�

��Lgfg (48)

which corresponds to Eq. (7) in Frieman-Chen's paper[3]. (In Frieman-Chen's equation (7), there is a term

qm(E¡E0) �v

@@"

where E is the macroscopic electric �eld and is in general di�erent from the E0 introduced when de�ningthe guiding-center transformation. In my derivation E0 is chosen to be equal to the macroscopic electric�eld thus the above term does not appear.) In expression (48), Lg is often called the unperturbed Vlasovpropagator in guiding-center coordinates (X; "; �; �).

Using the above results, Eq. (43) is written as

Lgfg= �Rfg; (49)

i.e.@fg@t

+@X@t� @fg@X

+@V@t� @fg@V

+(vkek+VE0) �@fg@X

+v ��v� @

@x

� ek

�� @fg@X

+@�@x

@fg@�

+@�@x

@fg@�

�¡@fg

@�

+qmE0 �

�v?B0

@fg@�

+e�v?

@fg@�

�=¡ q

m(�E+v� �B)�

� ek

�� @fg@X

¡ qm(v� �B) �

�e�v?

@fg@�

�¡ qm�E �

�v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

�: (50)

Perturbed Vlasov equation in guiding-center variables 9

It is instructive to consider some special cases of the above complicated equation. Consider the case that theequilibrium magnetic �eld B0 is uniform and time-independent, E0= 0, and the electrostatic limit �B= 0,then equation (50) is simpli�ed as

@fg@t

+ vkek �@fg@X

¡@fg@�

=¡ qm(�E)�

� ek

�� @fg@X

¡! spatial gradient drive (51)

¡ qm�E �

�v@fg@"

+v?B0

@fg@�

+e�v?

@fg@�

�¡! velocity space damping: (52)

3.1 Scale separation and two-scales expansion

Turbulence in tokamaks are usually of short perpendicular (to B0) spatial scale lengths of order �i, which ismuch smaller than the macroscopic scale length L0 (i.e., �= �i/L0 is a small parameter). Therefor physicalquantities fg can be separated into macroscopic and microscopic parts as

fg=Fg+ �Fg; (53)

where F is de�ned by

Fg�hfgiX?�Rfg d2X?Rd2X?

; (54)

which is the averaging of fg over (several times of) the short-scale perpendicular spatial scale. This is to say,Fg is constant over the short scale length �i in the perpendicular direction, i.e., the perpendicular spatial scalelength of Fg is much larger than �i. This long perpendicular scale length of Fg is denoted by L0. Equations(53) and (54) imply that

h�FgiX?=0: (55)

An example of this two-scales expansion in one-dimension case is given in Fig. 2.

0

0.10.20.30.40.50.60.70.80.91

1.1

0 0.2 0.4 0.6 0.8 1

x

f

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

x

F

δF

Figure 2. Given a f , then F is obtained from F = hf ix=Rxx+l

f(x)dx/Rxx+l

dx and �F is obtained from �F = f ¡F .Here l is a length comparable to the small scale-length of f .

The expansion for the electromagnetic �eld given by Eqs. (41) and (42) should also be considered to bein this two-spatial-scale expansion. Using this expression in Eq. (49), we obtain

LgFg+Lg�Fg= �RFg+ �R�Fg: (56)

Performing the perpendicular short scale length averaging on both sides of the above equation, we obtain

LgFg= h�R�FgiX?; (57)

10 Section 3

where use has been made of hLg�FgiX? = 0 and h�RFgiX? = 0. Subtracting the above equation from Eq.(56), we obtain

Lg�Fg=(�RFg+ �R�Fg¡h�R�FgiX?): (58)

3.2 Gyrokinetic orderingsThe following orderings assumed is often called the standard gyrokinetic orderings.

3.2.1 Ordering of macroscopic quantities

The spatial scale length L0 of macroscopic quantities is assumed to be much larger than the thermal iongyro-radius �i, i.e., �� i/L is a small parameter. De�ne the spatial scale length L0 as L0� Fg/ jrXFg j,then we have

�ijrXFg j �O(�1)Fg: (59)

This ordering is the result of the above two-scales expansion.The macroscopic quantities evolve on the long transport time scale, which is assumed to be

1

@Fg@t�O(�3)Fg; (60)

i.e., transport time scale is O(�¡3) longer than gyro-period.The equilibrium (macroscopic) E0�B0 �ow, vE0=E0�ek/B0=¡r�0�ek/B0, is assumed to be weak

withjvE0jvt

�O(�1); (61)

where vt is the thermal velocity.

3.2.2 Orderings of microscopic quantities

We consider low frequency perturbations with !/�O(�1), then

1@�Fg@t

�O(�1)�F : (62)

We consider small amplitude perturbations and adopt the following ordering for the microscopic �uctuations:

�FgFg� q��

T� j�Bj

B�O(�1); (63)

where �� is the perturbed scalar potential de�ned later in Eq. (68).The perturbation is a assumed to have a long wavelength (compared with �i) in the parallel direction

j�iek � rX�Fg j �O(�1)�Fg; (64)

[and have a short wavelength comparable to the ion gyro-radius in the perpendicular direction ** seems tobe not used in the derivation***

j�irX?�Fg j �O(�0)�Fg: (65)

Combining Eq. (64) and (65), we obtain

kkk?�ek � rX

rX?�O(�); (66)

i.e., the parallel wave number is one order smaller than the perpendicular wave-number.]In terms of the scalar and vector potentials �� and �A, the perturbed electromagnetic �eld is written as

�B=rx� �A; (67)

and

�E=¡rx��¡@�A

@t: (68)


Using the above orderings, it is ready see that �Ek is one order smaller than �E?, i.e.,

�Ek�E?

=O(�1): (69)

Further note that

�E?=¡r?��¡�@�A@t

�?; (70)

where the second term on the right-hand side is one order smaller than the �rst one.

3.3 Equation for macroscopic distribution function Fg

The evolution of the macroscopic quantity Fg is governed by Eq. (57), i.e.,

LgFg= h�R�FgiX?; (71)

where the left-hand side is written as

LgFg =@Fg@t

+@X@t� @Fg@X

+@V@t� @Fg@V

+ (vkek+VE0) �@Fg@X

+v � [(�B1+�B2)Fg]¡@Fg@�

+q

mE0 �

�v?B0

@Fg@�

+e�v?

@Fg@�

�and the right-hand side is written as

h�R�FgiX? = ¡ qm

�(�E+v� �B)�

� ek

�� @�Fg@X

�X?

¡ qm

�(v� �B) �

�e�v?

@�Fg@�

��X?

¡ qm

��E �

�v@�Fg@"

+v?B0

@�Fg@�

+e�v?

@�Fg@�

��X?

; (72)

which is of O(�2) (or O(�3)?), which indicates that the turbulence e�ects on the macroscopic quantitiesenters through terms of O(�2).

Expand Fg as Fg=Fg0+Fg1+ :::., where Fgi�Fg0O(�i). Then, the balance on order O(�0) gives

@Fg0@�

=0 (73)

i.e., Fg0 is independent of the gyro-angle �. The balance on O(�1) gives

vkek �@Fg0@X

+qmE0 �

�v?B0

@Fg0@�

�=

@Fg1@�

: (74)

Performing averaging over �,R0

2�(:::)d�, on the above equation and noting that Fg0 is independent of �, we

obtain �Z0

2�

d�vkek

�� @Fg0@X

+qm@Fg0@�

Z0

2�

d�E0 ��v?B0

�=

Z0

2�

d�@Fg1@�

(75)

Note that a quantity A=A(x) that is independent of v will depend on v when transformed to guiding-centercoordinates, i.e., A(x) = Ag(X; v). Therefore Ag depends on gyro-angle �. However, since �i/L � 1 forequilibrium quantities, the gyro-angle dependence of the equilibrium quantities can be neglected. Speci�cally,ek, B0 and can be considered to be independent of �. As to vk, we have vk=� 2("¡B0�)

p. Since B0 is

considered independent of �, so does vk. Using these results, equation (75) is written

vkek �@Fg0@X

+qm@Fg0@�

Z0

2�

d�E0 ��v?B0

�=0: (76)

Using E0=¡r�0, the above equation is written as

vkek �@Fg0@X

+qm@Fg0@�

Z0

2�

d�

�¡v? � r�0

B0

�=0; (77)

12 Section 3

Note that Z0

2�

d�1B0v? � rX�0� 0; (78)

where the error is of O(�2)�0, and thus, accurate to O(�), the last term of equation (77) is zero. Thenequation (77) is written as

vkek �@Fg0@X

=0; (79)

which implies that Fg0 is constant along a magnetic �eld line.

3.4 Equation for perturbed distribution function �Fg

Using Fg�Fg0, equation (58) is written as

Lg�Fg= �RFg0+ �R�Fg¡h�R�FgiX?|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }nonlinear terms

; (80)

where �R�Fg ¡ h�R�FgiX?are nonlinear terms which are of order O(�2) or higher, Lg�Fg and �RFg0 arelinear terms which are of order O(�1) or higher. The linear term �RFg0 is given by

�RFg0=¡qm

��E+

v� �Bc

�� ek

�� @Fg0@X||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

O(�2)

¡ qm�E �

�v@Fg0@"

+v?B0

@Fg0@�

�|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

O(�1)

; (81)

where use has been made of @Fg0/@�=0. Note that the �rst two terms are of order O(�2) while the secondtwo term are of order O(�1). The linear term Lg�Fg is written as

Lg�Fg�Lgf�Fg =@�Fg@t

+(vkek+VE0) �@�Fg@X

+v � [�B1+�B2]�Fg¡@�Fg@�|||||||||||||||||||||||||{z}}}}}}}}}}}}}}}}}}}}}}}}}

O(�1)

+qmE0 �

�v?B0

@�Fg@�

+e�v?

@�Fg@�

�; (82)

where use has been made of the assumption that the time partial derivative of the macroscopic quantitiesare of order O(�3) and thus can be dropped since we want to derive an equation for �Fg that is accurate toO(�2). The sub-index f in the notation Lgf means �frozen�, indicating the e�ect of equilibrium evolutionhas be neglected. Speci�cally, Lgf operator is given by

Lgf =@@t+(vkek+VE0) �

@@X

+v � [�B1+�B2]¡@@�

+qmE0 �

�v?B0

@@�

+e�v?

@@�

�: (83)

where �B1 and �B2 operators are given by expression (28) and (29). In Eq. (82), @�Fg/@� is of order O(�1)and all the other terms are of order O(�2).

Next, to reduce the complexity of algebra, we consider the easier case in which @Fg0/@�=0.

