A Comparison with Theory of Computation and Estimation of

Pitch-Angle Diffusion Coefficients from Simulations in Noisy Reduced

Magnetohydrodynamic Turbulence

CHANIDAPORN PLEUMPREEDAPORN, ANDREW P. SNODIN,ELVIN J. MOORE

Department of Mathematics

King Mongkut’s University of Technology North Bangkok

1518 Pracharat 1 Road, Bangkok 10800

THAILAND

Abstract: The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in var-

ious astrophysical and laboratory plasma contexts. In order to estimate the mean free path λ∥of a particle in

the direction parallel to the mean magnetic field, one can use theoretical expressions that include the pitch-angle

diffusion coefficient Dµµ. In this work we evaluate theories for Dµµ in the context of the noisy reduced magneto-

hydrodynamic (NRMHD) model where turbulent fluctuations are absent at large parallel wavenumbers. For most

turbulence models, the standard quasilinear theory predicts zero pitch-angle diffusion only for particles with a 90◦

pitch angle, but for NRMHD a range of pitch angles is affected, leading to infinite λ∥. We examine two theories

that include resonance broadening which yield finite λ∥and compare them with test-particle computer simulations

in which the parallel mean free path can be readily obtained. We find that both theories are quite accurate in some

regions of the parameter space considered, but neither is particularly good when the particle Larmor radius RL

becomes much smaller than the quasilinear resonance limit.

Key-Words: Diffusion, Pitch-Angle Scattering, Synthetic Magnetic Fields, Turbulence, Test Particle Simulations

Received: June 19, 2021. Revised: March 21, 2022. Accepted: April 21, 2022. Published: May 20, 2022.

1 Introduction

The transport of energetic charged particles (e.g., cos-

mic rays) in turbulent magnetic fields is an impor-

tant process in various astrophysical and laboratory

plasma contexts. A quantity of interest is the parallel

(to the mean magnetic field direction) diffusion co-

efficient, κ∥, which can be expressed in terms of the

parallel mean free path λ∥(e.g. [1]):

λ∥=3v

8Z1

−1

(1 −µ2)2

Dµµ(µ)dµ, (1)

where vis the particle velocity and µis the cosine

of its pitch angle with respect to the mean magnetic

field direction. This is expected to be valid when the

particle pitch-angle distribution function, f(µ, t), is

almost isotropic. The mean free path is related to the

diffusion coefficient via λ∥= 3κ∥/v. Parallel diffu-

sion theory is dominated by the quasilinear theory [3]

and its various extensions (e.g. [12, 11]), which focus

on obtaining expressions for the pitch-angle diffusion

coefficient Dµµ(µ).

For ease of reading, Table 1 gives definitions and

assumed values for variables and parameters used in

this paper.

Test particle computer simulations are a useful tool

in the study of charged particle transport, and in par-

Table 1: Definitions of variables, parameters and val-

ues. Lengths, velocities and magnetic field strengths

are given in dimensionless units as noted in the text.

Parameter Symbol Value

Parallel length scale l∥1

Perpendicular length scale l⊥1

Particle velocity v1

Pitch-angle cosine µ−1≤µ≤1

Larmor radius RLvariable

Gyrofrequency Ωvariable

Parallel wavenumber k∥variable

Perpendicular wavenumber k⊥variable

Mean magnetic field strength B01

Turbulence strength b/B0variable

Perpendicular bend-over scale ˆ

λ⊥2.4954

Dimensionless rigidity Rvariable

(RL/ˆ

λ⊥)

Magnetic power spectrum Svariable

Cut-off wavenumber kz=k∥K π/2, π/20

ξ=Kˆ

λ⊥ξ0.39198

3.9198

ticular can be used to test theoretical expressions for

particle diffusion coefficients. For test particle sim-

ulations, the parallel diffusion coefficient can be ob-

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tained as shown in equation (2)

