Propagation of radiation in fluctuating multiscale plasmas. I. Kinetic theory

Propagation of radiation in fluctuating multiscale plasmas. I. Kinetic theory

Yu. Tyshetskiy, Kunwar Pal Singh,a) A. Thirunavukarasu, P. A. Robinson, and Iver H. CairnsSchool of Physics, University of Sydney, NSW 2006, Australia

(Received 5 August 2012; accepted 31 October 2012; published online 29 November 2012)

A theory for propagation of radiation in a large scale plasma with small scale fluctuations is

developed using a kinetic description in terms of the probability distribution function of the radiation

in space, time, and wavevector space. Large scale effects associated with spatial variations in the

plasma density and refractive index of the plasma wave modes and small scale effects such as

scattering of radiation by density clumps in fluctuating plasma, spontaneous emission, damping, and

mode conversion are included in a multiscale kinetic description of the radiation. Expressions for the

Stokes parameters in terms of the probability distribution function of the radiation are used to enable

radiation properties such as intensity and polarization to be calculated. VC 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4767640]

I. INTRODUCTION

Plasma systems often involve multiscale radiation phe-

nomena, with large-scale propagation of radiation through

plasmas containing small-scale inhomogeneities. For example,

type III radio bursts involve propagation of radiation over dis-

tances of order an astronomical unit through a plasma that

contains fluctuations on scales down to tens of kilometers or

less,1–8 at least a million-fold difference in scales. Overall

energy transfer requires the largest scales to be included, but

in situ observations reveal that the wave fields are extremely

bursty and nonuniform on all scales down to the smallest

resolved.7–9 Moreover, stochastic growth theory (SGT)

implies that small-scale inhomogeneities are critical to these

multiscale fluctuations7–11 and to the dynamics of the electro-

magnetic radiation.12–14 Another class of examples occurs in

edge plasmas in laboratory plasma devices, where radio fre-

quency heating beams have to travel several meters through

turbulent inhomogeneities on scales of �0:01 m–a scale ratio

of up to 1000-fold.15–17

Much work has been done to apply small-scale analytic

results to study radiation phenomena in multiscale systems.

However, these typically involve approximations, many of

which have yet to be fully verified numerically. For example,

diffusive approximations to radiation propagation have been

made in analyses of type III solar radio burst time profiles

and frequency structure using SGT,12,13,18–23 and depolariza-

tion of solar radio bursts through multiple scattering has also

been studied.24 Unfortunately, direct simulation of situations

like those above is impractical, because of the need to simul-

taneously resolve fine scales while following radiation prop-

agation over much larger distances. Hence, one needs to

approximate the small-scale effects in a way that can be

incorporated into large-scale simulations.

One approach proposed to deal with large and small scale

effects simultaneously during radiation scattering is to use vari-

ous kinetic descriptions of the small-scale processes to produce

a Fokker-Planck equation for the overall large-scale evolution,

particularly in edge plasmas in fusion devices.15–17,25 Here we

develop a similar approach in a form particularly suited (but

not restricted) to solar-terrestrial applications, where linear

mode conversion on density fluctuations must be taken into

account,12,13 an effect that has not been previously incorpo-

rated in such treatments. In particular, we express the evolu-

tion of the radiation in terms of the probability distribution

function of radiation quanta in space, time, and wave vector.

The distribution function then enables calculation of the radi-

ation intensity and polarization via the mean Stokes parame-

ters.26,27 We assume the radiation propagates through an

inhomogeneous plasma according to standard wave disper-

sion and damping at the large scale while scattering and

undergoing linear mode conversion when encountering

plasma inhomogeneities. The small-scale effects of scatter-

ing and mode conversion of radiation quanta induced by

plasma fluctuations are then included in the kinetic equation

using collisional integrals that yield Fokker-Planck drift and

diffusion coefficients. Each analysis cited above15–17

included scattering and some other effects in forms suitable

for use in laboratory situations, often specialized to lower-

hybrid waves. However, we include linear mode conversion

(LMC), spontaneous emission, growth, and damping, which

were not all previously included; we also incorporate Lang-

muir waves, which are more relevant for many applications.

The aims of the current paper (Part I) are to develop a

method to study propagation of radiation in fluctuating mul-

tiscale plasmas. The numerical method is developed, and ini-

tial simulation results are presented in Part II (Ref. 28) to

verify its accuracy and speed. The structure of the present

paper is as follows. In Sec. II, we briefly discuss the propaga-

tion of radiation in plasmas with small scale fluctuations and

explain how this leads to a kinetic equation for the evolution

of the distribution function of quanta. This equation can

account for different mode dispersion (and hence different

ray paths), stochastic scattering of radiation off plasma den-

sity inhomogeneities, spontaneous emission, damping, and

mode coupling. Specific forms of the small-scale scattering

and linear mode conversion processes that give rise to drift

and diffusion terms in the kinetic equation are discussed in

Sec. III. In Sec. IV we use expressions for the Stokes param-

eters of a superposition of a pair of orthonormal transversea)Electronic mail: [email protected].

1070-664X/2012/19(11)/113303/18/$30.00 VC 2012 American Institute of Physics19, 113303-1

PHYSICS OF PLASMAS 19, 113303 (2012)

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modes,26,27 to calculate the probability distribution functions

(PDFs) of the Stokes parameters. These PDFs then enable

observable mean Stokes parameters to be calculated. Hence,

our kinetic description yields predictions for the observable

intensity and polarization of radiation measured far from the

source region.

II. RADIATION PROPAGATION IN ANINHOMOGENEOUS MEDIUM

In this section we discuss propagation of radiation in a

weakly magnetized plasma with small scale density inhomo-

geneities and large scale spatial variation of density and

magnetic field. We first outline the basic assumptions and

types of effects that can be included and then describe how

these lead to a kinetic equation for the evolution of the distri-

bution function of the quanta.

A. Radiation propagation

The electromagnetic radiation propagating in a fluctuat-

ing plasma is subject to stochastic scattering, spontaneous

emission, damping, and associated depolarization. A sche-

matic of the propagation of radiation from a source to an

observer in a fluctuating plasma is given in Fig. 1. The plasma

fluctuations are represented by randomly distributed density

clumps that scatter the radiation (these can be positive or neg-

ative in amplitude, i.e., enhancements or voids). The source of

the radiation is surrounded by randomly distributed plasma

density clumps. We make the following assumptions:

(i) The medium through which the radiation propagates

is an inhomogeneous weakly magnetized plasma.

(ii) The characteristic scales of plasma inhomogeneities

are large compared to radio wavelengths, except very

close to mode cutoffs. This justifies the use of geo-

metric optics, which correctly describes the far wave

field even though it breaks down near the cutoffs.29

(iii) Only transverse (T) waves (o and x modes) escape

from the source region and reach a remote observer,

so we are only interested in propagation of o and xmodes.

(iv) The o and x modes can exchange energy with each

other and with the local plasma waves [e.g., the Lang-

muir (L) mode] when scattering off plasma density

fluctuations, through mode coupling.12,30–38 Propaga-

tion and evolution of the local L waves are not consid-

ered in the present work because they are much slower

than the T waves and are typically heavily damped.

(v) The spatial scales of small-scale plasma inhomogene-

ities are assumed to be much smaller than the charac-

teristic scale of variations of the mean density and

magnetic field (large-scale gradients). This disparity

of scales permits a statistical description of the small-

scale effects (e.g., scattering and mode conversion on

small-scale plasma density fluctuations); these enter

the large-scale kinetic model via the corresponding

collision integrals, which yield drift and diffusion

coefficients.

To describe the propagation and properties of the o and

x modes, we introduce distribution functions fMðr; k; tÞ of

radiation quanta of the corresponding modes M¼ o, x, where

t denotes time, r is position, and k is wave vector. The ki-

netic equations for fM, derived in detail below, describe the

evolution of o and x radiation in the fluctuating plasma, sub-

ject to large- and small-scale effects such as refraction, scat-

tering, and linear mode conversion. As we discuss in detail

in Sec. II C, the general form of this kinetic equation is

@fM

@tþ @xM

@k� @fM@r� @xM

@r� @fM

@k

¼ StðfMÞ þ SM � cMfM; (1)

where xMðr; kÞ plays the role of a Hamiltonian when the

motion of quanta is viewed from a Hamiltonian-optics per-

spective, and SM is the radiation source (which is either a

specified input or the result of a radiation process that may

depend on the fM). The damping rate is cM, which includes

any collisional and resonant damping, and growth through

linear instabilities when negative. Concentrating on the

large-scale terms on the left of Eq. (1), there are a number of

contributions to the local rate of change @fM=@t. First, the

group velocity vg ¼ @xM=@k produces an advective rate of

change if there is a nonzero spatial gradient @fM=@r. Second,

if the mode properties are not spatially constant, the term

@xM=@r is nonzero and produces a refractive rate of change

if the radiation is not isotropic, i.e., if @fM=@k 6¼ 0. Likewise,

the source and damping terms on the right determine overall

energy input and loss from the system, aside from small-

scale effects embodied in StðfMÞ, which we discuss below.

B. Kinetic description

The structure of the kinetic model based on Eq. (1) is

shown schematically in Fig. 2. The large-scale propagation

and refraction of the natural electromagnetic modes in plasma

are defined by the large-scale plasma density and magnetic

field profiles; hence, the corresponding models for these

effects are incorporated via the large-scale Hamiltonians

source

clumps

observer

FIG. 1. Schematic of radiation propagation from a source to an observer in a

fluctuating plasma represented by randomly distributed density clumps

(which have positive or negative amplitude), shown as gray circles. Here the

radiation is shown as being emitted roughly isotropically, then being

refracted to the right on average (e.g., due to a large-scale density decrease

toward the right).

113303-2 Tyshetskiy et al. Phys. Plasmas 19, 113303 (2012)


xMðr; kÞ, where M¼ o, x, and r and k are canonically conju-

gate variables. For present purposes we express the depend-

ence of xMðr; kÞ on the plasma density and ambient

magnetic field using magnetoionic theory24 in Appendix A,

which yields the corresponding expressions for the refractive

indexes nMðr;x; hk;BÞ ¼ ck=x of the o and x modes in terms

of r, the wave frequency x, and the angle hk;B between the

wave vector k and local magnetic field B. These can be

inverted to find the Hamiltonians xM ¼ xðr; k; hk;BÞ for the

large-scale motion of o and x quanta.

