Download Gyrophase diffusion of charged particles in random magnetic fields

Mon. Not. R. Astron. Soc. 426, 880–891 (2012)
doi:10.1111/j.1365-2966.2012.21690.x
Gyrophase diffusion of charged particles in random magnetic fields
A. Shalchi
Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
Accepted 2012 July 9. Received 2012 July 9; in original form 2012 May 25
ABSTRACT
If charged particles propagate through a magnetized plasma, they experience different scattering effects. Well-known effects that have been studied previously include spatial diffusion and
pitch-angle scattering as well as stochastic acceleration. So-called gyrophase diffusion, which
describes stochastic changes of the gyrophase due to interaction with magnetic turbulence,
has been ignored in previous works. Therefore, we compute gyrophase diffusion coefficients
by employing quasi-linear theory as well as a second-order approach. We then compute the
velocity distribution function by solving the corresponding differential equation. These results
are important for testing previous assumptions employed in particle diffusion theories. As an
application, we compute velocity correlation functions and diffusion coefficients along and
across the mean magnetic field.
Key words: diffusion – magnetic fields – turbulence.
1 I N T RO D U C T I O N
In different physical scenarios, energetic particles propagate
through plasmas. Examples are charged particles in laboratory plasmas and fusion devices. In astrophysics and space science, solar energetic particles and cosmic rays interact with the interplanetary or
interstellar medium. Due to interaction with the plasma, charged energetic particles experience strong scattering. The magnetic fields
are especially important and control the different scattering processes. Most researchers assume that the particles move diffusively
and therefore diffusion coefficients can be computed and used to
characterize particle motion through the plasma. In astrophysics, for
instance, knowledge of these coefficients is important to describe
the propagation of cosmic rays in the Galaxy (see e.g. Büsching
& Potgieter 2008; Büsching et al. 2008), the solar modulation of
galactic cosmic particles (see e.g. Alania, Iskra & Siluszyk 2010;
Engelbrecht & Burger 2010; Hitge & Burger 2010; Wawrzynczak
& Alania 2010) and for understanding solar energetic particles (see
e.g. Dröge & Kartavykh 2009; Dröge et al. 2010). Furthermore,
analytical forms of the diffusion coefficients have to be known in
order to describe the acceleration of particles in interplanetary and
interstellar shock waves (see e.g. Kirk, Schneider & Schlickeiser
1988; Li, Zank & Rice 2003; Rice, Zank & Li 2003; Berezhko &
Völk 2007; Shalchi, Li & Zank 2010).
In the past, several authors have addressed the problem of charged
particle transport in different scenarios (see Schlickeiser 2002;
Shalchi 2009, for reviews). A major distinction has to be made between particle diffusion coefficients and particle distribution functions in phase space and in real space. In the first case, all parameters
depend on the particle velocity vector v and its position x. In the
E-mail: [email protected]
latter case the parameters depend only on the particle position x and
the speed v = |v|. By averaging over the particle’s pitch-angle cosine μ and gyrophase , one can connect parameters in real space
with those computed in phase space.
Shalchi (2011a) has developed a velocity-correlation-function
based theory for particle diffusion across a mean magnetic field
B 0 = B0 ez . In this theory the perpendicular diffusion coefficients
can be computed if the gyrophase diffusion coefficient is neglected.
Gyrophase diffusion has also been neglected in previous derivations
of the diffusive transport equation; see e.g. Schlickeiser (2002).
Therefore, we conclude that gyrophase diffusion could be an important process in particle diffusion theory. In the present paper
we investigate gyrophase diffusion by employing quasi-linear and
more advanced methods.
The organization of the present paper is as follows. In Section 2
we discuss fundamental equations such as the equation of motion
and the Fokker–Planck equation, as well as the diffusion equation.
In Section 3 we compute for the first time the quasi-linear Fokker–
Planck coefficient of gyrophase diffusion. In Section 4 quasi-linear
theory is replaced by a second-order approach. In Section 5 we
solve the Fokker–Planck equation to compute velocity distribution
functions. In Sections 6 and 7 we employ our results to compute
velocity correlation functions and diffusion coefficients along and
across the mean magnetic field. In Section 8 we summarize and
provide our conclusions.
2 F U N DA M E N TA L E Q UAT I O N S
2.1 The TGK formulation
In the present article we use Cartesian coordinates for the position
of the charged particle x. For the particle velocity it is convenient
C 2012 The Author
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
to introduce spherical coordinates, defined as
vx = v
vy = v
2.2 The transport equations
1 − μ2 cos (),
1 − μ2 sin (),
vz = vμ.
(1)
Here we have used the absolute value of the particle velocity v, the
pitch-angle cosine μ and the gyrophase . The relations (1) can be
rewritten as
v=
vx2 + vy2 + vz2 ,
vz
μ= vx2 + vy2 + vz2
vy
.
= arctan
vx
,
(2)
Please note that all quantities used here are time-dependent, i.e. v(t),
μ(t) and (t).
If charged particles propagate through a plasma, they experience scattering. These scattering effects can be described by using
mean square displacements xi xj or running diffusion coefficients dij (t). These parameters can be computed by employing
the well-established Taylor–Green–Kubo (TGK) formulation (see
Taylor 1922; Green 1951; Kubo 1957). In this formulation the diffusion coefficients can be computed by solving the following time
integrals (for a detailed derivation see e.g. Shalchi 2011b):
dτ ẋi (τ )ẋj (0).
(3)
0
If the two indices are equal (i = j), the running diffusion coefficient
can be expressed by using mean square displacements,
dii (t) =
Particle diffusion in phase space is described by the Fokker–Planck
equation (see e.g. Schlickeiser 2002):1
∂f
∂f
1 ∂
∂f
∂f
+ vμ
−
= S(x, t) + 2
p 2 Dij
.
(6)
∂t
∂z
∂
p ∂xi
∂xj
Here we have used the particle distribution function in phase space
f = f (x, y, z, v, μ, , t), the source function S(x, t) and the Fokker–
Planck coefficients Dij with i, j = x, y, v, μ, , which can be
computed by employing the TGK formula (5). It should be noted
that equation (6) is still incomplete, since some effects are neglected, such as the effect of adiabatic focusing (see e.g. Earl 1976;
Kunstmann 1979; Schlickeiser & Shalchi 2008). The scattering
effects described by the Fokker–Planck coefficients Dij contain perpendicular spatial diffusion Dxx , Dxy , Dyx , Dyy as well as velocity
diffusion Dμμ , Dvv , D , . . .. Schlickeiser et al. (2007) have shown
that there could also be mixed Fokker–Planck coefficients, e.g. Dμx .
In equation (6) we have used the unperturbed gyrofrequency
=
qB0 1 − v 2 /c2 ,
mc
1 d (xi )2 .
2 dt
(4)
For the drift coefficient i = j, the relation between velocity correlations and mean square displacements is more complicated (see
Shalchi 2011b and references therein).
The variable xi therein can be the particle position as well as a
velocity component. Thus, xi stands for xi = x, y, z, v, μ, .
If one assumes that the corresponding variable behaves diffusively, one can use diffusion coefficients to describe the particle
motion. In this case we have
∞
Dij = dij (t → ∞) =
dt
ẋi (t)ẋj (0) .
