Incoherent Scatter Spectral Theories—Part II: Modeling the Spectrum for Modes Propagating Perpendicular to ${\bf B}$

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 329

Incoherent Scatter Spectral Theories—Part II:Modeling the Spectrum for Modes Propagating

Perpendicular to BMarco A. Milla, Member, IEEE, and Erhan Kudeki, Member, IEEE

Abstract—Incoherent scatter (IS) spectral models for collisionaland magnetized F-region plasmas are developed based on a gen-eral framework described in part I of this paper and on thestatistics of simulated particle trajectories. In the simulations,a Langevin update equation is used to describe the motion ofcharge carriers undergoing Coulomb collisions. It is shown thatrandom displacements of oxygen ions in the F-region can be char-acterized as a Brownian-motion process with Gaussian distributeddisplacement vectors. Electron displacements, on the other hand,are non-Brownian, and their statistics exhibit a dependence onthe magnetic aspect angle. A numerical library of characteristicfunctions of electron displacements was constructed from thesimulation data obtained for a set of plasma parameters typical ofthe equatorial F-region. Spectral models for the IS radar signalsfrom F-region heights are constructed with one-sided Fouriertransforms of the characteristic functions of electron and iondisplacements. The models are valid at all magnetic aspect angles,including the zero aspect angle that corresponds to radar observa-tions perpendicular to the ambient geomagnetic field B.

Index Terms—Fokker–Planck collision model, incoherent scat-ter (IS), Langevin equation, remote sensing.

I. INTRODUCTION

THIS PAPER describes the development of an incoherentscatter (IS) spectral model for collisional and magnetized

F-region plasmas using a general approach described in acompanion paper by Kudeki and Milla [1]. The spectral modelis relevant for the 50-MHz IS radar (ISR) measurements ofthe equatorial F-region ionosphere conducted at the JicamarcaRadio Observatory near Lima, Peru.

At the Jicamarca Radio Observatory, high-resolution andhigh-accuracy F-region plasma drift measurements are con-ducted by using radar antenna beams pointed perpendicular toEarth’s magnetic field B. Such measurements, developed byKudeki et al. [2], are possible on a pulse-to-pulse basis—asopposed to autocorrelation function (ACF) measurements con-ducted with multipulse techniques—because the bandwidth ofthe ISR spectrum narrows considerably at small magnetic as-pect angles, i.e., when the radar beam is pointed perpendicular

Manuscript received March 30, 2009; revised November 26, 2009 andApril 20, 2010; accepted June 13, 2010. Date of publication October 7, 2010;date of current version December 27, 2010. This work was supported by theUniversity of Illinois, Urbana, under Grant ATM-0215246 from the NationalScience Foundation.

The authors are with the Department of Electrical and Computer Engineer-ing, University of Illinois at Urbana–Champaign, Urbana, IL 61801 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2010.2057253

to B. However, beam-weighted models of the ISR spectrumbased on the well-established collisionless IS theory (e.g., [3])do not fit the measured spectra accurately, as was first pointedout by Bhattacharyya [4]. The need for using a collisionalscattering theory in constructing the beam-weighted spectrawas first suggested by Sulzer and González [5] in a paper thatalso outlined how to properly include the effect of Coulomb col-lisions in the IS models. The collisional spectrum calculationsof Sulzer and González are based on the statistics of simulatedelectron trajectories randomized by Coulomb collisions. Inorder to simplify their simulations, Sulzer and González consid-ered only the strong geomagnetic field case when electron diffu-sion across the magnetic field lines can be neglected. This limitsthe validity of the Sulzer and González model at Jicamarca tomagnetic aspect angles larger than 0.1◦, whereas for observa-tions perpendicular to B, an extension of the model from 0.1◦

to 0◦ is needed. As shown in this paper, the IS spectrum variessubstantially over this aspect angle range, explaining why thecollisionless models developed in [2] did not fit the observa-tions well.

