Random walk of magnetic field-lines for different values of the energy range spectral
index
A. Shalchi & I. Kourakis
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik,
Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: October 9, 2007)
An analytical nonlinear description of field-line wandering in partially statistically magnetic systems was proposed recently. In this article the influence of the wave-spectrum in the energy range
onto field-line random walk is investigated by applying this formulation. It is demonstrated that
in all considered cases we clearly obtain a superdiffusive behaviour of the field-lines. If the energy
range spectral index exceeds unity a free-streaming behaviour of the field-lines can be found for all
relevant length-scales of turbulence. Since the superdiffusive results obtained for the slab model are
exact, it seems that superdiffusion is the normal behavior of field-line wandering.
PACS numbers: 47.27.tb, 96.50.Ci, 96.50.Bh
I.
INTRODUCTION
Understanding turbulence is an issue of major importance in space physics and astrophysics (see, for example, Refs. [1–6]). It has been demonstrated in several articles that stochastic wandering of magnetic fieldlines directly influences the transport of charged cosmic
rays (see, for example, Refs. [7–15]). Several theories
have been developed to describe field-line random-walk
(FLRW) analytically. The classic work of Jokipii (see
Ref. [16]), for instance, employed a quasilinear approach
for FLRW. In this theory the unperturbed field-lines are
used to describe field-line wandering by using a perturbation method. It has often been stated that this approach
is correct in the limit of weak turbulence where it is assumed that the turbulent fields are much weaker than
the uniform mean field (δBi ≪ B0 ). To achieve a more
reliable and general description of field-line wandering
Matthaeus et al. (see Ref. [17]) developed a nonperturbative statistical approach by combining certain assumptions about the properties of the field-lines (e.g. Gaussian
statistics) with a diffusion model. More precisely, in the
Matthaeus et al. theory of field-line wandering is has explicitly been assumed that field-line wandering behaves
diffusively.
An improved theory for FLRW, which is essentially a
generalization of the theory of Matthaeus et al., was recently developed by Shalchi & Kourakis (see Ref. [18]).
By explicitly assuming a diffusive behavior of the fieldlines, the Matthaeus et al. theory can be obtained from
the Shalchi & Kourakis approach as a special limit. However, it has also been demonstrated in Ref. [18] that for
slab/2D turbulence geometry, the field-lines behave superdiffusively. Thus, the Matthaeus et al. theory cannot be applied for slab/2D composite geometry. As also
demonstrated in Ref. [18], quasilinear theory is only correct for pure slab geometry or for small length scales.
In most past studies a constant spectrum in the energy range has been assumed (in this case the energy
range spectral index is equal to zero). It is the purpose
of this article to explore different values of the energy
range spectral index. The layout of this article goes as
follows. In Section 2, we discuss different forms of the
wave-spectrum which are appropriate for solar wind turbulence. In Section 3, we calculate the FLRW for pure
slab geometry for different values of the energy range
spectral index by applying the exact formulation for fieldline wandering. In Section 4, we employ the nonlinear
theory of Shalchi & Kourakis for FLRW, in order to deduce an analytic form for the field-line MSD for pure 2D
turbulence. These results can easily be combined with
the slab results to describe field-line wandering in the
slab/2D composite model (Section 5). In Sect. 6 we show
some numerical results and in Sect. 7 we summerize our
new results.
II.