In general, the term St(f) in Eq. (1) can incorporate the

stochastic effects of the small-scale processes shown in Fig. 2,

some of which are discussed in detail in Sec. III. These

include (i) stochastic scattering of o and x quanta off plasma

density fluctuations (see Sec. III C for details), which is the

combination of refractive scattering by density gradients and

linear mode conversion; (ii) stochastic scattering of o and xradiation from ambient magnetic field fluctuations; and (iii)

nonlinear effects such as three-wave processes Pþ Q$ M,

four-wave processes, and nonlinear scattering of waves by

plasma particles. In the present work we will focus on (i).

To specify any particular problem that one wishes to

solve, one must provide a number of pieces of information,

as shown at the top of Fig. 2: (i) The system geometry and

how the wave properties depend on position within it–disper-

sion and damping in particular. In particular, this requires

specifying the plasma density and magnetic field parameters:

magnitudes, inhomogeneity scales and profiles, and fluctua-

tion spectra. (ii) The source SM as a function of position and

time. (iii) The initial distribution functions fMðr; k; t ¼ 0Þ of

o and x quanta. These must be consistent with the mode cut-

offs, i.e., there are regions where the o and/or x mode cannot

propagate and are instead evanescent. (iv) Appropriate

boundary conditions for the distribution functions fM. Typi-

cally, open boundary conditions are a suitable assumption in

space-plasma applications, for example.

The outputs of the model are distribution functions

fMðr; k; tÞ of o and x quanta throughout the simulation domain,

which can then be used to calculate the observable properties of

the radiation, such as directivity, dynamic spectra, and polariza-

tion properties via evaluation of the mean Stokes parameters.

C. Kinetic equation

We construct the distribution function fMðr; k; tÞ of

wave quanta in mode M from the corresponding wave occu-

pation number NMðr; k; tÞ that defines the distribution of

quanta in r-k space39 (we assume slow variation of NM on

spatial and temporal scales k�1 and x�1, respectively). The

uncertainty relation between canonically conjugate variables

r and k in quantum mechanics implies that the uncertainty in

NM times the uncertainty in phase U of a quantum is of order

unity. We restrict our consideration to a semiclassical theory

by assuming NMðr; k; tÞ to be a well-defined quantity, which

implies complete uncertainty in the phase; i.e., the randomphase approximation.40 This theory describes processes in

which phases of the waves are unimportant but cannot be

used to describe phase-coherent processes.40

In the random phase approximation, the distribution

function of wave quanta in mode M can be written as

fMðr; k; tÞ ¼ NMðr; k; tÞPMðUÞ; (2)

with a phase distribution function PMðUÞ ¼ ð2pÞ�1.

To derive the evolution equation for fMðr; k; tÞ, we con-

sider fM ¼ fMðRq;Rp; tÞ, where Rq ¼ fr;Ug are the general-

ized coordinates and Rp ¼ fk; kUg are the generalized

momenta of quanta in the multidimensional coordinate-

momentum phase space (with kU being the generalized

momentum corresponding to the phase U). The quantities Rq

and Rp are canonically conjugate, with

_Rq ¼@xMðRq;Rp; tÞ

@Rp; (3)

_Rp ¼ �@xMðRq;Rp; tÞ

@Rq; (4)

where xMðRq;Rp; tÞ is the Hamiltonian. In cases where the

motion of quanta does not depend on U and _U / kU, i.e.,

xM ¼ xMðr; k; tÞ, we have

_U ¼ @xM

@kU¼ 0; (5)

_kU ¼ �@xM

@U¼ 0; (6)

_r ¼ @xM

@k� vg; (7)

_k ¼ � @xM

@r: (8)

Equations (5) and (6) imply that U and kU are constants of

motion; they hold for all wave modes for which xM is inde-

pendent of U and kU.

FIG. 2. Structure of the model based on kinetic equations for quanta (central

pink box). The top row of boxes (orange) show inputs such as geometry, me-

dium properties, and initial and boundary conditions which are required to

specify the problem to be solved. The left and right sides of the kinetic equa-

tion are shown in the boxes in the left (blue) and right (yellow) columns,

respectively. Model outputs that yield radiation properties are in the middle

column (green).



The continuity equation for fM in Ra � Va space is

@fM@tþ @

@RaðVafMÞ ¼ StðfMÞ þ SM � cMfM; (9)

where Va ¼ _Ra is the velocity of flow of fM in the phase

space, a ¼ q; p, and StðfMÞ contains the small scale effects

such as mode conversion and stochastic scattering of radiation

off plasma density inhomogeneities (see Sec. III for further

discussion). Assuming the characteristic spatial scale of

plasma fluctuations to be small compared to the characteristic

scales of global (large-scale) plasma gradients, we treat

plasma fluctuations as local scatterers of quanta and include

their effect in the StðfMÞ term via corresponding collision inte-

grals (see Sec. III). This allows us to retain only the large-

scale plasma structure in the Hamiltonian xM (Ref. 17).

Using Eqs. (3) and (4), we have

@Va

@Ra¼ @

@Rq

@xM

@Rpþ @

@Rp� @xM

@Rq

� �¼ 0; (10)

and Eq. (9) becomes

@fM

@tþ Va

@fM@Ra¼ StðfMÞ þ SM � cMfM; (11)

where a ¼ q; p. Finally, using Eqs. (5)–(8), we obtain the

kinetic equation (1) for fM.

The kinetic equations (1) for fo and fx are coupled via

the StðfMÞ terms if the latter depend on both fo and fx and

thus describe the coupled evolution of o and x modes in the

fluctuating plasma.

III. DRIFT AND DIFFUSION TERMS IN THE KINETICEQUATION

In this section we describe the processes that contribute

to the term StðfMÞ on the right side of the kinetic equation

(1). These include small scale processes, such as stochastic

scattering of quanta off plasma density and magnetic field

inhomogeneities (which is shown below to result in both

refractive scattering and linear mode conversion in the case

of scattering off density inhomogeneities) and nonlinear

wave-wave and wave particle processes. First, we briefly list

some processes that can contribute to StðfMÞ in Eq. (1), fol-

lowed by a detailed consideration of the process of stochastic

scattering of o and x waves by plasma density fluctuations,

which is critical to many of the applications cited in Sec. I.

A. Small-scale processes

In the following discussion, we assume that none of the

small-scale processes contributing to StðfMÞ or the other terms

on the right of Eq. (1) depend on the phases U of the quanta

[quanta are created/annihilated with random phases U that

have a uniform distribution PMðUÞ ¼ ð2pÞ�1], and the initial

distribution of U is uniform, which corresponds to incoherent

radiation sources. In applications this assumption is often jus-

tified by the fact that the radiation processes are not phase-

coherent, and/or source regions are much larger than the

coherence length and contain many independent radiating

regions, which thus produce incoherent waves overall.40

Because the large-scale Hamiltonian equations of motion imply

U ¼ constant and the small-scale processes are assumed to be

independent of U, the uniform initial PMðUÞ ¼ 1=ð2pÞ does

not change during the system evolution.

The term StðfMÞ on the right of Eq. (1) describes the

change of fM due to small scale processes such as stochastic

wave scattering off plasma density fluctuations ~Np (which is

due to a combination of refractive scattering and linear mode

conversion at density gradients), radiation scattering by mag-

netic field fluctuations, and nonlinear wave-wave and wave-

particle processes, as shown in Fig. 2. It can be written as

@fM@t

� �RS

þ @fM@t

� �LMC

þ @fM@t

� �B

þ @fM

@t

� �NL

; (12)

where the subscripts RS, LMC, B, and NL relate to refractive

scattering, linear mode conversion at density gradients, mag-

netic field fluctuations, and nonlinear processes, e.g., three-

wave processes and nonlinear scattering of waves by plasma

particles, respectively. In this paper, we focus on the first

two processes.

B. Stochastic scattering off plasma densityinhomogeneities: Refraction and mode conversion

Suppose plasma density inhomogeneities are randomly

distributed throughout the plasma, with some given statisti-

cal distribution of properties, as in Fig. 1. These inhomoge-

neities can be either positive or negative in amplitude, and

we will refer to them simply as clumps in both cases. We

next consider scattering of o and x mode quanta off such

clumps and derive the first two stochastic scattering terms on

the right of Eq. (12).

When an electromagnetic (T) wave encounters a density

clump it is refracted by the clump’s density gradient, which

is discussed in detail in Sec. III C. However, if the local

plasma frequency anywhere within the clump exceeds the Twave frequency, LMC can occur, and the incident T wave

loses part of its energy to the L wave it excites through

LMC. The amplitude of the T wave after LMC and reflection

from the density gradient is thus reduced by a factor of RT ,

with ETout ¼ RTETin, where R2T ¼ 1� gT!L½k; aðkT ;rMpÞ�

is the square of the reflection coefficient and gT!L is the

energy conversion efficiency of T ! L LMC, which depends

on the wave number k of the incident T wave, and the angle

aðk;rNpÞ between k of the incident T wave and plasma den-

sity gradient rNp in the clump.30,31

As refraction by density gradients and LMC happen

simultaneously upon collision of quanta with density clumps,

they are both included in the evaluation of the corresponding

collision integral describing the combined effect of stochastic

scattering of o and x quanta off plasma density fluctuations.

1. Scattering off a single density clump

Let us consider stochastic scattering of quanta of a given

mode M¼ o, x with initial wave vector k off a plasma density

clump. For this purpose, we define a scattering coordinate frame



with basis c1; c2; j, where j ¼ k=k is the unit vector along k,

c1 is a unit vector normal to j and lying in the plane of vec-

tors k and B, and c2 ¼ �c1 � j ¼ e/ (see Fig. 3). We define

drM ¼ FMðk; vkk0 Þdo0 as the cross-section for scattering

events (collisions) in which a quantum with initial wave vec-

tor k, directed into solid angle do, changes its wave vector to

k0 directed into solid angle do0 ¼ sin vkk0dvkk0d/0, where vkk0

is the scattering angle between k and k0 and /0 is the azi-

muthal angle of k0 in the j; c1; c2 coordinate frame defined

above and shown in Fig. 3. The differential cross section

FMðk; vkk0 Þ that characterizes the scattering process depends

on the mode M of the quantum being scattered and on the

characteristics of the density clump that scatters the quan-

tum; we assume that the scattering is azimuthally symmetric

with respect to direction of k, so FM does not depend on /0.Assuming that the scatterers (density clumps) are approx-

imately stationary, we can write the probability per unit time

for a quantum with initial wave vector k to experience the

above type of collision as NclvgMdrM, where Ncl is the local

number density of density clumps, and vgM is the group speed

of waves, i.e., the velocity of quanta, in mode M.

