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i

NASA/TP—2009–215906

Coupled Particle Transport and Pattern

Formation in a Nonlinear Leaky-Box Model

A.F. Barghouty

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

K.W. El-Nemr and J.K. Baird

The University of Alabama in Huntsville, Huntsville, Alabama

August 2009

National Aeronautics and

Space Administration

Marshall Space Flight Center • MSFC, Alabama 35812
ii

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076 –1320

443–757–5802

This report is also available in electronic form at

<https://www2.sti.nasa.gov>

Acknowledgments

A.F. Barghouty wishes to thank Dr. Steven Suess and Dr. Allen Gary, NASA Marshall Space Flight Center, 

and Professor Robert Naumann, The University of Alabama in Huntsville, for engaging discussions 

and insightful comments during the progress of this work.
iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................. 1

2. A COUPLED, TWO-PARTICLE FOKKER-PLANCK MODEL ..................................... 4

3. A COUPLED, NONLINEAR REACTIVE-DIFFUSIVE MODEL .................................. 7

4. THE TWO-PARTICLE FOKKER-PLANCK MODEL AS A NONLINEAR, 

REACTIVE DIFFUSIVE MODEL ................................................................................... 9

5. AN ILLUSTRATION ........................................................................................................ 12

6. SUMMARY AND CONCLUSIONS ................................................................................. 15

REFERENCES ....................................................................................................................... 16
iv

LIST OF FIGURES

1. Four steady-state solutions of the Thomas model for various relative 

diffusion and coupling strengths—the rest of the parameters appearing 

in equations (11) and (12) are unchanged. Solid curve depicts f ′

1 (E′) while 

the light curve depicts f ′

2 (E′). Light, jagged curve is the initial condition 

and dashed curve is the homogeneous solution. Sinusoidal steady-state 

patterns (Turing structures) can be seen for some values of the coupling 

strength parameter g and the relative diffusion coefficient D but not for others ............ 8

2. Three steady-state solutions, equation (16) transformed according 

to equation (25), as functions of E for three different values of magnetic 

turbulence spectral index, a. All three spectra, however, are characterized 

by the same turbulence level of D0=1. The turbulence index is seen 

to determine the relative ‘phase’ of the steady-state solution (structure) ...................... 12

3. Three steady-state solutions, equation (16) transformed according 

to equation (25),as functions of E for three different levels of the magnetic 

turbulence. The spectral index of the turbulence, a=5/3, is the same for all 

three solutions. All other parameters remain unchanged. The ‘structure’ is seen 

to become less pronounced (or, equivalently, move higher up in energy) with 

increasing level of turbulence ....................................................................................... 13

LIST OF TABLES

1. Correspondence between the two-particle Fokker-Planck model, equations (7) 

and (8), with that of the Thomas system, equations (11) and (12). Note that 

the Fokker-Planck model’s equations have been transformed twice and then 

normalized ................................................................................................................... 11
v

LIST OF ACRONYMS

ACR anomalous cosmic ray

EGCR extra galactic cosmic ray

GCR galactic cosmic ray

SEP solar energetic particle

TP Technical Publication
vi

NOMENCLATURE

A particle mass number

a′	 constant (Thomas model)

b′	 constant (Thomas model)

c speed of light

C(p) drift term in momentum space (Fokker-Planck model)

D diffusion coefficient (Fokker-Planck model)

D0 constant

E′ normalized energy

F coupling function (Thomas model)

f phase-space or number density function

f (p,t) isotropic part of the particle’s phase-space density

G coupling function (Thomas model)

K diffusion tensor

m0 unit mass

p particle’s momentum

Q source and sink term

q particle charge

V average propagation speed



V convection velocity

v particle speed

t time

l characteristic length scale

a spectral index of magnetic turbulence
vii

NOMENCLATURE (Continued)

a′ constant (Thomas model)

g coupling parameter (Thomas model)

k0 constant (diffusion coefficient)

k′ constant (Thomas model)

k|| spatial diffusion coefficient parallel to the magnetic field line

r constant (Thomas model)

t(p) momentum-dependent escape rate in the leaky-box model
viii
1

TECHNICAL PUBLICATION

COUPLED PARTICLE TRANSPORT AND PATTERN FORMATION

IN A NONLINEAR LEAKY-BOX MODEL

1. INTRODUCTION

In transport and acceleration of energetic (suprathermal) charged particles in space and 

astrophysical plasmas, the relevant length scales are typically much larger than the particle’s Larmor 

radius. As a result, the motion of these particles is effectively determined by the structure of the electromagnetic field via both its regular and stochastic components. In the absence of particle-particle 

collisions (the low density limit), the electromagnetic field is also responsible for accelerating the 

particles up to relativistic energies. 