DIFFERENT FORMS OF THE
WAVE-SPECTRUM
We consider a magnetostatic system which consists of
~ 0 = B0~ez ) and a turbulent coma uniform mean field (B
ponent (δBi ). Furthermore, we assume a negligible small
parallel turbulent component (δBz ≪ B0 ). In Ref. [19] a
two-component turbulence model has been proposed as
a realistic model for solar wind turbulence. In this model
we describe the turbulent fields as a superposition of a
slab model and a 2D model. As also demonstrated in
Ref. [18] (see also Sects. 3,4 and 5 of the current paper),
the key parameter which enters the fundamental formulas for field-line wandering, is the xx−component of the
correlation tensor in the wavenumber space, which can
be written as
slab ~
2D ~
Pxx (~k) = Pxx
(k) + Pxx
(k)
(1)
within the slab/2D composite model. In Eq. (1) we have
used the slab contribution
δ(k⊥ )
slab ~
Pxx
(k) = g slab (kk )
k⊥
(2)
and the 2D contribution
2D ~
Pxx
(k)
=g
2D
δ(kk )
kx2
(k⊥ )
1− 2 .
k⊥
k
(3)
TABLE I: The values of the normalization constants ci (i =
1, 2, 3). These expressions are correct for lslab ≪ Lslab .
case
In previous studies the forms
C(ν)
2
2
lslab δBslab
(1 + kk2 lslab
)−ν
g slab (kk ) =
2π
for the slab wave-spectrum, and
q<1
q=1
(4)
2C(ν)
2
2 2
l2D δB2D
(1 + k⊥
l2D )−ν
(5)
π
for the 2D wave-spectrum were used. Here we defined the
two bendover-scales lslab , l2D , the strengths of the turbu2
2
lent fields δBslab
, δB2D
, and the inertial range spectral
index 2ν. The energy range of the spectrum is defined
−1
−1
for kk ≪ lslab
and k⊥ ≪ l2D
. Clearly, both spectrum
forms are constant in the energy range. However, as discussed in several previous articles (see, for example, Ref.
[20]), we find in heliospheric observations a steeper spectrum (according to Ref. [20] the energy range spectral
index - cf. (6) below - should be q = 1.07). In the following we deduce and discuss analytical forms of the wave
spectrum for slab and 2D turbulence models.
1<q<2
Normalization constants ci
−1
4
4
+ 2v−1
c1 := 1−q
−1
slab
c2 := 14 ln Llslab
q−1
lslab
c3 := q−1
4
Lslab
g 2D (k⊥ ) =
A.
General form of the slab wavespectrum
According to solar wind observations, the following
form of the spectrum should be appropriate:
ci
2
g slab (kk ) =
lslab δBslab
2π

−1
0
if
kk < Lslab

−1
×
(k l )−q if L−1
slab ≤ kk ≤ lslab (6)
 k slab −2v
−1
(kk lslab )
if
lslab < kk .
In addition to the parameters used for the standard spectrum (see Eq. (4)) we have introduced the energy range
spectral index q. L−1
slab indicates the smallest parallel
wave-number which might be related to the bulk plasma
length scale Lslab . By taking into account the normalization condition
Z
h
i
2
2
2
slab ~
slab ~
δBslab = δBx + δBy = d3 k Pxx
(k) + Pyy
(k) (7)
we find for the normalization constant ci the values shown
in Table I. The values shown there are valid if the condition lslab ≪ Lslab is fulfilled. Note, that this restriction
simply means that the largest scale of turbulence (outer
scale, bulk plasma length scale) is much larger than the
slab bendover-scale of the turbulence. Also note, that
the first case of Table I (q < 1) also allows negative values of the spectral index corresponding to an increasing
spectrum in the energy range. In Fig. 1 the used spectrum is shown to illustrate the different regimes (energy
range, inertial range) and to show the difference between
previous spectra (with q = 0) and the spectrum used
in the current article (decreasing spectrum in the energy
range).
TABLE II: The values of the normalization constants di (i =
1, 2, 3). These expressions are correct for l2D ≪ L2D .
case
q<1
q=1
1<q<2
B.
Normalization constants di
−1
1
1
+ 2v−1
d1 := 1−q
−1
2D
d2 := ln Ll2D
q−1
d3 := (q − 1) l2D /L2D
General form of the 2D wavespectrum
For the 2D spectrum we can adopt the same form for
the spectrum as used in the last subsection for the slab
spectrum:
g 2D (k⊥ ) =
×
di
2
l2D δB2D
2π


0

−q
(k⊥ l2D )

 (k l )−2v
⊥ 2D
if
k⊥ < L−1
2D
−1
−1 (8)
if L2D ≤ k⊥ ≤ l2D
−1
if
l2D
< k⊥ .