κ∥=limt→∞ ⟨(∆z)2⟩

2t,(2)

where ∆z(t)is the displacement of a particle in the

parallel direction over a time t, and the angle brack-

ets denote an ensemble average over particles which

can be obtained by simulating the trajectories of a

large number of particles in turbulent magnetic fluc-

tuations up to a sufficiently large time where the limit

⟨(∆z)2⟩ ∝ tbecomes constant. This occurs in most

contexts where the magnetic field has a well-defined

integral length scale. One could apply an analogous

expression for the pitch-angle diffusion

Dµµ(µ) = lim

t→∞ (∆µ)2

2t,(3)

where ∆µ=µ(t)−µis the pitch-angle displace-

ment from its initial value µ. However, this form has

a severe difficulty since the pitch-angle µis bounded,

i.e. µ∈[−1,1], so that (∆µ)2≤4, and in the infi-

nite limit one obtains Dµµ(µ) = 0. In the literature,

this problem is remedied by taking the value of this

expression at a sufficiently large time (see, e.g. [17]),

or simply after one gyroperiod tLof a particle, as sug-

gested by [18]. An alternative approach to obtain Dµµ

involves following the evolution of the pitch-angle

distribution function f(µ, t)of the test particles and

assume that it obeys a pitch-angle diffusion equation

(and hence Fick’s law), so that Dµµ can be calculated

via the derivatives of fand the pitch-angle flux (e.g.

[4, 16, 2, 7]).

If we have a theoretical expression for Dµµ, we

can then compare it with the corresponding value of

Dµµ in computer simulations, or we can use Eq. 1 to

compare it with the parallel mean free path in the com-

puter simulations obtained via Eq. 2. In this work

we study pitch-angle diffusion in the noisy reduced

magnetohydrodynamic (NRMHD) model[10], a syn-

thetic model of homogeneous magnetic turbulence.

The model is interesting in that it has no magnetic

fluctuations at higher parallel wavenumber, kz> K.

According to the classical quasilinear scattering the-

ory [3], particles are in resonance with magnetic fluc-

tuations when kz= 1/(|µ|RL), where µis the cosine

of the pitch angle and RLis the particle’s Larmor ra-

dius. This implies that whenever RL<1/K, there is

no resonant scattering, so that the particle’s mean free

path λ∥will be infinite. Due to higher order nonlin-

ear effects, λ∥is actually finite, but can become very

large at small RL. This presents a problem for the the-

ory and the standard quasilinear approach, as detailed

below, cannot be used. We here consider a variant of

the second order quasilinear theory (SOQLT) [11] and

a special case of the weakly nonlinear theory (WNLT)

[12].

Below we provide details of the NRMHD model

and test particle simulations that we compare directly

with the SOQLT and WNLT theories in terms of Dµµ

and λ∥, as discussed above. This work is quite sim-

ilar to that of [9], but our computational simulations

are performed using a discrete magnetic field mesh,

rather than a locally-evaluated continuous magnetic

field. Here we also use a slightly different parame-

ter regime (stretching into the non-resonant scattering

region), evaluate SOQLT in detail and contrast Dµµ

results. This work is quite similar to [7], where the

widely-studied slab and two-component turbulence

models were considered.

2 Noisy RMHD Turbulence

The noisy reduced magnetohydrodynamic (NRMHD)

model can be applied to different systems such as fu-

sion plasma and the solar corona (for details of the

model, see, e.g. [10]). NRMHD is a model that char-

acterizes RMHD turbulence, which comes from a re-

duced set of MHD equations, that has 2D-like and 3D-

like properties [6]. The model was developed by [10],

and it has been applied in studies of the field line ran-

dom walk (related to test-particle transport, [15]) and

in some particle transport studies (see, e.g., [9, 13]).