In this paper, we describe our extension of the essentially1-D collisional IS spectrum model of Sulzer and González[5] into three dimensions and to all magnetic aspect angles.This is accomplished, as described in Section II, by using a3-D Langevin equation to simulate the trajectories of F-regionplasma particles and then by formulating the IS spectra for theregion in terms of the statistics of charge carrier displacementsin accordance with the general spectral framework discussed in[1]. In the 3-D Langevin equation, the effect of Coulomb colli-sions on charged particles is modeled by the action of a frictionforce and random diffusive forces. These forces are relatedto the friction and diffusion coefficients of the Fokker–Planckkinetic equation derived by Rosenbluth et al. [6]. As discussedin Section II, Langevin and Fokker–Planck equations providealternate descriptions of identical Markov processes, and asdescribed in Section III, the time-update form of the Langevinequation is ideally suited for producing simulated sample dataof such processes. The use of simulated particle trajectoriesin our IS spectral models will be discussed in Section IV.The procedure makes use of the characteristic functions ofthe particle displacement distributions (single-particle ACFs)and their one-sided Fourier transforms, known as Gordeyevintegrals, as described in [1].

In Section V, we present a detailed discussion of the statisticsof the simulated charge carrier trajectories that we produced.

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330 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011

We show that ion displacements in directions parallel andperpendicular to B can be well approximated by Gaussian prob-ability distribution functions (pdfs). This permits the character-ization of collisional ion dynamics as a simplified Brownian-motion process with constant friction and diffusion coefficientsand leads to a simple analytical expression for the ion Gordeyevintegral (e.g., [7] and [8]). We also show that electron displace-ments parallel to B have non-Gaussian pdfs. Hence, electronCoulomb collisions cannot be modeled as a Brownian-motionprocess, and due to the complex form of the friction anddiffusion coefficients of the Fokker–Planck collision model,no closed-form analytical expression could be identified forthe electron Gordeyev integral. Section V concludes with thedescription of a numerical library of single-electron ACFs foran oxygen plasma that had to be developed. The library spansa set of densities, temperatures, and magnetic fields needed forJicamarca F-region applications. In Section VI, we present anddiscuss the collisional F-region IS spectra derived with the toolsdeveloped and discussed in Sections III and IV. This paperconcludes in Section VII following a brief summary of ourresults.

II. IS SPECTRUM, LANGEVIN EQUATION, AND

FOKKER–PLANCK COLLISION MODEL

The problem of modeling and calculating the IS spectrumwas first addressed by Dougherty and Farley [9], Fejer [10],Salpeter [11], Hagfors [12], and others. Different methods ofderivation were followed, but identical results were obtainedin equivalent physical settings. The fundamental principles thatlink these different approaches constitute a general frameworkfor IS spectral theories. A detailed description of this frame-work is presented in our companion paper [1]. Aspects of theframework relevant to our work are discussed next.

A. General Framework of IS Spectral Models

In a plasma, electron and ion currents due to random thermalmotions produce the impressed electron and ion density fluctu-ations nte and nti, respectively, which can be regarded as sta-tistically independent random processes having the frequencyspectra ⟨

|nte,i(k, ω)|2⟩

= Ne,i

∫dτe−jωτ 〈ejk·Δre,i〉 (1)

expressed in terms of ambient densities Ne,i and random parti-cle displacements Δre,i, as described in [1]. Electrostatic fieldsgenerated by the imbalance between nte and nti, in turn, drivea conduction current so that the total current, including thedisplacement current, vanishes in the electrostatic approxima-tion. In the k–ω space, the spectrum of total electron densityfluctuations ne then exhibits a dependence on the spectra of nte

and nti, which is given by (e.g., [1] and [13])

⟨|ne(k, ω)|2

⟩=

|jωεo + σi(k, ω)|2⟨|nte(k, ω)|2

⟩|jωεo + σe(k, ω) + σi(k, ω)|2

+|σe(k, ω)|2 〈|nti(k, ω)|2〉

|jωεo + σe(k, ω) + σi(k, ω)|2(2)

where σe(k, ω) and σi(k, ω) are the electron and ion conduc-tivities and εo is the permittivity of free space. This expression,valid for stationary single-ion plasmas, is derived in [1] withoutmaking any assumption regarding the statistical distributions ofelectrons and ions.