L−1
2D indicates again the smallest wave-number and q is
again the energy range spectral index. Fulfilling the normalization condition
Z
h
i
2
2
2
2D ~
2D ~
δB2D = δBx + δBy = d3 k Pxx
(k) + Pyy
(k) (9)
we find for the normalization constant di the values
shown in Table II. Again the spectrum is correctly normalized for l2D ≪ L2D . In the following we consider
different values of the energy range spectral index q and
calculate the field-line mean square deviation analytically
for pure slab, pure 2D, and two-component turbulence.
III.
FLRW FOR PURE SLAB TURBULENCE
As shown in several previous papers (see, for example, Ref. [18]) the field-line mean square deviation can
be calculated exactly for pure slab geometry. For standard forms of the wave spectrum (see e.g. Eq. (4)),
where
qE = 0, we find the classical diffusive result:
D
2
= 2κF L |z| (see, for example, Ref. [16, 17]),
(∆x)
with the field-line diffusion coefficient κF L . Shalchi &
Kourakis (see Ref. [18]) derived the following ordinary
differential equation (ODE) for the mean square deviation and slab geometry
Z
E
2
d2 D
2
slab ~
(∆x)
d3 k Pxx
=
(k) cos(kk z)
dz 2
B02
Z ∞
8π
dkk g slab (kk ) cos(kk z).(10)
=
B02 0
It is obvious that we obtain superdiffusion (1 < b < 2)
for 0 < q < 1. Classical diffusion can only be obtained in
the limit q → 0 where we deduce from Eq. (14)
By employing the wave spectrum of Eq. (6) and the
integral transformation x = lslab kk this becomes
In this case 1−q is negative and (because of z/Lslab ≪
1) the second term in the rhs of Eq. (12) is dominant and
one gets
Z 1
E
2
d2 D
z
δBslab
2
−q
dx x cos x
(∆x)
≈ 4ci
dz 2
B02
lslab
xmin
Z ∞
z
(11)
+
dx x−2ν cos x
l
slab
1
D
B.
(∆x)
2
E
≈ 2πc1 lslab z
2
δBslab
.
B02
(16)
Steep spectrum form: the case 1 < q < 2
D
E z 2 δB 2
2
slab
(∆x) =
.
2 B02
(17)
This result if formally the same as the initial freestreaming result which can be found for small z values
(see, for example, Ref. [18]).
where we used xmin = lslab /Lslab . In Appendix A we
evaluate this expression in the limit lslab ≪ z ≪ Lslab .
We deduce there that
IV. FLRW FOR PURE 2D TURBULENCE
D
E
2
4ci
2
1−q δBslab q+1
lslab
z
(∆x)
=
q(q + 1)
B02
In this Section, we shall follow the nonlinear formal"
#
πq q(q + 1) z 1−q
ism for FLRW proposed by Shalchi & Kourakis (see Ref.
× Γ (1 − q) sin
(12). [18]). According to the results therein we have for pure
+
2
2(q − 1) Lslab
2D turbulence
Z
This expression is valid for 0 < q < 1 and for 1 < q < 2.
E
2
2
1
d2 D
2π ∞
2
For q = 1, Eq. (11) can be directly evaluated and we
=
(∆x)
dk⊥ g 2D (k⊥ )e− 2 h(∆x) ik⊥ .