The total magnetic field is expressed as



B(x) = B0ˆ

z +

b(x, y, z),

b⊥ˆ

z, (4)

where 

B0=B0

ˆzis the mean magnetic field, which

is taken to be aligned along the ˆ

z direction and to be

strong compared with a transverse fluctuating mag-

netic field 

b(x)with zero mean ⟨

b(x)⟩=

0. We as-

sume that the magnetic fields are static and homoge-

neous, which means that the field does not depend on

time and the statistical properties of the magnetic field

are invariant under translations. Although magnetic

fields will evolve over time, this time can be large

compared with the time taken by particles to become

diffusive and the variation can be neglected. This can

be justified in a variety of physical contexts, espe-

cially for highly relativistic particles, but in general

one could compare the length and timescales associ-

ated with the magnetic field to the particle’s velocity

magnitude.

The statistically homogeneous fluctuating field is

given in terms of wave vectors 

k, for which the po-

tential function in k-space can be written as [10]

a(

k) = CpA2D(k⊥)eiφ(

k)for |kz| ≤ K

0for |kz|> K, (5)

where φ(

k)is a random phase, A2D(k⊥)is the 2D

spectrum and Kis a cut-off wavenumber in kz. This

has a different random phase for every independent 

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(in 3D), and has a finite extent of the spectrum in kz.

The normalization constant Cis chosen to satisfy a

condition on the mean energy density in the magnetic

field. The properties of NRMHD are controlled by

the dimensionless parameters b/B0and Kˆ

λ⊥, where

bis the rms fluctuation strength (in units of mag-

netic field strength) and the other parameters are as

in Table 1. In computer simulations the magnetic

field is constructed on a three-dimensional grid of size

Nx×Ny×Nz, first in wavevector space, then ap-

plying an inverse FFT to each component. We use

Nx=Ny= 512 and Nz= 4096 grid points. Multi-

ple realisations of the field are constructed by taking

different random values of φ. The details of the syn-

thetic field construction can be found in [15].

3 Test Particle Simulations

For each particle in a realization we solve the (dimen-

sionless) Newton-Lorentz equations for the particle

trajectory x(t)and velocity v(t),

dv

dt =αv ×

B[x(t)] ,dx

dt =v, (6)

where α=qB0l⊥/(γmv0), with qthe particle

charge, γthe Lorentz factor, mthe particle rest mass

and v0the unit of velocity. Note that these equa-

tions imply that |v|is constant, or in other words, the

particle energy is conserved. For simplicity we take

|v|= 1 in units of v0,B0= 1 and then vary the pa-

rameter αto control the dimensionless Larmor radius

RL/l⊥=γmv0/(|q|B0).

We take a large number of particles with initial lo-

cations uniformly distributed in space over the mag-

netic field grid. The initial velocity distribution de-

pends on what quantity we wish to evaluate. To calcu-

late the parallel mean free path λ∥we take initially an

isotropic velocity distribution and then integrate the

equations of motion up to a time when Eq. 2 con-

verges for an ensemble of particle trajectories and

magnetic field realisations. In the case of evaluat-

ing Dµµ in simulation, we take a distribution of ve-

locities parallel to the mean field such that the dis-

tribution of pitch angle cosines µhas a constant gra-

dient, i.e. f(µ, 0) ∝µon some interval in [−1,1].

The details of obtaining Dµµ in simulations by cal-

culating ∂df/∂µ and the pitch-angle flux are given

in [7]. In the following, we denote this quantity as

Dµµ[FD], and the λ∥obtained directly from simula-

tions as λ∥[sim].

4 Pitch Angle Diffusion Theories

In the following we make a specific choice for the 2D

spectrum A(k⊥)in order to implement the model in

computer simulations and evaluate theoretical expres-

sions. We take (see [5])

A(k⊥) = A0

h1+(ˆ

λ⊥k⊥)2i7/3 ,(7)

where A0= 8b2ˆ

λ4

⊥/9.