3.4.1 Separate �Fg into adiabatic and non-adiabatic parts

Note that the term @�Fg/@� in Eq. (82) and the last two terms in Eq. (81) are of O(�1). Write �Fg as

�Fg= �Fa+ �G; (84)

where

�Fa=q

m��@Fg0@"

; (85)

which depends on the gyro-angle via �� and this term is often called adiabatic term. Plugging expression(84) into equation (80), we obtain

Lgf�G= �RFg0¡Lgf�Fa|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }linear terms

+ �R�Fg¡h�R�FgiX?|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }nonlinear terms

: (86)


Next, let us focus on the linear term on the right-hand side, i.e, �RFg0¡Lgf�Fa, and prove this term is ofO(�2) or higher (this means that @�Fa/@� cancels all the O(�1) terms in �RFg0). Lgf�Fa is written

Lgf�Fa =qm@Fg0@"

Lgf��+qm��Lgf

@Fg0@"

� qm@Fg0@"

Lgf��; (87)

where the error is of order O(�3). In obtaining the above expression, use has been made of ek �@Fg0/@X=0,@Fg0/@X=O(�1)Fg0, @Fg0/@�= 0, @Fg0/@�= 0, and the de�nition of �B1 and �B2 given in expressions(28) and (29). The expression (87) involves �� imposed by the Vlasov propagator Lgf. Since �� takesthe most simple form when expressed in particle coordinates (if in guiding-center coordinates, ��(x) =��(X¡ v� ek/), which depends on velocity coordinates and thus more complicated), it is convenient touse the Vlasov propagator Lgf expressed in particle coordinates. Transforming Lgf back to the particlecoordinates, expression (87) is written

Lgf�Fa =q

m

@Fg0@"

�@��

@tjx;v+v � rx��+

q

m(E0+v�B0) �

@�

@vjx�

=qm@Fg0@"

�@��@tjx;v+v � rx��

�(88)

=qm

@Fg0@"

�@��@tjx;v+v �

�¡�E¡ @�A

@tjx;v

��=

q

m

@Fg0@"

�@��

@tjx;v¡v � �E¡

@v � �A@t

jx;v�: (89)

=qm@Fg0@"

�@��@t

¡v � �E¡ @v � �A@t

�: (90)

Using this and expression (81), �RFg0¡Lg�Fa is written as

�RFg0¡Lgf�Fa = ¡ qm(�E+v� �B)�

� ek

�� @Fg0@X

¡ qm�E �

�v@Fg0@"

�¡ q

m

@Fg0@"

�@��@t

¡v � �E¡ @v � �A@t

�= ¡ q

m(�E+v� �B)�

� ek

�� @Fg0@X

¡ qm@Fg0@"

�@�@t¡ @v � �A

@t

�; (91)

where the two terms of O(�1) (the terms in blue and red) cancel each other, with the remain terms beingall of O(�2), i.e, the contribution of the adiabatic term cancels the leading order terms of O(�1) on the RHSof Eq. (86). The consequence of this is that, as will see in Sec. 3.5.1, on order O(�1), �G is independent ofthe gyro-angle. Therefore separating �F into adiabatic and non-adiabatic parts also means separating �Finto gyro-angle dependent and gyro-angle independent parts.

3.4.2 To Frieman-Chen form

Let us write the linear terms in expression (91) into the Frieman-Chen form. The �E+v� �B term in Eq.(91) can be further written as

�E+v� �B = ¡rx��¡@�A@t

+v�rx� �A� ¡rx��+v�rx� �A;

where the error is of O(�2). Using the vector identity v�rx� �A=(r�A) �v¡ (v � r)�A and noting v isconstant for rx operator, the above equation is written

�E+v� �B=¡rx��+rx(�A �v)¡ (v � rx)�A (92)

Note that Eq. (30) indicates that rx�� rX��, where the error is of O(�2), then the above equation iswritten

�E+v� �B=¡rX��+rX(�A �v)¡ (v � rX)�A (93)

14 Section 3

Further note that the parallel gradients in the above equation are of O(�2) and thus can be dropped. ThenEq. (93) is written

�E+v� �B=¡rX?��+rX?(�A �v)¡ (v? � rX?)�A:

=¡rX?�L¡v? � rX?�A; (94)

where �L is de�ned by�L= ��¡v � �A: (95)

Using expression (94), equation (91) is written

�RF0¡Lg�Fa=¡qm

h(¡rX?�L¡v? � rX?�A)�

ek

i� @Fg0@X

¡ qm@�L@t

@Fg0@"

; (96)

where all terms are of O(�2).

3.5 Equation for the non-adiabatic part �GPlugging expression (96) into Eq. (86), we obtain

Lgf�G=¡ qm

h(¡rX?�L¡v? � rX?�A)�

ek

i� @Fg0@X

¡ qm@�L@t

@Fg0@"

+�R�Fg¡h�R�FgiX? (97)

3.5.1 Expansion of �G

Expand �G as�G= �G0+ �G1+ :::;

where �Gi � O(�i+1)Fg0, and note that the right-hand side of Eq. (97) is of O(�2), then, the balance onorder O(�1) requires

@�G0

@�=0; (98)

i.e., �G0 is gyro-phase independent. The balance on order O(�2) requires

@�G0

@t+ vkek �

@�G0

@X+v � [�B1+�B2]�G0

= ¡ qm

h(¡rX?�L¡v? � rX�A)�

ek

i� @Fg0@X

¡ qm@�L@t

@Fg0@"

+�R�Fg¡h�R�FgiX?: (99)

3.5.2 Gyro-averaging

De�ne the gyro-average operator h:::i� by

hhi�=(2�)¡1Z0

2�

hd�; (100)

where h = h(X; �; "; �) is an arbitrary function of guiding-center variables. [For a �eld quantity that isindependent of the velocity in particle coordinates, i.e., h=h(x), it is ready to see that the above averagingis a spatial averaging over a gyro-ring]

Gyro-averaging Eq. (99), we obtain

@�G0@t

+

�vkek �

@�G0

@X

�+ hv � [�B1+�B2]�G0i�

=¡ qm

h¡rX?h�Li��

ek

i� @Fg0@X

¡ qm@ h�Li�@t

@Fg0@"

+ h�R�Fgi�¡hh�R�FgiX?i�; (101)

where use has been made of h(v? � rX)�Ai�� 0, where the error is of order higher than O(�2). Note thatvk = � 2("¡B0�)

p. Since B0 is approximately independent of �, so does vk. Using this, the �rst gyro-

averaging on the left-hand side of the above equation is written�vkek �

@�G0

@X

��

= hvkeki �@�G0@X

= vkek �@�G0

@X(102)


The second gyro-averaging appearing on the left-hand side can be written

hv � [�B1+�B2]�G0i�=VD � rX�G0; (103)

where VD is the magnetic curvature and gradient drift (I do not derive Eq. (103), to be done). Then Eq.(101) is written�

@@t+ vkek � rX+VD � rX

��G0

=¡ qm

h¡rX?h�Li��

ek

i� @Fg0@X

¡ qm@ h�Li�@t

@Fg0@"

+ h�R�Fgi�¡hh�R�FgiX?i�: (104)

3.5.3 Simpli�cation of the nonlinear term

Next, we try to simplify the nonlinear term h�R�Fgi� appearing in Eq. (104), which is written as

h�R�Fgi� =

��R

�qm��

@Fg0@"

+ �G0

��

=

�qm�R

��

@Fg0@"

��

+ h�R�G0i� (105)

First, let us focus on the �rst term, which can be written as

�R

��@Fg0@"

�� ¡ q

m@Fg0@"

��E+

v� �Bc

�� ek

�� @��@X

¡ qm@Fg0@"

�v� �Bc

��e�v?

@��@�

�¡ q

m

@Fg0@"

�E ��v@��

@"+v?B0

@��

@�+e�v?

@��

@�

�+q

m��E �v@

2Fg0@"2

= ¡ qm

��E+

v� �Bc

�� rv��+

qm��E �v@

2Fg0@"2

=qm��E �v@

2Fg0@"2

(106)

Using the above results, the nonlinear term h�R�F i� is written as

h�R�F i�=qm

��E �v@

2Fg0@"2

��

+ h�R�G0i� (107)

Accurate to O(�2); the �rst term on the right-hand side of the above is zero. [Proof:��E �v@

2Fg0@"2

��

=

�@2Fg0@"2

��r�� v��

=@2Fg0@"2

hv � r(��)2i�

� @2Fg0@"2

hv? � r(��)2i�� 0; (108)

where use has been made of hv? �rX��i��0, where the error is of O(�2). Using the above results, expression(107) is written as

h�R�Fgi�= h�R�G0i�: (109)

Using the expression of �R given by Eq. (44), the above expression is written as

h�R�G0i� = ¡ qm

��E+

v� �Bc

�� ek

��

� @�G0@X

¡ qm@�G0

@"h�E �vi�¡

qm@�G0@�

��E � v?

B0

��

(110)

16 Section 3

where use has been made of @�G0/@�=0. Using Eq. (94), we obtain

¡ q

m

D(�E+v� �B)�

� ek

�E�=q

mrX?h�Li��

ek: (111)

The other two terms in Eq. (110) can be proved to be zero. [Proof:

¡ qm@�G0

@"h�E �vi� =

qm@�G0

@"hv � rx�i�

� qm@�G0

@"hv? � rx�i�

� q

m

@�G0

@"hv? � rX�i�

� 0 (112)

¡ qm@�G0@�

��E � v?

B0

��

=qm@�G0

@�

�1B0v? � rx�

��

� qm@�G0

@�

�1B0v? � rX�

��

� 0 (113)

] Using the above results, the nonlinear term is �nally written as

h�R�G0i�=qmrX?h�Li��

ek� rX�G0: (114)

Using this in Eq. (109), we obtain

h�R�Fgi�=qmrX?h�Li��

ek� rX�G0; (115)

which is of O(�2). Using this form, it can be proved that hh�R�Fgi�iX? is of O(�3) (to be proved later).

3.5.4 Final equation for the non-adiabatic part of the perturbed distribution function

Using the above results, the gyro-averaged kinetic equation for �G0 is �nally written as�@@t+�vkek+VD¡

qmrX?h�Li��

ek

�� rX

��G0

=qm

hrX?h�Li��

ek

i� rXFg0¡

qm@ h�Li�@t

@Fg0@"

: (116)

where �L = �� ¡ v � �A, h:::i� is the gyro-phase averaging operator, VD is the equilibrium guiding-centerdrift velocity, and �G0= �G0(X; "; �) is gyro-angle independent and is related to the perturbed distributionfunction �Fg by

�Fg=qm��

@Fg0@"

+ �G0; (117)

where the �rst term is called the adiabatic term and this term depends on the gyro-phase � via ��. Equation(116) is the special case (@Fg0/@�j"= 0) of the Frieman-Chen nonlinear gyrokinetic equation given in Ref.[3]. Note that the nonlinear terms only appear on the left-hand side of this equation and all the terms onthe right-hand side are linear.