(5)
0
Several authors have suggested that the particles do not move diffusively (see e.g. Zimbardo 2005; Shalchi & Kourakis 2007). In
the present paper, however, we assume normal diffusion to ensure
mathematical tractability.
The parameters Dij used here depend on the pitch-angle cosine
μ and could also depend on the gyrophase . The usual diffusion
coefficients, however, only depend on the absolute values of the
velocity, i.e. κ ij = κ ij (v), whereas Dij = Dij (μ, v) or Dij = Dij (μ,
, v). The μ-dependent diffusion coefficients are usually called the
Fokker–Planck coefficients.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society (7)
with particle charge q and mass m, mean magnetic field B0 and
speed of light c.
By averaging equation (6) over the pitch-angle cosine μ and
gyrophase , we can derive a diffusion equation of the form (again
see Schlickeiser 2002 for details)
∂
∂F
∂F
=
κij
(8)
∂t
∂xi
∂xj
with i, j = x, y, z and
t
dij (t) =
881
F (x, y, z, v, t) =
1
4π
2π
+1
d
0
dμ f (x, y, z, v, μ, , t).
−1
(9)
Here we have again neglected the source term S(x, t) and stochastic
acceleration effects. During this derivation, we have also neglected
gyrophase diffusion by assuming
D , Dμ , Dμ .
(10)
One of the purposes of the present article is to test this assumption.
Also important is the relation between the diffusion coefficients
κ ij in equation (8) and the Fokker–Planck coefficients Dij used in
equation (6). For the coefficients of perpendicular diffusion we have
1 +1
dμ Dij (μ), i, j = x, y,
(11)
κij =
2 −1
and for the parallel diffusion coefficient we have the relation
v 2 +1
(1 − μ2 )2
.
(12)
κzz =
dμ
8 −1
Dμμ (μ)
Different derivations of the latter formula have been presented by
Jokipii (1966), Earl (1974), Shalchi (2006) and Shalchi (2011a).
The spatial diffusion coefficients κ ij and κ zz do not depend on the
gyrophase Fokker–Planck coefficients D , Dμ and Dμ . In the
present paper we will compute those coefficients to explore the
influence of gyrophase diffusion on spatial diffusion coefficients.
1
Later we will average over the spatial coordinates x, y and z. Therefore, it
is not important whether we are using the coordinates of the particles or the
coordinates of the guiding centre for the investigations of the current article.
882
A. Shalchi
3 Q UA S I L I N E A R F O K K E R – P L A N C K
C O E F F I C I E N T S O F G Y RO P H A S E D I F F U S I O N
The simplest approach to compute Fokker–Planck coefficients is
the application of quasi-linear theory (QLT; see Jokipii 1966). In
this case we compute the particle diffusion coefficients along the
unperturbed particle trajectory.
3.1 The magnetostatic approximation
Stochastic acceleration is caused by turbulent electric fields δ E.
Since those fields are usually much weaker than the turbulent magnetic fields δ B, we neglect momentum diffusion in the present
article. Within this approximation the particle velocity and momentum are conserved, i.e. p = const and v = const. Thus, we find from
equation (2)
v̇z
,
v
˙ = vx v̇y − vy v̇x ≡ vx v˙y − vy v˙x ,
vx2 + vy2
v⊥2
μ̇ =
(13)
for the time derivatives of μ and , respectively. The particle accelerations v̇x , v̇y , v̇z can be replaced by employing the Newton–
Lorentz equation, which yields for purely magnetic fluctuations
δBz
δBy
,
v̇x = vy + vy
− vz
B0
B0
δBx
δBz
v̇y = −vx + vz
,
− vx
B0
B0
δBy
δBx
v̇z = vx
.
− vy
(14)
B0
B0
By combining equations (13) and (14) we can easily derive
vy δBx
vx δBy
μ̇ = ,
−
v B0
v B0
˙ = − 1 + δBz + vz vx δBx + vy δBy
.
B0
B0
B0
v⊥2
and the Fokker–Planck coefficients of velocity diffusion can be
computed by using
∞
dt gi (t)gj∗ (0) ,
(20)
Dij = 0
with i, j = μ, . Please note that equation (19) is in coincidence
with the expressions derived by Schlickeiser (2002), who used leftand right-handed polarized magnetic field components instead of
Cartesian coordinates δBx , δBy and δBz used here.
In the following, we compute the velocity diffusion coefficients
by specifying the properties of the turbulent magnetic fields.
3.2 Magnetostatic slab turbulence
In the present paragraph we employ the magnetostatic slab model, in
which the turbulent magnetic field satisfies by definition δBi (x, t) =
δBi (z). This model is not very realistic, since real turbulence is
anisotropic. A more realistic model would be the model of twocomponent turbulence where we assume a superposition of slab
modes and two-dimensional modes. Quasi-linear theory used in
the present section provides no scattering for the two-dimensional
modes. Therefore, we can justify the application of the slab model.
Due to the solenoidal constraint ∇ · δ B(z) = 0, we have δBz =
0. Thus we find for the force terms in the slab model
(vx δBy − vy δBx ),
gμ (t) =
B0 v
vz
g (t) =
(vx δBx + vy δBy ).
(21)
B0 v⊥2
If we apply quasi-linear theory, we can use
vx = v 1 − μ2 cos (0 − t),
vy = v 1 − μ2 sin (0 − t),
(22)
vz = vμ,
If we set δBi = 0, corresponding to the unperturbed system, we find
and δBi (x, t) = δBi (z = vμt) on the right-hand side of equation (21). To proceed we assume axisymmetric slab turbulence.
With
(23)
Rij (x, t) := δBi (x, t)δBj∗ (0, 0)
μ̇ = 0,
and Rxy = Ryx = 0, the force correlation functions become
˙ = −,
(15)
(16)
∗
(0) =
g (t)g
and therefore
+ sin (0 − t) sin (0 )Ryy ],
μ = μ0 = constant,
= 0 − t.
(17)
2 μ
g (t)gμ∗ (0) =
[− cos (0 − t) sin (0 )Rxx
B02
In the unperturbed system, the particle’s pitch angle is conserved
while the perpendicular motion occurs with = 0 − t with the
initial gyprophase 0 . If the scattering coefficients are computed,
one has to remove the unperturbed particle orbit. We define the force
terms as suggested by Schlickeiser (2002):
gμ := μ̇,
˙ + .
g := (18)
+ sin (0 − t) cos (0 )Ryy ],
2 μ ∗
gμ (t)g
cos (0 − t) sin (0 )Ryy
(0) =
B02
− sin (0 − t) cos (0 )Rxx ] ,
2 (1 − μ2 )
gμ (t)gμ∗ (0) =
[cos (0 − t) cos (0 )Ryy
B02
+ sin (0 − t) sin (0 )Rxx ].