4.1 Quasilinear theory

QLT was originally developed by Jokipii [3]. It is the

simplest approach for describing spatial diffusion. In

QLT the actual particle trajectory in the NRMHD field

is approximated by the trajectory of a particle in the

mean field, i.e. it is assumed that the turbulent fluctu-

ations are relatively weak. With this approximation, a

particle is taken to rotate in the direction perpendicu-

lar to the mean field and to move with constant veloc-

ity along the mean field, i.e., a particle’s trajectory is

assumed to be a helical motion with a constant Larmor

radius. We employ this concept to compute the pitch-

angle diffusion coefficient Dµµ(µ)for NRMHD tur-

bulence as in the following expression [9]:

DQLT

µµ (µ) = πΩ2

2v|µ|R2

LKB2

0R∞

0P+∞

n=−∞ k⊥A(k⊥)

×HK−



nΩ

vµ 



n2J2

n(W)dk⊥.(8)

where W=RLk⊥p1−µ2, H the Heaviside step

function and Jnthe Bessel function of order nof the

first kind. We denote this estimate by Dµµ[QLT].

4.2 Second-order quasilinear theory

The 90◦scattering problem is a well-known problem

in diffusion theory (see, e.g., [14]). The SOQLT pro-

vided a solution of this problem of finding a diffusion

formula for µ= 0, i.e., at a pitch angle of 90◦. Here

for NRMHD we adopt the simplified form of SOQLT

pitch angle diffusion coefficient as in [9],

DSOQLT

µµ (µ) = 2πΩ2

LKB2

+∞

n=0 ZK

0Z∞

Rn(µ, kz)dkz

×k−2

⊥

A(k⊥)

2πn2J2

n(W)dk⊥,

(9)

with resonance function Rn

Rn(µ, kz) = √π

v|kz|

bexp −(µRLkz+n)2B2

(RLkzb)2,

(10)

with the various other quantities as defined previ-

ously. We denote this estimate by Dµµ[SOQLT].

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4.3 Weakly non-linear theory

As shown in [9], the weakly nonlinear pitch-angle

diffusion coefficient can be obtained from the theory

[12]. The WNLT formula for Dµµ(µ)that we use in

our calculations is

DWNLT

µµ =8

9ξR4|µ|

0Z∞

(1 + x2)7/3

+∞

n=1

n2J2

nRxp1−µ2

×arctan n+RL|µ|K

RLD⊥k2

⊥/2 

−arctan n−RL|µ|K

RLD⊥k2

⊥/2 dx.

(11)

In (11), we take the form of D⊥(µ)to be

D⊥(µ) = (ν+ 1)|µ|νκ⊥,(12)

where κ⊥is the particle perpendicular diffusion coef-

ficient and νis an arbitrary parameter (0≤ν < 1).

D⊥is the pitch-angle dependent perpendicular diffu-

sion coefficient, and the form used here was originally

proposed by [14, 9] based on 2D field simulation re-

sults and might not be the most appropriate form for

NRMHD.

In our calculations, we have tested a range of val-

ues of νin the resonant regime RL>|n|

µK and found

that ν= 0 usually gave either the best value for the

diffusion coefficients or an acceptable value. In this

case,

D⊥=κ⊥,(13)

which is independent of µ. This model requires an

estimate for κ⊥. In our WNLT calculations, we take

κ⊥from computer simulations.

5 Results

In our numerical calculations for the NMRHD model,

we used the values of parameters summarized in Ta-

ble 1. As shown in Tables 2, 3 and 4, we carried

out test-particle simulations for 12 sets of values of

b/B0and RL/l⊥and computed the λ∥[sim]/l⊥and

λ∥[FD]/l⊥values. For the same 12 sets of values,

we calculated the λ∥/l⊥values from the SOQLT and

WNLT theories.