2
2
dz
B
0 0
find a logarithmic behavior of the MSD. In the following,
(18)
we shall further simplify Eq. (12), by distinguishing the
It
can
easily
be
seen
that
we
obtain
a
nonlinear
formulaD
E
ranges 0 < q < 1 and 1 < q < 2. We stress that we
2
tion, since the field-line MSD (∆x) can also be found
are interested in the large z range, although we note that
the condition lslab ≪ z ≪ Lslab is assumed to hold evon the rhs of Eq. (18). By employing the spectrum of
erywhere (since Lslab is related to the size of the plasma
Eq. (8) and the integral transformation x = k⊥ l2D we
“box”). It can easily be shown (see, for example, Ref.
find
[18]) that for z ≪ lslab we find a free streaming solution
Z 1
E
2
2 2
δB2D
d2 D
2
of the form
dx x−q e−γ x
(∆x)
≈
d
i
2
2
dz
B0
xmin
D
E z 2 δB 2
Z ∞
2
slab
(∆x) =
.
(13)
2 2
2
−2ν
2 B0
dx x
e−γ x
+
(19)
1
A.
Smooth spectrum form: the case 0 < q < 1
In this case 1 − q is positive and by taking into account
z/Lslab ≪ 1 the first term in the rhs of Eq. (12) is
dominant and we obtain
E
D
πq 2
4c1
2
1−q q+1 δBslab
(∆x) ≈
(14)
.
lslab
Γ (1 − q) sin
z
2
q(q + 1)
B0
2
In generalDthe mean
E square deviation of the field-lines has
2
the form (∆x) = az b . According to Eq. (14) we find
for the slab model and for the values of the energy range
spectral index considered the characteristic exponent
b = q + 1.
(15)
where we used xmin = l2D /L2D and
D
E
2
(∆x)
γ2 =
.
2
2l2D
(20)
In Appendix B we evaluate this expression and we find
by assuming that γ 2 ≪ 1 (i.e., the field-line MSD cannot
exceed the maximum turbulence square length scale L22D )
!(q−1)/2
E
2
(∆x)2
di 1−q δB2D
d2 D
2
(∆x)
l
≈
dz 2
2 2D B02
2

!(1−q)/2 
(∆x)2
1
−
q
2
.
× Γ
+
(21)
2
q−1
2L22D
The formula can be applied so long as 0 < q < 2, except for q = 1. In the latter case, Eq. (19) can be
directly evaluated and we find a logarithmic behavior of
the MSD. In the following, we shall further simplify Eq.
(21), separately considering
0 < q < 1 and for
the cases
1 < q < 2. The relation (∆x)2 ≪ L22D is assumed to
hold everywhere.
A.
The case 0 < q < 1
In this
case 1 − q is positive and thus, because of
(∆x)2 ≪ L22D , the first term in Eq. (21) is dominant
E d
2
d2 D
1 1−q δB2D
2
(∆x)
≈
l
2D
dz 2
2
B02
4
3−q
(23)
and
2
2
d1 1−q δB2D
(1−q)/2 (3 − q)
a=
l2D
2
Γ
2
B02
4(1 + q)
1−q
2
2/(3−q)
.
(24)
Obviously we find
4
<b<2
3
(25)
which is interpreted as superdiffusion. A diffusive behavior (b = 1) cannot be obtained. We can recover the
result of Ref. [18] by considering the limit q → 0 where
we find b → 4/3. Even for q → 0 we find a superdiffusive
behaviour of FLRW.
B.
FLRW FOR SLAB/2D COMPOSITE
GEOMETRY
According to solar wind observations, it is more realistic than plainly adopting a pure slab or pure 2D model,
to consider a 20% slab/80% 2D composite model (see,
for example, Ref. [19]). In this case, one rigorously obtains a 2nd-order ODE [cf. (10), (18)], whose rhs is the
sum of the slab and 2D contributions, the relative weight
of which is determined by the corresponding turbulence
2
2
strength, i.e., δBslab
/δB 2 and δB2D
/δB 2 .
We shall now attempt to evaluate the asymptotic behavior of the field-line MSD in this hybrid (composite)
model.
!(q−1)/2 (∆x)2
1−q
Γ
.