Expressions for the conductivity σs(k, ω) and the spectrumof thermal fluctuations 〈|nts(k, ω)|2〉 of each plasma speciescan be independently derived from plasma kinetic equations.However, if each species is in thermal equilibrium—i.e., if itsvelocity distribution is Maxwellian—then it can be verifiedthat σs(k, ω) and 〈|nts(k, ω)|2〉 are related by the fluctuation-dissipation or generalized Nyquist theorem (e.g., [14] and[15]) as

ω2q2s

k2

⟨|nts(k, ω)|2

⟩= 2KTsRe {σs(k, ω)} (3)

where qs and Ts are the charge and temperature of particlesof type s, respectively, and K is the Boltzmann constant.Kudeki and Milla [1] make use of this relation as follows. First,the spectrum of thermal fluctuations is derived from a micro-scopic model of particle dynamics, and then, the fluctuation-dissipation theorem is used to determine the real part of theconductivity of the corresponding particle species. Next, theimaginary part of σs(k, ω) is obtained by applying Kramers–Kronig relations. As a result, it is found that⟨

|nts(k, ω)|2⟩

Ns= 2Re {Js(ω)} (4)

σs(ω,k)jωεo

=1 − jωJs(ω)

k2h2s

(5)

where Ns is the mean species density

hs =

√εoKTs

Nsq2s

(6)

is the corresponding Debye length, and

Js(ω) ≡∞∫

0

dτe−jωτ 〈ejk·Δrs〉 (7)

is a Gordeyev integral, a one-sided Fourier transform of thecharacteristic function 〈ejk·Δrs〉 of random particle displace-ments Δrs = rs(t + τ) − rs(t) occurring over intervals τ ’s inthe absence of collective interactions (see [1]). This character-istic function is also termed as the single-particle ACF, since itcan be interpreted as the normalized correlation of the signalscattered by a single particle exposed to a radar pulse. In thisway, the problem of determining the IS spectrum is reduced tothe estimation of single-particle ACFs and the evaluation of thecorresponding Gordeyev integrals.

B. Single-Particle ACF and Langevin Equation

The single-particle ACF 〈ejk·Δr〉 for a species in a givenplasma can be calculated directly from

〈ejk·Δr〉 =∫

d(Δr)ejk·Δrf(Δr, τ) (8)

MILLA AND KUDEKI: INCOHERENT SCATTER SPECTRAL THEORIES—PART II 331

where f(Δr, τ) is the pdf of the particle displacement Δrat a time delay τ . This is a straightforward calculation aslong as f(Δr, τ) is known either analytically or numerically.The standard procedure for determining f(Δr, τ) starts withsolving the Boltzmann kinetic equation for f(r, t) with a propercollision operator, e.g., the Fokker–Planck kinetic equation ofRosenbluth et al. [6] for Coulomb collisions, subject to aninitial condition (e.g., [8])

f(r, 0) = δ(r). (9)

Although analytical solutions of simplified versions of theFokker–Planck kinetic equation are available (e.g., [16] and[17]), determining f(Δr, τ) would be a daunting task when thefull equation is considered.

An alternative approach to calculating 〈ejk·Δr〉 in the caseof Coulomb collisions involves producing, as the result ofsome simulation procedure, suitable sets of particle displace-ment data Δr for the same stochastic process described bythe Fokker–Planck kinetic equation. Assuming the process tobe ergodic, a sufficiently large set of simulated samples ofparticle trajectories r(t) can be used to compute 〈ejk·Δr〉 andany other statistical function of displacements Δr over timedelays τ ’s. Random trajectory simulations, of course, requirethe availability of a stochastic equation describing how theparticle velocities

v(t) ≡ drdt

(10)

may evolve under the influence of Coulomb collisions.Restricting v(t) to be a Markovian random process—

meaning that past values of v would be of no help in predictingits future values if the present value is available—constrainsthe stochastic evolution equation of v(t) by the very strictself-consistency conditions discussed by Gillespie [18], [19] toacquire the form of a Langevin equation

dv(t)dt

= A(v, t) + C(v, t)W(t), (11)

where vector A(v, t) and matrix C(v, t) consist of arbitrarysmooth functions of arguments v and t. W(t) is a randomvector having statistically independent Gaussian white noisecomponents. A more natural way of expressing the Langevinequation is to cast it in an update form, namely

v(t + Δt) = v(t) + A(v, t)Δt + C(v, t)Δt1/2U(t), (12)

where Δt is an infinitesimal update interval and U(t) is avector composed of independent zero-mean Gaussian randomvariables with unity variance, i.e.,