Of all quanta in mode M within a phase volume

dX ¼ d3rd3k=ð2pÞ3 centered at r, k, the number leaving dXby changing their k to k0 directed into do0, regardless of the

size of k0, is

½d3rk2dkdo=ð2pÞ3� fMðr; k; hkrÞNclðrÞ� vgMðr; k; hkrÞFMðk; vkk0 Þdo0; (13)

where we have used d3k ¼ k2dkdo. On the other hand, an

influx of quanta into dX occurs due to scattering from

other parts of the phase space; as a result, of all quanta

within dX0 ¼ d3rd3k0=ð2pÞ3 ¼ d3rk02dk0do0=ð2pÞ3 the num-

ber given by Eq. (14) would acquire wave vector directed

into do with any k, if LMC did not occur:

½d3rk02dk0do0=ð2pÞ3� fMðr; k0; h0krÞNclðrÞ� vgMðr; k0; h0krÞFMðk0; vk0kÞdo; (14)

where h0kr is the azimuthal angle of k0 in the “global”

(aligned with the magnetic field) coordinate frame er; eh; e/,

i.e., h0kr is the angle between k0 and B.

If LMC can occur during the scattering events, i.e., if the

clump is sufficiently dense (which requires positive clump

amplitude) that x2pðrÞ=x2 > 1 somewhere within it, where x

is the frequency of the incident quanta and h is the angle

between radiation quanta and the normal to the surface), there

is nonzero probability for those quanta to be converted into

quanta of a different mode, say L, during scattering (more

generally, xp should be replaced by the relevant mode cutoff

frequency if the approximation of isotropic scatterers is being

made for nondegenerate modes). Therefore, there will be a

deficiency of quanta incident into d3rd3k=ð2pÞ3 from all other

elementary phase volumes d3rd3k0=ð2pÞ3, compared to the

case (14) when LMC does not occur [since we are only

accounting for the balance of quanta within d3rd3k=ð2pÞ3, it

is unimportant that quanta leaving d3rd3k=ð2pÞ3 may also

undergo LMC and disappear; all that matters is that they leave

d3rd3k=ð2pÞ3]. To account for this reduction in the number of

quanta arriving in d3rd3k=ð2pÞ3, the number of quanta arriv-

ing from d3rk02dk0do0=ð2pÞ3 to d3rk2dkdo=ð2pÞ3, given by

Eq. (14), must be multiplied by the probability that a given

quantum does not undergo LMC to turn into an L-mode quan-

tum, which is 1� PLMCT!Lðk0; ak0;rNp

Þ, where PLMCT!Lðk0; ak0;rNp

Þis the probability for a T mode quantum (T¼ o, x) to turn into

an L mode (Langmuir) quantum via T ! L LMC. The latter

is the energy efficiency gT!L of T ! L LMC, which depends

on k0 and on the angle ak0;rNpbetween k0 and the density gra-

dient rNp of the clump that refracts k0 into k. In the isotropic

scattering approximation, the incident and final angles of the

scattered quanta are equal, and ak0;rNp¼ ðp� vkk0 Þ=2, seen in

Fig. 4.

It is important to note that gT!L introduced here is not

an angle averaged efficiency of LMC34 but is the efficiency

of LMC for a given value of ak0;rNp.30,31,35–38 Averaging

FIG. 3. Orientations of the magnetic field B, wave vectors k, k0, and unit

vector j ¼ k=k; here er; eh, and e/ are unit vectors in the r, h, and / direc-

tions, respectively, hkr is angle between er and k, and h0kr is angle between

er and k0. The unit vector c1 is normal to j and lies in the plane of k and B,

c2 ¼ e/, vkk0 is the angle between k and k0, and /0 is the azimuthal angle of

k0 in the j; c1; c2 frame. The line k0N is parallel to k, while k0M is perpen-

dicular to the eh � e/ plane.

FIG. 4. Refraction of k0 into k by a density clump, whose local density gra-

dient’s orientation is uniquely defined by the orientations of k0 and k. The

angle of incidence is ak0 ;rNp. Since the angle of incidence equals the angle

of reflection, we have ak0 ;rNp¼ ðp� vk0kÞ=2.



over all possible scattering angles is carried out below in Eq.

(15), where we sum the quanta arriving into d3rd3k=ð2pÞ3from all phase space volumes dX0 ¼ d3rd3k0=ð2pÞ3 centered

at all possible k0 (but with the same d3r).

We assume that the scattering of o and x quanta off den-

sity clumps is an elastic process, so k ¼ k0 in the above rates.

Under this assumption the rate of change of the number of

M-mode quanta in d3rd3k=ð2pÞ3 due to the combined effect

of refractive scattering and LMC on a single plasma density

fluctuation (we call this combined effect “stochastic

scattering”) is the difference between the rates given by Eqs.

(13) and (14), integrated over d3k0 with k0 ¼ k, i.e.,

d3rd3kdfMðr; k; hkr; tÞ

dt

� ��ss:single

¼ d3rd3kNclðrÞð

do0FMðk; vkk0 Þ

� fvgMðr; k; h0krÞ fMðr; k; h0kr; tÞ

� ½1� PLMCT!Lðk; vkk0=2Þ�

�vgMðr; k; hkrÞfMðr; k; hkr; tÞg; (15)

where do0 ¼ sinh0krdh0krd/0k, and h0kr and /0k are zenith and az-

imuthal angles of k0 in the er; eh; e/ frame in Fig. 5.

2. Scattering off multiple density clumps

The scattering rate (15) is due to stochastic scattering

off a single density clump, whereas we have many clumps

within a small volume element d3r centered at r, generally

with different shapes, density profiles, and amplitudes (posi-

tive or negative). Therefore, to obtain the average rate of

change of the number of M mode quanta in d3rd3k=ð2pÞ3due to the combined effect of refractive scattering and LMC

on all density fluctuations within d3r, the rate (15) must be

averaged over the distribution of all clumps within d3r; we

denote this average by h…iclumps. We assume that all density

clumps are spherically symmetric and have the same density

profile, differing only in their amplitudes ~Np0, with ~Np0 > 0

for density enhancements and ~Np0 < 0 for density voids.

Although the clump shapes are assumed identical, the scat-

tering cross-sections seen by the quanta being scattered off

them are not identical, being different for clumps with differ-

ent amplitudes. Averaging over clumps under these assump-

tions thus reduces to averaging over clump amplitudes,

h…iclumps ¼Ðð…ÞPcð ~Np0Þd ~Np0, where Pcð ~Np0Þ is the

distribution of the model clump amplitudes within d3r,

which must be chosen to approximate the distribution Pð ~NpÞof actual plasma density fluctuations. The corresponding

relation between Pcð ~Np0Þ and Pð ~NpÞ is discussed in detail in

Appendix C.

Dropping the common multiplier d3rd3k in Eq. (15) and

rearranging terms, we see that the collisional integral corre-

sponding to the process of stochastic scattering off plasma

density fluctuations ðdfM=dtÞjSS breaks into two terms: the

first, ðdfM=dtÞjRS, describes pure refractive scattering without

LMC, and the second, ðdfM=dtÞjLMCT!L, describes the effect of

LMC on the scattering, i.e.,

dfM

dt

� ��SS

¼ dfM

dt

� ��RS

þ dfM

dt

� ��LMC

T!L

; (16)

with

dfM

dt

� ��RS

¼ NclðrÞð

do0hFMðk; vkk0 Þiclumps

�fvgMðr; k; h0krÞfMðr; k; h0kr; tÞ�vgMðr; k; hkrÞfMðr; k; hkr; tÞg; (17)

dfM

dt

� ��LMC

T!L

¼ �NclðrÞð

do0hPLMCT!Lðk; vkk0=2Þ

� FMðk; vkk0 ÞiclumpsvgMðr; k; h0krÞ� fMðr; k; h0kr; tÞ: (18)

The integration in Eqs. (17) and (18) is most conven-

iently performed in the scattering coordinate frame c1; c2; j[see Figs. 3 and 5, in which do0 ¼ sin vkk0dvkk0d/0, and h0kr is

related to vkk0 and /0 by

cos h0kr ¼ cos hkr cos vkk0 þ sin hkr sin vkk0 cos /0: (19)

In order to calculate the collisional integrals in Eqs. (17)

and (18) for refractive scattering and LMC in an isotropic

FIG. 5. Coordinate systems, showing the key vectors and angles discussed in

the text. (a) Spherical spatial coordinates r, h, and also showing hkz; the ambi-

ent magnetic field is in the direction er . (b) Cartesian spatial coordinates z, x,

and also showing hkz; the ambient magnetic field is in the vertical direction.



approximation [with h0kr ¼ h0krðhkr; vkk0 ;/0Þ via Eq. (19)], we

need to define FMðk; vkk0 Þ and PLMCT!Lðk; vkk0=2Þ. Accordingly,

we consider refractive scattering and linear mode conversion

of o and x modes on plasma density clumps in more detail in

Subsection III C.

C. Refractive scattering by density clumps

Diffusion of quanta due to stochastic refractive scatter-

ing of radiation quanta off plasma density fluctuations is

analogous to the problem of diffusion of a light gas in a

heavy gas,32 with quanta being the light gas and clumps

being the heavy gas. Hence, we derive the corresponding

collision integral in a similar way. In the zeroth-order

approximation, the scattering process is elastic and the pho-

ton momentum k only changes direction, not magnitude, as a

result of the scattering off a density clump. The scatterers are

assumed to be at rest, with a given distribution of sizes.

A key point here is that we assume that the geometric

optics approximation holds for the ray paths. This is valid so

long as the wavelength of the radiation is small compared to

the size of the scatterers,33,41 an approximation that is valid

in the case of the solar-terrestrial applications that we pri-

marily envisage. If this inequality was not the case, diffrac-

tion would need to be included,41 a point that has been

investigated in some cases of Alfv�en, drift-wave, and other

turbulence, for example.25,42

The problem of calculating the differential cross-section

FMðk; vkk0 Þ of this scattering process, considered in detail in

Appendix B, is azimuthally symmetric with respect to the

direction of k and is simplified by the existence of two local

integrals of motion for the quanta: energy and angular mo-

mentum with respect to the center O of the density clump.

Using these conserved integrals of motion, the scattering

problem can be solved analytically for a given density pro-

file, as discussed in Appendix B. The differential cross-

section FMðk; vkk0 Þ is defined by

FMðk; vkk0 Þ ¼qMðk; vkk0 Þ

sin vkk0

�� @qMðk; vkk0 Þ@vkk0

��: (20)

The function qMðk; vkk0 Þ, which characterizes the relation

between the impact parameter q and the scattering angle vkk0

in the considered scattering process, is obtained from the so-

lution of the particular refractive scattering problem for

mode M quanta on density clumps of a chosen profile. An

example of such a solution for refractive scattering off den-

sity clumps with an inverse-square density profile is given in

Appendix B, yielding FMðk; vkk0 Þ in the form (B45).