With equation (15), we derive
vx δBy
vy δBx
−
,
v B0
v B0
δBz
vz
δBx
δBy
,
+ 2 vx
+ vy
g = −
B0
B0
B0
v⊥
2 μ2
[cos (0 − t) cos (0 )Rxx
− μ2 )
B02 (1
gμ = (19)
(24)
Now we average over the initial gyrophase 0 by applying the
operator
2π
1
d0
2π 0
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
883
to equation (24). By taking into account the fact that Rxx = Ryy do
not depend on 0 , we find
∗
g (t)g
(0) =
2 μ2
cos (t)Rxx ,
− μ2 )
B02 (1
2 μ
g (t)gμ∗ (0) = − 2 sin (t)Rxx ,
B0
2 μ
∗
(0) =
sin (t)Rxx ,
gμ (t)g
B02
2 (1 − μ2 )
cos (t)Rxx .
gμ (t)gμ∗ (0) =
B02
The remaining function Rxx can be written as
∞
dk g slab (k )eivμk t ,
Rxx = 4π
(25)
(26)
0
with the turbulence spectrum of the slab fluctuations gslab (k ). Now
we combine equations (20), (25) and (26). Furthermore, we employ
∞
dt sin (t)eivμk t = 0
(27)
0
and
∞
dt cos (t)eivμk t
(28)
0
=
π
δ(vμk + ) + δ(vμk − ) ,
2
(29)
where we have used the Dirac delta δ(z). We derive
∞
2π2 2 μ2
dk g slab (k )
D = 2
B0 (1 − μ2 ) 0
× δ(vμk + ) + δ(vμk − ) ,
Dμ = 0,
2π2 2 (1 − μ2 ) ∞
dk g slab (k )
B02
0
× δ(vμk + ) + δ(vμk − ) .
(30)
Thus, we find the relations
μ2
Dμμ .
(1 − μ2 )2
(31)
Previous results derived for the pitch-angle Fokker–Planck coefficient Dμμ in magnetostatic slab turbulence can be applied to compute the gyrophase diffusion coefficient D . The ratio D /Dμμ
is shown in Fig. 1 versus the pitch-angle cosine μ.
Gyrophase diffusion exceeds pitch-angle scattering for pitch angles satisfying μ2 > (1 − μ2 )2 . We can define a characteristic
pitch-angle cosine μc which satisfies μ2c = (1 − μ2c )2 . One can
easily derive
√
5−1
.
(32)
μc =
2
For |μ| > μc gyrophase diffusion is stronger and for |μ| < μc
pitch-angle scattering is dominant. In Fig. 1 we have shown the
characteristic pitch-angle cosine μc also.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society By evaluating the k integral in equation (30), the quasi-linear pitchangle diffusion coefficient for magnetostatic turbulence reads
2π2 2 (1 − μ2 ) slab
.
(33)
g
k
=
Dμμ =
v |μ|
B02 v |μ|
Equation (33) can easily be evaluated for different forms of the slab
spectrum gslab (k ). In the present paper we use the model spectrum
suggested by Shalchi & Weinhorst (2009), namely
C(s, q) 2
δBslab lslab
2π
k lslab q
×
(s+q)/2 .
1 + (k lslab )2
(34)
Here we allow different values of the energy range spectral index
q as well as the inertial range spectral index s. Furthermore, we
have used the slab bend-over scale lslab , which denotes the turnover
from the energy range to the inertial range of the spectrum. In
equation (34) we have used the normalization function
s+q
(35)
s−1 2 q+1 ,
C(s, q) =
2
2 2
with the Gamma function (z).
For this form of the spectrum we derive
Dμ = Dμ = 0,
D =
3.3 Quasi-linear results for a standard spectrum
g slab (k ) =
Dμ = 0,
Dμμ =
Figure 1. The ratio D /Dμμ versus the pitch-angle cosine μ (solid line).
For |μ| > μc , gyrophase diffusion is stronger than pitch-angle diffusion. For
|μ| < μc , pitch-angle scattering is dominant.
δB 2
Dμμ
= πC(s, q)(1 − μ2 ) slab
B02
×
|μR|s−1
(s+q)/2 .
1 + (μR)2
(36)
Here we have used the parameter R ≡ v/(lslab ), which is a measure
for the particle momentum/rigidity. In Fig. 2 we have shown the
ratio Dμμ / for δB = B0 , s = 5/3, q = 0 and different values
of R. For the values chosen for the two spectral indexes we have
C(s = 5/3, q = 0) ≈ 0.1188. According to our result, we always
have Dμμ . In Fig. 3 we have shown the ratio D / for
the same turbulence and particle parameters. As demonstrated, we
have D for smaller values of |μ|. If the pitch-angle cosine
is close to ±1, however, the gyrophase diffusion coefficient is much
larger than the unperturbed gyrofrequency . The latter result is in
contradiction with the assumption (10).
884
A. Shalchi
consequence of a different pitch-angle dependence of the scattering
parameter Dμμ .
The second-order theory has been applied by Tautz et al. (2008) to
describe parallel diffusion in isotropic turbulence, in agreement with
computer simulations. Furthermore, the theory has been employed
to describe the propagation of ultrahigh-energy cosmic rays in the
Galaxy by Shalchi et al. (2009a).
4.1 Analytic forms of the second-order coefficient
Shalchi et al. (2009b) have used SOQLT to compute a second-order
pitch-angle Fokker–Planck coefficient analytically. For low-energy
particles those authors derived
Dμμ (|μ| ≤ δB/B0 )
= E(1 − μ2 ) × (δB/B0 + |μ|)s + (δB/B0 − |μ|)s
Figure 2. The ratio Dμμ / versus the pitch-angle cosine μ computed by
employing standard quasi-linear theory. The curves were calculated using
different values of particle rigidity, namely R = 0.1 (dotted line), R = 1
(dashed line), R = 10 (dash–dotted line) and R = 100 (solid line).
(37)
and
Dμμ (|μ| ≥ δB/B0 )
= E(1 − μ2 ) × (|μ| + δB/B0 )s − (|μ| − δB/B0 )s .
(38)
Here we have used the strength of the turbulent magnetic field δB,
the strength of the mean magnetic field B0 and the inertial range
spectral index s. For |μ| = δB/B0 , the two solutions are equal. The
parameter E used here reads
π
δB
E
= C(s, q)R s−1
,
2s
B0
Figure 3. The ratio D / versus the pitch-angle cosine μ computed by
employing standard quasi-linear theory. The curves were calculated using
different values of particle rigidity, namely R = 0.1 (dotted line), R = 1
(dashed line), R = 10 (dash–dotted line) and R = 100 (solid line).
(39)
with C(s, q) from equation (35). The above results are valid for
particles with Larmor radii smaller than the turbulence bend-over
scale corresponding to lower particle energies/rigidities with R =
RL /lslab 1. The parameters Dμ , Dμ and D can again be
computed using equation (31).
In Figs 4 and 5 we have shown pitch-angle and gyrophase diffusion coefficients by employing equations (37), (38) and (31). For
the turbulence parameters we have used s = 5/3 and q = 0. For the
particle rigidity, represented by parameter R, we have considered
different values, as well as for turbulence strength δB/B0 . Again we
find that for most cases Dμμ , except for strong turbulence.
For the gyrophase diffusion coefficient we find a result similar to
4 S E C O N D - O R D E R R E S U LT F O R A
S TA N DA R D S P E C T RU M
From equation (36) and Fig. 2, we can see that Dμμ (μ = 0) = 0 for
s > 1. This behaviour is problematic and leads to the well-known
90◦ problem of cosmic-ray diffusion theory (see e.g. Shalchi 2009
for a review). To solve this problem, Shalchi (2005) has developed
a second-order quasi-linear theory (SOQLT). In the latter theory
one takes into account fluctuations of the particle trajectory. If those
fluctuations are taken into account, one finds a broadened resonance
function instead of the delta function of magnetostatic QLT: see
e.g. equation (29). Thus, second-order theory and other non-linear
theories are also known as resonance-broadening theories.