4 Characteristic curves of Frieman-Chen nonlinear gyrokineticequation

The Frieman-Chen nonlinear gyrokinetic equation takes the following form:

@�G0

@t+�vkek+VD ¡

qmrX h�Li� �

ek

�� rX�G0=

qm

hrXh�Li� �

ek

i� rXF0¡

qm@ h�Li�@t

@F0@"

¡ qmSl2;

(118)

Characteristic curves of Frieman-Chen nonlinear gyrokinetic equation 17

where �G0= �G0(X; "; �), which is gyro-phase independent, �L= ��¡v � �A, and the term Sl2 is due to the� dependence of F0, which is not considered in this note. Examining the left-hand side of Eq. (118), it isready to �nd that the characteristic curves of this equation are given by the following equations:

dXdt

= vkek+VD¡qmrXh�Li��

ek; (119)

d"dt

=0; (120)

d�dt

=0: (121)

(It is instructive to notice that the kinetic energy " is conserved along the characteristic curves while thereal kinetic energy of a particle is usually not conserved in a perturbed electromagnetic �eld. This may bean indication that Frieman-Chen equation neglects the velocity space nonlinearity.) For notation ease, wede�ne the perturbed drift �VD by

�VD=¡qmrXh�Li��

ek: (122)

and the total guiding-center velocity VG by

VG=dXdt

= vkek+VD+ �VD: (123)

Equations (119)-(121) for the characteristics are, however, not in the form that can be readily evolvednumerically because there is no time evolution equation for vk, which appears explicitly in Eq. (119). It isready to realize that the equation for vk is implicitly contained in the combination of the equations for " and�. Next, we derive the equation for vk.

4.1 Time evolution equation for vkUsing the de�nition �= v?2 /(2B0), equation (121), i.e., d�/dt=0, is written as

ddt

�v?2

2B0

�=0; (124)

which is written as1

B

d

dt(v?2 )+ v?

2 d

dt(1

B0)= 0; (125)

which can be further written asddt(v?2 ) =2�

ddt(B0): (126)

Using the de�nition of the characteristics, the right-hand side of the above equation can be expanded, giving

ddt(v?2 ) =2�

�@B0@t

+dXdt� rXB0+

d"dt

@B0@"

+d�dt

@B0@�

�; (127)

where dX/dt, d"/dt, and d�/dt are given by Eq. (119), (120), and (121), respectively. Using Eqs. (119)-(121) and @B0/@t=0, equation (127) is reduced to

ddt(v?2 )= 2�

�dXdt� rXB0

�: (128)

On the other hand, equation (120), i.e., d"/dt=0, is written as

ddt(v2)= 0; (129)

which can be further written asddt(vk2)=¡ d

dt(v?2 ): (130)

Using Eq. (128), the above equation is written as

ddt(vk)=¡

�vk

�dXdt� rXB0

�; (131)

18 Section 4

which is the equation for the time evolution of vk. This equation involves dX/dt, i,e., the guiding-centerdrift, which is given by Eq. (119). Equation (131) for vk can be simpli�ed by noting that the Frieman-Chenequation is correct only to the second order, O(�2), and thus the characteristics need to be correct only tothe �rst order O(�) and higher order terms can be dropped. Note that, in the guiding-center drift dX/dtgiven by Eq. (119), only the vkek term in is of order O(�0), all the other terms are of O(�1). Using this,accurate to order O(�1), equation (131) is written as

ddt(vk) =¡�ek � rXB0; (132)

which is the time evolution equation ready to be used for numerically advancing vk. Note that only themirror force ¡�ek � rB appears in Eq. (132) and there is no parallel acceleration term qvk�Ek/m in Eq.(132). This is because �Ek=¡b � r��¡ @�Ak/@t is of order O(�2) and (**check**the terms involving Ekare of O(�3) or higher and thus have been dropped in the process of deriving Frieman-Chen equation.)

5 Gyrokinetic equation in forms amenable to numerical simulation

In the case of isotropic F0, Frieman-Chen's gyrokinetic equation for �G0 is given by�@

@t+(vkek+VD+ �VD) � rX

��G0

=¡�VD � rXF0¡qm@ h�Li�@t

@F0@"

; (133)

where �G0 is the gyro-phase independent part of the perturbed distribution �F , and is related to �F by

�F =q

m��

@F0@"

+ �G0; (134)

where the �rst term is called �the adiabatic term�, which depends on gyro-phase � via ��. In Eq. (133),�L= ��¡v � �A.

5.1 Eliminate @h��i�/@t term on the right-hand side of GK equationNote that the coe�cient before @F0/@" in Eq. (133) involves the time derivative of h��i�, which is problem-atic if treated by using explicit �nite di�erence in particle simulations (I test the algorithm that treats thisterm by implicit scheme, the result roughly agrees with the standard method discussed in Sec. 5.6). It turnsout that @ h��i�/@t can be readily eliminated by de�ning another gyro-phase independent function �f by

�f =q

mh��i�

@F0@"

+ �G0: (135)

Then, in terms of �f , the perturbed distribution function �F is written as

�F =q

m(��¡h��i�)

@F0@"

+ �f: (136)

Using Eq. (135) and Eq. (133), the equation for �f is written as�@@t+(vkek+VD+ �VD) � rX

��f

¡ qm@F0@"

�@@t+(vkek+VD+ �VD) � rX

�h��i�

¡ qmh��i�


�@F0@"

=¡�VD � rXF0¡qm@ h�Li�@t

@F0@"

(137)

Gyrokinetic equation in forms amenable to numerical simulation 19

Noting that @F0/@t= 0, ek � rF0= 0, rF0�O(�1)F0, we �nd that the third line of the above equation isof order O(�3) and thus can be dropped. Moving the second line to the right-hand side and noting thath�Li�= h��¡v � �Ai�, the above equation is written as�

@@t+(vkek+VD+ �VD) � rX

��f

=¡�VD � rXF0

¡ qm

�¡@ hv � �Ai�

@t¡ (vkek+VD+ �VD) � rXh��i�

�@F0@"

; (138)

where two @ h�i� / @t terms cancel each other. Equation (138) corresponds to Eq. (A8) in Yang Chen'spaper[2] (where the �rst minus on the right-hand side is wrong and should be replaced with q/m; one q ismissing before @(v � �A)/@t in A9).

The blue term in expression (136) will give rise to the so-called �polarization density� when integrated inthe velocity space (discussed in Sec. 5.6). The reason for the name �polarization� is that (��¡h��i�) is thedi�erence between the local value and the averaged value on a gyro-ring, expressing a kind of �separation�.This �separation� is obviously proportional to the gyro-radius and hence larger for massive particles with largevelocity. In tokamak plasma, thermal ion gyro-radius is mi/me

p� 60 times larger than thermal electron

gyro-radius. Therefore the ion polarization density is much larger than the electron polarization densityand the latter is often dropped in simulations except for studying electron temperature gradient (ETG)turbulence.

5.2 Eliminate @h�v � �Ai�/@t term on the right-hand side of GK equationSimilar to the method of eliminating @ h��i�/@t, we de�ne another gyro-phase independent function �h by

�h= �f ¡ qmhv � �Ai�

@F0@"

: (139)

then Eq. (138) is written in terms of �h as�@@t+(vkek+VD+ �VD) � rX

��h

+qm@F0@"

��@@t+ vkek+VD+ �VD

�� rX

�hv � �Ai�

+q

mhv � �Ai�

��@

@t+ vkek+VD+ �VD

�� rX

��@F0@"

�=¡�VD � rXF0

¡ qm

�¡@ hv � �Ai�

@t¡ (vkek+VD+ �VD) � rXh��i�

�@F0@"

; (140)

Noting that @F0/@t= 0, ek � rF0= 0, rF0�O(�1)F0, we �nd that the third line of the above equation isof order O(�3) and thus can be dropped. Moving the second line to the right-hand side and noting thath�Li�= h��¡v � �Ai�, the above equation is written as�


��h

=¡�VD � rXF0

¡ q

m[(vkek+VD+ �VD) � rX(hv � �A¡ ��i�)]

@F0@"

; (141)

where two @ hv � �Ai�/@t terms cancel each other and no time derivatives of the perturbed �elds appear onthe right-hand side. Noting that �VD given by Eq. (122) is perpendicular to rX hv � �A¡ ��i� and thus theblue term in Eq. (141) is zero, then Eq. (141) simpli�es to�


��h

=¡�VD � rXF0

¡ qm[(vkek+VD) � rX(hv � �A¡ ��i�)]

@F0@"

: (142)

20 Section 5

5.2.1 For special case �A� �Akek

Most gyrokinetic simulations approximate the vector potential as �A��Akek. In this case, accurate to O(�2),equation (142) can be further written as�


��h

=¡�VD � rXF0

¡ qm[¡VG � rXh��i�+ h�Aki�vkek � rX(vk)+ vkVG � rX(h�Aki�)]

@F0@"

; (143)

Using Eq. (222), we obtain rX(vk)=¡� (rB0)/vk. Using this, equation (143) is further written as�@@t+(vkek+VD+ �VD) � rX

��h

=¡�VD � rXF0

¡ q

m[¡VG � rXh��i�¡h�Aki��ek � rB0+ vkVG � rX(h�Aki�)]

@F0@"

; (144)

which agrees with the so-called pk formulation given in GEM code manual (the �rst line of Eq. 28), whichuses pk= vk+ qhAki�/m as an independent variable.

5.3 Summary of split of the distribution functionIn the above, the perturbed part of the distribution function, �F , is split at least three times in order to(1) simplify the gyrokinetic equation by splitting out the adiabatic response and (2) eliminate the timederivatives, @�� / @t and @�A / @t, on the right-hand. To avoid confusion, I summarize the split of thedistribution function here. The total distribution function F is split as

F =F0+ �F ; (145)

where F0 is the equilibrium distribution function and �F is the perturbed part of the total distributionfunction. �F is further split as

�F = �h+qm(��¡h��i�)

@F0@"

+qmhv � �Ai�

@F0@"

; (146)

where �h satis�es the gyrokinetic equation (142) or (144). In Eq. (146), the red term gives rise to the so-called polarization density (discussed in Sec. 5.6). The analytic dependence of this term on �� is utilized insolving the Poisson equation. The blue term also has an analytic dependence on �A, which, however, willcause numerical problems in particle simulations (so-called �cancellation problem� in gyrokinetic simulations)if it is utilized in solving the Ampere equation.