2
2
(22)
E
D
2
By making the ansatz (∆x) = a z b we can solve this
ODE analytically. It can easily be demonstrated that
b=
V.
A.
In this case we can combine Eqs. (14) and (22) into:
E
πq 2
d2 D
2
1−q q−1 δBslab
=
4c
l
z
(∆x)
Γ
(1
−
q)
sin
1
slab
dz 2
B02
2
!(q−1)/2 2
(∆x)2
1−q
1−q δB2D
Γ
(29),
+ d1 l2D
B02
2
2
where negligible contributions were omitted in the rhs.
Obviously this equation has the form
E
(q−1)/2
d2 D
2
(∆x) = αz q−1 + β (∆x)2
.
(30)
2
dz
By applying the ansatz (∆x)2 = az b we find
ab(b − 1)z b−2 = αz q−1 + βa(q−1)/2 z b(q−1)/2
(31)
where the definitions of α and β are obvious. It is
straightforward to prove that, since b < 2, the second
term in the rhs is dominant for z → ∞. The slab contribution can therefore be neglected, so we can use Eqs.
(23) and (24) also within the two-component model.
The case 1 < q < 2
B.
In this case q−1 is negative and the second term within
brackets in Eq. (21) is dominant and we have
E
2
d2 D
d3 δB2D
2
(∆x)
=
dz 2
q − 1 B02
l2D
L2D
1−q
.
(26)
By using Table II for d3 this can be simplified to
E δB 2
d2 D
2
2D
(∆x) =
2
dz
B02
In this case we can simply add the two contributions
(Eqs. (17) and (28)):
D
E z 2 δB 2
2
(32)
(∆x) =
2 B02
2
2
δB 2 = δBslab
+ δB2D
.
(27)
VI.
(∆x)
2
E
2
=
z
2
2
δB2D
B02
which is again the initial free-streaming result.
The case 1 < q < 2
where we have set
and we finally find
D
The case 0 < q < 1
(33)
NUMERICAL EVALUATION OF THE
INTEGRALS
(28)
In this article several analytical results are derived
asymptotically. It is the purpose of this section to test
some of these analytical results by evaluating the integral of Eq. (11) numerically. In Figs. 2-4 we compare
numerical results with the analytical results derived in
Sect. 2, for different values of the energy range spectral
index, namely for q = 0 (see Fig. 2), q = 0.5 (see Fig. 3),
and q = 1.07 (see Fig. 4). Shown
D areErunning diffusion
TABLE III: In this table, the results obtained
for the parameters a and b, having adopted the form (∆x)2 = az b for the
field-line mean square deviation, are presented. In all (but
one) cases, we find either superdiffusion (1 < b < 2) or freestreaming (b = 2) of the field-lines. Diffusion (b = 1) can only
be found for slab geometry and q = 0.
2
coefficients defined as Dxx = (∆x) /(2z) versus the
parallel position z.
As illustrated, in the first two cases there is always
a good agreement between numerical and analytical results (except for intermediate length scale where we have
z ≈ lslab ). In the latter case a small discrepancy between
the analytical and numerical results can be seen. The
reason for this difference is that q = 1.07 ≈ 1 and therefore the analytical solution is no longer very accurate.
All results shown in Figs. 2-4 are for pure slab geometry.
For slab/2D composite geometry the numerical evaluation of the corresponding integrals is difficult due to the
nonlinear character of the ODE. Such an investigation
will be subject of future work.
Another possibility to test our results would be a comparison with simulations of FLRW. Such a comparison
would also allow a test of our nonlinear theory for FLRW
used to derive the results for 2D and slab/2D turbulence.
Such simulations and a comparison between them and
our analytical findings will also be subject of future work.
VII.