We like to emphasize that the 90◦ problem is a consequence of the
application of quasi-linear theory. Furthermore, the problem of zero
scattering at μ = 0 was derived for slab turbulence. However, for
full three-dimensional turbulence the situation can be even more
problematic (see e.g. Tautz, Shalchi & Schlickeiser 2006, 2008),
since the vanishing scattering at 90◦ can lead to singularities if
the parallel mean free path is calculated. These singularities are a
Figure 4. The ratio Dμμ / versus the pitch-angle cosine μ computed by
employing second-order quasi-linear theory. The results were calculated
using different values of particle rigidity, namely R = 0.1 (dotted line) and
R = 1 (dashed line) for intermediate strong turbulence δB/B0 = 1. We have
also shown the results for weaker turbulence δB/B0 = 0.5 (dash–dotted line)
and stronger turbulence δB/B0 = 2 (solid line). In the latter two cases we
have used R = 0.1.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
885
Note that in this notation we have omitted the v dependence. By
averaging the Fokker–Planck equation (6) over x, we find
∂G
∂
∂
∂G
∂G
∂G
=
+
Dμμ
+
D
,
(43)
∂t
∂ ∂μ
∂μ
∂
∂
where we have again neglected the source term, stochastic acceleration and other effects such as adiabatic focusing. Furthermore, we
used Dμ = Dμ = 0 due to the assumption of axisymmetry.
For quasi-linear and second-order scattering we have Dμμ
(μ, ) = Dμμ (μ) and D (μ, ) = D (μ) and, therefore,
∂G
∂
∂G
∂ 2G
∂G
(44)
=
+
Dμμ
+ D
.
∂t
∂ ∂μ
∂μ
∂2
To solve this equation we employ the Ansatz
+∞
G(μ, , t) = 2π
Figure 5. The ratio D / versus the pitch-angle cosine μ computed by
employing second-order quasi-linear theory. The results were calculated
using different values of particle rigidity, namely R = 0.1 (dotted line) and
R = 1 (dashed line) for intermediate strong turbulence δB/B0 = 1. We have
also shown the results for weaker turbulence δB/B0 = 0.5 (dash–dotted line)
and stronger turbulence δB/B0 = 2 (solid line). In the latter two cases we
have used R = 0.1.
G(μ, , t = 0) = 2π
Hm (μ, t = 0)eim(−0 )
m=−∞
= 4πδ (μ − μ0 )
+∞
eim(−0 )
m=−∞
4.2 The strong turbulence limit
The parameter D describes pitch-angle diffusion at 90◦ . For the two
velocity diffusion coefficients we, therefore, derive
Dμμ = (1 − μ2 )D,
D
+∞
= 4πδ (μ − μ0 ) δ ( − 0 )
For strong turbulence, δB B0 , we can easily derive the isotropic
form Dμμ = (1 − μ2 )D from equation (38) with the parameter
π
δB s+1
D
= C(s, q)R s−1
.
(40)
s
B0
(41)
This strong turbulence solution is much simpler than the quasilinear result (36) and the second-order formula for arbitrary δB/B0 :
see equations (37) and (38). Therefore, we use equation (41) in the
present paper to solve the Fokker–Planck equation.
5 SOLUTION OF THE FOKKER-PLANCK
E Q UAT I O N
Above we have computed the Fokker–Planck coefficients of velocity diffusion Dμμ and D . In the following, we use those results to
calculate the velocity distribution function by solving the Fokker–
Planck equation.
(45)
which satisfies G( = 2π) = G( = 0). The factor e−im0 , with
initial gyrophase 0 , was introduced so that
the quasi-linear result. For pitch angles satisfying μ ≈ ±1, we have
D , in contradiction to the assumption (10).
μ2
=
D.
1 − μ2
Hm (μ, t)eim(−0 +t) ,
m=−∞
(46)
for the initial distribution function. Here we have used
Hm (μ, t = 0) = 2δ (μ − μ0 ) ,
(47)
with the initial pitch-angle cosine μ0 . In equation (45) we have also
used the time- and pitch-angle dependent function H m (μ, t), which
has to satisfy the differential equation
∂
∂Hm
∂Hm
=
Dμμ
− m2 D Hm .
(48)
∂t
∂μ
∂μ
Even if we would neglect gyrophase diffusion by setting D = 0,
solving this equation would be difficult; see e.g. Shalchi (2011a).
Therefore, we simplify this formula by employing the second-order
results derived in the strong turbulence limit; see equation (41).
Next, we employ the Ansatz
Hm (μ, t) = Km (μ, t)em
2 Dt
(49)
,
and equation (48) becomes for second-order diffusion
∂
m2
∂Km
∂Km
=D
(1 − μ2 )
−
DKm .
∂t
∂μ
∂μ
1 − μ2
(50)
To proceed we use a further Ansatz, namely
Km (μ, t) =
∞
αlm Plm (μ)e−ωl t ,
(51)
l=0
and equation (45) becomes
5.1 The velocity distribution function in the general case
In the following we compute the velocity distribution function,
defined as2
G(μ, , t) := d3 x f (x, μ, , t).
(42)
2
Here we have averaged over all x. Therefore, it is not important whether
x are particle coordinates or guiding-centre coordinates.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society G(μ, , t) = 2π
∞
+∞ αlm Plm (μ)
m=−∞ l=0
×e−ωl t em
2 Dt
eim(−0 +t) .
(52)
The functions Plm (μ) have to satisfy the ordinary differential equation (ODE)
m2
∂
∂P m
ωl m
Pl +
(1 − μ2 ) l −
P m = 0.
(53)
D
∂μ
∂μ
1 − μ2 l
886
A. Shalchi
This ODE has the form of a Sturm–Liouville equation, where Plm (μ)
are the eigenfunctions and ωl are the eigenvalues.
Equation (53) can be identified with the general Legendre equation. Therefore, we conclude that the eigenfunctions have to be
identified with the associated Legendre functions and the eigenvalues are given by ωl = l(l + 1)D with −l ≤ m ≤ l. Thus, we derive
for the velocity distribution function (52) the following form:
G(μ, , t) = 2π
∞ +l
αlm Plm (μ)
l=0 m=−l
2
×e−[l(l+1)−m ]Dt eim(−0 +t) .
(54)
The μ and dependence has the form ∼ Plm (μ)eim . Therefore, it
is convenient to introduce spherical harmonics, defined as
(2l + 1)(l − m)! m
m
Pl [cos ()] eim .
(55)
Yl (, ) =
4π(l + m)!
Here we have used the pitch-angle by using μ = cos (). Spherical harmonics are well-known in electrodynamics as well as in
quantum mechanics. By using those functions, we can use the Ansatz
G(μ, , t) = 4π
+l
∞ Ylm (, ) Yl∗m (0 , 0 )
l=0 m=−l
2
×e−[l(l+1)−m ]Dt eimt
(56)
for the velocity distribution function G(μ, , t). It can easily be
demonstrated that this form satisfies the initial conditions (46):
G(μ, , t = 0) = 4π
∞ +l
Ylm (, ) Yl∗m (0 , 0 )
l=0 m=−l
= 4πδ [cos () − cos (0 )] δ ( − 0 )
= 4πδ (μ − μ0 ) δ ( − 0 ) .