5.3.1 Velocity space moment of q

mhv � �Ai�@F0@"

Consider the approximation �A� �Akek, then the blue term in Eq. (146) is written as

qmhvk�Aki�

@F0@"

: (147)

For electrons, the FLR e�ect can be neglected and then the above expression is written

qmvk�Ak

@F0@"

: (148)

The zeroth order moment (number density) is then written as

qm�Ak

Zvk@F0@"

dv; (149)

which is zero if F0 is Maxwellian. The parallel current is given by

�jk=q2

m�Ak

Zvk2@F0@"

dv: (150)

If F0 is a Maxwellian distribution, then


5.4 Generalized split-weight scheme for electronsFor electrons, due to their small Larmor radius, the di�erence ��¡h��i� can be neglected. Then Eq. (146)is reduced to

�F = �h+qmhv � �Ai�

@F0@"

; (151)

where the adiabatic response is cancelled out. The generalized split-weight scheme introduces again theadiabatic response but with a free small parameter �g. Speci�cally, �h is split as

�h= �hs+ �gqm��

@F0@"

: (152)

Substituting this expression into Eq. (142), we obtain the following equation for �hs:�@@t+(vkek+VD+ �VD) � rX

��hs

+�gq

m

@F0@"

�@

@t+(vkek+VD+ �VD) � rX

��

+�gqm��


�@F0@"

=¡�VD � rXF0


@F0@"

: (153)

Noting that @F0/@t=0, ek � rF0=0, rF0�O(�1)F0, we �nd that the third line of the above equation is oforder O(�3) and thus can be dropped. Moving the second line to the right-hand side, the above equation iswritten as �


��hs

=¡�VD � rXF0

¡ qm

�(vkek+VD) � rXhv � �A¡ ��i�+ �g

�@��@t

+VG � rX��

��@F0@"

: (154)

Equation (154) agrees with Eq. (39) in the GEM manual. Note that the right-hand side of Eq. (154) containsa nonlinear term VG �rX��, which is di�erent from the original Frieman-Chen equation, where all nonlinearterms appear on the left-hand side. For the special case of �g= 1 (the default and most used case in GEMcode, Yang Chen said �g=/ 1 cases are sometimes not accurate, so he gave up using it since 2009), the aboveequation can be simpli�ed as (again nelecting the di�erence between �� and h��i�):�


��hs

=¡�VD � rXF0

¡ q

m

�VG � rXhv � �Ai�+

@��

@t

�@F0@"

: (155)

Since the original Frieman-Chen equation already splits the perturbed distribution into adiabatic part andnonadiabatic part, equation (155) actually goes back to the original Frieman-Chen equation. The onlydi�erence is that q

mhv � �Ai�@F0@" is further split from the perturbed distribution function. Considering this,

equation (155) can also be obtained from the original Frieman-Chen equation (133) by write �G0 as

�G0= �hs+qmhv � �Ai�

@F0@"

; (156)[In this case �F is written as

�F = �hs+qm��

@Fg0@"

+qmhv � �Ai�

@F0@"

; (157)

22 Section 5

] Substituting expression (156) into equation (133), we obtain the following equation for �hs:�@@t+(vkek+VD+ �VD) � rX

��hs

+qm@F0@"


�hv � �Ai�

+qmhv � �Ai�


�@F0@"

=¡�VD � rXF0¡qm@ h��¡v � �Ai�

@t@F0@"

; (158)

Noting that @F0/@t=0, ek � rF0=0, rF0�O(�1)F0, we �nd that the third line of the above equation is oforder O(�3) and thus can be dropped. Moving the second line to the right-hand side, the above equation iswritten as �


��hs

=¡�VD � rXF0

¡ qm

�@ h��i�@t

+ [(vkek+VD+ �VD) � rX]hv � �Ai��@F0@"

; (159)

which agrees with Eq. (155).

5.5 Comments on how to split the distribution functionIn particle simulations, the seemingly trivial thing on how to split the distribution function is always con-sidered to be a big deal. Separating the perturbed part from the equilibrium part is considered to be a bigdeal and gives rise to the famous name ��f particle method�, in contrast to the conventional particle methodwhich is now called full-f particle method. Summarizing the above result, we know that the total distributionfunction F is split in the following form:

F = F0+ �F

= F0+ �h+qm(��¡h��i�)

@F0@"

+qmhv � �Ai�

@F0@"

; (160)

and only �h is actually evolved by using markers and its moment in the phase-space is evaluated via Monte-Carlo integration. The blue and red terms in the above expression depends on the perturbed �eld and theirvalue can be obtained from the values on the grid and the velocity integration can be performed analytically.However, in some cases, the phase space integration of the blue terms must be evaluated using markers, i.e.,using Monte-Carlo method, to avoid the inaccurate cancellation between the integration of these parts andthe integration of �h (which are computed using Monte-Carlo method). When will the inaccurate cancellationis signi�cant depends on the problem being investigated and thus can only be determined by actual numericalexperiments. Many electromagnetic particle simulation experiments indicate that the parallel current carriedby the blue term must be evaluated via Monte-Carlo method, otherwise inaccurate cancellation between thisterm and �h will give rise to numerical instabilities.

5.6 Poisson's equation and polarization densityPoisson's equation is written as

¡ "0r2��= qi�ni+ qe�ne; (161)

where ¡"0r2�� is called the space-charge term. Since we consider modes with kk� k?, the space-chargeterm is approximated as r2��r?2 ��+rk2��r?2 ��. Then Eq. (161) is written as

¡ "0r?2 ��= qi�ni+ qe�ne: (162)

This approximation eliminates the parallel plasma oscillation from the system. The perpendicular plasmaoscillations seem to be partially eliminated in system with gyrokinetic ions and drift-kinetic electrons. Thereare the so-called H modes that appear in the gyrokinetic system which have some similarity with the plasmaoscillations but with a much smaller frequency H� (kk/k?)(�D/�s)!pe. (electrostatic shear Alfven wave)


Using expression (146), the perturbed ion density �ni is written as

�ni =

Z�Fdv

=

Z�hdv+

Z �qm(��¡h��i�)

@F0@"

�dv+

Z �qmhv � �Ai�

@F0@"

�dv; (163)

where the last term (in blue) is approximately zero for isotropic F0 and this term is usually dropped insimulations that assume isotropic F0 and approximate �A as �Akek. The second term (in red) in expression(163) is the so-called the polarization density np, i.e.,

�npi(x) =

Zqimi

(��¡h��i�)@Fi0@"

dv; (164)

which has an explicit dependence on �� and is usually moved to the left hand of Poisson's equation whenconstructing the numerical solver of the Poisson equation, i.e., equation (162) is written as

¡ "0r?2 ��¡ qiZ

qimi

(��¡h��i�)@Fi0@"

dv= qi�ni0+ qe�ne; (165)

where �ni0 = �ni ¡ �npi =R�hi dv, which is evaluated by using Monte-Carlo markers. Since some parts

depending on �� are moved from the right-hand side to the left-hand side of the �eld equation, numericalsolvers (for ��) based on the left-hand side of Eq. (165) probably behaves better than the one that is basedon the left-hand side of Eq. (162), i.e., ¡"0r?2 ��.

The polarization density is part of the perturbed density that is extracted from the right-hand side andmoved to the left-hand side. The polarization density will be evaluated analytically, which is independentof Monte-Carlo markers, whereas the remained density on the right-hand side will be evaluated using Mote-Carlo markers. The two di�erent methods of evaluating the two di�erent parts of the total perturbed densitycan possibly introduce signi�cant systematic errors especially if the two terms are expected to cancel eachother and give a small quantity that is much smaller than either of the two terms. This is one pitfall forPIC simulations that extract some parts from the source term and move them to the left-hand side. Toremedy this, we can introduce an additional term on the right-hand that gives the di�erence between thepolarization density evaluated by the two methods. This is often called the cancellation scheme. It turnsout the cancellation scheme is not necessary for Eq. (165), but for the �eld solver for Ampere's equation(discussed later), this cancellation scheme is necessary in order to obtain stable results.

Next, let us perform the gyro-averaging and velocity integration in expression(164). Since �� is inde-pendent of velocity coordinates, the �rst term (adiabatic term) in expression (164) is trivial and the velocityintegration can be readily performed (assume F0 is Maxwellian), givingZ

qm(��)

@F0@"

dv:

=qm(��)

Z �¡mTfM

�dv:

=¡qn0T��: (166)

Next, we try to perform the gyro-averaging and the velocity integration of the second term in expression(164). In order to perform the gyro-averaging, we expand �� in wave-number space as

��(x)=

Z��kexp(ik �x)

dk(2�)3

; (167)

and we need to express x in terms of the guiding center variables (X; v) since the gyro-averaging is takenby holding X rather than x constant. The guiding-center transformation gives

x=X+ �(x;v)�X¡v�ek(X)

(X)(168)

24 Section 5

Using this, the gyro-average of �� is written as

h��i� =

�Z��kexp(ik �x)

dk(2�)3

��

=

�Z��kexp(ik � (X+ �))

dk(2�)3

��

=

Z��kexp(ik �X)hexp(ik � �)i�

dk

(2�)3

=

Z��kexp(ik �X)

�exp�¡ik �v�

ek(X)

(X)

��

dk(2�)3

: (169)

When doing the gyro-averaging,X is hold constant and thus ek(X) is also constant. Then it is straightforwardto de�ne the gyro-angle �. Let k? de�ne one of the perpendicular direction e1, i.e., k?=k?e1. Then anotherperpendicular basis vector is de�ned by e2=ek� e1. Then v? is written as v?= v?(e1cos�+ e2sin�), whichde�nes the gyro-angle �. Then the expression in Eq. (169) is written as

¡ik �v�ek(X)

(X)= ¡ik � v?(e1cos�+ e2sin�)�

ek(X)

(X)

= ¡ik � v?(X)

(¡e2cos�+ e1sin�)

= ¡ik?v?

sin�: (170)

Then the gyro-averaging in expression (169) is written as�exp�¡ik �v�

ek(X)

(X)

��

=

�exp�¡ik?v?

sin�

��

=12�

Z0

2�

exp�¡ik?v?

sin�

�d�

= J0

�k?v?

�: (171)

where use has been made of the de�nition of the zeroth Bessel function of the �rst kind. Then h��i� inexpression (169) is written as

h��i�=Z��kexp(ik �X)J0

�k?v?

�dk(2�)3

: (172)

Next, we need to perform the integration in velocity space, which is done by holding x rather thanX constant.Therefore, it is convenient to transform back to particle coordinates. Using X = x + v � ek(x)

(x), expression

(172) is written as

h��i�=Z��kexp(ik �x)J0

�k?v?

�exp�ik �v�

ek

�dk(2�)3

: (173)

Then the velocity integration is written asZh��i�

@F0@"

dv =


ZJ0

�k?v?