The fluctuating component of the field can scatter particles especially when the particle’s Larmor radius is comparable to the wavelength of the scattering hydromagnetic wave, ‘resonant scattering.’ If scattering is frequent and strong, it can isotropize the particle’s density function and, in the 

diffusive limit, the motion of those particles can then be described statistically. The resulting kinetic 

equation1,2 assumes a Fokker-Planck equation of the form 

∂ f

∂t = ∇ ⋅(K∇f ) − ∇ ⋅ 

(V f ) , (1) 

where f is the phase-space or number density function of the particle, K is a diffusion tensor, and 

V

is a convection velocity. The diffusion tensor can be formally3 written in terms of the stochastic component of the magnetic field. With the addition of other terms like energy loss, e.g., due to ionization 

and synchrotron radiation, gain terms, e.g., due to the acceleration mechanism or mechanisms, and 

source and sink terms, equation (1) forms the basis of many phenomenological and computational 

kinetic studies of the transport and acceleration of energetic charged particles in space and astrophysical plasmas. 

In case of solar energetic particles (SEPs),4 recent and more realistic transport models5,6

incorporate particle-wave coupling that requires self-consistent7 solutions of the particle kinetic and 

wave propagation equations. Particles are scattered and accelerated by waves they themselves generate and/or amplify.8

 SEP observations not only support but appear to require this coupling between 

particles and waves in space.9 For SEPs, as well as for various other energetic particles in disparate 

astrophysical situations—anomalous cosmic rays (ACRs) at the heliospheric termination shock,10 

galactic cosmic rays (GCRs) at supernovae shocks,11,12 and even extra galactic cosmic rays (EGCRs) 

at galactic shocks13—diffusive shock acceleration14 has been demonstrated to be a more efficient 
2

(compared for example to Fermi’s original mechanism) and ‘natural’ process in accelerating suprathermal charged particles in space and astrophysical plasmas. 

In addition to acceleration, large-scale transport in space plasmas is for most particles also 

diffusive since the diffusion length due to the fluctuating magnetic field tends to be smaller than the 

characteristic length of the field.3,15 The magnetic field lines themselves can also wander diffusively 

and enhance the diffusion of the particles, affecting further their overall transport and acceleration 

characteristics.16–20

From a modeling perspective, coupling via any of the various terms that describe the particles’ acceleration, interactions, propagation, and even source terms, is likely to result in a nonlinear 

reactive-diffusive system of equations. Such systems are prone to diffusion-driven instabilities, which 

can give rise to steady-state (Turing) structures. These structures are brought about not necessarily 

due to one process (acceleration, transport, etc.) or another individually, but rather from the mutual 

interplay of these processes. Turing theory, or morphogenesis21 based on diffusion-driven instability, 

has already found a number of suggestive applications in astrophysics,22–28 statistical physics,29 and 

condensed matter physics.30 This is in addition to the more ‘natural’ applications to chemical and 

biological systems.31

Motivated in part by recent data32 in which SEP characteristics appear to show patterns that 

are difficult to explain phenomenologically, this Technical Publication (TP) explores the effects of 

a specific form of coupling—via a nonlinear source term—on the coupled particles’ characteristics. 

For purposes of this study, a reduced form of equation (1) ignoring explicit spatial dependence 

but including diffusion in momentum space can be simplified33,34 to the following Fokker-Planck 

form (the leaky-box limit): 

∂ f

∂t = 1

p

2

∂

∂ p

p

2

D ∂ f

∂ p

⎛

⎝

⎜ ⎞

⎠

⎟ − 1

p

2

∂

∂ p

p

2 ( Cf ) − f

τ

+ sources − sinks , (2)

where f (p,t) is understood to be the isotropic part of the particle’s phase-space density, p is the 

particle’s momentum, D the diffusion coefficient in momentum space, C describes the systematic momentum loss and gain rates, and t is an escape term in lieu of spatial diffusion assuming 

homogeneous spatial transport, i.e., in the limit ∇ ⋅(K∇f ) → − f / τ where () denotes spatial averaging of some kind. 