The values for λ∥[sim]/l⊥in Table 2 were com-

puted from the test-particle simulation results for κ∥

using

λ∥=3

vκzz.(14)

The values for λ∥[FD]/l⊥and λ∥[SOQLT]/l⊥were

computed using Eq. (1) from values for the pitch an-

gle coefficients Dµµ(µ)computed from the flux and

derivative, and SOQLT (Eq. (9)) formulas. The val-

ues for λ∥[WNLT]/l⊥in Table 4 was computed using

Eq. (11) from values for the pitch-angle coefficients

Dµµ(µ). The errors in Tables 2, 3 and 4 are relative

errors computed from the formula

λ∥[·]error =

λ∥[sim]/l⊥−λ∥[·]/l⊥



λ∥[sim]/l⊥

.(15)

Table 2: Comparison of the values of parallel mean

free path (λ∥) computed from test-particle simulations

with estimates λ∥[FD].

b/B0RL/l⊥λ∥[sim]/l⊥λ∥[FD]/l⊥λ∥[FD]error

0.3 0.37 7478.19 7480.52 0.0003

0.3 1.93 560.19 561.27 0.0019

0.3 3.73 935.34 927.22 0.0087

0.3 10.0 7801.61 7837.17 0.0046

0.5 0.2 292.08 297.13 0.0173

0.5 0.5 154.36 154.30 0.0004

0.5 1.93 92.54 92.07 0.0051

0.5 10.0 1652.40 1650.90 0.0009

1.0 0.2 7.91 7.91 0.0006

1.0 0.5 9.23 9.23 0.0003

1.0 1.93 14.13 14.12 0.0005

1.0 10.0 272.59 272.62 0.0001

Table 3: Comparison of the values of parallel mean

free path (λ∥) computed from test-particle simulations

with estimates λ∥[SOQLT]based on Eq. (9)

b/B0RL/l⊥λ∥[SOQLT]/l⊥λ∥[SOQLT]error

0.3 0.37 4.90 ×1014 6.55 ×1010

0.3 1.93 195.75 0.6506

0.3 3.73 394.68 0.5780

0.3 10.0 2503.04 0.6792

0.5 0.2 5.20 ×1017 1.78 ×1015

0.5 0.5 1738.47 10.2624

0.5 1.93 46.82 0.4940

0.5 10.0 837.05 0.4934

1.0 0.2 10208.36 1289.5638

1.0 0.5 5.71 0.3814

1.0 1.93 9.50 0.3277

1.0 10.0 211.14 0.2254

As shown by the small errors in the λ∥[FD]er-

ror column in Table 2, the flux derivative method

gives very good agreement with the test-particle sim-

ulations. These results show that the flux deriva-

tive method, which is based on interpreting Dµµ as

a Fokker-Planck coefficient, gives very good results

in all cases.

The SOQLT has been discussed by Reimer and

Shalchi [9] who concluded that the SOQLT was not

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Table 4: Comparison of the values of parallel mean

free path (λ∥) computed from test-particle simulations

with estimates λ∥[WNLT]based on Eq. (11)

b/B0RL/l⊥λ∥[WNLT]/l⊥λ∥[WNLT]error

0.3 0.37 124.33 0.9834

0.3 1.93 789.43 0.4090

0.3 3.73 1572.02 0.6807

0.3 10.0 7229.86 0.0733

0.5 0.2 13.65 0.9530

0.5 0.5 37.20 0.7590

0.5 1.93 134.82 0.4569

0.5 10.0 1473.36 0.1084

1.0 0.2 2.45 0.6903

1.0 0.5 5.09 0.4485

1.0 1.93 14.04 0.0064

1.0 10.0 215.46 0.2096

useful. They argued that as their theory predicts a

dependence exp(B2

0/b2)(see Eq. (10)) it would give

very large parallel mean free paths for b < B0. How-

ever, their simulation results did not seem to have

huge parallel mean free paths and therefore their con-

clusion that SOQLT was not useful seemed reason-

able. In contrast, in the cases shown in Table 2, we

usually consider small b/B0and our results show very

large λ∥values when b/B0<0.5. Also, our results

are actually better than expected for SOQLT as the

λ∥[SOQLT]values are of a similar order of magni-

tude to the λ∥[sim]values for all values of b/B0, ex-

cept in the small RL= 0.2,0.37 cases. A possi-

ble explanation for the results is that RL/l∥<2/πis

the condition for the QLT non-resonant regime. The

standard QLT would have Dµµ = 0 for all µand so

infinite parallel mean free path (see Eq. (1)). In the

SOQLT the QLT resonant wavenumber of the parti-

cles is broadened and this has the effect of smoothing

the QLT result so that it looks more like the simulation

result. However, the RL= 0.2cases are far below the

QLT resonant scattering regime and therefore also the

SOQLT regime. In this regime, the simulation result

is determined by higher-order nonlinear scattering.