(57)
Therefore, equation (56) is the correct solution of equation (44).
It describes the particle’s velocity distribution for second-order
Fokker–Planck coefficients. On requiring the validity of equation (41), this form is exact. In Fig. 6 we plot equation (56) for
different times t.
5.2 Pitch-angle and gyrophase isotropization
In the previous section we have already considered the limit t → 0
to show that our solution (56) satisfies the correct initial conditions
(46). In the following we consider the limit t → ∞. This will
help us to understand the physics of solution (56). In the limit of
2
late times the exponential function in our solution e−[l(l+1)−m ]Dt
tends to zero for most values of m and l. The main contribution for
t → ∞ is
G(μ, , t → ∞) ≈ 4πY00 (, ) Y0∗0 (0 , 0 )
+ 4π Y1−1 (, ) Y1∗−1 (0 , 0 ) eit
+ Y11 (, ) Y1∗1 (0 , 0 ) e−it
× e−Dt .
(58)
To evaluate this expression we have to know some of the spherical harmonics. From text books (see e.g. MacRobert 1948; Müller
1966), we obtain
1 1
,
Y00 (, ) =
2 π
1 3
Y10 (, ) =
cos (),
2 π
1
3
Y1−1 (, ) =
sin ()e−i ,
2 2π
−1
3
sin ()ei ,
Y11 (, ) =
2
2π
1 15
−2
sin2 ()e−2i ,
Y2 (, ) =
4 2π
1 15
2
sin2 ()e2i .
Y2 (, ) =
(59)
4 2π
Therefore, we can rewrite equation (58) to find
G(μ, , t → ∞) ≈ 1 + 3 1 − μ2 1 − μ20
× cos ( − 0 − t)e−Dt .
(60)
In the limit t → ∞ the velocity distribution function tends to 1.
Thus, equation (56) describes a pitch-angle isotropization process
as well as a gyrophase isotropization process. The next contributing
term already decays exponentially with ∼e−Dt . The characteristic
time-scale of the two isotropization processes is τ = D−1 .
In Fig. 6 we have visualized the isotropization processes by
plotting equation (56) for different times. One can easily see the
isotropization in pitch angle and gyrophase.
5.3 The velocity distribution function by neglecting
gyrophase diffusion
To provide a solution that is simpler from a mathematical point of
view, we solve equation (44) by considering different limits. If we
neglect the gyrophase scattering contribution in equation (53), we
obtain
∂
∂P m
Dμμ l
= 0.
(61)
ωl Plm +
∂μ
∂μ
Figure 6. The distribution function G(μ, , t) versus μ and for different
times t. To generate these plots we have employed equation (56). As time
passes, we can clearly see the pitch-angle and gyrophase isotropization
process.
The solution for the isotropic Dμμ = (1 − μ2 )D is well known
(see e.g. Shalchi 2011a). In this case the eigenvalues are ωl = l(l +
1)D and the m-independent eigenfunctions Plm (μ) = Pl (μ) can be
identified with Legendre polynomials. In this case, equation (52)
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
887
We can understand the physics of equation (68) also.
becomes
+∞
G(μ, , t) = 2π
∞
(2l + 1)Pl (μ0 )Pl (μ)
m=−∞ l=0
×e−l(l+1)Dt eim(−0 +t)
= 2πδ ( − 0 + t)
∞
(2l + 1)Pl (μ0 )Pl (μ)
l=0
×e−l(l+1)Dt .
(62)
For negligible gyrophase diffusion this results in the exact velocity distribution function. We can easily understand the physics of
equation (62).
(i) The factor δ( − 0 + t) means that the gyrophase is =
0 − t, corresponding to an unperturbed gyrorotation. This result
is trivial, since we have neglected gyrophase diffusion in the present
subsection.
−l(l+1)Dt
is well known
(ii) The term ∞
l=0 (2l + 1)Pl (μ0 )Pl (μ)e
and has been derived earlier (see e.g. Shalchi 2011a). It describes
the pitch-angle isotropization process due to pitch-angle scattering.
For some applications such as the calculation of parallel velocity
correlation functions, gyrophase diffusion is negligible and equation (62) provides the correct solution. For Dt > 1, equation (62)
becomes
G(μ, , t → ∞) ≈ 2πδ ( − 0 + t) 1 + 3μμ0 e−2Dt . (63)
The characteristic time-scale of the isotropization process is
1
τμ =
.
2D
(64)
For times t τ μ , the pitch-angle distribution function is nearly
isotropic.
In the following we present a more detailed discussion of the second
effect, since a quantitative description has not been presented before.
We can easily simplify equation (68) by considering two limits.
In the first case we assume D t 1 and we find
G(μ, , D t 1) = 4πδ (μ − μ0 )
+∞
eim(−0 +t)
m=−∞
= 4πδ (μ − μ0 ) δ ( − 0 + t) ,
In the current paragraph we neglect the pitch-angle scattering contribution in equation (48). We derive
(65)
(69)
corresponding to an unperturbed motion.
For late times, D t > 1, we only obtain a contribution to the
sum over m in equation (68) if m is small:
G(μ, , D t 1)
= 4πδ(μ − μ0 ) × [1 + 2 cos ( − 0 + t)e−D t ].
(70)
The cos function describes the gyrorotation of the particle. The
exponential factor e−D t describes the isotropization process. The
characteristic time-scale of the gyrophase isotropization is
τ =
1
.
D (μ)
(71)
It should be noted that this time is pitch-angle-dependent. If we
employ the second-order result (41) for D , we find τ =
(1 − μ2 )/(μ2 D).
In the limit t → ∞, we derive
G(μ, , t → ∞) = 4πδ (μ − μ0 )
5.4 The velocity distribution function neglecting
pitch-angle diffusion
∂Hm
= −m2 D Hm ,
∂t
(i) The factor δ(μ − μ0 ) means that the pitch-angle cosine is
given by μ = μ0 and is therefore conserved. This corresponds to an
unperturbed motion of the particle along the mean magnetic field.
This result is again trivial, since we have neglected pitch-angle
diffusion in the present
subsection.
−m2 D t im(−0 +t)
e
, however, is new.
(ii) The term +∞
m=−∞ e
For D t → ∞ this factor tends to 1. Thus, this factor describes a
gyrophase isotropization process.
(72)
and the velocity distribution function no longer depends on the
gyrophase. This process corresponds to a gyrophase isotropization
process.
It is interesting to note that we only have this process if D = 0.
For very small values of μ, gyrophase diffusion is negligible since
we find D (μ ≈ 0) ≈ 0. In this case, the gyrophase is conserved.
which can easily be solved by
Hm (μ, t) = Hm (μ, t = 0)e−m
2D
t
(66)
.
With this result, equation (45) becomes
G(μ, , t) = 2π
+∞
Hm (μ, t = 0)e−m
2D
t
m=−∞
(67)
×eim(−0 +t) .
