Gyrokinetic Fokker-Planck Collision Operator
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Li, B., and D. Ernst. “Gyrokinetic Fokker-Planck Collision
Operator.” Physical Review Letters 106.19 (2011). © 2011
American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.106.195002
Publisher
American Physical Society
Version
Final published version
Accessed
Thu May 26 11:22:51 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/65931
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PRL 106, 195002 (2011)
week ending
13 MAY 2011
PHYSICAL REVIEW LETTERS
Gyrokinetic Fokker-Planck Collision Operator
B. Li and D. R. Ernst
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 1 March 2011; published 9 May 2011)
The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for
plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes
both the test-particle and field-particle contributions, and automatically conserves particles, momentum,
and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both fieldparticle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects
can be precomputed. The field-particle operator at each time step requires the evaluation of a single twodimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than
conserving model operators.
DOI: 10.1103/PhysRevLett.106.195002
PACS numbers: 52.65.Ff, 52.30.Gz, 52.65.Tt
Collisions play an important role in plasma turbulence
and can strongly influence turbulent transport in magnetic
fusion experiments. Within terms of the order of the inverse Coulomb logarithm, large angle Coulomb collisions
and long time correlations, which relax on the short
time scale of plasma oscillations, can be neglected [1].
Considering small-angle Coulomb collisions results in
the Fokker-Planck collision operator [1,2]. The linearized
full collision operator includes the test-particle and fieldparticle operators, and satisfies Boltzmann’s H theorem
and the conservation of particles, momentum, and energy
[3,4]. The field-particle operator, which involves the
Rosenbluth potentials of the non-Maxwellian distribution,
has generally been considered intractable. It is often approximated by ‘‘conserving’’ terms that restore the conservation of low-order moments to the test-particle operator.
In situations involving frequencies much less than the
gyrofrequency, where the equilibrium varies weakly on
the gyroradius (Larmor radius) spatial scale, the FokkerPlanck equation can be averaged over the particle gyration,
reducing its dimensionality from six to five phase space
dimensions. When finite gyroradius effects are accounted
for, this transformation results in the gyrokinetic equation
[5–7]. This equation describes low-frequency turbulence
driven by gradients of the plasma density, temperature, and
flows across the magnetic field, in magnetically confined
laboratory plasmas. However, a gyrokinetic version of the
exact Fokker-Planck collision operator for this equation
has not been developed.
Significant effort has been made to construct a model
gyrokinetic operator which includes the effect of finite
gyroradius [5], and conserves particles, momentum, and
energy. The test-particle terms of this operator, including
energy scattering, were implemented in the gyrokinetic
code GS2 [8], and shown to strongly damp short wavelength
trapped electron modes. Most recently, new conserving
terms were formulated to ensure non-negative entropy
production [9,10]. Upon implementation in the GS2 code
0031-9007=11=106(19)=195002(4)
[11], this model operator yields a similar damping of short
wavelength entropy modes, in general providing physical
dissipation at short wavelengths. Quantitative accuracy is
suggested in comparisons with analytic theory for the
damping of electromagnetic waves by resistivity [11].
However, calculation of the classical collisional ion heat
transport across the magnetic field, using the same model
operator, reveals a 50% discrepancy [10] relative to the
linearized full Fokker-Planck operator. This shows that
present model operators still do not have all of the properties of the full collision operator.
In this Letter, we present the gyrokinetic linearized full
Fokker-Planck collision operator for arbitrary mass ratio,
including both the test-particle and field-particle contributions. The gyrokinetic operator we obtain accounts for
finite gyroradius effects in both the test-particle and fieldparticle terms within the usual gyrokinetic ordering. The
gyrophase averages resulting from the field-particle operator are independent of the guiding-center (gyroaveraged)
distribution, and can therefore be precomputed. Once precomputed, these gyroaverages can be efficiently reused in
successive time steps in gyrokinetic simulations. Further,
the field-particle operator at each time step is reduced to
the evaluation of a single two-dimensional velocity integral
over the evolving guiding-center distribution. In contrast,
recent model operators typically require the evaluation of
several more complex integrals for the conserved moments. Thus the exact operator is not only more accurate,
but much less expensive to evaluate than conserving model
operators.