�exp�ik �v�

ek

�@F0@"

dvdk(2�)3

: (174)

Similar to Eq. (170), except for now at x rather than X, ik �v� ek

is written as

ik �v�ek= i

k?v?

sin�: (175)

Since this is at x rather than X, k?, v?, and are di�erent from those appearing in expression (170).However, since this di�erence is due to the variation of the equilibrium quantity ek/ in a Larmor radius,and thus is small and is ignored in the following.

Plugging expression (175) into expression (174) and using dv= v?dv?dvkd�, we obtainZh��i�

@F0@"

dv=


ZJ0

�k?v?

�exp�ik?v?

sin��@F0@"

v?dv?dvkd�dk(2�)3

: (176)


Note that @F0/@" is independent of the gyro-angle � in terms of guiding-center variables. When transformedback to particle coordinates, X contained in @F0/@" will introduce � dependence via X= x+ v� ek

. This

dependence on � is weak since the equilibrium quantities can be considered constant over a Larmor radiusdistance evaluated at the thermal velocity. Therefore this dependence can be ignored when performing theintegration over �, i.e., in terms of particle coordinates, @F0/@" is approximately independent of the gyro-angle �. Then the integration over � in Eq. (176) can be performed, yieldingZ

h��i�@F0@"

dv =


ZZJ0

�k?v?

��Z0

2�

exp�ik?v?

sin��d�

�@F0@"

v?dv?dvkdk(2�)3

=


�ZZJ0

�k?v?

�2�J0

�k?v?

�@F0@"

v?dv?dvk

�dk(2�)3

; (177)

where again use has been made of the de�nition of the Bessel function.

5.6.1 Special case: F0 is Maxwellian

In order to perform the remaining velocity integration in expression (177), we assume that F0 is a Maxwelliandistribution given by

F0= fM =n0(X)

(2�T (X)/m)3/2exp�¡mv22T (X)

�=

n0

(2�)3/2vt3exp�¡v22vt

2

�; (178)

where vt= T /mp

, then@F0@"

=¡mTfM: (179)

Again we will ignore the weak dependence of n0(X) and T0(X) on v introduced by X=x+v� ek/ whentransformed back to particle coordinates (for su�ciently large velocity, the corresponding Larmor radius willbe large enough to make the equilibrium undergo substantial variation. Since the velocity integration limitis to in�nite, this will de�nitely occur. However, F0 is exponentially decreasing with velocity, making thoseparticles with velocity much larger than the thermal velocity negligibly few and thus can be neglected).

Plugging (179) into expression (177), we obtainZh��i�

@F0@"

dv = ¡mT

n0

(2�)3/2


ZZ2�J0

2

�k?v?

�exp

¡vk2+ v?

2

2vt2

!1

vt3v?dv?dvk

dk(2�)3

;

= ¡mT

n0

(2�)1/2

Z��kexp(ik � x)

Z0

1Z¡1

¡1J02

�k?vt

v?

�exp

¡vk2+ v?

2

2

!v? d v? d vk

dk(2�)3

;

(180)

where vk= vk/vt, v?= v?/vt. UsingZ¡1

1exp�¡x

2

2

�dx= 2�

p(181)

the integration over vk can be performed, yieldingZh��i�

@F0@"

dv = ¡mTn0


Z0

1J02

�k?vt

v?

�exp�¡v?

2

2

�v? d v?

dk(2�)3

(182)

Using (I veri�ed this by using Sympy)Z0

1J02(ax)exp

�¡x

2

2

�xdx= exp(¡a2)I0(a2); (183)

where I0(a) is the zeroth modi�ed Bessel function of the �rst kind, expression (182) is writtenZh��i�

@F0@"

dv=¡mTn0

Z��kexp(ik �x)exp(¡b)I0(b)

dk

(2�)3; (184)

where b= k?2 vt2/2. Then the corresponding polarization density is written asZq

mh��i�

@F0@"

dv=¡qn0T

Z��kexp(ik �x)exp(¡b)I0(b)

dk

(2�)3: (185)

26 Section 5

In Fourier space, the adiabatic term in expression (166) is written as

¡ qn0T��=¡qn0

T


dk(2�)3

: (186)

Plugging expression (184) and (186) into expression (164), the polarization density np is �nally written as

np = ¡qn0T

Z��kexp(ik �x)[1¡ exp(¡b)I0(b)]

dk(2�)3

: (187)

= ¡qn0T

Z��kexp(ik �x)[1¡¡0]

dk(2�)3

; (188)

where¡0= exp(¡b)I0(b): (189)

Expression (188) agrees with the result given in Yang Chen's notes.

5.6.2 Pade approximation¡0 de�ned in Eq. (189) can be approximated by the Pade approximation as

¡0�1

1+ b: (190)

The comparison between the exact value of ¡0 and the above Pade approximation is shown in Fig. 3.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

Γ0

k⊥ρ

exa t

Pade approximation

Figure 3. Comparison between the exact value of ¡0= exp(¡(k?�)2)I0((k?�)2) and the corresponding Pade approx-imation 1/(1+ (k?�)2).

Using the Pade approximation (190), the polarization density np in expression (188) can be written as

np � ¡qn0T


k?2 �2

1+ k?2 �2

dk(2�)3

: (191)

5.6.3 Long wavelength approximation of the polarization densityIn the long wavelength limit, k?�� 1, expression (191) can be further approximated as

np � ¡qn0T

Z��kexp(ik �x)k?2 �2

dk

(2�)3;

=qn0T

�2r?2 ��: (192)

Then the corresponding term in the Poisson equation is written asq"0np =

q2n0"0T

�2r?2 ��

=�2

�D2 r?

2 ��; (193)


where �D is the Debye length de�ned by �D2 = T"0/(n0q2). For typical tokamak plasmas, the thermal ion

gyroradius �i is much larger than �D. This indicates that the term in expression (193) for ions is muchlarger than the space charge term r2�� r?2 �� +rk2�� r?2 �� in the Poisson equation. Therefore thespace charge term can be neglected for ion-scale modes, indicating that gyrokinetic plasmas are intrinsicallycharge-neutral on the ion-scale. In other words, the ion polarization shielding is dominant compared withthe Debye shielding for ion-scale low-frequency micro-instabilities. For electron-scale modes, since �e��D,the space charge term is comparable to the polarization density and should be included.

5.6.4 Polarization density expressed in terms of Laplacian operator

The polarization density expression (192) is for the long wavelength limit, which partially neglects FLRe�ect. Let us go back to the more general expression (191). In the wave-number space, the Poisson equationis written

¡"0r?2 ��= qi�ni+ qe�ne: (194)

Write �ni=npi+ �ni0, where �npi is the ion polarization density, then the above expression is written

¡"0r?2 ��¡ qinpi= qi�ni0+ qe�ne: (195)

The Fourier transforming in space, the above equation is written

¡ "0k?2 ��¡ qinpi= qi�ni0+ qe�ne; (196)

where npi is the Fourier transformation (in space) of the polarization density npi and similar meanings for��, �ni0, and �ne. Expression (191) implies that npi is given by

npi=¡qini0Ti

��k?2 �i

2

1+ k?2 �i

2 : (197)

Using this, equation (196) is written

¡"0k?2 ��¡ qi�¡qini0

Ti

k?2 �i

2

1+ k?2 �i

2��

�= qi�ni

0+ qe�ne; (198)

Multiplying both sides by (1+ k?2 �i2)/"0, the above equation is written

¡(1+ k?2 �i2)k?2 ��¡qi"0

�¡qini0

Ti(k?2 �i

2)��

�=1"0(1+ k?

2 �i2)(qi�ni

0+ qe�ne): (199)

Next, transforming the above equation back to the real space, we obtain

¡(1¡ �i2r?2 )r?2 ��¡qi"0

�qini0Ti

�i2r?2 ��

�=1"0(1¡ �i2r?2 )(qi�ni0+ qe�ne): (200)

Neglecting the Debye shielding term, the above equation is written

¡��i2

�Di2 r?2 ��

�=1"0(1¡ �i2r?2 )(qi�ni0+ qe�ne); (201)

which is the equation actually solved in many gyrokinetic codes, where �Di2 = "0Ti/(qi2ni0).

6 SummaryEquations (142) and (188) are the primary results derived in this note. For reference ease, let us summarizethe results. The total distribution function F is split as

F =F0+ �F ; (202)

where F0 and �F are the equilibrium part and perturbed part of the distribution function, respectively. �Fis further split as

�F = �h+q

m(��¡h��i�)

@F0@"

+q

mhv � �Ai�

@F0@"

; (203)

28 Section 6

where �h= �h(X; �; ") satis�es the following nonlinear gyrokinetic equation (142), i.e.,�@@t+(vkek+VD+ �VD) � rX

��h

= ¡�VD � rXF0


@F0@"

: (204)

where VD is the equilibrium guiding-center drift velocity and �VD is the perturbed drift velocity given by(122), i.e.,

�VD=qmrX hv � �A¡ ��i��

ek: (205)

Here the independent variables for the distribution functions F0 and �h are (X; �;") withX being the guiding-center position, � = v?

2 /(2B0), and "= v2/2. In the above, �� and �A is the perturbed electric potentialand vector potential, h:::i� is the gyro-averaging operator over the gyro-phase �, ek=B0/B0, =B0q/mwith q and m being the particle charge and mass, respectively.

When integrated in velocity space, the red term in expression (203) gives rise to the polarization densitynp, i.e.,

np=

Zqm(��¡h��i�)

@F0@"

dv; (206)

which, for Maxwellian equilibrium distribution, can be written in wave-number space as

np=¡qn0T

Z��kexp(ik �x)[1¡¡0]

dk(2�)3

; (207)

where ¡0= exp(¡b)I0(b), b=k?2 vt2/2, and I0(b) is the zeroth modi�ed Bessel function of the �rst kind. Thisexpression is useful for gyrokinetic simulations that use spectral methods in solving the Poisson equation.

These notes were initially written when I visited University of Colorado at Boulder (Sept.-Nov. 2016),where I worked with Dr. Yang Chen, who pointed out that most gyrokinetic simulations essentially employFrieman-Chen's nonlinear gyrokinetic equation. Therefore a careful re-derivation of the equation to knowthe gyrokinetic orderings and physics included in the model is highly desirable, which motivates me to writethis note.