To satisfy the initial conditions (46), we employ equation (47) to
obtain
G(μ, , t) = 4πδ (μ − μ0 )
×
+∞
e−m
2D
t
eim(−0 +t) .
m=−∞
(68)
Please note that this result is correct for a gyrophase-independent
D . However, to derive this result we have not specified the pitchangle dependence of the gyrophase diffusion coefficient.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society 5.5 The gyrophase-averaged Fokker–Planck equation
Above we have averaged the Fokker–Planck equation (6) over all
particle positions x to compute the velocity distribution function. In
the present subsection we average equation (6) over the gyrophase
to derive the gyrophase-averaged Fokker–Planck equation. We
defined the averaged distribution function as
2π
1
d f (x, y, z, μ, , t).
(73)
g(x, y, z, μ, t) :=
2π 0
Here we have again omitted the v dependence. In the present paper,
we have employed quasi-linear theory as well as a second-order
approach. Within both theories we found that the Fokker–Planck
coefficients do not depend on the gyrophase . Thus we can easily average the Fokker–Planck
2π equation (6) over the gyrophase by
employing the operator 21π 0 . We derive for the averaged equation
∂f
1 ∂
∂g
∂g
+ vμ
= S(x, t) + 2
p 2 Dij
,
(74)
∂t
∂z
p ∂xi
∂xj
888
A. Shalchi
now with i, j = x, y, z, μ, v. This result agrees with that derived by
Schlickeiser (2002). This is an important conclusion, since we have
shown that the assumption D , Dμ , D,μ is, in general, not
valid. The latter assumption has been used in Schlickeiser (2002).
Since quasi-linear and second-order diffusion coefficients do not
depend on the gyrophase, equation (74) is valid, confirming the
calculations presented in Schlickeiser (2002).
With the general solution (56) and by using the orthogonality relation (77), we find after straightforward algebra
6 V E L O C I T Y C O R R E L AT I O N F U N C T I O N S
vz (t)vz (0) =
In the present section we use the results of the previous sections
to compute velocity correlation functions along and across the
mean magnetic field. This section complements the work of Shalchi
(2011a).
Equation (79), as well as equation (80), agrees with the results derived previously (see e.g. Shalchi 2011a). In the present paper, however, we have derived this result by taking gyrophase diffusion into
account. As demonstrated, we always find an exponential decorrelation if the pitch-angle Fokker–Planck coefficient is isotropic,
Dμμ = (1 − μ2 )D. Gyrophase diffusion does not influence parallel
transport of charged particles.
6.1 General relations
The velocity correlation function can be computed by employing
+1
2π
+1
2π
1
dμ
dμ
d
d0
vi (t)vj (0) =
0
(4π)2 −1
0
0
−1
×vi (μ, )vj (μ0 , 0 )G(μ, μ0 , , 0 , t),
(75)
with the distribution function found in equation (56). For the velocities therein we can employ equation (1). It is convenient to express
the velocities by spherical harmonics. With equation (59) we find,
for the velocity components,
2π −1
Y1 (0 , 0 ) − Y11 (0 , 0 ) ,
vx (0 , 0 ) = v
3
2π −1
Y1 (0 , 0 ) + Y11 (0 , 0 ) ,
vy (0 , 0 ) = iv
3
π 0
Y (0 , 0 ) ,
vz (0 , 0 ) = 2v
3 1
2π ∗−1
Y1 (, ) − Y1∗1 (, ) ,
vx (, ) = v
3
2π ∗−1
Y1 (, ) + Y1∗1 (, ) ,
vy (, ) = −iv
3
π ∗0
(76)
Y (, ) .
vz (, ) = 2v
3 1
By combining equations (75) and (76) we can compute the velocity
correlation functions by employing the orthogonality relation of the
spherical harmonics:
2π
+1
(, ) = δll δmm .
dμ
d Ylm (, ) Yl∗m
(77)
−1
0
In the following we compute all nine velocity correlation functions.
6.2 The parallel velocity correlation function
In the first step we compute the velocity correlation function along
the mean magnetic field v z (t)v z (0). By combining equations (76)
with (75), we derive
+1
2π
+1
2π
v2
vz (t)vz (0) =
dμ
dμ0
d
d0
12π −1
0
0
−1
×Y10 (0 , 0 ) Y1∗0 (, )
×G(μ, μ0 , , 0 , t).
(78)
vz (t)vz (0) =
v 2 −2Dt
e
.
3
(79)
In the next section we will demonstrate that the parallel mean free
path is given by λ = 2D. Thus we can express the parallel velocity
correlation function as
v 2 −vt/λ
e
.
3
(80)
6.3 The mixed velocity correlation functions
We define the mixed velocity correlation functions as v z (t)v x (0),
v z (t)v y (0), v x (t)v z (0) and v y (t)v z (0). By again combining equations (76) and (75) we derive, for instance,
√ 2 +1
2π
+1
2π
2v
vz (t)vx (0) =
dμ
dμ0
d
d0
24π −1
0
0
−1
∗0
× Y1 (, ) Y1−1 (0 , 0 )
− Y1∗0 (, ) Y11 (0 , 0 )
×G(μ, μ0 , , 0 , t).
(81)
The indexes are different in the spherical harmonics. According to
the orthogonality relation (77), this velocity correlation function is
equal to zero. The same result can be found for all other mixed
velocity correlations. Thus, we find
vz (t)vx (0) = 0,
vz (t)vy (0) = 0,
vx (t)vz (0) = 0,
vy (t)vz (0) = 0.
(82)
All velocity correlations involving velocity components along and
across the mean magnetic field are zero.
6.4 The perpendicular velocity correlation functions
Here we compute the velocity correlation functions across the
mean magnetic field v x (t)v x (0), v x (t)v y (0), v y (t)v x (0) and
v y (t)v y (0). This work complements the work of Shalchi (2011a), in
which perpendicular velocity correlation functions have been computed by neglecting gyrophase diffusion. In the following we repeat
the calculations presented by Shalchi (2011a) by taking account of
this effect.
By again combining equations (75) and (76), we derive
+1
2π
+1
2π
v2
vx (t)vx (0) =
dμ
dμ0
d
d0
24π −1
0
0
−1
× Y1∗−1 (, ) Y1−1 (0 , 0 )
+ Y1∗1 (, ) Y11 (0 , 0 )
(83)
×G(μ, μ0 , , 0 , t).
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
By using equation (56) to replace the distribution function G(μ,
μ0 , , 0 , t) therein, we find, after employing the orthogonality
relation (77),
889
and employing the formulae (79) and (85) derived above for the
velocity correlation functions, we deduce
v2
,
6D
2
v
D
,
κxx =
3 D 2 + 2
v2
,
κxy =
3 D 2 + 2
v2
,
κyx = −
3 D 2 + 2
v2
D
(89)
.
κyy =
3 D 2 + 2
Please note that the result for κ zz is in perfect coincidence with
equation (12).
For the mean free paths defined as λij = 3κ ij /v we find
κzz =
vx (t)vx (0) =
v2
cos (t) e−Dt .
3
(84)
To compute the other velocity correlations is straightforward: we
find
v2
cos (t) e−Dt ,
3
v2
vx (t)vy (0) =
sin (t) e−Dt ,
3
v2
vy (t)vx (0) = − sin (t) e−Dt ,
3
v2
vy (t)vy (0) =
cos (t) e−Dt .