For completeness, the linearized full collision operator
is first derived in a symmetric form, utilizing Rosenbluth
potentials. This simplifies both the test-particle and fieldparticle terms. The Fokker-Planck collision operator gives
the time rate of change in the distribution function fa of
species a due to Coulomb P
collisions with particles of
species b [1,3], ð@fa =@tÞc ¼ b Cðfa ; fb Þ. It is often convenient to express the operator in terms of the Rosenbluth
195002-1
Ó 2011 American Physical Society
week ending
PHYSICAL REVIEW LETTERS
13 MAY 2011
PRL 106, 195002 (2011)
R 3 0 0
R 3 0 0
potentials GðvÞ ¼ d v fb u and HðvÞ ¼ d v fb =u,
v2 @2 fa1
ma @fa1
0
0
0
C
ðf
;
f
Þ
¼
1
v
E a1 b0
k
s
where f ¼ fðvÞ, f ¼ fðv Þ, and u ¼ jv v j is the rela2 @v2
mb
@v
tive velocity [2,12]. Employing the properties of the
@f
m
Rosenbluth potentials r2v H ¼ 4fb and r2v G ¼ 2H,
þ D v a1 þ 4 a fb0 fa1 ;
(5)
@v
mb
the full Fokker-Planck collision operator may be written
in the symmetric form [5]
where the frequencies D ¼ ð=v3 ÞdG0 =dv, k ¼
s ¼ ð=vÞdH0 =dv.
For
a
ð=v2 Þd2 G0 =dv2 ,
1 @ 2 G @ 2 fa
ma
Cðfa ; fb Þ= ¼
þ 4
fa fb
Maxwellian background fb0 , the frequencies can be
2 @vi @vj @vi @vj
mb
expressed as [4,10] k ¼ 2ðnb =v3 ÞðxÞ, D ¼
m @H @fa
ðnb =v3 ÞerfðxÞ k =2, s ¼ x2 k , where erfðxÞ is the
þ 1 a
;
(1)
error function, ðxÞ ¼ ½erfðxÞ xerf 0 ðxÞ=ð2x2 Þ is the
mb @vi @vi
Chandrasekhar function, and x ¼ v=vTb .
where ¼ 4Z2a Z2b e4 ln=m2a and the sum over repeated
Because of the symmetry in Eq. (1), the differential form
indices i and j is understood. The form Eq. (1) is symof the field-particle operator is similar to the test-particle
metric in the sense that exchanging the distribution
operator and the Lorentz operator on G1 can be removed
function and Rosenbluth potentials results in the same
by employing the property r2v G1 ¼ 2H1 . Thus the fielddifferential form. For small departures from thermodyparticle operator Eq. (3) can be rewritten as
namic equilibrium, the distribution functions for all species
1 d2 fa0 @2 G1
1 dfa0
@2 G1
are nearly Maxwellian, f ¼ f0 þ f1 , where f0 ¼
Cðf
2H
;
f
Þ=
¼
þ
a0 b1
1
ðn=3=2 v3T Þ expðv2 =v2T Þ is Maxwellian for a species of
2 dv2 @v2
2v dv
@v2
density n, the perturbed distribution f1 f0 , and vT ¼
m @H1 dfa0
m
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ 4 a fa0 fb1 :
þ 1 a
2T=m is the thermal speed. It follows from Eq. (1) that
mb @v dv
mb
the linearized full Fokker-Planck collison operator consists
(6)
of the sum of the test-particle operator [5]
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
For the Maxwellian fa0 with vTa ¼ 2Ta =ma , the field1 @2 G0 @2 fa1
ma
particle
operator Eq. (6) becomes [4,13]
Cðfa1 ; fb0 Þ= ¼
þ 4
f f
2 @vi @vj @vi @vj
mb b0 a1
2v2 @2 G1
2v
ma @H1
ma @H0 @fa1
Cðf
;
f
Þ=ðf
Þ
¼
1
a0 b1
a0
þ 1
;
(2)
mb @v
v2Ta
v4Ta @v2
mb @vi @vi
2
m
2 H1 þ 4 a fb1 :
(7)
and the field-particle operator
mb
vTa
1 @2 G1 @2 fa0
m
The collision operator in the gyrokinetic equation is
Cðfa0 ; fb1 Þ= ¼
þ 4 a fa0 fb1
2 @vi @vj @vi @vj
mb
obtained via the transformation from guiding-center to
m @H1 @fa0
particle coordinates, application of the collision operator,
þ 1 a
;
(3)
and transformation back to guiding-center coordinates,
mb @vi @vi
followed by the average over the gyroangle [5,6,9]. Thus
where the Rosenbluth
the gyrokinetic linearized full Fokker-Planck collision opR 3 0 0 potentials of Maxwellian
R 3 0 0distribution G0 ðvÞ ¼ d v fb0 u and H0 ðvÞ ¼ R d v fb0 =u are
erator consists of the gyrokinetic test-particle operator
0 u and
isotropic Rin velocity space; G1 ðvÞ ¼ d3 v0 fb1
0 =u evaluated using the perturbed distriCgk ðha ; fb0 Þ ¼ heiLa Cðha eiLa ; fb0 Þi;
(8)
H1 ðvÞ ¼ d3 v0 fb1
bution are in general anisotropic in velocity space.