Appendix A Diamagnetic �ow

The perturbed distribution function �F given in Eq. (136) contains two terms. The �rst term is gyro-phasedependent while the second term is gyro-phase independent. The perpendicular velocity moment of the �rstterm will give rise to the so-called �E�B0 �ow (seems wrong, need checking) and the second term will giverise to the so-called diamagnetic �ow. Let us discuss the diamagnetic �ow �rst. For this case, it is crucialto distinguish between the distribution function in terms of the guiding-center variables, fg(X;v), and thatin terms of the particle variables, fp(x;v). In terms of these denotations, equation (136) is written as

�Fg=qm(��¡h��i�)

@F0g@"

+ �fg: (208)

Next, consider the perpendicular �ow U? carried by �fg. This �ow is de�ned by the corresponding distrib-ution function in terms of the particle variables, �fp, via,

nU?=

Zv?�fp(x;v)dv; (209)

where n is the number density de�ned by n=R�fpdv. Using the relation between the particle distribution

function and guiding-center distribution function given by Eq. (7), i.e.,

�fp(x;v) = �fg(x¡ �;v); (210)

Diamagnetic flow 29

equation (209) is written as

nU?=

Zv?�fg(x¡ �;v)dv: (211)

Using the Taylor expansion near x, �fg(x¡ �;v) can be approximated as

�fg(x¡ �;v)� �fg(x;v)¡ � � r�fg(x;v): (212)

Plugging this expression into Eq. (211), we obtain

nU?�Zv?�fg(x;v)dv¡

Zv?� � r�fg(x;v)dv (213)

As mentioned above, �fg(x;v) is independent of the gyro-angle �. It is obvious that the �rst integration iszero and thus Eq. (213) is reduced to

nU?=¡Zv?� � r�fg(x;v)dv (214)

Using the de�nition �=¡v� ek/, the above equation is written

nU? =

Zv?v� ek

� r�fg(x;v)dv

=

Zv?� ek�r�fg(x;v)

��v?dv:

=

Zv?H �v?dv; (215)

whereH=ek

�r�fg(x;v), which is independent of the gyro-angle � because both ek(x)/(x) and �fg(x;v)

are independent of �. Next, we try to perform the integration over � in Eq. (215). In terms of velocity spacecylindrical coordinates (vk; v?; �), v? is written as

v?= v?(xcos�+ ysin�); (216)

where x and y are two arbitrary unit vectors perpendicular each other and both perpendicular to B0(x). Hcan be written as

H=Hxx+Hyy; (217)

where Hx and Hy are independent of �. Using these in Eq. (215), we obtain

nU? =

Zv?(xcos�+ ysin�) v?(Hxcos�+Hysin�)dv

=

Zv?2 [x(Hxcos2�+Hysin�cos�) + y(Hxcos�sin�+Hysin2�)] dv: (218)

Using dv= v?dvkdv?d�, the above equation is written as

nU? =

Z¡1

1dvk

Z0

1v?dv?

Z0

2�

v?2 [x(Hxcos2�+Hysin�cos�)+ y(Hxcos�sin�+Hysin2�)]d�

=

Z¡1

1dvk

Z0

1v?dv?

Z0

2�

v?2 (xHxcos2�+ yHysin2�)d�

=

Z¡1

1dvk

Z0

1v?dv?[v?

2 (xHx�+ yHy�)]

=

Z¡1

1dvk

Z0

1v?dv?[v?

2H�]

=

Z¡1

1dvk

Z0

1v?dv?[v?

2 ek�r�fg(x;v)�]

=ek�r

Z¡1

1dvk

Z0

1v?dv?�fg(x;v)

v?2

22�

=ekm

�r�p?; (219)

30 Appendix A

where

�p? �Z¡1

1dvk

Z0

1v?dv?�fg(x;v)

mv?2

22�

=

Z�fg(x;v)

mv?2

2dv; (220)

is the perpendicular pressure carried by �fg(x;v). The �ow given by Eq. (219) is called the diamagnetic �ow.

Appendix B Transform gyrokinetic equation from (X; �; "; �) to(X; �; vk; �) coordinates

The gyrokinetic equation given above is written in terms of variables (X; �; ";�), where � is the gyro-phase.Next, we transform it into coordinates (X0; �0; vk; �0) which is de�ned by8>>><>>>:

X0(X; �; "; �)=X�0(X; �; "; �) = ��0(X; �; "; �)=�

vk(X; �; "; �)=� 2("¡ �B0(X))p (221)

Use this de�nition and the chain rule, the gradient operators in (X; �; "; �) variables are written, in termsof (X0; �0; vk; �0) variables, as

@@X

��;";�

=@X0

@X� @@X0

��0;vk;�0

+@�0

@X@@�0

��X0;vk;�0

+@vk@X

@@vk

��X0;�0;�0

+@�0

@X@@�0

��X0;�0;vk

=@@X0

+0@@�0

¡ �vk

@B0@X

@@vk

+0@@�0

(222)

and

@

@"

��X;�;�

=@X0

@"

@

@X0+@�0

@"

@

@�0+@vk@"

@

@vk+@�0

@"

@

@�0

= 0@

@X0+0

@

@�0+

1

vk

@

@vk+0

@

@�0(223)

Then, in terms of independent variable (X0; �0; vk; �0), equation (133) is written�@@t+(vkek+VD+ �VD) �

@@X0

��G0¡ (vkek+VD+ �VD) �

�vk

@B0dX

@�G0

@vk

=¡�VD ��@F0@X0

¡ �vk

@B0dX

@F0@vk

�¡ qm@ h�Li�@t

@F0@vk

1vk; (224)

where �VD and h�Li� involve the gyro-averaging operator h:::i�. The gyro-averaging operator in (X0; �0; vk;�0) coordinates is similar to that in the old coordinates since the perpendicular velocity variable � is identicalbetween the two coordinate systems. Also note that the perturbed guiding-center velocity �VD is given by

�VD=ek�rX h��i�

B0+ vk

h�B?i�B0

; (225)

where @ /@X (rather than @ /@X0) is used. Since ��(x) = ��g(X; �0; �0), which is independent of vk, thenEq. (222) indicates that @��/@X= @��/@X0.

Dropping terms of order higher than O(�2), equation (224) is written as�@@t+(vkek+VD+ �VD) �

@@X0

��G0¡ ek � �rB0

@�G0

@vk

=¡�VD ��@F0@X0

�+

��VD � �rB0¡

qm@ h�Li�@t

�@F0@vk

1vk; (226)

Transform gyrokinetic equation from (X; �; "; �) to (X; �; vk; �) coordinates 31

Similarly, in terms of independent variable (X0; �0; vk; �0), equation (138) is written as�@@t+(vkek+VD+ �VD) �

@@X0

��f ¡ ek � �rB0

@�f@vk

=¡�VD ��@F0@X0

�+ �VD �

��vkrB0

@F0@vk

�¡ qm

�¡@ hv � �Ai�

@t¡�vkek+VD+ vk

h�B?i�B0

�� rXh��i�

�@F0@vk

1vk; (227)

The guiding-center velocity in the macroscopic (equilibrium) �eld is given by

vkek+VD=B0?

Bk0? vk+

�Bk0

? B0�rB0+1

B0Bk0? E0�B0 (228)

whereB0?=B0+B0

vkr�b; (229)

Bk?�b �B?=B

�1+

vkb � r�b

�; (230)

Using Bk0? �B0, then expression (228) is written as

vkek+VD= vkb+vk2

r�b||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

curvature drift

+�

B0B0�rB0||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }rB drift

+1

B02E0�B0|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

E�B drift

; (231)

where the curvature drift, rB drift, and E0�B0 drift can be identi�ed. Note that the perturbed guiding-center velocity �VD is given by

�VD=ek�rX h��i�

B0+ vk

h�B?i�B0

: (232)

Using the above results, equation (227) is written as�@@t+(vkek+VD+ �VD) �

@@X0

��f ¡ ek � �rB0

@�f@vk

=¡�VD ��@F0@X0

�+

�ek�rXh��i�

B0+ vk

h�B?i�B0

��vkrB0

@F0@vk

�¡ qm

"¡@ hv � �Ai�

@t¡ vkek+

vk2

r� b+ �

B0B0�rB0+

1

B02E0�B0+ vk

h�B?i�B0

!� rXh��i�

#@F0@vk

1vk;

(233)

Collecting coe�cients before @F0/@vk, we �nd that the two terms involving rB0 (terms in blue and red)cancel each other, yielding�

@@t+(vkek+VD+ �VD) �

@@X0

��f ¡ ek � �rB0

@�f@vk

=¡�VD ��@F0@X0

�+qm

"mqvkh�B?i�B0

� (�rB0) +@ hv � �Ai�

@t+

vkb +

vk2

r � b +

1

B02E0 � B0 + vk

h�B?i�B0

!�

rXh��i�

#@F0@vk

1vk; (234)

This equation agrees with Eq. (8) in I. Holod's 2009 pop paper (gyro-averaging is wrongly omitted in thatpaper) and W. Deng's 2011 NF paper. Equation (234) drops all terms higher than O(�2) and as a resultthe coe�cient before @�f /@vk contains only the mirror force, i.e.,

dvkdt

=¡ek � �rB0; (235)

32 Appendix B

which is independent of any perturbations.

Appendix C Transform gyrokinetic equation from (��; �A) to (�E;�B)

C.1 Expression of �B? in terms of �ANote that

�B? = r� �A¡ (ek � r� �A)ek= r� (�A?+ �Akek)¡ [ek � r� (�A?+ �Akek)]ek (236)

Correct to order O(�), �B? in the above equation is written as (ek vector can be considered as constantbecause its spatial gradient combined with �A will give terms of O(�2), which are neglected)

�B? � r� �A?+r�Ak� ek¡ [ek � r� �A?+ ek � (r�Ak� ek)]ek (237)= r� �A?+r�Ak� ek¡ (ek � r� �A?)ek (238)

Using local cylindrical coordinates (r; �; z) with z being along the local direction of B0, and two componentsof A? being Ar and A�, then r�A? is written as

r� �A?=�¡@�A�

@z

�er+

�@�Ar@z

�e�+

1r

�@@r(r�A�)¡

@�Ar@�

�ek: (239)

Note that the parallel gradient operator rk�ek �r=@ /@z acting on the the perturbed quantities will resultin quantities of order O(�2). Retaining terms of order up to O(�), equation (239) is written as

r� �A?�1r

�@@r(r�A�)¡

@�Ar@�

�ek; (240)

i.e., only the parallel component survive, which exactly cancels the last term in Eq. (238), i.e., equation(238) is reduced to

�B?=r�Ak� ek: (241)

C.2 Expression of �Bk in terms of �A

�Bk = ek � r� �A= ek � r� (�A?+ �Akek) (242)

Accurate to O(�1), �Bk in the above equation is written as (ek vector can be considered as constant becauseits spatial gradient combined with �A will give O(�2) terms, which are neglected)

�Bk � ek � r� �A?+ ek � (r�Ak� ek)= ek � r� �A? (243)

[Using local cylindrical coordinates (r; �; z) with z being along the local direction of B0, and two componentsof �A? being �Ar and �A�, then r� �A? is written as

r� �A?=�¡@�A�

@z

�er+

�@�Ar@z

�er+

1r

�@@r(r�A�)¡

@�Ar@�

�ek (244)

Note that the parallel gradient operator rk�ek �r=@ /@z acting on the the perturbed quantities will resultin quantities of order O(�2). Retaining terms of order up to O(�), equation (239) is written as

r� �A?�1r

�@@r(r�A�)¡

@�Ar@�

�ek; (245)

Transform gyrokinetic equation from (��; �A) to (�E; �B) 33

Using this, equation (243) is written as

�Bk=1r

�@@r(r�A�)¡

@�Ar@�

�: (246)

However, this expression is not useful for GEM because GEM does not use the local coordinates (r; �; z).]