The stability of a system of two coupled equations based on equation (2) when treated as a 

nonlinear, reactive-diffusive system, and its implications on the characteristics of the coupled particles the system describes is being the focus here. Noncolliding particles can (mathematically) be 

coupled through any one of the terms in equation (2). Particle-wave coupling through the transport 

coefficients while the more physically compelling effect is difficult to reduce to amenable forms for 

purposes of this study. Instead, a mathematically well-studied model is relied on in which coupling is 

affected via a nonlinear source term. 
3

The idealized physical picture here is one in which two particles are in a spatially homogeneous region with their momenta subject to both stochastic and systematic changes. The two particles are assumed coupled via a common, nonlinear source term. It is essentially a leaky-box limit 

description with an added caveat with regard to the nonlinear source term. While assumptions are 

not made about the physical processes that result in such a coupling in the source term, a specific 

form that is known to support steady-state structures under some conditions will be used. 

The specific goals here are (1) to demonstrate that even a simple leaky-box type model, but 

with nonlinear coupling affected in some manner, can support steady-state structures under certain conditions, (2) to delineate the general physical conditions associated with these structures, and 

(3) to quantify their effects on modeled or observed particle characteristics. 

While the applicability and potential implications of the formal concepts of Turing structures 

and diffusion-driven instabilities to energetic particles in space plasmas is only heuristically demonstrated here with the help of an idealized physical model, the insight gained may be of wider use and 

significance, especially to similar systems implementing more realistic coupling via, e.g., particlewave interactions. Such applications of Turing morphogenesis theory can potentially have significant 

organizing benefits in space physics. 

This TP is organized as follows: The two-particle Fokker-Planck model is described in section  2 and the prototypical nonlinear, reactive-diffusive system known as the Thomas system is 

introduced in section 3. In section 4, explicit correspondence between the idealized two-particle Fokker-Planck model and the Thomas system is derived. Physical significance of this correspondence is 

discussed in section 5 and sample application presented. Section 6 offers a brief summary and some 

concluding remarks. 
4

2. A COUPLED, TWO-PARTICLE FOKKER-PLANCK MODEL

The acceleration and transport of two distinct charged-particle species are idealized as they 

interact with a collisionless, magnetized plasma. The particles are described by their respective distribution functions, f

1,2



p,  ( x,t), that evolve in momentum, space, and time due to their interactions 

with the plasma magnetic field. The magnetic field is assumed to have a regular and a stochastic 

component. The details of the regular component will be ignored here. The stochastic component 

is characterized by a spectral index and an energy level. The plasma is assumed to be spatially 

homogeneous, i.e., its spatial parameters change over timescales much larger than those that characterize f

1,2



p,  ( x,t). This allows us to ignore explicit spatial variations in the distribution function, i.e., 

f(



p,



x,t) ≈ f(



p,t), and hence in all the particle properties described by it. 

Also assumed is that magnetic turbulence scatters the particles sufficiently fast to give rise to 

diffusion both in coordinate and in momentum space. In a spherically symmetric momentum space 

assuming scattering off ‘hard spheres,’ the diffusion coefficient is typically written as7,34–36

D( p) = V 2

p

2

9κ ||( p) , (3)

where V is an average propagation speed of the scattering centers (Alfvén speed). k||, the spatial diffusion coefficient parallel to the magnetic field line, is taken to be proportional to the rigidity (momentum p per charge q) of the particle as 

κ ||( p) =κ 0

p

q

⎛

⎝

⎜ ⎞

⎠

⎟

2−α

, (4) 

where k0 is a constant inversely proportional to the level or energy of the magnetic turbulence (but 

establishes the energy scale) while a is the spectral index of the turbulence. Diffusion in momentum 

(or energy) is an expression37 of what is known as Fermi second-order acceleration process wherein 

the incremental change in momentum, ∆p/p∝(v /c)

2

, where v is the particle speed and c is the speed 

of light. 