The WNLT results in Table 4 are for the special

case of ν= 0. In their recent paper on WNLT, Reimer

and Shalchi [9] studied the dependence of the diffu-

sion coefficients on νfor 0≤ν < 1and found that

the WNLT results agreed well with the simulation val-

ues with suitable choices for ν.

The plots in Figs. 1, 2, 3 show the values of the

pitch angle diffusion coefficients calculated from the

FD, SOQLT, QLT and WNLT(ν= 0) methods for

the 12 cases in Tables 2 and 4. In each figure, panels

(a)-(l) show plots of Dµµ(µ)values calculated from

the diffusion equation integration method Dµµ[FD]

(solid lines) ([2, 7]), the second-order quasilinear the-

ory Dµµ[SOQLT](dashed lines), the quasilinear the-

ory Dµµ[QL](dotted lines) and the weakly nonlinear

theory with ν= 0 Dµµ[WNLT](dot-dash lines).

For the QLT non-resonant small RLcases shown in

panels (a) and (e), Dµµ[WNLT]deviates significantly

from Dµµ[FD].

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Dµµ

SOQLT

QLT

WNLT

(a) RL=0.37275937,b/B0=0.3

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

−0.002

0.000

0.002

0.004

0.006

0.008

Dµµ

SOQLT

QLT

WNLT

(b) RL=1.93069773,b/B0=0.3

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.0000

0.0005

0.0010

0.0015

0.0020

Dµµ

SOQLT

QLT

WNLT

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

Dµµ

SOQLT

QLT

WNLT

(d) RL=10.0,b/B0=0.3

Figure 1: Plots of pitch-angle diffusion coef-

ficients in case b/B0= 0.3.Dµµ[FD](solid

lines), Dµµ[SOQLT](dashed lines), Dµµ[QLT](dot-

ted lines) and Dµµ[WNLT(ν= 0)] (dash-dot lines)

(see Tables 2, 4)

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0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Dµµ

SOQLT

QLT

WNLT

(e) RL=0.2,b/B0=0.5

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Dµµ

SOQLT

QLT

WNLT

(f) RL=0.5,b/B0=0.5

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

−0.01

0.00

0.01

0.02

0.03

0.04

0.05

Dµµ

SOQLT

QLT

WNLT

(g) RL=1.93069773,b/B0=0.5

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.0000

0.0002

0.0004

0.0006

0.0008

Dµµ

SOQLT

QLT

WNLT

(h) RL=10.0,b/B0=0.5

Figure 2: Plots of pitch-angle diffusion coef-

ficients in case b/B0= 0.5.Dµµ[FD](solid

lines), Dµµ[SOQLT](dashed lines), Dµµ[QLT](dot-

ted lines) and Dµµ[WNLT(ν= 0)] (dash-dot lines)

(see Tables 2, 4)

6 Discussion

In this work we have compared different theo-

ries for calculating the pitch-angle diffusion coeffi-

cient, Dµµ(µ), with test-particle simulation results for

NRMHD turbulence and assessed their ability to pre-

dict the parallel mean free path λ∥using Eq. (1). As

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

−0.2

−0.1

0.0

0.1

0.2

0.3

Dµµ

SOQLT

QLT

WNLT

(i) RL=0.2,b/B0=1.0

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

−0.05

0.00

0.05

0.10

0.15

Dµµ

SOQLT

QLT

WNLT

(j) RL=0.5,b/B0=1.0

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Dµµ

SOQLT

QLT

WNLT

(k) RL=1.93069773,b/B0=1.0

0.0 0.2 0.4 0.6 0.8 1.0

|µ|

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Dµµ

SOQLT

QLT

WNLT

((l) RL=10.0,b/B0=1.0

Figure 3: Plots of pitch-angle diffusion coef-

ficients in case b/B0= 1.0.Dµµ[FD](solid

lines), Dµµ[SOQLT](dashed lines), Dµµ[QLT](dot-

ted lines) and Dµµ[WNLT(ν= 0)] (dash-dot lines)