Ui(t) = N (0, 1), (13)

where N (μ, σ2) denotes the normal random variable with meanμ and variance σ2. The formal equivalence of (11) and (12)requires that the ith component of W(t) be defined as

Wi(t) = limΔt→0

(Δt)−1/2N (0, 1) = limΔt→0

N (0, 1/Δt) (14)

compatible with the requirement that

〈Wi(t + τ)Wi(t)〉 = δ(τ). (15)

Note that the Langevin equation (11) describing a Markov-ian process has the form of Newton’s second law of motion,with the terms on the right representing forces per unit massexerted on simulated particles. Considering the Lorentz forceon a charged particle in a magnetized plasma with a constantmagnetic field B, and not violating the strict format of (11),we can modify the equation by adding a term qv(t) × B/mto its right-hand side. We would then have an additional termqv(t) × BΔt/m on the right-hand side of the update equation(12) as well.

Another relevant fact is that a special type of Markov processcharacterized by a linear A(v, t) = −βv and a constant matrixC = D1/2I, independent of v and t, is known as Brownian-motion process (e.g., [16] and [20]), which is often invokedin simplified models of collisional plasmas (e.g., [7], [8],and [17]). In these models, friction and diffusion coefficients,namely, β and D, respectively, are constrained to be related by

D =2KT

mβ (16)

for a plasma in thermal equilibrium.

C. Link Between Fokker–Planck and Langevin Equations

In return for having restricted v(t) to the space of Markovianprocesses, we have gained a stochastic evolution equation (11)with a plausible Newtonian interpretation and with the potentialof taking us beyond Brownian-motion-based collision models.

Letting v(t) to be Markovian turns out to have a second con-sequence of importance in this paper, namely, the evolution ofprobability density f(v, t) of a random variable v(t) is knownto be governed, when v(t) is Markovian, by the Fokker–Planckkinetic equation having a “friction vector” and a “diffusiontensor,” respectively⟨

ΔvΔt

⟩c

=A(v, t) (17)

⟨ΔvΔv�

Δt

⟩c

= C(v, t)C�(v, t) (18)

specified in terms of the input functions of the Langevin equa-tion. This intimate link between the Langevin equation andthe Fokker–Planck kinetic equation—in describing Markov-ian processes from two different but mutually compatibleperspectives—was first pointed out by Chandrasekhar [16] and,more recently, by Gillespie [19]. Since the Fokker–Planckfriction vector and diffusion tensor for equilibrium plasmaswith Coulomb interactions have already been worked out byRosenbluth et al. [6] as⟨

ΔvΔt

⟩c

= − β(v)v (19)

⟨ΔvΔv�

Δt

⟩c

=D⊥(v)

2I +

(D‖(v) − D⊥(v)

2

)vv�

v2(20)


in terms of scalar functions β(v), D‖(v), and D⊥(v) to be spec-ified hereinafter, it follows that the Langevin update equation tobe used in Coulomb collision particle trajectory simulations canbe written as

v(t + Δt) = v(t) +q

mv(t) × BΔt − β(v)Δtv(t)

+√

D‖(v)ΔtU1v‖ +

√D⊥(v)

Δt

2(U2v⊥1 + U3v⊥2) (21)

including a dc magnetic field term, with v‖(t), v⊥1(t), andv⊥2(t) denoting an orthogonal set of unit vectors parallel andperpendicular to the particle trajectory, and with Ui denotingzero-mean Gaussian random variables with unit variance.

D. Friction and Diffusion Coefficients

Expressions for the Fokker–Planck friction vector and diffu-sion tensor for a particle moving in a magnetized plasma havebeen derived by different authors (e.g., [21] and [22]). However,when the plasma is “weakly” magnetized, i.e., if the particleDebye length is smaller than the mean gyroradius, magneticfield effects in the friction vector and diffusion tensor can be ne-glected [21], and the friction coefficient β(v) and velocity spacediffusion coefficients D‖(v) and D⊥(v) introduced before canbe specified in (21), as formulated in the standard collisionmodel of Rosenbluth et al. [6]. For a Maxwellian plasma, thesecoefficients take the forms given by Spitzer [23]. Specifically,if a particle of type s colliding against a background of parti-cles of type s′ is considered, the Spitzer coefficients take theforms [24]