In addition to diffusion in momentum space, also assumed is that the particles are subject to 

systematic momentum loss and gain processes like acceleration due to a shock (the so-called Fermi 

first-order acceleration) or deceleration due to passage through the plasma. However, the exact form 

of these terms is not crucial to our purposes here and will collectively be referred to as the ‘drift’ term 

in momentum space, C(p). 

Upon acceleration or deceleration, the particles are assumed to be able to escape the 

finite momentum region with a momentum-dependent escape rate, t(p)=l

2

/k||(p), with l being a 
5

characteristic length scale. This escape-rate term approximates diffusion in coordinate space in the 

homogeneous or leaky-box limit. Finally, the particles are assumed to share a common, but may be 

nonlinear, source and sink term, Q. 

The above idealized particle transport picture can be described by the following set of 

Fokker-Planck equations in momentum space, 

∂f

1

∂t = 1

p

2

∂

∂p

p

2

D1( p)

∂f

1

∂p

⎡

⎣

⎢ ⎤

⎦

⎥ − 1

p

2

∂

∂p

p

2

C1( p) f

1 ⎡

⎣ ⎤⎦ − f

1

τ 1( p)

+Q f

1, f ( 2) (5) 

and

∂f

2

∂t = 1

p

2

∂

∂p

p

2

D2( p)

∂f

2

∂p

⎡

⎣

⎢ ⎤

⎦

⎥ − 1

p

2

∂

∂p

p

2

C2( p) f

2 ⎡

⎣ ⎤⎦ − f

2

τ 2( p)

+Q f

1, f ( 2) . (6) 

Note that while the above system is an idealization of charged particle transport in space plasmas,33,34,38 the simplifying assumptions made, i.e., leaky-box limit, should not take much away from 

the relevance and/or direct applicability of what follows to more realistic modeling involving other 

coupling venues. Moreover, outside of the assumption of a common nonlinear source term, there 

are no more (basic) assumptions in this two-particle model than what is typically made in a good 

number of similar physical models based on the leaky-box approximation. 

To make contact with the so-called Thomas model to be discussed below, the above system 

is written in terms of the classical kinetic energy of the particle, E=p/(2m), as 

∂f

1

∂t = ∂2

∂E2 D′

1

(E) f

1 ⎡⎣ ⎤⎦ − ∂

∂E

C′

1(E) f1 ⎡⎣ ⎤⎦ − f

1

τ1(E)

+Q f

1,f ( 2) (7) 

and

∂f

2

∂t = ∂2

∂E2 D′

2(E) f2 ⎡⎣ ⎤⎦ − ∂

∂E

C′

2(E) f2 ⎡⎣ ⎤⎦ − f

2

τ 2(E)

+Q f

1,f ( 2) . (8) 

The diffusion and drift coefficients in energy (in their canonical Fokker-Planck forms) are 

now written in terms of those appearing in equations (5) and (6) in momentum space as (i=1,2): 

D′

i (E) = dE

dp

⎛

⎝

⎜ ⎞

⎠

⎟

2

Di( p) (9) 
6

and

C′

i

(E) = dE

dp

⎛

⎝

⎜ ⎞

⎠

⎟Ci

( p) . (10)

Assumptions about the nature of particle-particle coupling in the above two-particle FokkerPlanck model will only be made by analogy with that appearing in the Thomas system, i.e., through 

the source term. No other coupling is assumed. Also, defining the appropriate initial and boundary 

conditions will be delayed until the model is transformed to a form suitable for direct comparison 

with the Thomas system, which is introduced and discussed in section 3. 
7

3. A COUPLED, NONLINEAR REACTIVE-DIFFUSIVE MODEL

The Thomas model31 is a coupled, nonlinear reactive diffusive model known to support 

Turing structures in its steady-state solutions. For two species described by densities f ′

1(E′,t) and

f ′

2 (E′,t) , assumed to be in nondimensionalized form, the model is described by the following coupled equations: 