The assumption of azimuthal symmetry about k strictly

limits the scattering analysis in the present subsection to the

unmagnetized case, although the results will provide a good

approximation so long as the cyclotron frequency is much

less than the plasma frequency. More generally, however, it

would be necessary to compute the scattering separately for

the o and x modes, accounting for the fact that the magnetic

field reduces the symmetry of the problem and gives the

modes distinct dispersion relations. We stress, though, that

the large-scale propagation analysis in the present paper

retains magnetic effects through the separate dispersion of

the modes.

D. Linear mode conversion on density clumps

We now consider LMC between o, x (electromagnetic),

and L (Langmuir) modes, occurring at plasma density pertur-

bations. In principle we should take into account both the

direct and inverse processes of mode conversion fo; xg $ L,

as both these processes affect the distribution functions of oand x modes. Indeed, if the level of o or x radiation in some

region changes, then the direct mode conversion fo; xg ! Lshould change the level of L radiation, which in turn influen-

ces o and x mode radiation levels via the inverse processes

L! fo; xg. However, because the volume averaged effi-

ciency of energy transfer in both the direct and inverse LMC

processes is very low,34 this self-influence of o and x modes

mediated by the L mode is extremely weak in most cases and

can be neglected, especially as the L waves are usually heav-

ily damped except when in resonance with a driver such as

an electron beam. Thus we retain only the direct LMC proc-

esses fo; xg ! L, regarding them simply as energy sinks for

o and x mode waves. The corresponding rate of annihilation

of o and x mode quanta due to this process is described by

the sink term (18). To evaluate Eq. (18), we must obtain

PLMCT!Lðk; vkk0=2Þ, in addition to FMðk; vkk0 Þ obtained above.

Consider a T (¼o, x) wave incident on a plasma density

gradient and undergoing LMC into a Langmuir (L) wave.

The LMC occurs around the cutoff point xmc for the incident

wave (as shown in Fig. 6) where the local plasma frequency

equals the frequency of the incident T wave.35 Therefore, the

LMC process T ! L can occur only on density enhance-

ments ( ~Np0 > 0) and not on density voids ( ~Np0 < 0). In the

vicinity of this cutoff point, we approximate the clump den-

sity profile with the linear ramp shown in Fig. 6

Nlinp ðxÞ ¼

Np0 þ ðx=lÞNcr if x � 0;Np0 if x < 0;

�(21)

FIG. 6. Schematic of T ! L linear mode conversion on a density gradient.

The density profile is shown by the green line. The T-wave (red), incident at

angle hT to the surface x¼ 0, gets reflected by the density ramp and turns at

xturn, beyond which the field of the T-wave is evanescent. Through this evan-

escent field, a fraction of the T-wave’s energy leaks through to the mode

conversion point xmc where it couples to and excites the L-wave, which then

propagates to the right at an angle hL to the surface x¼ 0.



where l�1 ¼ ½rSðrsÞ=SðrsÞ�rs¼rmc, rmc is the distance from the

clump center to the cutoff point xmc corresponding to the crit-

ical plasma density Ncr ¼ x2�0me=e2; SðrsÞ defines the

clump’s radial density profile, and rs is the distance from the

clump center O (more generally, one could use the expres-

sions for the cutoff of the o and x modes if approximating

their interactions as isotropic).

The efficiency of T ! L LMC, defined as the ratio of

energy transfered to the L wave to the energy of the incident

T wave, has been calculated for the approximate density pro-

file (21). It has been shown36,37 that the LMC efficiency is

same for the direct (T ! L) and inverse (L! T) LMC, so

we can use existing results for that of L! T LMC30,34 to

find the efficiency of T ! L LMC. It has also been shown

that magnetization effects do not change the efficiency sig-

nificantly,31,34,38 so we neglect magnetization for simplicity.

Neglecting all magnetization effects (including setting the

cutoffs of the dispersion relations to be equal) would imply

that the efficiencies of T ! L LMC are the same for o and xmodes,31,38 but these cutoffs can be kept distinct while con-

tinuing to approximate the scattering as isotropic.

It has been shown30 that for g0 ¼ x2p0=x

2 6¼ 0 and

xp0 ¼ ðe2Np0=�0meÞ1=2the electron plasma frequency, the

mode conversion efficiency gLMCL!T (and hence gLMC

T!L) is pri-

marily a function of the parameter

q ¼ ðk0lÞ2=3 ð1� g0Þ sin2 hL

cb; (22)

where k0 ¼ x=c; c is the plasma adiabatic index, b ¼ kBTe

=mec2; Te is the plasma temperature, kB is Boltzmann’s con-

stant, and hL is the angle of incidence of the L wave onto the

linear density ramp, i.e., the angle between kL and rNlinp . The

angle hL is related to the angle hT between kT andrNlinp via34

sin2 hL ¼ cb sin2 hT : (23)

Using Eqs. (22) and (23) we obtain gLMCT!L ¼ gT!LðqÞ, with

q ¼ q0 sin2 hT ; (24)

where q0 ¼ ð1� g0Þðxl=cÞ2=3. We see that q and gT!LðqÞ

depend on x; xp0, l, and hT .

In the case Np0 ¼ 0 ðg0 ¼ 0Þ, when the T wave is inci-

dent on a linear density ramp from vacuum, the LMC effi-

ciency averaged over T polarizations is35

gT!LðqÞ 1:33q exp � 4

3q3=2

� �: (25)

In the case Np0 > 0 ðg0 6¼ 0Þ, when the T wave is inci-

dent onto a linear density ramp from the background plasma,

which is the case relevant here, the form of gT!LðqÞ is more

easily obtained numerically and was shown in Figs. 1 and 2

of Ref. 30 and in Fig. 1 of Ref. 34. The effect of increasing

g0 is primarily to decrease the cutoff value q0 of gðqÞ while

increasing the peak value of gðqÞ at q < q0. Consider two

distinct cases: (i) g0 < 1 (but not 1) and (ii) g0 � 1. In the

first case, the functional form of gðqÞ is similar to that of the

vacuum case [cf., Eq. (25)], only with decreased q0 and

increased peak value gmax 0:5. However, for g0 � 1 the

mode conversion efficiency differs markedly: the mode con-

version for g0 � 1 is strongly cut off at q ¼ q0 ¼ 0:79, and

the maximum conversion efficiency occurs at lower q, with

gmax 0:7. Interestingly, gT!LðqÞ in this case (g0 � 1) is

well approximated by34

gðqÞ 4gmax

q

q0

1� q

q0

� �; (26)

for 0 q q0, where gmax is the maximum value of gðqÞ. In

general gmax has a weak dependence on x; xp0, and l,34 but

we will assume, following Ref. 34, that gmax is a constant,

gmax 0:7. With this assumption and noting that q=q0

¼ sin2 hT , we see from Eq. (26) that for g0 � 1 the mode con-

version efficiency g only depends on hT , with

gðqÞ gmax sin2 ð2hTÞ: (27)

Finally, noting hT ¼ ðp� vkk0 Þ=2 (Fig. 4), we find

PLMCT!Lðk; vkk0=2Þ ¼ gLMC

T!L; (28)

gmax sin2 vkk0 ; (29)

in the case g0 � 1, with gmax 0:7. We see that within the

above approximations (linear density ramp, unmagnetized

plasma, and g0 ¼ x2p0=x

2 � 1), PLMCT!L is only a function of vkk0

and does not depend on k. The LMC efficiency (29) is symmet-

ric around its maximum gmax at vkk0 ¼ p=2 (i.e., at incidence

angle hT ¼ p=4) and falls to zero for vkk0 ¼ 0 ðhT ¼ p=2Þ and

vkk0 ¼ p ðhT ¼ 0Þ.The approximation (29) for PLMC

T!Lðk; vkk0=2Þ in Eq. (18)

is only valid for quanta with g0 ¼ x2p0=x

2 � 1, i.e., for

quanta undergoing LMC within (or in the vicinity of) their

source region where x2p0=x

2 � 1 can occur. As quanta with a

given x escape from their source region, their g0 typically

becomes small as xp0 decreases, ultimately approaching the

vacuum case g0 ¼ 0, and the approximation (29) must be

replaced with that of Eq. (25). However, there are relatively

few plasma density fluctuations away from the source region

that can cause LMC, compared to the source region.12

Hence, the role of LMC away from the source is relatively

minor and can be either neglected or approximated via Eq.

(29), which is valid in the source region where most LMC

occurs, without introducing significant error.

E. Average over density inhomogeneities

To obtain the rate of change (16) of the distribution

function due to refractive scattering and LMC on plasma

density fluctuations within a physically small volume d3r,

we take the corresponding rates of refractive scattering and

LMC on a single clump and average them over all clumps

within d3r. We perform the averaging over clumps using the

following approximations:

(i) Magnetization effects that cause anisotropy in the

scattering problem at microscales are neglected. This



permits the o and x modes to be treated as nonidenti-

cal in some respects (e.g., they have different cutoffs)

but does not allow for effects of the anisotropy of

their dispersion on the scattering problem.

(ii) All clumps have the same density profile SðrsÞ, where

rs is the distance from the clump center, differing

only in their amplitudes, which can be negative. The

example of an inverse-square clump density profile

SðrsÞ ¼ ðRc=rsÞ2 is considered in Appendix B.

(iii) The distribution of clump amplitudes Pcð ~Np0Þ is a

Gaussian with zero mean and standard deviation r ~N p0

Pcð ~Np0; r ~N p0Þ ¼ 1ffiffiffiffiffiffi

2pp

r ~N p0

exp �~N

2

p0

2r2~N p0

!: (30)

Consider the refractive scattering term (18) first. Aver-

aging FMðk; vkk0 Þ over ~Np0 gives

hFMiclumps ¼ð1�1

FMPcð ~Np0Þd ~Np0; (31)

¼ð0

�1FM�Pcð ~Np0Þd ~Np0

þð1

0

FMþPcð ~Np0Þd ~Np0; (32)

with FM6ðk; vkk0 Þ defined by the clump density profile, and

þ and � signs corresponding to scattering off density

enhancements/voids, respectively. For an inverse-square pro-

file, FM6ðk; vkk0 Þ is (see Appendix B)

FM6ðk; vkk0 Þ ¼~x2

p0

c2k2pR2

c

�� psin vkk0

p7vkk0

v2kk0 ð2p7vkk0 Þ2

��; (33)

where ~x2p0 ¼ e2j ~Np0j=�0me; FMþ corresponds to density

enhancements ( ~Np0 > 0), and FM� corresponds to density

depletions ( ~Np0 < 0). Noting that for a Gaussian Pcð ~Np0Þgiven by Eq. (30), one has

ð0

�1j ~Np0jPcð ~Np0Þd ~Np0 ¼

ð10

j ~Np0jPcð ~Np0Þd ~Np0; (34)

¼r ~N p0ffiffiffiffiffiffi

2pp : (35)

We then obtain

hFMðk; vkk0 Þiclumps e2r ~N p0

�0me

pR2cffiffiffiffiffiffi

2pp

c2k2

pv2

kk0 sin vkk0

� pþ vkk0

ð2pþ vkk0 Þ2þ p� vkk0

ð2p� vkk0 Þ2

( );

(36)

where we note that 0 vkk0 p so 0 sin vkk0 1. The

first and second terms in Eq. (36) correspond to refractive

scattering off density voids ( ~Np0 < 0) and enhancements

( ~Np0 > 0), respectively.