3
vx (t)vx (0) =
(85)
The velocity correlation functions derived here are similar to those
employed by Bieber & Matthaeus (1997). In the present paper,
however, we derived these forms by employing the Fokker–Planck
equation. Furthermore, we now know the characteristic time-scale
for the velocity correlations in the perpendicular direction, which
is, according to equation (85),
2λ
1
≡
.
D
v
τ⊥ =
(86)
The results are also similar to the forms derived by Shalchi (2011a),
who neglected the effect of gyrophase diffusion. In the present paper
we have taken this effect into account. In the following we will use
equations (85) to compute particle diffusion coefficients across the
mean magnetic field.
7.1 Spatial diffusion coefficients
Here we use the velocity correlation functions derived in the previous section to compute the spatial diffusion coefficients κ xx , κ xy ,
κ yx , κ yy and κ zz . As shown above, the mixed diffusion coefficients
are zero: κ xz = κ yz = κ zx = κ zy = 0. To perform these calculations
we employ the TGK formula (5), which yields
∞
dt
vi (t)vj (0) .
(87)
0
Please note that we employ this formula now to compute the spatial
diffusion coefficient κ ij in real space and not in phase space as in
the previous sections. By combining the TGK formula (87), using
the relations
∞
0
∞
0
D
,
+ 2
= 2
,
D + 2
dt cos (t)e−Dt =
dt sin (t)e−Dt
D2
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society λxx =
λxy =
λyx =
λyy =
RL
v
=
,
2D
2D/ D/ ,
RL
(D/ )2 + 1
1
,
RL
(D/ )2 + 1
1
,
−RL
(D/ )2 + 1
D/ .
RL
(D/ )2 + 1
(90)
Here we have used the unperturbed Larmor radius of the charged
particle, defined as RL = v/. The latter parameter is a measure for
the particle momentum.
7.2 The weak scattering limit
We can define the weak scattering limit as D , i.e. the pitchangle diffusion coefficient at 90◦ is much smaller than the gyrofrequency of the particle. In this case, equation (89) becomes
RL2
D,
3
vRL
.
κA = κxy = −κyx =
(91)
3
The result for the drift coefficient κ A is well-known in the literature
(see e.g. Burger & Visser 2010). The weak scattering limit for the
perpendicular diffusion coefficient is discussed below.
κ⊥ = κxx = κyy =
7 DIFFUSION COEFFICIENTS ALONG AND
AC RO S S T H E M E A N M AG N E T I C F I E L D
κij =
λ =
(88)
7.3 Comparison with the Bieber & Matthaeus model
The formulae (89) and (90) have a similar structure to the formulae
derived earlier by Bieber & Matthaeus (1997), Dosch, Shalchi &
Weinhorst (2009) and Shalchi (2011a). In the following we compare
those results with the formulae derived above. Furthermore, we
compare the underlying physical and mathematical assumptions.
An approach to solve the problem of perpendicular diffusion and
to calculate the off-diagonal elements of the diffusion tensors (κ xy ,
κ yx ) was presented by Bieber & Matthaeus (1997). The Bieber &
Matthaeus model is also based on the TGK formula (87). By assuming an exponential decorrelation of velocities, Bieber & Matthaeus
found results that have the same form as our results; see equations
(85) and (89). Bieber & Matthaeus noted correctly that the set of
equations (89) is formally the same as for hard-sphere scattering in
890
A. Shalchi
a magnetized medium (Gleeson 1969). In the Bieber & Matthaeus
model, however, the decorrelation time τ ⊥ is an unknown parameter. Thus, those authors tried to estimate τ ⊥ by assuming that the
field-line random-walk is the source of decorrelation of orbits. This
assumption is natural, since it is usually assumed that field-line wandering is an important effect in the theory of cross-field transport. On
the other hand, several authors employed a Chapman–Kolmogorov
equation description of compound transport of cosmic rays due to
random-walking magnetic field lines (see e.g. Webb et al. 2006;
Shalchi & Kourakis 2007; Webb et al. 2009). This approach is very
systematic and can easily be used to demonstrate that perpendicular
transport is subdiffusive for most cases if charged particles follow
magnetic field lines. In the approach presented above, we have considered the limit t → ∞ and therefore subdiffusive effects cannot be
described. Thus, it is natural to expect that field-line random-walk
do not have an influence on the perpendicular diffusion coefficients
derived in the present article.
In any case, the velocity-correlation-function based approach presented here provides the perpendicular correlation time automatically. We found
π
D
δB s+1
1
= C(s, q)R s−1
=
,
(92)
τ⊥
s
B0
based on equation (40). Please note that this result is only valid
for certain conditions (see previous sections of the present paper).
Very important here is the restriction R 1 that has been applied
to derive the second-order pitch-angle diffusion coefficient. Thus,
our results cannot be applied for high-energy particles.
Formula (92) for the perpendicular decorrelation time is in disagreement with the time-scale proposed by Bieber & Matthaeus
(1997), who derived
3 κFL
1
=
τ⊥
2 RL
(93)
with the diffusion coefficient of field-line wandering κ FL .
7.4 Comparison with the Newton–Lorentz equation approach
for cross-field scattering
Another approach for describing cross-field diffusion has been developed by Shalchi & Dosch (2008). These authors combined the
Newton–Lorentz equation with some ideas used in non-linear diffusion theory. This theory has been investigated analytically in Dosch
et al. (2009). There it was demonstrated that, according to the
Newton–Lorentz equation, cross-field transport can be described
as a superposition of perpendicular scattering due to field-line wandering and an additional (microscopic) diffusion. The latter effect is
described by formulae that are very similar to those derived above;
e.g. Dosch et al. (2009) found for low particle energies
v 2 δBx 2
v/λ
.
(94)
κ⊥ =
3
B0
(vλ )2 + 2
In the present paper we derived
κ⊥ =
v2
v/(2λ )
,
3 v/(2λ ) 2 + 2
(95)
by using the relations found in (89). Apart from the factor of 2 in the
descorrelation time and the ratio δBx /B0 in (94), the two results are
equal. It seems that the perpendicular diffusion coefficients derived
in the present paper as well as in Shalchi (2011a) describe the
scattering of charged particles away from a single magnetic field
line. Sometime those diffusion coefficients are called microscopic
diffusion coefficients.
Dosch & Shalchi (2009) have combined the approach based on
the Newton–Lorentz equation with quasi-linear theory. They found
for the Fokker–Planck coefficient of perpendicular scattering
D⊥ = D⊥FLRW + D⊥GR ,
(96)
where we used the well-known coefficient of perpendicular diffusion due to field-line random-walk (see Jokipii 1966) and a new
(microscopic) coefficient D⊥GR . The latter term is a gyro-resonant
term, given by
2π2 v |μ| slab
GR
k =
g
D⊥ =
vμ
B02
=
(vμ)2
Dμμ .
2 1 − μ2
(97)
Here we used the spectrum of the slab modes gslab (k ) and the pitchangle Fokker–Planck coefficient Dμμ . If we employ the isotropic
form Dμμ = (1 − μ2 )D as above, we derive
(vμ)2
D.
(98)
2
We can easily derive the diffusion coefficient of cross-field scattering
1 +1
R2
(99)
dμ D⊥GR = L D.
κ⊥GR =
2 −1
3
D⊥GR =
The latter formula is in agreement with the weak scattering limit
derived above (see equation 91 of the present paper). This result
suggested that formulae (90) describe microscopic diffusion, i.e.
the diffusion of the particle away from the magnetic field line.