and the gyrokinetic field-particle operator
It is convenient to write the collision operator in terms
of spherical velocity variables v ¼ jvj, pitch angle ¼
(9)
Cgk ðfa0 ; hb Þ ¼ heiLa Cðfa0 ; hb eiLb Þi;
vk =v, and gyrophase defined with respect to the magH
with the gyrophase average h i ¼ d=2. The pernetic field B ¼ Bb, where the velocity v ¼ v? ðe^ 1 cos þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
turbed guiding-center distribution h is independent of the
e^ 2 sinÞ þ vk b and v? ¼ v 1 2 . Then the testgyroangle, Ls ¼ ðv bÞ k? =s and s ¼ Zs eB=ms c
particle operator Eq. (2) can be rewritten as the sum of
for
species s. The binormal basis vector e^ 1 may be
the Lorentz operator (pitch-angle scattering) [5]
chosen in the direction of the perpendicular wave number
so that k? ¼ ke^ 1 . Then Ls ¼ k sinv? =s and L0s ¼
@
@
1
@2
ð1 2 Þ
þ
CL ðfa1 ; fb0 Þ ¼ D
f
k sin0 v0? =s .
@ 1 2 @2 a1
2 @
It is well known that the gyroaverage of the test-particle
(4)
operator results in finite gyroradius terms that are proporand energy scattering
tional to k2 v2 =2 [5,9,10]. Inserting the test-particle
195002-2
PRL 106, 195002 (2011)
PHYSICAL REVIEW LETTERS
operator into the formula Eq. (8), the gyroaverage of the
Lorentz operator Eq. (4) yields [9,10]
Cgk
L ðha ; fb0 Þ ¼
D @
@
ð1 2 Þ ha
@
2 @
2
k v2
D ð1 þ 2 Þ
ha ;
42a
(10)
and the gyroaverage of energy scattering Eq. (5) gives
v2 @2 ha
ma @ha
@h
gk
þ D v a
s 1 v
CE ðha ; fb0 Þ ¼ k
2
2 @v
mb @v
@v
þ 4
ma
k2 v2
fb0 ha k ð1 2 Þ
ha :
mb
42a
(11)
For the calculations of the field-particle operator, it is
convenient to normalize the velocities to the thermal speed,
i.e., v=vTa ! v and the wave number k to the thermal
gyroradius a ¼ vTa =a , i.e., ka ! k, then La ¼
Lb = ¼ kv? sin, where ¼ a =b ¼ Za mb =ðma Zb Þ.