Now consider averaging over clumps in the LMC term

given by Eq. (18). Here, we use the approximation (29) for

PLMCT!L, according to which PLMC

T!L > 0 only for enhancements

( ~Np0 > 0) and PLMCT!L ¼ 0 for voids ( ~Np0 < 0), and PLMC

T!L for

enhancements does not depend on the clump amplitude ~Np0,

which simplifies averaging over ~Np0 in Eq. (18). For clumps

with inverse-square density profiles SðrsÞ ¼ ðRc=rsÞ2 and

Gaussian distribution of amplitudes (30) we obtain, using

FMþ from Eq. (33),

hPLMCT!Lðk; vkk0=2ÞFMðk; vkk0 Þiclumps

gmax sin 2ðvkk0 Þð1

0

FMþ Pcð ~Np0Þd ~Np0; (37)

¼ gmax

e2r ~N p0

�0me

pR2cffiffiffiffiffiffi

2pp

c2k2

p sin vkk0

v2kk0

p� vkk0

ð2p� vkk0 Þ2: (38)

The limit of no plasma density fluctuations [NpðrÞ¼ Np0ðrÞ and ~NpðrÞ ¼ 0] corresponds to r ~N p0

¼ 0, and in

this limit both the refractive scattering term (17) with

hFMðk; vkk0 Þiclumps given by Eq. (36), and the LMC term (18)

with hPLMCT!Lðk; vkk0=2ÞFMðk; vkk0 Þiclumps given by Eq. (38) are

zero, as expected.

F. Fokker-Planck approximation

The kinetic equation (1) with the integral terms (17) and

(18) describing refractive scattering and linear mode conver-

sion of T quanta, respectively, on plasma density clumps, is an

integrodifferential equation for fM, which is in general difficult

to solve and analyze. However, under certain assumptions, the

problem can be greatly simplified by approximating this equa-

tion as a differential Fokker-Planck equation. Here we discuss

this approximation and derive the corresponding drift-

diffusion approximation of (17) and (18).

The function FMðk; vkk0 Þ characterizes refractive scatter-

ing of wave quanta off plasma density clumps (which are

assumed spherically symmetric here) in an isotropic approxi-

mation and is fully characterized by the clump radial density

profile ~NpðrsÞ ¼ ~Np0SðrsÞ, where rs is the distance from the

center of the clump, SðrsÞ is the function describing

the clump shape, and ~Np0 is the amplitude of the clump (see

Appendix B). If SðrsÞ decreases sufficiently fast with rs, then

FMðk; vkk0 Þ is strongly peaked (in fact, singular) at vkk0 ¼ 0

and falls off rapidly as vkk0 increases; in other words, the

small-angle scatterings of quanta off such clumps dominate

over large-angle scatterings. In this case, the main contribu-

tion into the integrals over vkk0 in Eqs. (17) and (18) comes

from small vkk0 . This fact enables us to approximate the inte-

grals in Eqs. (17) and (18) by expanding their kernels to sec-

ond order in vkk0 around vkk0 ¼ 0 and then performing the

integration over do0 ¼ sin vkk0dvkk0d/0.From Eq. (19) at small vkk0 , we have

h0kr ¼ hkr � vkk0 cos /0 þ v2kk0

2

cos hkr

sin hkrsin2 /0 þ Oðv3

kk0 Þ: (39)

Substituting Eq. (39) into vgMðr; k; h0krÞfMðr; k; h0kr; tÞ and

keeping the terms up to v2kk0 , we obtain



vgMðh0krÞfMðh0krÞ ¼ vgMðhkrÞfMðhkrÞ

� vkk0 cos/0@

@hkr½vgMðhkrÞfMðhkrÞ�

þv2kk0

2

coshkr

sinhkrsin2 /0

@

@hkr½vgMðhkrÞfMðhkrÞ�

�

þ cos2 /0@2

@h2kr

½vgMðhkrÞfMðhkrÞ�: (40)

Substituting Eq. (40) into Eqs. (17) and (18) and integrating

over /0, we obtain the following approximations:

dfMdt

� ��RS

¼ NclðrÞhGrsMð ~Np0;Rc; kÞiclumps

� psin hkr

@

@hkrsin hkr

@

@hkr½vgMfM�

� ; (41)

dfM

dt

� ��LMC

T!L

¼ �NclðrÞhGLMCM;1 ð ~Np0;Rc; kÞiclumps vgM fM

�NclðrÞhGLMCM;2 ð ~Np0;Rc; kÞiclumps

� psin hkr

@

@hkrsin hkr

@

@hkr½vgM fM�

� ;

(42)

with

GrsMð ~Np0;Rc; kÞ ¼

ðp

0

FMðk; vkk0 Þðv2kk0=2Þ

� sin vkk0dvkk0 ; (43)

GLMCM;1 ð ~Np0;Rc; kÞ ¼

ðp

0

PLMCT!Lðk; vkk0=2Þ

� FMðk; vkk0 Þ sin vkk0 dvkk0 ; (44)

GLMCM;2 ð ~Np0;Rc; kÞ ¼

ðp

0

PLMCT!Lðk; vkk0=2Þ

� FMðk; vkk0 Þðv2kk0=2Þ sin vkk0dvkk0 : (45)

The first term on the right of Eq. (42) corresponds to damp-

ing via mode conversion, while the other terms in Eqs. (41)

and (42) give rise to drift and diffusion.12,13,21,22,31,34 The

procedure for averaging over the ensemble of density clumps

in the physically small volume d3r (small compared to the

system size, but containing many clumps) in Eqs. (41) and

(42) is discussed in Sec. III E.

We see from Eqs. (41) and (42) that the combined effect

of elastic scattering of o and x quanta off plasma density

clumps and the T ! L linear mode conversion on plasma

density clumps (only those of positive amplitude) can be

approximated as angular diffusion of quanta fo;x in k-space,

described by the right side of Eq. (41) and the second term in

the right side of Eq. (42), combined with annihilation of fo;x

(due to conversion of o and x quanta into damped local

plasma waves), described by the first term in Eq. (42).

The averages in Eqs. (41) and (42) depend on the shape

of density clumps. For example, we again consider clumps

with the inverse-square profile SðrsÞ ¼ ðRc=rsÞ2. Substituting

Eqs. (36) and (38) that correspond to this profile into Eqs.

(43)–(45) and integrating over vkk0 , we find

hGrsMð ~Np0;Rc; kÞi ~N p0

¼e2r ~N p0

�0me

pR2c

c2k2

p2

�3=2

� ln 3� 2

3

� �; (46)

hGLMCM;1 ð ~Np0;Rc; kÞi ~N p0

¼e2r ~N p0

�0me

pR2c

c2k2

gmax

2

ffiffiffip2

rg1; (47)

hGLMCM;2 ð ~Np0;Rc; kÞi ~N p0

¼e2r ~N p0

�0me

pR2c

c2k2

gmax

2

p2

�3=2

g2; (48)

where g1 ¼ 2Sið2pÞ�Sið4pÞ 1:34414; g2 ¼ ln2þCið2pÞ�Cið4pÞþ 2p½Sið2pÞ�Sið4pÞ� 0:21169. Here Ci and Si

are the cosine and sine integrals.43

IV. RADIATION PROPERTIES

We now derive the distributions PSAðSAÞ of the Stokes

parameters SA ¼ I;Q;U, and V of the EM radiation observed

far from the source region using an antenna, and the corre-

sponding mean Stokes parameters as would be measured by

the antenna, in terms of the distribution functions fMðr; k; tÞof the o and x quanta. This derivation follows those in Refs.

26 and 27.

A. Radiation measurement by antenna

We make the following assumptions about the antenna:

(i) The ambient magnetic field at the location of the

antenna is weak, so the electromagnetic waves

detected by the antenna are purely transverse.

(ii) The antenna simultaneously measures two orthogonal

components of the wave electric field in its plane.

(iii) The antenna is tuned to detect only a narrow fre-

quency band near a chosen frequency x ¼ xA and a

narrow range of wave vectors k kA, which we

assume to be perpendicular to the antenna plane.

(iv) The measured instantaneous components of the wave

electric field are cross-multiplied and recorded over the

radiation coherence time sc which is assumed to last

for many wave periods (i.e., sc � x�1), to obtain the

corresponding instantaneous Stokes parameters,26,27

which are then averaged over a time sav � sc to obtain

the “running mean” Stokes parameters. We assume

that the time sav, over which the mean Stokes parame-

ters are evaluated, is short compared to the characteris-

tic time scale s of evolution of fMðr; k;U; tÞ, so that

x�1 � sc � sav � s.26,27

Each quantum incident on the antenna represents a wave

packet of length lq � csc, where c is the speed of light. Of all

quanta incident on the antenna, the antenna detects only

those with a given frequency xM ¼ x and wavevector k to

which the antenna is tuned. In other words, out of all quanta

in mode M located at r ¼ rA only the quanta with fixed val-

ues of k and hkr are observed, where hkr is defined by the ori-

entation of the antenna’s line of sight relative to the radius

vector from the center of the system (e.g., the Sun), and k ¼kM is defined from the antenna’s tuned frequency x via the



dispersion relation for the mode M at the antenna; i.e.,

x ¼ xMðrA; kM; hkrÞ. Therefore, at any instant in time, the

antenna simultaneously detects all quanta that occupy the

phase space volume dXA ¼ d3rAd3kA=ð2pÞ3 centered at

rA; kA, where d3rA ¼ cscrA; rA is the antenna’s cross-

section area, and d3kA is defined by the range of wave vec-

tors that the antenna with a given receiving frequency x and

orientation can detect. The number of these quanta is

NMðrA; kM; hkr; tÞ ¼ 2p fMðrA; kM; hkr; tÞdXA: (49)

The instantaneous electric field corresponding to a

single quantum of a mode M within dXA is

EM1ðrA; kM; tÞ ¼ EM1eMe�iðxMt�kM �rAþUjÞ; (50)

where xM ¼ xMðrA; kÞ; Uj is the random phase of the quan-

tum, EM1 is the amplitude, and eM is the polarization vector

of the wave associated with the quantum. The waves repre-

sented by Eq. (50) correspond to the individual quanta within

dXA sharing the same amplitude E1M, and the same polariza-

tion vector eM (as the waves are all copropagating, of the

same frequency, and belong to the same mode) and only dif-

fer in phase due to the random initial phases Uj of individual

quanta.