C.3 Expressing the perturbed drift in terms of �E and �B

The perturbed drift �VD is given by Eq. (122), i.e.,

�VD=¡qmrXh�Li��

ek: (247)

Using �L= ��¡v � �A, the above expression can be further written as

�VD = ¡ qmrX h��¡v � �Ai��

ek

=qm

ek�rX h��i�¡

qm

ek�rXhvk�Aki�¡

qm

ek�rXhv? � �A?i�: (248)

Accurate to order O(�), the term involving �� isq

m

ek�rXh��i� =

ekB0�hrX��i�

�ekB0�hrx��i�

�ekB0��¡�E¡ @�A

@t

��

�ekB0�h¡�Ei�

� VE ; (249)

which is the �E�B0 drift. Accurate to O(�), the hvk�Aki� term on the right-hand side of Eq. (248) is written

¡ qm

ek�rXhvk�Aki� � ¡ q

m1hek�rX(vk�Ak)i�

� ¡ qm

1hek�rx(vk�Ak)i�

� ¡ qm

vkhek�rx(�Ak)i�

= vkh�B?i�B0

; (250)

which is due to the magnetic �uttering and this is actually not a real drift. In obtaining the last equality,use has been made of Eq. (241), i.e., �B?=rx�Ak� ek.

Accurate to O(�), the last term on the right-hand side of expression (248) is written

¡ qm

ek�rX hv? � �A?i� � ¡ 1

B0hek�rX(v? � �A?)i�

� ¡ 1B0hek�rx(v? � �A?)i�

= ¡ 1B0hek� (v?�rx� �A?+v? � rx�A?)i�

= ¡ 1B0h(ek � rx� �A?)v?+ ek�v? � rx�A?i�

Using equation (243), i.e., �Bk= ek � r� �A?, the above expression is written as

¡ qm

ek�rXhv? � �A?i� = ¡ 1

B0h�Bkv?+ ek�v? � rx�A?i�

� ¡ 1

B0h�Bkv?+ ek�v? � rX�A?i�

� ¡ 1B0h�Bkv?i�¡

1B0ek�hv? � rX�A?i�:

� ¡ 1

B0h�Bkv?i�: (251)

34 Appendix C

where use has been made of hv? � rX�A?i�� 0, where the error is of O(�)�A?. The term h�Bkv?i�/B0 isof O(�2) and thus can be neglected (I need to verify this).

Using Eqs. (249), (250), and (251), expression (248) is �nally written as

�VD�¡qmrXh�Li��

ek=h�Ei�� ek

B0+ vk

h�B?i�B0

: (252)

Using this, the �rst equation of the characteristics, equation (119), is written

dX

dt= vkek+VD+ �VD (253)

= vkek+VD+h�Ei�� ek

B0+ vk

h�B?i�B0

� VG (254)

C.4 Expressing the coe�cient before @F0/@" in terms of �E and �B

[Note that@�A?@t

=¡(�E?+r?��); (255)

where @�A?/@t is of O(�2). This means that �E?+r?�� is of O(�2) although both �E? and �� are of O(�).]Note that

@ hv � �Ai�@t

= vk@ h�Aki�

@t+v? �

@ h�Ai�@t

= vk@ h�Aki�

@t+ hv? � (¡�E¡r��)i�

� vk@ h�Aki�

@t¡hv? � �Ei� (256)

where use has been made of hv? � r��i � 0, This indicates that hv? � �Ei� is of O(�1)�E. Using Eq. (256),the coe�cient before @F0/@" in Eq. (138) can be further written as

¡ qm

�¡@ hv � �Ai�

@t¡�vkek+VD¡

qm

ek�rX hv � �Ai�

�� rX h��i�

�=¡ q

m

�¡vk

@ h�Aki�@t

+ hv? � �Ei�¡�vkek+VD¡

qm

ek�rXhv � �Ai�

��¡�E¡ @�A

@t

��

��¡ q

m

�¡vk

@ h�Aki�@t

+ hv? � �Ei�¡�vkek+VD¡

q

m

ek�rXhv � �Ai�

�� h¡�Ei�+ vk

�@Ak@t

��

�=¡ q

m

hhv? � �Ei�+

�vkek+VD¡

qm

ek�rX hv � �Ai�

�� h�Ei�

i�¡ q

m

�hv? � �Ei�+

�vkek+VD+ vk

h�B?iB0

�� h�Ei�

�: (257)

Using Eq. (257) and (132), gyrokinetic equation (138) is �nally written as�@@t+

�vkek+VD+

h�Ei�� ekB0

+ vkh�B?i�B0

�� rX

��f

=¡�h�Ei�� ek

B0+ vk

h�B?i�B0

�� rXF0

¡ q

m

�hv? � �Ei�+

�vkek+VD+ vk

h�B?i�B0

�� h�Ei�

�@F0@"

: (258)

Transform gyrokinetic equation from (��; �A) to (�E; �B) 35

Appendix D Drift-kinetic limitIn the drift-kinetic limit, hv? ��Ei�=0, h�Bkv?i�=0, and h�hi�=�h, where �h is an arbitrary �eld quantity.Using these, gyrokinetic equation (258) is written as�

@@t+

�vkek+vD+vE+ vk

�B?B0

�� rX

��f

=¡�vE+ vk

�B?B0

�� rXF0¡

qm

��vkek+vD+ vk

�B?B0

�� E

�@F0@"

: (259)

D.1 Linear caseNeglecting the nonlinear terms, drift-kinetic equation (259) is written�

@@t+(vkek+vD) � rX

��f

=¡�vE+ vk

�B?B0

�� rXF0¡

qm[(vkek+vD) � �E]

@F0@"

: (260)

Next let us derive the parallel momentum equation from the linear drift kinetic equation (this is needed inmy simulation). Multiplying the linear drift kinetic equation (260) by qvk and then integrating over velocityspace, we obtain

@�jk@t

= ¡qZdvvk(vkek+vD) � rX�f

¡ q

Zdvvk

�vE+ vk

�B?B0

�� rXF0¡

qmq

Zdvvk[(vkek+vD) � �E]

@F0@"

: (261)

Equation (261) involve rX�f and this should be avoided in particle methods whose goal is to avoid directlyevaluating the derivatives of �f over phase-space coordinates. On the other hand, the partial derivatives ofvelocity moment of �f are allowed. Therefore, we would like to make the velocity integration of �f appear.Note that rX�f here is taken by holding ("; �) constant and thus vk is not a constant and thus can not bemoved inside rX. Next, to facilitate performing the integration over vk, we transform the linear drift kineticequation (260) into variable (X; �; vk).

D.2 Transform from (X; �; ") to (X; �; vk) coordinatesThe kinetic equation given above is written in terms of variable (X; �; "). Next, we transform it intocoordinates (X0; �0; vk) which is de�ned by

X0(X; �; ") =X; (262)

�0(X; �; ") = �; (263)and

vk(X; �; ")= 2("¡ �B0(X))p

: (264)Use this, we have

@�G0@X

j�;" =@X0

@X@�G0

@X0+@�0

@X@�G0@�0

+@vk@X

@�G0@vk

=@�G0@X0

j�;vk+0@�G0

@�0¡ �vk

@B0dX

@�G0@vk

; (265)

and@F0@"

=@F0@�0

@�0

@"+@F0@vk

@vk@"

= 0@F0@�0

+@F0@vk

@vk@"

=@F0@vk

1vk

(266)

36 Appendix D

Then, in terms of variable (X0; �; vk), equation (260) is written

@�f

@t+(vkek+vD) � r�f ¡ ek � �rB

@�f

@vk

=¡�vE+ vk

�B?B0

�� rF0+

�vEvk

+�B?B0

�� rB @F0

@vk¡ qm

��ek+

vDvk

�� E

�@F0@vk

; (267)

where r� @/@X0j�;vk.

D.3 Parallel momentum equationMultiplying the linear drift kinetic equation (267) by qvk and then integrating over velocity space, we obtain

@�jk@t

+ q

Zdvvk(vkek+vD) � rX�f ¡ q

Zdvvkek � �rB

@�f@vk

=¡qZdvvk

�vE+ vk

�B?B0

�� rXF0+ q

Zdvvk

�vEvk

+�B?B0

�� rB @F0

@vk

¡ qmq

Zdvvk

��ek+

vDvk

�� E

�@F0@vk

: (268)

Consider the simple case that F0 does not carry current, i.e., F0(X; �; vk) is an even function about vk. Thenit is obvious that the integration of the terms in red in Eq. (268) are all zero. Among the rest terms, onlythe following term

¡ qmq

Zdvvk[(vkek) � �E]

@F0@vk

1vk

(269)

explicitly depends on �E. Using dv=2�Bdvkd�, the integration in the above expression can be analyticallyperformed, giving

¡ qmq

Zdvvk[(vkek) � �E]

@F0@vk

1vk

=¡q2

m

Z2�Bdvkd�vk�Ek

@F0@vk

=¡q2

m

Z2�Bd��Ek

Zvk@F0@vk

dvk

=¡q2

m

Z2�Bd��Ek

�0¡

ZF0dvk

�=q2

m�Ekn0: (270)

Using these results, the parallel momentum equation (268) is written

@�jk@t

=q2

m�Ekn0¡ q

Zdvvk(vkek+vD) � rX�f + q

Zdvvkek � �rB

@�f@vk

¡qZdvvk

�vk�B?B0

�� rF0+ q

Zdvvk

��B?B0

�� rB @F0

@vk; (271)

where the explicit dependence on �E is via the �rst term q2n0�Ek/m, with all the the other terms beingexplicitly independent of �E (�f and �B implicitly depend on �E).

Equation (271) involve derivatives of �f with respect to space and vk and these should be avoided in theparticle method whose goal is to avoid directly evaluating these derivatives. Using integration by parts, theterms involving @/@vk can be simpli�ed, yielding

@�jk@t

=q2

m�Ekn0¡ q

Zdvvk(vkek+vD) � rX�f ¡ q(ek � rB0)

Z��fdv

¡qZdvvk

�vk�B?B0

�� rF0¡ q

��B?B0

�� (rB0)

Z�F0 dv; (272)

Drift-kinetic limit 37

De�ne p?0=Rmv?