Inserting the field-particle operator Eq. (7) into the formula
Eq. (9), the gyrokinetic field-particle operator then takes
the form
m
Cgk ðfa0 ; hb Þ=ðfa0 Þ ¼ heiLa Pi þ 4 a hb heiðLb La Þ i;
mb
(12)
where the normalized quantity
@2 G 1
ma @H1
;
(13)
2H
2
1
v
1
mb
@v
@v2
R
the Rosenbluth
G1 ¼ d3 v0 h0b expðiL0b Þu and
R 3 0 0 potentials
H1 ¼ d v hb expðiL0b Þ=u. The integral representation of
Bessel functions gives
P ¼ 2v2
heiðLb La Þ i ¼ J0 ½kv? ð1 Þ:
(14)
Since the guiding-center distribution h0 is independent of
gyroangle 0 , we have
Z
heiLa Pi ¼ 2 d2 v0 h0b ðv0 ; 0 ÞIðv; ; v0 ; 0 ; kÞ; (15)
where the normalized gyrophase integral
I d I d0
Iðv; ; v0 ; 0 ; kÞ ¼
expðiL0b iLa ÞU; (16)
2
2
U ¼ 2v2
@2 u 2
ma
@ 1
2
1
;
v
2
u
mb @v u
@v
(17)
u ¼ ½v2 þ v02 2v? v0? cosð 0 Þ 2vk v0k 1=2 : (18)
The velocity derivatives in Eq. (17) can be expressed as
2v2 @2 u=@v2 ¼ ðv2 þ v02 Þ=u u=2 ðv2 v02 Þ2 =ð2u3 Þ
and 2v@ð1=uÞ@v ¼ 1=u ðv2 v02 Þ=u3 , so that
week ending
13 MAY 2011
m 1 u
U ¼ v2 þ v02 1 a mb u 2
m
ðv2 v02 Þ2 1
þ 1 a ðv2 v02 Þ :
mb
2
u3
(19)
The finite gyroradius effects described by I in gyrophase
integral Eq. (16) can be precomputed independently of the
evolving distribution function h0 , given the velocity grid
(v, , v0 , 0 ) and wave number k. Thus the field-particle
operator at each time step in gyrokinetic simulations is
reduced to the evaluation of a single two-dimensional
velocity integral, Eq. (15). Note that the quantity U defined
in Eq. (17) depends only on the difference 0 through
the relative velocity u which is periodic and even in
0 . In the limit k ¼ 0, the gyrophase integral
Eq. (16) reduces to
I0 ¼ Iðv; ; v0 ; 0 ; k ¼ 0Þ ¼
I d0
Uð0 Þ
2
(20)
The relative
velocity given by Eq. (18) can be rewritten as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u ¼ 1 2 sin2 , where ¼ ð þ 0 Þ=2, 2 ¼
ðv? þ v0? Þ2 þ ðvk v0k Þ2 , and 2 ¼ 4v? v0? =2 . Thus I0 ,
the gyrophase integral I evaluated at wave number k ¼ 0,
may be expressed in terms of complete
integrals K
H elliptic
and E [1,14] using the results
d0 =u ¼ 4KðÞ=,
H 0
H
d u ¼ 4EðÞ, d0 =u3 ¼ 4EðÞ=½3 ð1 2 Þ.
The periodic function U may be decomposed into a
P
0
Fourier series Uð 0 Þ ¼ n cn einð Þ where the coefficient c0 is equal to I0 . Using
H the integral representation
of Bessel functions
d expðikv? sin inÞ ¼
2Jn ðkv? Þ, the gyrophase integral Eq. (16) then becomes
X
I ¼ cn Jn ðkv? ÞJn ðkv0? Þ:
(21)
n
Since the Fourier coefficients cn are real for even functions, the gyrophase integral Eq. (16) is real and can be
rewritten as
I¼
I d I d0
cosðkv0? sin0 kv? sinÞU: (22)
2
2
The gyrophase integral in the form of Eq. (22) can be
accurately precomputed numerically using an adaptive
multidimensional integration algorithm [15]. For large
kv? and kv0? , the gyrophase integral can be asymptotically
approximated using the saddle point method as well. For
unlike particle collisions such as electron-ion collisions,
the collision operators can be simplified using a large mass
ratio mi =me 1 expansion [2]. However, for like particle
and ion-impurity collisions, the exact operator is necessary.
Figure 1 shows the typical wave number dependence of
the gyrophase integral I=I0 versus k for self-collisions.
Here ¼ vT = is the thermal gyroradius. Since the integrand in the gyrophase integral becomes oscillatory
for large k values, the wave number dependence is
195002-3
PRL 106, 195002 (2011)
PHYSICAL REVIEW LETTERS
1
ξ=0.5
ξ=0.8
ξ=1
I/I0
0.5
0
-0.5
0
2
4
6
8
10
12
kρ
FIG. 1 (color online). Typical wave number dependence
of gyrophase integral I=I0 for self-collisions. Here v0 ¼ 1,
0 ¼ 0, v ¼ 0:5, ¼ 1 (solid line), ¼ 0:8 (dashed line),
¼ 0:5 (dash-dotted line).
characterized by oscillatory decay and the largest integral
value is at k ¼ 0. For v? ¼ 0 or v0? ¼ 0 the gyrophase
integral becomes I=I0 ¼ J0 ðkv0? Þ or J0 ðkv? Þ. Thus in
Fig. 1 for the case ¼ 1 with v? ¼ 0 and v0? ¼ 1, the
zeros of gyrophase integral I=I0 ¼ J0 ðkÞ occur at k 2:4; 5:5; . . . the roots of Bessel function of order n ¼ 0,
verifying the accuracy of numerical integration.