∂f ′

1

∂t′

= ∂2 f ′

1

∂E′

2 + γ F f ′

1, f ′2 ( ) (11)

and

∂f ′

2

∂t′

= D

∂2 f ′

2

∂E′

2 + γ G f ′

1

, f ′

2 ( ) , (12)

where D is the relative diffusion coefficient and g is a coupling parameter. Functions F and G carry 

the ‘kinetics’ in this system. For the Thomas system, these are given by 

F f ′

1, f ′2 ( ) = a′ − f ′1 ( ) − H f ′1, f ′2 ( ) , (13)

G f ′

1, f ′2 ( ) = α′ b′ − f ′2 ( ) − H f ′1, f ′2 ( ) , (14)

where

H f ′

1, f ′2 ( ) = ρ f ′

1 f ′2

1+ f ′

1 +κ′ f ′2

. (15)

Parameters a′, b′, a′, r, and k ′ are constants. The variable E′ ∈ [0,4] is taken to be a normalized 

‘energy’ variable. 

There are a number of conditions that the system described above must first satisfy before it 

can support Turing structures in steady state. These are related to the stability of the homogeneous 

solution to the introduction of small perturbations in the absence of diffusion, i.e., for t→∞ with 

D = 0. 

The reader is referred to reference 31 for a thorough treatment. For the particular set a′ = 92, 

b′=64, a′ =1.5, r=18.5, and k′=0.1, it can be shown39 that the above system must satisfy D 

> 10 

and g 

> 8 before it can support any structure. However, satisfying these two critical values for the relative diffusion and strength of the coupling does not in itself guarantee that the steady-state solutions 

of equations (11) and (12) exhibit Turing structures. 
8

Figure 1 shows four steady-state numerical solutions40 of equations (11) and (12) for four 

different sets of (g,D) values but for the same values for the rest of the parameters mentioned above. 

Here, the initial conditions were taken to be random variations about the homogeneous solutions, 

i.e., when F=G=0, and the boundary conditions are such that ∂f ′

1 / ∂E′= ∂f ′2 / ∂E′ = 0 at E′ = (0,4). 

Note that all four pairs of (g,D) satisfy the critical conditions that D 

> 10 and g > 8, yet only two of 

them, (10,10) and (12,9.8) actually show the sinusoidal structure. Note also that f

1(E′) and f2(E′) are 

in phase over the entire domain of E′. 

For the case of (12,9.8), which will be used for illustration later on, the sinusoidal structure is 

well approximated by the relation 

f ′(E′) = a cos(ωE′) + b , (16)

with a=2.28, b=9.8 for f ′

1(E′) , and a=0.43, b=9.3 for f ′2(E′), with w=p for both. 

14

12

10

8

6

14

12

10

8

6

0 1 2 3 4 1 2 3 4

γ =10; D=10 γ =10; D=9.8

γ =11; D=9.8 γ =12; D=9.8 ƒ′

1(E′); ƒ′

2(E′)

E ′

Figure 1. Four steady-state solutions of the Thomas model for various relative diffusion 

and coupling strengths—the rest of the parameters appearing in equations (11) 

and (12) are unchanged. Solid curve depicts f ′

1 (E′) while the light curve depicts 

f ′

2 (E′). Light, jagged curve is the initial condition and dashed curve is the 

homogeneous solution. Sinusoidal steady-state patterns (Turing structures) 

can be seen for some values of the coupling strength parameter g and the 

relative diffusion coefficient D but not for others.
9

4. THE TWO-PARTICLE FOKKER-PLANCK MODEL AS A NONLINEAR, 

REACTIVE DIFFUSIVE MODEL

The system described by equations (7) and (8) must first be transformed to a constant diffusion, zero drift system and normalized before it can be made to correspond to the system in equations (11) and (12). Upon this second transformation, particle 1 will have a diffusion coefficient of 

unity and particle 2 will have a diffusion coefficient of D>1. The transformed Fokker-Planck drift 

coefficients will both be zero. The transformed escape terms are structurally unaffected. The source 

term, which is taken to depend only on the independent variables, will assume its form by analogy 

with the Thomas system. It is in this regard that in this model, the particle-particle coupling is considered to be affected only via the source term. 