β(v) =∑s′

(1 +

ms

ms′

)Ns′Γs/s′

1C2

s′

ψ(zs′)v

(22)

D‖(v) =∑s′

2Ns′Γs/s′ψ(zs′)

v(23)

D⊥(v) =∑s′

2Ns′Γs/s′φ(zs′) − ψ(zs′)

v(24)

where Cs =√

KTs/ms is the thermal speed of particles withmass ms. Additionally

Γs/s′ ≡ q2sq2

s′

4πε2om2s

ln Λs/s′ (25)

in terms of a “Coulomb logarithm” ln Λs/s′ (see hereinafter),while

zs ≡ v√2Cs

(26)

in terms of particle speed v, and

ψ(z) ≡ φ(z) − zφ′(z)2z2

, (27)

where

φ(z) ≡ 2√π

z∫0

e−x2dx (28)

TABLE ITYPICAL PLASMA PARAMETERS FOR AN EQUATORIAL

F-REGION IONOSPHERE

is the well-known error function. Finally, defining a plasmaDebye length

hD ≡(∑

s

1h2

s

)−1/2

=

(∑s

Nsq2s

εoKTs

)−1/2

(29)

and a minimum impact parameter

bmin ≡ qsqs′

12πεomss′C2ss′

, (30)

where mss′ ≡ msmx′/(ms + ms′) is a reduced mass andC2

ss′ ≡ C2s + C2

s′ , we have

Λs/s′ ≡ hD

bmin(31)

which is known as the plasma parameter and is proportional tothe number of particles of type s inside a Debye cube.

E. Summary

The general framework of the IS theory formulates theelectron density spectrum in terms of electron and ion Gordeyevintegrals, functions that depend on the statistics of individualparticle trajectories. Since collisions in the plasma modifythese trajectories, they have an impact on the statistics of theparticle displacements and also on the corresponding Gordeyevintegrals. It is through the estimation of these functions thatthe effect of Coulomb collisions is considered in our IS spec-trum model. Although the statistics of particle trajectories,i.e., velocity and displacement density distributions, can beobtained by solving numerically the Boltzmann equation withthe Fokker–Planck collision operator, we chose to utilize adifferent approach based on the simulation of plasma parti-cle trajectories using Langevin update equations. In the nextsection, we describe our computer simulations that provide uswith sample electron and ion trajectories in magnetized andcollisional F-region plasmas.

III. COMPUTER SIMULATIONS OF

PARTICLE TRAJECTORIES

Consider the motion of a particle governed by the Langevinupdate equation (21). Denoting by v[n] the particle velocityvector at a discrete time nΔt, the updated velocity after asufficiently small time interval Δt can be determined by using

v[n + 1] = v[n] +q

mv[n] × BΔt − β(v)Δtv[n]

+√

D‖(v)ΔtU1[n]v‖ +

√D⊥(v)

Δt

2(U2[n]v⊥1 + U3[n]v⊥2)

(32)


TABLE IIREPRESENTATIVE PHYSICAL PARAMETERS OF A PLASMA. DEFINITIONS AND SAMPLE VALUES ARE PROVIDED. THE VALUES WERE COMPUTED

FOR AN O+ PLASMA WITH PARAMETERS GIVEN IN TABLE I. NOTE THAT THE PLASMA AND GYROFREQUENCY FORMULAS ARE GIVEN

FOR ANGULAR FREQUENCIES, BUT THE SAMPLE VALUES ARE GIVEN IN HERTZ

where m and q denote the mass and the electric charge of thesimulated particle (which can be either an electron or an ion).The different terms in (32) on the right include the changes inthe velocity produced by the magnetic and collisional forcesdiscussed in Section II. In addition, the particle position r[n]can be calculated from the velocities v[n] using

r[n + 1] = r[n] +v[n + 1] + v[n]

2Δt, (33)

another update equation that can be obtained by approximatingthe integral of the velocity vector over a time interval Δt usingthe trapezoidal rule.