As all quanta in mode M located within dXA at time

instant t are simultaneously detected by the antenna, the total

electric field due to these quanta, measured by the antenna,

is the superposition of all the NM waves Eq. (50)

EMðrA; kMtÞ ¼ EM1eMe�iðxt�kM �rAÞXNM

j¼1

expðiUjÞ: (51)

Since the phases Uj in Eq. (51) are random so is the result-

ing field EMðrA; tÞ in Eq. (51). Henceforth, we assume

NM � 1.

The polarization vector eM of a mode M defines the

shape and orientation of the polarization ellipse of that

mode. Let us define unit vectors t and a to lie along the main

axes of the polarization ellipse defined by eM. Assuming that

the mode is purely transverse (see the antenna assumptions

above), we have

eM ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T2M þ 1

p ð TM i Þ�

t

a

�; (52)

where magnetoionic theory gives39

TMðrA;x; hkrÞ ¼� 1

2Y2 sin2 hkr � rMD

Yð1� XÞ cos hkr; (53)

D2 ¼ 1

4Y4 sin 4hkr þ ð1� XÞ2Y2 cos2 hkr; (54)

ro ¼ 1; rx ¼ �1; hkr is the angle between k and local mag-

netic field B (which is along the local er and so is radial),

and X ¼ xpðrAÞ2=x2 and Y ¼ xceðrAÞ=x.

If both o and x quanta reach the antenna simultaneously,

then the total vector field measured by the antenna is the

superposition of both modes, which then can be written using

Eqs. (51) and (52) as

Etot ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

T2o þ 1

p ðTo i Þ�

t

a

�Eo1 exp½iðxt� ko � rAÞ�

�XNo

j¼1

expðiUjÞ

þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiT2

x þ 1p ðTx i Þ

�t

a

�Ex1 exp½iðxt� kx � rAÞ�

�XNx

l¼1

exp ðiUlÞ: (55)

Here Eo1 and Ex1 are the amplitudes of the waves correspond-

ing to individual quanta of the o and x modes, respectively

(defined below), and the wave vectors ko and kx are collinear

(as they are both orthogonal to the antenna plane), but are not

equal in magnitude in general, ko 6¼ kx (unless the antenna

region is unmagnetized and the o and x modes are degenerate

with the same dispersion). Note that the o and x modes are

orthonormal, so their polarization vectors are orthogonal, and

we can use the same basis vectors t and a for both.

The amplitudes EM1 (M¼ o, x) in Eq. (55) of the waves

corresponding to individual quanta within dXA can be esti-

mated by assuming that the spectral energy of the wave

packet associated with a quantum localized within dXA is

also localized within dXA. The energy of the quantum �hxM

equals the total wave packet energy

�hxMðr; k; tÞ ¼ð

wEMðr; k; tÞdXA; (56)

where wEM is the total spectral energy density of the electro-

magnetic field of the wave packet associated with the quan-

tum. This can be approximated by

�hxM �wEMðr; k; tÞdXA; (57)

where �wEM ¼ �0�Mðr; k; tÞjEM1j2 is the mean energy of the

wavepacket averaged over dXA; �Mðr; k; tÞ ¼ n2Mðr; k; tÞ is

the relative dielectric permittivity of the medium, nM is the

refractive index, and jEM1j is the average (characteristic) am-

plitude of the wave packet associated with the quantum.

Assuming the electric energy is half the total energy,39,40 we

find

E2M1

�hxM

�0n2MdXA

: (58)

B. Stokes parameters

Since t and a are orthogonal to k for transverse o and xwaves, and the observation plane is orthogonal to k, the

“instantaneous” Stokes parameters measured by the antenna

over the coherence time sc � x�1 are26,27

I ¼ jEttotj

2 þ jEatotj

2; (59)



Q ¼ jEttotj

2 � jEatotj

2; (60)

U ¼ 2 ReðEttotE

a totÞ; (61)

V ¼ �2 ImðEttotE

a totÞ; (62)

where Ettot and Ea

tot are the components of Etot along t and a

and asterisks denote complex conjugation. Using these defi-

nitions and Eq. (55), we have

I ¼ E2o1no þ E2

x1nx

þ2ToTx þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðT2o þ 1ÞðT2

x þ 1Þp

� Eo1Ex1Refnox exp ½iðko � kxÞ � rA�g; (63)

Q ¼ T2o � 1

T2o þ 1

E2o1no þ

T2x � 1

T2x þ 1

E2x1nx

þ2ToTx � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðT2o þ 1ÞðT2

x þ 1Þp

� Eo1Ex1 Refnox exp ½iðko � kxÞ � rA�g; (64)

U ¼ 2To � Txffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðT2o þ 1ÞðT2

x þ 1Þp� Im fnox exp ½iðko � kxÞ � rA�g; (65)

V ¼ 2To

T2o þ 1

E2o1ReðnoÞ þ

2Tx

T2x þ 1

E2x1ReðnxÞ

þ2To þ Txffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðT2o þ 1ÞðT2

x þ 1Þp

� Eo1Ex1Refnox exp½iðko � kxÞ � rA�g: (66)

Here no; nx, and nox are random quantities defined by

no ¼XNo

j¼1

XNo

j0¼1

exp½iðUj � Uj0 Þ�; (67)

nx ¼XNx

l¼1

XNx

l0¼1

exp½iðUl � Ul0 Þ�; (68)

nox ¼XNo

j¼1

XNx

l¼1

exp½iðUj � UlÞ�; (69)

with Uj; Uj0 ; Ul, and Ul0 being the independent random

phases of the individual waves.

Since o and x modes are orthonormal, we have

ToTx ¼ �1,39 and the above expressions reduce to

I ¼ E2o1no þ E2

x1nx; (70)

Q ¼ T2o � 1

T2o þ 1

ðE2o1no � E2

x1nxÞ

� 4jTojT2

o þ 1Eo1Ex1fnr

ox cos ½ðko � kxÞ � rA�

þniox sin ½ðko � kxÞ � rA�g; (71)

U ¼ 2jTojTofni

ox cos ½ðko � kxÞ � rA�

�nrox sin ½ðko � kxÞ � rA�g; (72)

V ¼ 2To

T2o þ 1

ðE2o1no � E2

x1nxÞ

þ2jTojTo

T2o � 1

T2o þ 1

Eo1Ex1fnrox cos ½ðko � kxÞ � rA�

þniox sin ½ðko � kxÞ � rA�g; (73)

where nrox ¼ ReðnoxÞ and ni

ox ¼ ImðnoxÞ, and no; nx, and nox

are defined in Eqs. (67)–(69). Thus the measured “instantaneous”

Stokes parameters are random quantities, which have a coher-

ence time sc.

It can be shown that the probability distribution func-

tions PðnoÞ; PðnxÞ; and Pðnr;ioxÞ for large enough No;x are

PðnMÞ ¼2

NMexp � 2nM

NM

� �HðnMÞ; (74)

for NM � 1 and

Pðnr;ioxÞ ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiNoNx

p exp � 2jnr;ioxjffiffiffiffiffiffiffiffiffiffiffiffiffi

NoNx

p� �

; (75)

for NoNx � 1. The approximations (74) and (75) are very

accurate for NM � 10. Using these approximations and Eqs.

(70)–(73) for the measured instantaneous Stokes parameters

I, Q, U, V, one can derive their probability densities P(I),P(Q), P(U), and P(V) via

PðSAÞ ¼ð

PðnoÞPðnxÞPðnroxÞPðni

oxÞ

� d½SA � SAðno; nx; nrox; n

ioxÞ�dnodnxdnr

oxdniox;

(76)

where SA denotes the Stokes parameters I, Q, U, and V, d½…�is the Dirac delta function, and the functions SAðno; nx;nr

ox; nioxÞ are defined by Eqs. (70)–(73). In deriving Eq. (76),

we have assumed that the random variables no; nx; nrox, and

niox are independent, so their joint probability is a product

of individual probabilities: Pðno; nx; nrox; n

ioxÞ ¼ PðnoÞPðnxÞ

PðnroxÞPðni

oxÞ.Equation (76) yields

PðIÞ ¼ 2

NoE2o1 �NxE2

x1

� exp � 2I

NoE2o1

� �� exp � 2I

NxE2x1

� �� ; (77)

for I � 0. It can be easily verified thatÐ

PðIÞdI ¼ 1. In the

special case NoE2o1 ¼NxE2

x1, Eq. (77) reduces to

PðIÞ ¼ 4I

N2oE4

o1

exp � 2I

NoE2o1

� �: (78)

Obtaining P(Q), P(U), and P(V) using Eq. (76) is much

more complicated than calculating P(I) and involves rather



cumbersome integration. However, we do not need to know

the distributions P(I), P(Q), P(U), and P(V) to calculate the

mean Stokes parameters �I; �Q; �U , and �V , defined by averag-

ing the instantaneous Stokes parameters (70)–(73) over the

time sav � sc [which is equivalent to the ensemble averag-

ing of the random instantaneous Stokes parameters (70)–

(73)]. Instead, to obtain the mean Stokes parameters�I; �Q; �U ; �V , we average Eqs. (70)–(73) and note that

no ¼ð1�1

noPðnoÞdno ¼No

2; (79)

nx ¼ð1�1

nxPðnxÞdnx ¼Nx

2; (80)

nr;iox ¼

ð1�1

nr;ioxPðnr;i

oxÞdnr;iox ¼ 0; (81)

with PðnoÞ; PðnxÞ, and Pðnr;ioxÞ defined in Eqs. (74) and (75).

After this averaging we obtain

�I ¼No

2E2

o1 þNx

2E2

x1; (82)

�Q ¼ T2o � 1

T2o þ 1

No

2E2

o1 �Nx

2E2

x1

� �; (83)

�U ¼ 0; (84)

�V ¼ 2To

T2o þ 1

No

2E2

o1 �Nx

2E2

x1

� �: (85)

Note that �I of Eq. (82) matches with �I ¼Ð

IPðIÞdI with P(I)defined by Eq. (77), as expected. This verifies the averaging

procedure leading to Eqs. (82)–(85).