The transformation requires that41

E′

1

(E) = 1

D′

1

(E) ∫ dE (17)

and

E′

2 (E) = D

D′

2

(E) ∫ dE . (18)

The transformed drift coefficients satisfy (i =1,2): 

C′′

i E′i ( ) = (1or D)

D′

i

C′

i − 1

2

dD′

i

dE

⎡

⎣

⎢ ⎤

⎦

⎥ , (19)

which can be made to vanish if 

C′

i (E) = 1

2

dD′

i

dE (E) . (20)

Requiring that f ′

i

 dE ′

i

=f

i

 dEi, the transformed distribution functions will then take the form 

f ′

i E′i ( ,t) = D′

i

(1 or D) f

i (E,t) . (21)
10

Note, however, that the above transformations apply to each particle separately since, in 

general, E ′

1≠E ′2. Since a coupled, two-particle system is used, the requirement is that 

E′′

1 (E) = E′′2 (E) = E′(E) . (22)

To that end, a typical form for the diffusion coefficient is assumed (cf., eqs. (3) and (9)): 

D′

i (E) = D0,i

E2−α /2 , (23)

where, for i =1,2, D0,i can be written as,

D0,i =D0

2

ε1m0

ε 2Ai

qi

⎛

⎝

⎜

⎞

⎠

⎟

2−α

, (24)

with m0 being a unit mass (e.g., amu) that is normalized to unity here, A is mass number, 

e1=(1 –a/4)/(1–a/2), e2=e1 –1, and D0 is now a constant independent of energy as well as particle 

species. Since D0∝1/k0, equation (3), and k0 is, in turn, inversely proportional to the strength of the 

turbulence, D0 is taken to characterize the level of magnetic turbulence. Normalization here then 

simply means D0=1. With this normalization, the particles are characterized simply by their massto-charge ratios and the fluctuating magnetic field is characterized by its spectral index and energy 

level. Again, a is the spectral index of the magnetic turbulence as in equation (4) and q is the particle 

charge. 

The transformed energy E′ that applies self-consistently to both particles must then take the 

form 

E′(E) = 4α−1

D0,1D0,2

Eα/4 . (25)

Self-consistency also requires a transformation of the time variable as 

t′(t) = t

D0,1

D

0,2 ( ) . (26)

Direct contact is then made with the Thomas system, equations (11) and (12), if the two-particle 

Fokker-Planck system, equations (7) and (8), is transformed according to table 1. For simplicity, 

a single common source, i.e., Q1≈Q2=Q, is assumed since in the Thomas system, a′b′≈a′.
11

Table 1. Correspondence between the two-particle Fokker-Planck 

model, equations (7) and (8), with that of the Thomas system, 

equations (11) and (12). Note that the Fokker-Planck model’s 

equations have been transformed twice and then normalized.

Quantity in the Fokker-Planck Model Quantity in the Thomas System

ƒi ƒ ʹi (i = 1,2)

E Eʹ

t tʹ

D′′1 1

D′′2 D

C′′i 0 (i = 1,2)

τ1 γ

τ1/ τ2 a′

Q≈Q1 γ (aʹ-H)

Q≈Q2 γ (aʹb′-H)

Finally, since the ratio of the distribution functions in steady state is of interest, the following 

relation will be used: 

f

1(E)

f

2(E)

= 1

D

D 0,2

D0,1

f ′

1 [E′(E)]

f ′

2 [E′(E)]

. (27)

It is clear that any structure in the ratio f ′

1 / f ′2(E′) will appear as a transformed structure in 

the ratio f ′

1 / f ′2(E) . The transformed structure in turn depends on two other factors: The first term 

in equation (27) simply multiplies it by a constant. The second term, though it does not depend on 

energy, it does depend on the coupled particles’ mass-to-charge ratios, which appear in the transformation of energy, equation (25), as well as on the magnetic turbulence (characterized by D0 and a). 

As such, this factor also affects the transformed structure. 

Hence, whenever a structure exists in E′-space due to particle-particle coupling, the transformed structure in E-space is expected to be shaped both by the property of the magnetic turbulence 

as well as by the two particles themselves through their mass-to-charge ratios. Next, the dependence 

of the structure on both of these factors is explored in more detail. 
12

5. AN ILLUSTRATION

Using the two-particle Fokker-Planck model described above and assuming that the two 

particles are coupled via their common source term (cf., table 1), the steady-state solutions of the 

Thomas system can simply be treated by proxy as the steady-steady solutions of the doubly transformed and normalized two-particle Fokker-Planck system. Hence, equation (16) can be used and 

transformed according to equation (25) to study the dependence of the structure on the attributes of 

the coupled particles and those of the magnetic turbulence. 