SUMMARY AND CONCLUSION
We have investigated the random walk of magnetic
field-lines for a more general spectrum, than the one employed in previous works. By exploring pure slab, pure
2D, and two-component turbulence models, we have calculated the field-line mean square deviation by applying
the analytical description for FLRW proposed by Shalchi
& Kourakis (see Ref. [18]). A superdiffusive behaviour
is found in all cases considered. In Table III the results
obtained in this article are summarized. The only case
where one obtains diffusion is for pure slab geometry and
q = 0. As shown in this article the energy range spectral
index is a key-input parameter if FLRW is described.
In the two-component turbulence model, which has
been considered as a realistic model for solar wind turbulence (see Ref. [19]), we already find a weakly superdiffusive behavior
if q = 0. For larger values of q we have
D
E
2
(∆x) ∼ z 4/(3−q) . If the energy range spectral index
exceeds unity we find the same solution as in the initial
free-streaming regime. Obviously, the energy range spectral index has a very strong influence on FLRW behavior.
¿From a theoretical point of view the results for pure
slab geometry deduced in Section 3 and numerically in
Sect. 6
are very interesting and important because of two reasons:
• for pure slab turbulence the parameter < (∆x)2 >
Geometry spectral index a
b
2
slab
0<q<1
slab
1<q<2
2D
0<q<1
2D
1<q<2
slab/2D
0<q<1
slab/2D
1<q<2
δB
4c1
Γ (1
l1−q slab
2
q(q+1) slab B0
2
1 δBslab
2
2 B0
2
d1 1−q δB2D
2
2 2D
B0
2
1 δB2D
2
2 B0
2
1−q δB2D
1 2D
2
B0
2
h
h
l
d l
1 δB
2
2 B0
− q) sin
πq
2
2(1−q)/2 Γ
i2/(3−q)
1−q
2(1−q)/2 Γ
i2/(3−q)
1−q
2
2
q+1
2
4/(3 −
2
4/(3 −
2
can be calculated exactly. No theory nor any ad
hoc assumption have to be applied.
• In all cases except q = 0 we find superdiffusion of
FLRW.
Since in reality 20 % of the fluctuations can be represented by slab modes (see Ref. [19]) it is self-evident to
assume that superdiffusion and not (classical or Markovian) diffusion is the regular case in astrophysical turbulence.
If we merge from pure slab geometry to the slab/2D
composite model a (nonlinear) theory has to be applied
and an exact description of FLRW in no longer possible.
By applying the ODE deduced by Shalchi & Kourakis
(see Ref. [18]) we have shown that the superdiffusivity
becomes even stronger in comparison to the pure slab
results.
It must be the subject of future work to apply these
new results on realistic systems, such as solar wind turbulence. An important example is perpendicular transport
of charged cosmic rays which is directly controlled by
the FLRW, since charged particles are tied to magnetic
field-lines (see, for example, Ref. [15]).
APPENDIX A: EVALUATION OF THE
INTEGRAL FOR SLAB GEOMETRY
For slab turbulence the field-line MSD is given by Eq.
(11). Taking into account the relation
Z ∞
i
1 h −iπµ/2
e
Γ (µ, +iu) + e+iπµ/2 Γ (µ, −iu) ,
dx xµ−1 cos x =
2
u
(A1)
according to Gradshteyn & Ryzhik (see Ref. [21], page
430, Eq. 3.761.7 therein), for µ < 1 (implying here
q > 0), where we have
R ∞employed the incomplete Gamma
function Γ(µ, x) = x dt tµ−1 e−t (see Eq. 8.35 in the
latter reference), and approximating Γ(µ, x), for small
values of the argument x, as
xµ
Γ(µ, x ≪ 1) ≈ Γ (µ) 1 −
(A2)
µΓ(µ)
(see Eq. 8.354.2 in the same reference) we find
Z ∞
π 1
dx xµ−1 cos x ≈ Γ (µ) cos
µ − uµ .