2 F0/2dv and �p?=Rmv?

2 �f /2dv, then the above equation is written

@�jk@t

=q2

m�Ekn0¡ q

Zdvvk(vkek+vD) � rX�f ¡ q(ek � rB0)

�p?mB0

¡qZdvvk

�vk�B?B0

�� rF0¡ q

��B?B0

�� (rB0)

p?0mB0

; (273)

Next, we try to eliminate the spatial gradient of �f by changing the order of integration. The second termon the right-hand side of Eq. (273) is written

¡qZdvvk(vkek) � rX�f ;

=¡qZ2�B0 dvkd�vk

2ek � rX�f

=¡q2�B0 ek � rX

Zvk2�fdvkd�

=¡qB0 ek � rX

�1

mB0

Zmvk

2�fdv

�=¡qB0 ek � rX

��pkmB0

�; (274)

where �pk=Rmvk

2�fdv. Similarly, the term ¡qRdvvk

�vk

�B?B0

�� rXF0 is written as

¡qZdvvk

�vk�B?B0

�� rXF0

=¡qZ2�B0 dvkd�

�vk2�B?B0

�� rXF0

=¡q��B?B0

��B0rX

Z(vk2F02�dvkd�)

=¡q��B?B0

��B0rX

�1B

Z(mvk

2F0dv)

�=¡q�B? � rX

�pk0mB0

�(275)

where pk0=Rmvk

2F0dv. Similarly, the term ¡qRdvvkvD �rX�f can be written as the gradient of moments

of �f . Let us work on this. The drift vD is given by

vD=B0

vk

r�bBk? vk+

�Bk

?B0�rB0: (276)

where Bk?=B0

¡1+

vk

b � r�b

�(refer to my another notes). Using b � r�b� 0, we obtain Bk

?�B. ThenvD is written

vD=vk2

r�b+ �

b�rB0:

Using this and dv=2�B0 dvkd�, the term ¡qRdvvkvD � rX�f is written as

¡qZdvvkvD � rX�f = ¡q

Z2�B0 dvkd�vk

vk2

r�b+ �

b�rB0

!� rX�f

= ¡q2�B01(r�b ) � rX

Zvk3�fdvkd�¡q2�B0

1(b�rB0) � rX

Zvk��fdvkd�

= ¡qB01

(r�b ) � rX

�1

B0

Zvk3�fdv

�¡ qB0

1

(b�rB0) � rX

�1

B0

Zvk��fdv

�;

= ¡m(r�b ) � rX

�1B0

Zvk3�fdv

�¡m(b�rB0) � rX

�1B0

Zvk��fdv

�; (277)

38 Appendix D

which are the third order moments of �f and may be neglect-able (a guess, not veri�ed). Using the aboveresults, the linear parallel momentum equation is �nally written

@�jk@t

=e2ne0m

�Ek+ eB0b � rX

��pkmB0

�+ e(b � rB0)

�p?mB0

+e�B? � rX

�pk0mB0

�+ e

��B?B0

�� (rB0)

p?0mB0

:

¡m(r�b ) � rX

�1B0

Zvk3�fdv


�1B0

Zvk��fdv

�(278)

De�ne

D0=r�

pk0mB0

�+rB0B0

p?0mB0

; (279)

which, for the isotropic case (pk0= p?0= p0), is simpli�ed to

D0=rp0mB0

: (280)

then Eq. (278) is written as

@�jk@t

=e2n0m

�Ek+ e�B? �D0

+ eB0b � rX

��pkmB0

�+ e(b � rB0)

�p?mB0

¡ m(r�b ) � rX

�1B0

Zvk3�fdv


�1B0

Zvk��fdv

�: (281)

D.4 Special case in uniform magnetic �eldIn the case of uniform magnetic �eld, the parallel momentum equation (278) is written as

@�jk@t

=qmqEkne0¡ q ek � rX(�pk)¡ q

�B?B0

� rXpk0: (282)

D.5 Electron perpendicular �owUsing the gyrokinetic theory and taking the drift-kinetic limit, the perturbed perpendicular electron �ow,�Ve?, is written (see Sec. A or Appendix in Yang Chen's paper[2])

ne0�Ve?=ne0B0

�E�b||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }E�B �ow

¡ 1eB0

b�r�p?e|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }diamagnetic �ow

(283)

where ne0 is the equilibrium electron number density, �pe? is the perturbed perpendicular pressure ofelectrons.

D.5.1 Drift kinetic equation

Drift kinetic equation is written

@f@t+

�vkb~ +vD+

�E�bB0

�� rf +

�¡ em�Ek¡ �b~ � rB

�@f@vk

=0; (284)

where f = f(x; �; vk; t), � =mv?2 /B0 is the magnetic moment, b~ = b + �B?/B0, b =B0/B0 is the unit

vector along the equilibrium magnetic �eld, vD= vD(x; �; vk) is the guiding-center drift in the equilibriummagnetic �eld. �E and �B are the perturbed electric �eld and magnetic �eld, respectively.

Drift-kinetic limit 39

D.5.2 Parallel momentum equation

Multiplying the drift kinetic equation () by vk and then integrating over velocity space, we obtainZ@fe vk@t

dv+

Zvk

�vkb~ +vD+

�E�bB0

�� rf edv+

Zvk

�¡ em�Ek¡ �b~ � rB

�@fe@vk

dv=0; (285)

which can be written as

@Jke@t

+

Zvk

�vkb~ +vD+

�E� ekB0

�� rfe dv+

Zvk

�¡ em�Ek¡ �b~ � rB

�@fe@vk

dv=0; (286)

Using dv=B¡12�mdvkd�, the last term on the RHS of the above equation is writtenZZvk

�¡ em�Ek¡ �b~ � rB

�@fe@vk

dv

=

ZZvk

�¡ em�Ek¡ �b~ � rB

�@fe@vk

2�Bmdvkd�

=¡ em�Ek2�

Bm

ZZvk@fe@vk

dvkd�¡ (b~ � rB)2�Bm

Z�

Zvk@fe@vk

dvkd�

=¡ em�Ek2�

Bm

Z �vkfej¡1+1¡

Zfedvk

�d�¡ (b~ � rB)2� B

m

Z�

�vkfej¡1+1¡

Zfedvk

�d�

=em�Ek2�

Bm

ZZfedvkd�+(b~ � rB)2� B

m

Z�

Zfedvkd�

=em�Ekne+

ZZ�(b~ � rB)fedv (287)

� em�Ekne

ZZvk(vkb~) � rf edv

=

ZZb~ � r(vk2f e)dv

=

ZZb~ � r(vk2f e)2�

B

mdvkd�

=b~ � r�ZZ

vk2f e2�

1mdvkd�

�B0

=b~ � r�pkB0

�B0

=b~ � r�pk0+ �pk

B0

�B0

=b~ � r�pk0B0

�B0+b~ � r

��pkB0

�B0

�b � r�pk0B0

�B0+

�B?B0

� r�pk0B0

�B0+b � r

��pkB0

�B0

�b � r(pk0)+�B?B0

� r(pk0) +b � r(�pk) (288)

=�B?B0

� r(pk0) +b � r(�pk)

where use has been made of b � rpk0=0.

@�Jek@t

=¡ em�Ekne¡

�B?B0

� r(pk0)¡b � r(�pk) (289)

40 Appendix D

Using Eq. () in Eq. (), we obtain

�0eem�Ekne+b � r�r� �E=¡�0e

��B?B0

� r(pek0)+b � r(�pek)�

(290)

��

¡b � r�r� �E=¡�0e�¡q�B?

B0� rXpk0+

qmqEkne0¡ q ek � rX(�pk)

�: (291)

dddddZZvk

�vkb~ +vD+

�E�bB0

�� rf edv

=

ZZ �vkb~ +vD+

�E�bB0

�� r(vkf e)dv

=

ZZvkb~ � r(vkf e)dv+

ZZvD � r(vkf e)dv+

ZZ�E�bB0

� r(vkf e)dv

=

ZZb~ � r(vk2f e)dv+

ZZvD � r(vkf e)dv+

ZZ�E�bB0

� r(vkf e)dv

=

ZZb~ � r(vk2f e)2�

Bmdvkd�+

ZZvD � r(vkf e)2�

Bmdvkd�+

ZZ�E�bB0

� r(vkf e)2�Bmdvkd�

=

Zb~ � r

�vk2f e2�

1

mdvkd�

�B+


B

mdvkd�+

ZZ�E�bB0

� r(vkf e)2�B

mdvkd�

=b~ � r� pkB

�B+


Bmdvkd�+

ZZ�E�bB0


=b~ � r� pkB

�B+

ZZ1

mb� (�rB) � r(vkf e)2�

B

mdvkd�+

ZZ1

mb� (mvk2�) � r(vkf e)2�

B

mdvkd�

+

ZZ�E�bB0


Appendix E Modern view of gyrokinetic equation

The modern form of the nonlinear gyrokinetic equation is in the total-f form. The modern way of derivingthe gyrokinetic equation is to use transformation methods to eliminate gyro-phase dependence of the totaldistribution function and thus obtain an equation for the resulting gyro-phase independent distributionfunction (called gyro-center distribution function).

The resulting equation for the gyro-center distribution function is given by (see Baojian's paper)�@

@t+X_ � r+ v_k

@

@vk

�f(X; vk; �; t)= 0; (292)

where

R_ =VD+ekB0�hr��i�+ vk

h�B?i�B0

(293)

v_k=¡1mB?

Bk? � (qrh��i+ �rB0)¡

qm

@ h�Aki�@t

: (294)

Here the independent variables are gyro-center position X, magnetic moment � and parallel velocity vk.

Modern view of gyrokinetic equation 41

The gyro-phase dependence of the particle distribution can be recovered by the inverse transforma-tion of the transformation used before. The pull-back transformation gives rise to the polarization densityterm. (phase-space-Lagrangian Lie perturbation method (Littlejohn, 1982a, 1983), I need to read these twopapers.).

In the traditional iterative method of deriving the gyrokinetic equation, the perturbed distribution func-tion has two terms, one of which depends on the gyro-angle, the other of which does not. The former willgive rise to the the polarization density and the latter is described by the gyrokinetic equation.

Bibliography

[1] P J Catto. Linearized gyro-kinetics. Plasma Physics, 20(7):719, 1978.[2] Yang Chen and Scott E. Parker. Particle-in-cell simulation with vlasov ions and drift kinetic electrons. Phys. Plasmas,

16:52305, 2009.[3] E. A. Frieman and Liu Chen. Nonlinear gyrokinetic equations for low�frequency electromagnetic waves in general plasma

equilibria. Physics of Fluids, 25(3):502�508, 1982.

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