Figure 2 shows three transformed solutions as functions of E for three different values of 

the spectral index: a Kolmogorov spectrum (a=5/3), a Kraichnan spectrum (a=3/2), and a third 

characterized by (a=4/3). All three spectra, however, are characterized by the same turbulence level 

D0=1. Particle 1 has an A/q=1.5 while particle 2 has A/q=1. 

E

= 5/3

3/2

4/3

0.01 0.1 1 10 100

5

4

3

2

1

0

Figure 2. Three steady-state solutions, equation (16) transformed according to equation (25), 

as functions of E for three different values of magnetic turbulence spectral index, a.

All three spectra, however, are characterized by the same turbulence level of D0=1. 

The turbulence index is seen to determine the relative ‘phase’ of the steady-state 

solution (structure).

The dependence of the steady-state solution (or structure) on the spectral index of the magnetic turbulence is seen; two different spectral indices can give rise to two different structures. The 

transformed sinusoidal structures appear to be simply out of phase with respect to each other as a
13

varies. This can be readily surmised from equation (25) and the fact that ratios of the solutions are 

plotted rather than the solutions themselves. The ‘phase’ of the structure appears to be predicated 

by the spectral index a. 

The physical significance of this dependence can be appreciated as follows. If the same particles are now decoupled but their transport is still assumed to take place in the same two magnetic 

fields, the ratio of their steady-state density functions will either be a monotonically decreasing or 

a monotonically increasing function of E. This is readily seen in the steady-state solutions of the 

now decoupled equations (11) and (12). The decoupled solutions for a given a will share a common 

inverse power-law behavior in E, i.e., f(E)∝ E–h, where h ∝ a. As such, no ‘structure’ in the ratio of 

the two functions should be present. 

Figure 3 shows three transformed sinusoidal solutions as functions of E for three different 

levels of the magnetic turbulence. Here, the spectral index of a=5/3 is the same for all three solutions. All other model parameters as well as the particles’ charge-to-mass ratios are the same as those 

used for figure 2. It can be seen that as the turbulence strength increases (i.e., higher D0 since D0 is 

directly proportional to the turbulence strength, equation (3), while k|| is inversely proportional to it, 

equation (4)), the structure becomes less pronounced. Equivalently, the structure appears to move 

to higher energy regions with stronger turbulence, and vice versa. Given that the structure in this 

model is simply a transformed sinusoidal function, this behavior is straightforward to appreciate 

both mathematically and physically, as follows. 

ƒ1/ƒ2

E

D0 = 5

1

0.5

0.01 0.1 1 10 100

5

4

3

2

1

0

Figure 3. Three steady-state solutions, equation (16) transformed according to equation (25), 

as functions of E for three different levels of the magnetic turbulence. The spectral 

index of the turbulence, a=5/3, is the same for all three solutions. All other 

parameters remain unchanged. The ‘structure’ is seen to become less pronounced 

(or, equivalently, move higher up in energy) with increasing level of turbulence.
14

As the turbulence level increases, the strength of the diffusion coefficient in momentum or 

energy space increases. In the coupled case, even though their relative diffusion remains the same, 

they both will have correspondingly shorter ‘mean free paths’ in energy space with increasing turbulence strength. If the two coupled particles are assigned a ‘correlation length’ in E-space, then when 

their ‘mean free paths’ become much shorter than this ‘correlation length,’ they can be considered 

effectively decoupled. In this case, any structure in E-space is not expected to be seen. With higher 

energy, however, the particles can have longer ‘mean free paths’ and hence if these happen to be comparable to or larger than their ‘correlation length’ the particles will remain coupled to each other and 

a structure can be expected in this higher energy region. 

Finally, the dependence of the transformed sinusoidal structure on the mass-to-charge ratio 

of the particles has been examined. This dependence was found to be much weaker but similar, 

as in equation (25), to the effects of the strength of the magnetic turbulence. As a result, particles 

with somewhat different mass-to-charge ratios (e.g., SEP He+2-to-p versus SEP Fe+14-to-O+2) are 

expected to show rather similar structures for the same magnetic turbulence whenever these structures can be attributed to particle-particle coupling. 