2
µ
u
(A3)
By applying this formula onto Eq. (11) one gets
Gamma-functions Γ(a, z)
E
2
d2 D
1−q 2 2
di δB2D
2
q−1
(∆x)
γ
Γ
, γ xmin
≈
dz 2
2 B02
2
1−q 2
− γ q−1 Γ
,γ
2
1
2
2ν−1
− ν, γ
.
(B1)
+ γ
Γ
2
By taking into account the restriction
E
2
d2 D
2
1−q δBslab q−1
1 ≪ γ 2 ≪ x−2
(B2)
(∆x)
z
=
4c
l
i
min
slab
dz 2
B02
#
"
1−q
πq 1
z
and by employing (see, for example, Ref. [22])
+
(A4).
× Γ (1 − q) sin
2
q − 1 Lslab
Γ(a, z ≫ 1) ≈ z a−1 e−z → 0
The result can easily be integrated to obtain
za
Γ(a, z ≪ 1) ≈ Γ(a) −
(B3)
D
E
2
a
4ci
2
1−q δBslab q+1
(∆x)
=
l
z
q(q + 1) slab B02
# this expression can be simplified to
"
πq q(q + 1) z 1−q
(A5).
× Γ (1 − q) sin
+
E
2
2
2(q − 1) Lslab
(1−q)/2
d2 D
2
di δB2D
1−q
2
q−1
(∆x)
+
≈
γ
Γ
γ 2 x2min
(B
2
2
dz
2 B0
2
q−1
This is the result for the field-line MSD for lslab ≪ z ≪
Lslab . A further simplification of this formula can only
To simplify this expression we have to distinguish bebe achieved if we specify the parameter q. This is done
tween different limits of the parameter q.
in the main part of this paper.
ACKNOWLEDGMENTS
APPENDIX B: EVALUATION OF THE
INTEGRAL FOR 2D GEOMETRY
Here we evaluate the integrals of Eq. (19). According
to Ref. [21], this integral can be expressed by incomplete
This research was supported by Deutsche Forschungsgemeinschaft (DFG) under the Emmy-Noether Programme (grant SH 93/3-1). As a member of the
Junges Kolleg A. Shalchi also aknowledges support by the
Nordrhein-Westfälische Akademie der Wissenschaften.
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FIGURE CAPTIONS
Figure 1: The power spectrum used for the slab model.
For the 2D model a spectrum with the same form is used.
Figure 3: Numerical results for the running fieldline diffusion coefficient Dxx in comparison with the
asymptotic behaviour derived analytically for the slab
model (q = 0.5). Shown are the analytical result for the
initial free streaming regime (z ≪ lslab , dashed line),
the analytical result for z ≫ lslab (solid line), and the
numerical results (dotted line).
Figure 4: Numerical results for the running field-line
diffusion coefficient Dxx in comparison with the asymptotic behaviour derived analytically for the slab model
(q = 1.07). Shown are the analytical result for the
initial free streaming regime (z ≪ lslab , dashed line),
the analytical result for z ≫ lslab (solid line), and the
numerical results (dotted line). Note that in this case
the analytical solutions for z ≪ lslab and z ≫ lslab are in
coincidence.
∼ k−q
||
0
10
Spectrum
Figure 2: Numerical results for the running fieldline diffusion coefficient Dxx in comparison with the
asymptotic behaviour derived analytically for the slab
model (q = 0). Shown are the analytical result for the
initial free streaming regime (z ≪ lslab , dashed line),
the analytical result for z ≫ lslab (solid line), and the
numerical results (dotted line).
2
10
−2
10
−4
10
Energy range
In
−6
10
−4
10
L−1
slab
−2
10
0
10
lsla
0
Dxx/lslab
10
−1
10
−2
10
−1
10
0
10
1
10
2
10
3
10
z/l
slab
1
10
0
Dxx/lslab
10
−1
10
−2
10
−1
10
0
10
1
10
z/lslab
2
10
3
10
2
10
1
Dxx/lslab
10
0
10
−1
10
−2
10
−1
10
0
10
1
10
z/l
slab
2
10
3
10