Finally, using the definition NM ¼ NMdX [where

NM ¼ ð2pÞfM is the occupation number] and the estimate

(58) of E2M1, we obtain

�I ¼ p�hxfon2

o

þ fx

n2x

� �; (86)

�Q ¼ T2o � 1

T2o þ 1

p�hxfon2

o

� fx

n2x

� �; (87)

�U ¼ 0; (88)

�V ¼ 2To

T2o þ 1

p�hxfo

n2o

� fx

n2x

� �; (89)

where all quantities are measured at r ¼ rA and kM is found

from xMðrA; kM; hkr; tÞ ¼ x with x and hkr defined by the

antenna frequency and wave-vector tuning. Equations (86)–(89)

thus express the observed mean Stokes parameters in terms of

the distribution functions fo and fx.

The measured degree of polarization is

�Q2 þ �U

2 þ �V2

�I2

¼ 1� 4fofx

nx

nofo

�2

þ no

nxfx

�2

þ 2fofx

: (90)

In the limit of zero plasma magnetization in the antenna

region (e.g., when the antenna is in free space far from the

source), we have T2o ¼ 1 (circularly polarized EM modes),

and hence the above results yield �Q ¼ �U ¼ 0, as expected

for superposition of circularly polarized waves with random

initial phases. Moreover, in this limit, no ¼ nx (same disper-

sion for both modes at the antenna), and the degree of polar-

ization of received radiation becomes

�Q2 þ �U

2 þ �V2

�I2

!no¼nx

¼ fo � fxfo þ fx

� �2

: (91)

Thus, in the special case of unmagnetized plasma at the

antenna’s location and equal fluxes of o and x quanta, the

received radiation received is unpolarized, as expected.

V. SUMMARY AND CONCLUSIONS

We have developed a kinetic theory for propagation of

electromagnetic radiation through multiscale fluctuating

plasmas, based on kinetic equations for the distribution func-

tions of radiation quanta, in which large scale effects such

as dispersion and refraction are included via the dispersion

relations of wave modes, including growth and damping.

The small scale effects of scattering and LMC of radiation

quanta due to plasma density fluctuations are included via

collision integrals. The kinetic equation is then recast in the

form of a Fokker-Planck equation, with scattering and linear

mode conversion at plasma density fluctuations giving rise to

damping, drift, and diffusion terms.

Subject to appropriate initial and boundary conditions,

the equations obtained describe the evolution of the distribu-

tion functions of radiation propagating in a large scale

plasma with small scale fluctations. The observable intensity

and polarization properties of the radiation are described by

the Stokes parameters, which can be written in terms of the

distributions of o and x quanta.

The numerical implementation of the approach devel-

oped here is given in Part II,28 which describes and verifies

an algorithm to solve the kinetic equation.

ACKNOWLEDGMENTS

The authors thank D. B. Melrose for useful discussions.

The Australian Research Council supported this work.

APPENDIX A: PLASMA REFRACTIVE INDEX

The large scale evolution (due to dispersion and refrac-

tion) of the electromagnetic radiation quanta distributions is

governed by the refractive index of the plasma through

which the quanta propagate. A cold collisionless plasma per-

meated by a magnetic field B supports o and x mode electro-

magnetic waves. In the coordinate system in Fig. 5, with the

local magnetic field B along the z axis, and k at angle hkz to

B in the x-z plane, the refractive indexes of the o and xmodes are

n2o;x ¼ ðB 6 FÞ=2A; (A1)

F ¼ ðB2 � 4ACÞ1=2; (A2)



where the þ and � signs in (A1) correspond to the o and xmodes, respectively,39

A ¼ S sin2 hkz þ P cos2 hkz; (A3)

¼ 1� X � Y2 þ XY2 cos2 hkz

1� Y2; (A4)

B ¼ ðS2 � D2Þ sin2 hkz þ PSð1þ cos2 hkzÞ; (A5)

¼ 2ð1� XÞ2 � 2Y2 þ XY2ð1þ cos2 hkzÞ1� Y2

; (A6)

C ¼ PðS2 � D2Þ; (A7)

¼ ð1� XÞ½ð1� XÞ2 � Y2�1� Y2

; (A8)

S ¼ 1� X � Y2

1� Y2; (A9)

D ¼ � XY

1� Y2; (A10)

P ¼ 1� X; (A11)

X ¼ x2p=x

2 and Y ¼ xce=x, where xce ¼ eB=me is the elec-

tron cyclotron frequency.

APPENDIX B: REFRACTIVE SCATTERING OFFSPHERICAL DENSITY CLUMPS IN THE ISOTROPICAPPROXIMATION

We consider the scattering of quanta with momenta k off

a spherically symmetric density clump within the volume d3r.

It is convenient to consider the scattering in the c1; c1; k coor-

dinate frame with the center coinciding with the center of the

density clump in Fig. 7. We call this the local scattering coor-

dinate frame exs; eys

; ezs. This frame is fixed when we consider

the scattering of quanta located at a given (large-scale) r for a

given (large-scale) k. We represent all quantities associated

with the quanta in the scattering problem by the subscript s, to

avoid confusion with the corresponding quantities in the

large-scale problem of evolution of the distribution function

governed by the LHS of the kinetic equation. The position of

a quantum in the scattering problem is denoted by rs, which

has rectangular coordinates ðxs; ys; zsÞ and spherical coordi-

nates ðrs; hs;/sÞ in the local scattering coordinate frame

shown in Fig. 7. The momentum of a quantum is denoted by

ks with coordinates ðkxs; kys

; kzsÞ or ðks; hks

;/ksÞ, respectively.

The initial vector ks before the scattering is equal to k with

spherical coordinates (k, 0, 0), and the final ks after the scatter-

ing is equal to k0 with spherical coordinates ðk; h0ks;/0ksÞ,

where h0ks¼ vkk0 , and /0ks

corresponds to /0 in Fig. 3.

The equations of motion of quanta are

_rs ¼@xM

@ks; (B1)

_ks ¼ �@xM

@rs; (B2)

with

_rs ¼ _xsexsþ _yseys

þ _zsezs; (B3)

_ks ¼ _kxsexsþ _kys

eysþ _kzs

ezs; (B4)

@xM

@ks¼ @xM

@kseksþ 1

ks

@xM

@hks

ehksþ 1

ks sin hks

@xM

@/ks

e/ks; (B5)

� @xM

@rs¼ � @xM

@rsers� 1

rs

@xM

@hsehs� 1

rs sin hs

@xM

@/s

e/s: (B6)

The transformations between the bases feks; ehks

; e/ksg;

fers; ehs

; e/sg, and fexs

; eys; ezsg are given by

eks

ehks

e/ks

0B@

1CA¼

sinhkscos/ks

sinhkssin/ks

coshks

coshkscos/ks

coshkssin/ks

�sinhks

�sin/kscos/ks

0

0B@

1CA

�exs

eys

ezs

0B@

1CA; (B7)

ers

ehs

e/s

0B@

1CA¼

sinhs cos/s sinhs sin/s coshs

coshs cos/s coshs sin/s � sinhs

� sin/s cos/s 0

0B@

1CA

�exs

eys

ezs

0B@

1CA: (B8)

FIG. 7. Refractive scattering of quanta with initial wave vector k and impact

factor q off spherically symmetric plasma density clumps in the isotropic

scattering approximation (i.e., with magnetization effects neglected in the

local scattering problem, except via their effect on the wave cutoff frequen-

cies. Axes xs; ys, and zs of the scattering coordinate system are chosen to be

along c1; c2, and j, respectively. The scattering angle vkk0 ¼ jp� 2h0j is

defined to be the angle between initial k (before scattering) and final k0 (after

scattering). The scattering is azimuthally symmetric around k.



Using Eqs. (B7) and (B8) in Eqs. (B5) and (B6), respec-

tively, and substituting these and Eqs. (B3) and (B4) into

Eqs. (B1) and (B2), we obtain equations of motion

_xs ¼@xM

@kssin hks

cos /ksþ 1

ks

@xM

@hks

cos hkscos /ks

� 1

ks sin hks

@xM

@/ks

sin /ks; (B9)

_ys ¼@xM

@kssin hks

sin /ksþ 1

ks

@xM

@hks

cos hkssin /ks

þ 1

ks sin hks

@xM

@/ks

cos /ks; (B10)

_zs ¼@xM

@kscos hks �

1

ks

@xM

@hks

sin hks ; (B11)

_kxs¼ � @xM

@rssin hs cos /s �

1

rs

@xM

@hscos hs cos /s

þ 1

rs sin hs

@xM

@/s

sin /s; (B12)

_kys¼ � @xM

@rssin hs sin /s �

1

rs

@xM

@hscos hs sin /s

� 1

rs sin hs

@xM

@/s

cos /s; (B13)

_kzs¼ � @xM

@rscos hs þ

1

rs

@xM

@hssin hs: (B14)

In the isotropic approximation to the plasma density

fluctuations, we have @xM=@hks¼ @xM=@/ks

¼ 0, and azi-

muthal and spherical symmetry of the problem due to iso-

tropy of the medium and spherical symmetry of the density

clumps implies that @xM=@hs ¼ @xM=@/s ¼ 0. Thus we

obtain the following isotropic approximation to scattering of

quanta off spherically symmetric clumps:

_xs ¼@xM

@kssin hks

cos /ks; (B15)

_ys ¼@xM

@kssin hks

sin /ks; (B16)

_zs ¼@xM

@kscos hks

; (B17)

_kxs¼ � @xM

@rssin hs cos /s; (B18)

_kys¼ � @xM

@rssin hs sin /s; (B19)

_kzs¼ � @xM

@rscos hs: (B20)

Using the following relations

xs ¼ rs sin hs cos /s; (B21)

ys ¼ rs sin hs sin /s; (B22)

zs ¼ rs cos hs; (B23)

kxs¼ ks sin hks

cos /ks; (B24)

kys¼ ks sin hks

sin /ks; (B25)

kzs¼ ks cos hks

: (B26)

Equations (B15)–(B20) with Eqs. (B21)-(B26) yield 6 differ-

ential equations of motion for rs; hs;/s; ks; hks; and /ks

.