The correction to the weak scattering limit for the drift coefficient
can be estimated by expanding equation (90) for D :
D2
vRL
1− 2 .
(100)
κA =
3
Obviously there is no term that is linear in D. Therefore, we conclude
that quasi-linear theory should provide a drift coefficient of zero.
The well-known limit κ A = vRL /3 can already be obtained from the
unperturbed orbit (see e.g. Burger & Visser 2010).
8 S U M M A RY A N D C O N C L U S I O N
In the present paper we have investigated the effect of gyrophase
diffusion in the theory of charged particle transport in turbulent
plasmas. The most important tasks were as follows.
(i) To compute the gyrophase Fokker–Planck coefficients Dμ ,
Dμ and D by using standard quasi-linear theory (SQLT: Jokipii
1966) as well as second-order quasi-linear theory (SOQLT: Shalchi
2005). We have also explored the validity of the assumption D , which has been used previously to derive the cosmic-ray transport equation.
(ii) To solve the Fokker–Planck equation to compute the velocity
distribution function. To do this we have employed second-order
results for the Fokker–Planck coefficients in the strong turbulence
limit. This solution can be used to describe the time-dependent
process of pitch-angle and gyrophase isotropization. These results
are useful to visualize the process of charged particle scattering.
(iii) To compute velocity correlation functions v i (t)v j (0) as
well as spatial diffusion coefficients κ ij by using our analytical
solution of the Fokker–Planck equation. The results provide an
alternative description of cross-field diffusion and drifts.
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society Gyrophase diffusion
The present work is important for understanding the whole process of particle diffusion. Some of the assumptions that have been
used previously without justifications were confirmed in the current article. Furthermore, our new results, especially equation (56),
are useful for visualizing two important effects in transport theory,
namely isotropization in the pitch angle and in the gyrophase.
In the last two sections we have revisited the problem of velocity correlations and spatial diffusion. The work presented there
complements the work done in Shalchi (2011a) and is also related
to the work of Bieber & Matthaeus (1997). Shalchi (2011a) has
computed velocity correlations and spatial diffusion coefficients by
neglecting the effect of gyrophase diffusion. In the present paper we
have taken into account gyrophase diffusion. The resulting theory is
an alternative to traditional theories for cross-field diffusion based
on the assumption that guiding centres follow field lines (see e.g.
Shalchi 2009 for a review) and theories based on solutions of the
Newton–Lorentz equation (see Shalchi & Dosch 2008). One aim of
future work will be to understand better the relation between those
previous theories and velocity-correlation-function based theories
such as the approach presented in Shalchi (2011a) and in the current
article.
AC K N OW L E D G M E N T S
Support by the Natural Sciences and Engineering Research Council (NSERC) of Canada is acknowledged. Furthermore, I thank
Professor Reinhard Schlickeiser for very fruitful discussions.
REFERENCES
Alania M. V., Iskra K., Siluszyk M., 2010, Adv. Space Res., 45, 1203
Berezhko E. G., Völk H. J., 2007, ApJ, 661, L175
Bieber J. W., Matthaeus W. H., 1997, ApJ, 485, 655
Burger R. A., Visser D. J., 2010, ApJ, 725, 1366
Büsching I., Potgieter M. S., 2008, Adv. Space Res., 42, 504
Büsching I., de Jager O. C., Potgieter M. S., Venter C., 2008, ApJ, 678, L39
Dosch A., Shalchi A., 2009, MNRAS, 394, 2089
Dosch A., Shalchi A., Weinhorst B., 2009, Adv. Space Res., 44, 1326
Dröge W., Kartavykh Y. Y., 2009, ApJ, 693, 69
Dröge W., Kartavykh Y. Y., Klecker B., Kovaltsov G. A., 2010, ApJ, 709,
912
Earl J. A., 1974, ApJ, 193, 231
Earl J. A., 1976, ApJ, 205, 900
C 2012 The Author, MNRAS 426, 880–891
C 2012 RAS
Monthly Notices of the Royal Astronomical Society 891
Engelbrecht N. E., Burger R. A., 2010, Adv. Space Res., 45, 1015
Gleeson L., 1969, Planet. Space Sci., 17, 31
Green M. S., 1951, J. Chem. Phys., 19, 1036
Hitge M., Burger R. A., 2010, Adv. Space Res., 45, 18
Jokipii J. R., 1966, ApJ, 146, 480
Kirk J. G., Schneider P., Schlickeiser R., 1988, ApJ, 328, 269
Kubo R., 1957, J. Phys. Soc. Jpn., 12, 570
Kunstmann J. E., 1979, ApJ, 229, 812
Li G., Zank G. P., Rice W. K. M., 2003, J. Geophys. Res., 108, 1082
MacRobert T. M., 1948, Spherical Harmonics: An Elementary Treatise on
Harmonic Functions with Applications, 2nd edn. Dover Press, London
Müller C., 1966, Spherical Harmonics, Lecture Notes in Mathematics, 1st
edn. Springer, Berlin
Rice W. K. M., Zank G. P., Li G., 2003, J. Geophys. Res., 108, 1369
Schlickeiser R., 2002, Cosmic Ray Astrophysics. Springer-Verlag, Berlin
Schlickeiser R., Shalchi A., 2008, ApJ, 686, 292
Schlickeiser R., Dohle U., Tautz R. C., Shalchi A., 2007, ApJ, 661, 185
Shalchi A., 2005, Phys. Plasmas, 12, 52905
Shalchi A., 2006, A&A, 448, 809
Shalchi A., 2009, Astrophysics and Space Science Library, Vol. 362,
Nonlinear Cosmic Ray Diffusion Theories. Springer-Verlag, Berlin
Shalchi A., 2011a, Adv. Space Res., 47, 1147
Shalchi A., 2011b, Phys. Rev. E, 83, 046402
Shalchi A., Dosch A., 2008, ApJ, 685, 971
Shalchi A., Kourakis I., 2007, A&A, 470, 405
Shalchi A., Weinhorst B., 2009, Adv. Space Res., 43, 1429
Shalchi A., Škoda T. A., Tautz R. C., Schlickeiser R., 2009a, Phys. Rev. D,
80, 023012
Shalchi A., Škoda T. A., Tautz R. C., Schlickeiser R., 2009b, A&A, 507,
589
Shalchi A., Li G., Zank G. P., 2010, Ap&SS, 325, 99
Tautz R. C., Shalchi A., Schlickeiser R., 2006, J. Phys. G: Nucl. Particle
Phys., 32, 809
Tautz R. C., Shalchi A., Schlickeiser R., 2008, ApJ, 685, L165
Taylor G. I., 1922, Proc. Lond. Math. Soc., 20, 196
Wawrzynczak A., Alania M. V., 2010, Adv. Space Res., 45, 622
Webb G. M., Zank G. P., Kaghashvili E. Kh., le Roux J. A., 2006, ApJ, 651,
211
Webb G. M., Kaghashvili E. Kh., le Roux J. A., Shalchi A., Zank G. P.,
Li G., 2009, J. Phys. A: Math. Theor., 42, 235502
Zimbardo G., 2005, Plasma Phys. Controlled Fusion, 47, B755
This paper has been typeset from a TEX/LATEX file prepared by the author.