Some recent SEP data32 appear to have qualitatively similar ‘structures.’ The observed ironto-oxygen ratio as a function of energy from two separate but similar solar particle events in 2002, 

for example, seem to follow each other closely up to a certain ‘critical’ energy after which they appear 

to diverge and separate widely. SEP iron and oxygen ions in these two large solar events are believed 

to have been accelerated by similar shocks driven by coronal mass ejections. One possible explanation42 for this high-energy ‘variability’ invokes the changing angle between the magnetic field line 

and a direction normal to the shock as the shock propagates away from the lower corona and into 

the interplanetary space. 

While this particular explanation for this observed SEP variability may be of limited applicability to the larger SEP database,5,43 other ‘variabilities’ in SEP characteristics as well (e.g., elemental 

composition and charge state) still await more quantitative descriptions. 
15

6. SUMMARY AND CONCLUSIONS

Motivated in part by modeling requirements in the description of energetic charged particles 

transport in space plasmas, an idealized two-particle Fokker-Planck model has been developed in 

order to explore the mathematical implications of particle-particle coupling on the coupled particles’ 

characteristics. Such implications are expected to be manifest in the steady-state solutions of the 

systems as patterns or Turing structures. 

In this two-particle leaky-box model, the particles’ motion in momentum or energy space is 

entirely determined by the magnetic field’s regular and stochastic components. Plasma properties are 

assumed fixed relative to particle properties. 

Particle-particle coupling is assumed to be entirely due to a common nonlinear source term, 

but without deriving or promoting a specific form. The two-particle Fokker-Planck model is then 

made to correspond to the so-called Thomas model—a coupled, nonlinear, reactive-diffusive system 

that is known to support Turing structures in its steady-state solutions. 

Relying on proxy steady-state solutions of the Thomas system, the dependence of the solutions on both the particles’ and the magnetic field attributes is analyzed in some detail. 

Particle-particle coupling (via a nonlinear common source term per the Thomas model) is 

seen to give rise to patterns (Turing structures) in the ratio of the particles’ steady-state density functions. The structures are found to depend sensitively on the strength and spectral index of the magnetic turbulence, but only weakly on the coupled particles’ mass-to-charge ratios. The spectral index 

of the turbulence is seen to determine the ‘phase’ of the structure, while the structure itself appears 

to become less pronounced and to move to higher energy regions with higher levels of turbulence. 

While some distance away from direct correspondence to data, this study using a nonlinearly 

coupled leaky-box model demonstrates the possibility of describing observed structures in energetic 

particle characteristics as being due to reaction-diffusion-driven instabilities arising from, in this particular demonstration, particle-particle coupling. Putting this potentiality on firmer grounds—both 

mathematically and in connection with accepted phenomenology—could have a powerful organizing 

effect on an already rich but increasingly complex phenomenon. 
16

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Coupled Particle Transport and Pattern Formation in a Nonlinear 

Leaky-Box Model

A.F. Barghouty, K.W. El-Nemr,* and J.K. Baird*

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 90

Availability: NASA CASI (443–757–5802)

*Department of Chemistry, The University of Alabama in Huntsville, Huntsville, AL 35899 

Prepared by the Science & Exploration Research Office, Science & Mission Systems Office

M–1265

Technical Publication

NASA/TP—2009–215906

diffusion-driven instability, Turing structures, acceleration of particles, plasma turbulence, leaky-box model, 

cosmic rays, solar energetic particles

01–08–2009

UU 28

NASA

U U U

Effects of particle-particle coupling on particle characteristics in nonlinear leaky-box type descriptions of the acceleration and transport of energetic particles in space plasmas are examined in the framework of a simple two-particle 

model based on the Fokker-Planck equation in momentum space. In this model, the two particles are assumed coupled via a common nonlinear source term. In analogy with a prototypical mathematical system of diffusion-driven 

instability, this work demonstrates that steady-state patterns with strong dependence on the magnetic turbulence but 

a rather weak one on the coupled particles’ attributes can emerge in solutions of a nonlinearly coupled leaky-box 

model. The insight gained from this simple model may be of wider use and significance to nonlinearly coupled leakybox type descriptions in general. 

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