We can define the angular momentum of a quantum

with respect to the center O of the density clump as

ms ¼ rs � ks: (B27)

Using the equations of motion for rs; hs;/s; ks; hks; and /ks

,

we can show that _ms ¼ 0 in the isotropic case of quanta scat-

tering off spherically symmetric density clumps, i.e., ms is an

integral of motion. Another integral of motion is the energy

xM of the quantum. The existence of these two integrals of

motion greatly simplifies the integration of the equations of

motion of the quanta in the scattering problem. The constant

ms implies that /s and /ksare constant along the trajectories

which are planar. This reduces the system to just 4 equations

of motion for rs; hs; ks; hks. Moreover, the two equations for ks

and hkscan be replaced with jmsj ¼ jms0j and xM ¼ xM0,

where ms0 and xM0 are the initial values of ms0 and xM0 of a

quantum before the scattering, i.e., far away from the density

clump, as shown in Fig. 7.

The initial angular momentum of a quantum far from

the clump is ks0qeys, where q is the impact parameter shown

in Fig. 7 and ks0 is the magnitude of ks before the scattering.

Since k ¼ ks; ks0 ¼ k. Hence, conservation of angular mo-

mentum gives

rsksðcos hs sin hks� sin hs cos hks

Þ ¼ kq: (B28)

Conservation of energy for a quantum, under the assumption

that the plasma magnetization is ignorable, gives

x2pðrsÞ þ c2k2

s ¼ x2p0 þ c2k2; (B29)

where xp0 is the plasma frequency of the background plasma

far away from the density clump, and k ¼ ks0 is again the

magnitude of ks before the scattering.

Using Eq. (B28), we obtain equations of motion for

rs; hs; ks; hks

_rs ¼@xM

@kscos ðhks

� hsÞ; (B30)

rs_hs ¼

@xM

@kssin ðhks

� hsÞ; (B31)

c2k2s ¼ c2k2 þ x2

p0 � x2pðrsÞ; (B32)

sin ðhks� hsÞ ¼

kqksrs

: (B33)

Noting that cos ðhks� hsÞ ¼ 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� sin2 ðhks

� hsÞq

and_hs ¼ dhs=dt ¼ ðdhs= _rsÞdrs, we find



dhs ¼ 6kqrs

k2 1� q2

r2s

� �þ

x2p0 � x2

pðrsÞc2

" #�1=2drs

rs: (B34)

Integration over rs then yields

h0 ¼ð1

rsmin

drs

rs

kq=rsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 1� q2

r2s

�þ x2

p0�x2

pðrsÞc2

r ; (B35)

where rs min is defined via

kqksrs min

¼ 1; (B36)

or, equivalently,

kqrs min

� �2

¼ k2 þx2

p0 � x2pðrs minÞ

c2: (B37)

To obtain the scattering angle vkk0 ¼ jp� 2h0j from Eq.

(B35), we need a model for xpðrsÞ of the spherically sym-

metric density inhomogeneity in the background plasma.

Let us assume that the plasma density profile due to a

clump is of the form NpðrsÞ ¼ Np0 þ ~NpðrsÞ, where Np0 is the

background plasma density, and ~NpðrsÞ is the contribution of

the density clump. Suppose that this contribution is of the

form ~NpðrsÞ ¼ ~Np0SðrsÞ, where ~Np0 is the amplitude of the

density perturbation (which can be negative) and SðrsÞ is

the profile of the spherically symmetric density perturbations.

Here we consider a particular example of inverse square

clump shape and calculate the corresponding refractive scat-

tering cross-sections of T-wave quanta off such clumps.

If we assume the clump density profile of the form~NpðrsÞ ¼ ~Np0SðrsÞ,with

SðrsÞ ¼ ðRc=rsÞ2; (B38)

where Rc is the characteristic clump radius, then

x2pðrsÞ ¼ x2

p0 6 ~x2p0ðRc=rsÞ2; (B39)

in the clump, where ~x2p0 ¼ e2j ~Np0j=�0m, and the 6 signs are

for clumps with positive and negative amplitudes, respec-

tively. Substituting (B39) into (B37), solving for rs and

choosing the non-negative root, we obtain

rsmin¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq26

~x2p0

k2c2R2

c

s: (B40)

Note that since the total plasma density must be non-

negative, NpðrsÞ ¼ Np0 þ ~Np0SðrsÞ � 0 restricts negative

amplitudes so that rsminis always real and non-negative.

With rsmindefined by Eq. (B40), the integration in Eq.

(B35) with x2pðrsÞ from Eq. (B39) yields

h0 ¼pq

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq26

~x2p0

c2k2 R2c

q ; (B41)

and the scattering angle vkk0 is then

vkk0 ¼ p 1� 16~x2

p0

c2k2

Rc

q

� �2" #�1=2��:

�� (B42)

The differential cross-section FMðk; vkk0 Þ in the isotropic

case considered here is [cf., Eq. (20)]

FMðk; vkk0 Þ ¼qðk; vkk0 Þ

sin vkk0

@qðk; vkk0 Þ@vkk0

��:�� (B43)

Since q � 0, we have

q

�� @q@vkk0

�� ¼��q @q@vkk0

�� ¼�� 12 @q2

@vkk0

�� ¼�� 1

2@vkk0=@q2

��: (B44)

Using this with vkk0 ¼ vkk0 ðq2Þ from Eq. (B42), we finally

obtain for FMðk; vkk0 Þ for SðrsÞ ¼ ðRc=rsÞ2

FMðk; vkk0 Þ ¼~x2

p0

c2k2pR2

c

�� psin vkk0

p7vkk0

v2kk0 ð2p7vkk0 Þ2

��: (B45)

APPENDIX C: RELATION BETWEEN DISTRIBUTIONOF CLUMP AMPLITUDES AND DISTRIBUTIONS OFPLASMA DENSITY FLUCTUATIONS

In this appendix, we derive the relation between the dis-

tribution Pcð ~Np0Þ of clump amplitudes and the distribution

Pð ~NpÞ of plasma density fluctuations in an elementary vol-

ume d3r. This relation is then used to define Pcð ~Np0Þ from

given parameters of the mean and variance of Pð ~NpÞ for the

specified shape SðrsÞ of density clumps.

We assume that the plasma density profile in the scatter-

ing problem is NpðrsÞ ¼ Np0 þ ~NpðrsÞ, where Np0 is the back-

ground plasma density and ~NpðrsÞ is associated with density

clumps or voids. We consider an ensemble of spherically sym-

metric plasma density clumps in a volume d3r, with ampli-

tudes ~Np0 and shape specified by the function SðrsÞ. In

general, the shapes and amplitudes of clumps in the ensemble

are different, but we simplify here by assuming that all the

clumps have the same shape SðrsÞ, with only their amplitudes~Np0 being different. This allows us to reduce the averaging

over the clump ensemble to averaging over ~Np0 only.

We define two characteristics of the clump ensemble in

d3r: the local number density NclðrÞ of clumps within d3r,

and the local distribution function Pcð ~Np0; rÞ of amplitudes~Np0. These are obtained from the assumed mean and var-

iance of local distribution of plasma density fluctuations

Pð ~Np; rÞ and from the assumed shape function SðrsÞ. To do

this, consider a volume d3r, filled with spherical clumps of

the same shape SðrsÞ and different amplitudes ~Np0. Let us

make the approximation that the clumps do not overlap, i.e.,

that their number density satisfies NclVb < 1, where Vb ¼4pR3

b=3 is the characteristic clump volume and Rb is the

characteristic clump radius, beyond which the associated

density perturbation is insignificant. This assumption means

that any point within d3r belongs to no more than one clump.

Let us choose a random point in d3r and calculate the



probability that the plasma density at that point is in the

range ½Np;Np þ dNp�, or, equivalently, that the deviation ~Np

of plasma density from the mean background value Np0 is

within the range ½ ~Np; ~Np þ d ~Np�. On one hand, this probabil-

ity is, by definition,

Pð ~NpÞd ~Np: (C1)

On the other hand, this is equal to the probability for the point

to be within one of Ncld3r spherical shells defined by the values

of ~Np and d ~Np, each shell centered at one of the clump centers,

and bounded by the surfaces at which ~NpðrsÞ ¼ ~Np (the inner

boundary of the shell) and ~NpðrsÞ ¼ ~Np þ d ~Np (the outer

boundary of the shell), where ~NpðrsÞ ¼ ~Np0SðrsÞ. The volume

of the shell defined above is

Vshell ¼ 4 pr2shellð ~Np; ~Np0Þ

��drshellð ~Np; ~Np0Þ��; (C2)

where rshellð ~Np; ~Np0Þ is the radius of the shell defined by

~NpðrshellÞ ¼ ~Np0SðrshellÞ ¼ ~Np: (C3)

From the second part of the previous equation we obtain the

thickness of the shell drshell

drshell ¼d ~Np

~Np0jdSðrsÞ=drsj

��rs¼rshellð ~N p; ~N p0Þ

: (C4)

The volume of the shell is thus

Vshell ¼ 4p r2shellð ~Np; ~Np0Þ

� d ~Np

~Np0jdSðrsÞ=drsj

��rs¼rshellð ~N p; ~N p0Þ

; (C5)

which depends on the amplitude ~Np0 of the clump.

Each shell is centered at a density clump, so the number

of such shells within the volume d3r is equal to the number

of density clumps in this volume, Ncld3r. The probability for

a randomly selected point within d3r to be within one of

these shells is equal to the ratio of the total volume of all

such shells within d3r, averaged over the ensemble of clump

amplitudes ~Np0,

Ncld3r

ðVshellð ~Np0ÞPcð ~Np0Þd ~Np0; (C6)

to the total volume d3r. Comparing with (C1), we have

Pð ~NpÞd ~Np ¼ 4pNcl

ðr2

shellð ~Np; ~Np0Þ

� Pcð ~Np0Þd ~Np0

~Np0jdSðrsÞ=drsjrs¼rshellð ~N p; ~N p0Þ

!d ~Np: (C7)

The limits of integration in Eq. (C7) are defined by

0 rshellð ~Np; ~Np0Þ Rb: (C8)

Equation (C7) defines the distribution Pð ~NpÞ of plasma

density fluctuations within d3r in terms of the number

density Ncl of clumps in d3r, the distribution Pcð ~Np0Þ of

clump amplitudes, and the clump shape function SðrsÞ. The

relation (C7) can also be used for the inverse problem, i.e.,

to define the number density Ncl and the parameters of

Pcð ~Np0Þ of an ensemble of clumps with a given shape SðrsÞ,in order for them to mimic the required distribution Pð ~NpÞ of

the actual plasma density fluctuations within d3r. Note that

(C7) does not allow one to uniquely deduce the form of

Pcð ~Np0Þ, which thus must be postulated. For Pcð ~Np0Þ postu-

lated to be the Gaussian (30), the problem thus reduces to

finding just the two parameters Ncl and r ~N p0, which can be

done uniquely using Eq. (C7) given a required distribution of

density fluctuations Pð ~NpÞ and a clump shape SðrsÞ.

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