(see Tables 2, 4)

was seen in previous work [7], we find that Dµµ[FD]

(from test-particle simulations) provides λ∥in this

way that is very close to the value obtained via Eq. (2)

in simulations, as shown in Table 2 (i.e. the differ-

ences between the third and fourth columns are very

small). On this basis, we take Dµµ[FD]as reference

value that theories should try to replicate.

The weakly non-linear theory (WNLT), as in

Eq. 11, was found to give λ∥closest to simulations

when taking a µ−independent D⊥, with ν= 0. This

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DOI: 10.37394/23206.2022.21.32

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Volume 21, 2022

does not seem to be consistent with D⊥used in the-

ory [12] or found in previous simulations [8]. Further

study into an appropriate form of D⊥could help us

to understand this discrepancy. WNLT only seems

to work well for RL/l⊥≲2/π≈0.64, which is

the quasilinear limit of parallel resonant scattering.

Below this limit, Dµµ[WNLT]seems too large when

compared with Dµµ[FD], especially near µ= 0, as

seen in Fig. 2(a) and Fig. 2(b), so the parallel mean

free path is underestimated. When RLis large, as in

Figs. 1(d), 2(d) and 3(d), the WNLT theory shifts the

QLT value up near µ= 0 to a value close to the FD

result, so does very well with predicting λ∥.

The second-order quasilinear theory (SOQLT),

calculated here in detail for the first time for NRMHD

turbulence, gives λ∥to within a factor of a few for

large RLand is still quite accurate for RL= 0.5

when b/B0= 1 (where we might expect the theory

to work). In other cases, particularly at small RL, the

results are unacceptably large, as shown in Table 4.

Here we have adopted the same form of SOQLT as

in [9], who concluded that the SOQLT was not use-

ful. They argued that as their theory predicts a depen-

dence exp(B2

0/b2)(see Eq. (10)) it would give very

large parallel mean free paths for b<B0. However,

their simulation results did not seem to have huge par-

allel mean free paths and therefore their conclusion

that SOQLT was not useful seemed reasonable. It is

possible an alternative, and more complicated, vari-

ant of the theory from [11] could be applied to give

greater accuracy.

In summary, for the parameters used here, at least

one of the theories for Dµµ will provide a λ∥value

within a factor of a few of the simulation value, pro-

vided RLis not too much smaller that 1/K, the quasi-

linear resonance limit. Although the original QLT of

Eq. 8 provides Dµµ of similar shape to the simula-

tions, it always yields an infinite parallel mean free

path due to the troublesome resonance gap that NM-

RHD turbulence presents. The two other theories pre-

sented here broaden the resonance and close up the

gap, but not in a way that is broadly consistent with

the test-particle simulations. Thus NRMHD turbu-

lence remains a challenge for parallel diffusion theo-

ries. Since WNLT looks promising, a detailed exam-

ination of D⊥in computer simulations may provide

clues on how to proceed.

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Contribution of individual authors to

the creation of a scientific article

Chanidaporn Pleumpreedaporn performed test par-

ticle simulations, evaluated theories and analysed

results.

Andrew P. Snodin contributed to writing and testing

the numerical tools, analysis of results and writing

the text.

Elvin J. Moore assisted with the writing and the

preparation of the final version.

Sources of funding for research

presented in a scientific article or

scientific article itself

No funding was received for the research pre-

sented in this scientific article.

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Volume 21, 2022