The Debye length D characterizes charge shielding,
and effectively limits the range of interaction between
charges, limiting the maximum impact parameter for
small-angle deflections in the plasma [1]. This excludes
the value u ¼ 0 for the case v ¼ v0 from the calculation,
since the classical distance of closest approach e2 =ðmu2 =2Þ
should be smaller than the Debye length. If the function
Uð 0 Þ is smooth with no jumps, as is the case when v
and v0 differ significantly, then the Fourier series converges
quickly so that the gyrophase integral may be approximated by a small number of terms in Eq. (21). Thus in
Fig. 1 for the case ¼ 0:8 with v? ¼ 0:3 and v0? ¼ 1, the
gyrophase integral I=I0 J0 ðkv? ÞJ0 ðkv0? Þ. In gyrokinetic
simulations, the full expression in Eq. (22) can be readily
precomputed. The gyrokinetic linearized Fokker-Planck
operator is finally
@ha
gk
¼ Cgk
L ðha ;fb0 Þ þ CE ðha ;fb0 Þ
@t c
2 2ma
kv?
a
þ a pffiffiffiffi
h J
1
b
m b b 0 a
Z
2 0 0
0 0
0 0
þ d v hb ðv ; ÞIðv;;v ; ;kÞ expðv2 =v2Ta Þ;
(23)
where the frequency a ¼ na =v3Ta and the sum over all
species b including b ¼ a is understood.
week ending
13 MAY 2011
In summary, the gyrokinetic linearized full FokkerPlanck collision operator for arbitrary mass ratio has
been obtained, including both field-particle and testparticle contributions. Finite gyroradius effects are included in both field-particle and test-particle terms.
These effects were obtained via the transformation from
guiding-center to particle coordinates, application of the
collision operator, and transformation back to guidingcenter coordinates, averaging over the gyroangle. The
full operator automatically conserves particles, momentum, and energy, while ensuring positive entropy production. We find that the finite gyroradius effects can
be precomputed independently of the evolving distribution function, leaving a single two-dimensional velocity integral over the evolving distribution function.
Accordingly, this operator is both more accurate and
appears to be less computationally expensive than model
operators which approximate the field-particle terms with
conserving terms.
We thank Peter J. Catto for helpful discussions, and
Steven G. Johnson for help with numerical integration
using the CUBATURE code. This work was supported by
the U.S. Department of Energy Grants No. DE-FC0299ER54966 and No. DE-FG02-91ER54109.
[1] M. N. Rosenbluth, W. MacDonald, and D. Judd, Phys.
Rev. 107, 1 (1957).
[2] R. D. Hazeltine and F. L. Waelbroeck, The Framework of
Plasma Physics (Perseus Books, Massachusetts, 1998).
[3] F. L. Hinton, in Handbook of Plasma Physics, edited by
M. N. Rosenbluth and R. Z. Sagdeev (North-Holland,
Amsterdam, 1983), Vol. 1, p. 147.
[4] P. Helander and D. J. Sigmar, Collisional Transport in
Magnetized Plasma (Cambridge University Press,
Cambridge, 2002).
[5] P. J. Catto and K. T. Tsang, Phys. Fluids 20, 396
(1977).
[6] T. M. Antonsen and B. Lane, Phys. Fluids 23, 1205
(1980).
[7] E. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).
[8] D. R. Ernst et al., in Proc. 20th IAEA Fusion
Energy Conference, Chengdu, China (IAEA Paper
No. IAEA-CN-149/TH/1-3, 2006). Available as http://
www-pub.iaea.org/MTCD/Meetings/FEC2006/th_1-3.pdf.
[9] I. G. Abel et al., Phys. Plasmas 15, 122509 (2008).
[10] P. J. Catto and D. R. Ernst, Plasma Phys. Controlled Fusion
51, 062001 (2009).
[11] M. Barnes et al., Phys. Plasmas 16, 072107 (2009).
[12] M. N. Rosenbluth, R. D. Hazeltine, and F. L. Hinton, Phys.
Fluids 15, 116 (1972).
[13] S. P. Hirshman and D. J. Sigmar, Phys. Fluids 19, 1532
(1976).
[14] F. L. Hinton, Phys. Plasmas 15, 042501 (2008).
[15] A. C. Genz and A. A. Malik, J. Comput. Appl. Math. 6,
295 (1980).
195002-4