We have developed a computer program for the simulationof F-region particle trajectories governed by (32) and (33). Inthe simulations, a homogeneous plasma in the presence of auniform magnetic field B ≡ Boz is considered. The plasmahas electron density Ne and electron and ion temperatures Te

and Ti, respectively. The values of Bo, Ne, Te, and Ti thatare typical of the equatorial F-region ionosphere are listed inTable I. For completeness, a list of some representative plasmaparameters is given in Table II.

The F-region simulation runs as follows. First, we randomlypick some initial velocity v[0] and set the starting positionof the particle to the origin (i.e., r[0] = 0). Next, the valuesof friction and diffusion coefficients for the given particlespeed are calculated. The velocity increments resulting fromthe action of each of the simulated forces (magnetic, friction,and diffusion) in a time interval Δt are then computed. Forthe calculation of the diffusive forces, we have to randomlypick three normal numbers. These values are computed usingthe Ziggurat method for random generation of Gaussianvariables [25]. Once all the velocity increments are calculated,we proceed to compute the new value of the velocity vectorusing (32), and subsequently, the position of the particle isupdated using (33). The algorithm is then repeated in a loop.The resulting series of velocities and positions are stored insequences of length M (typically equal to 217 samples). About104 of these sequences are generated, which provides more than109 simulated velocities and positions for a single particle in agiven plasma configuration. Note that the starting velocity andposition of each new sequence are set equal to the last velocityand position of the previous sequence. Only the initial velocityof the first sequence is randomly chosen, and the remaining

samples of the series are calculated using the update velocityequation (32).

A sample trajectory of an electron moving in an O+ plasmawith density Ne = 1012 m−3 and temperatures Te = Ti =1000 K is shown in Fig. 1. The intensity of the magnetic fieldwas taken as 25 μT. On the top, the left panel shows the particletrajectory in 3-D space, while the panel on the right is the pro-jection of the trajectory on the xy plane (i.e., the plane perpen-dicular to B). On the bottom, the displacement of the electronin the direction parallel to B is displayed as a function of time.As shown, the electron moves approximately in a spiral paththat is randomized by the action of collisions. As the electronaccelerates and decelerates at random rates, not only does theguiding center of its trajectory randomly drift in space, but alsothe diameter of the particle orbits changes as a function of time.Because of this, there are time intervals in which the electronorbits have smaller, or larger, radii than the mean gyroradiusρe. For instance, the trajectory shown in Fig. 1 corresponds toa period when the orbit radius is smaller than ρe (which, in thisexample, is approximately 40 mm). We can also see that, in theplane perpendicular to B, the particle never returns to the sameposition after a gyroperiod. Additionally, note that there is alarge difference between the distances covered by the electronin the directions parallel and perpendicular to B. These charac-teristics of the simulated electron trajectories have implicationsfor the shape of the computed single-particle ACFs and theircorresponding Gordeyev integrals presented later in thispaper.

The simulation procedure just outlined was motivated bythe earlier work of Sulzer and González [5]. Both simulationstudies make use of Spitzer friction and diffusion coefficientsin order to include the effects of Coulomb collisions in ISspectral models. However, the equations of motion employedby Sulzer and González [5] differed from our Langevin-based3-D update procedure. Sulzer and González considered theeffect of Coulomb collisions on particle displacements only inthe direction of the ambient field B, neglecting the diffusionand random walk effects across the field lines. That limits theapplicability of their results to magnetic aspect angles largerthan about 0.1◦, in relation to the 50-MHz Jicamarca radar ob-servations. Our simulation results, by contrast, enable ACF andGordeyev integral calculations for all aspect angles, includingthe direction of exact perpendicularity to the geomagnetic fieldB. Furthermore, our Langevin-based update procedure does


Fig. 1. Sample trajectory of an electron moving in an O+ plasma with density Ne = 1012 m−3 and temperatures Te = Ti = 1000 K. The presence of anuniform magnetic field B parallel to the z-axis was considered in the simulation. The intensity of the field was assumed to be 25 μT. On the top, the left panelshows the particle trajectory in 3-D space, while the panel on the right is the projection of the trajectory on the xy plane (i.e., the plane perpendicular to B). Onthe bottom, the displacement of the electron in the direction parallel to B is displayed as a function of time. The trajectory was sampled every Δt = 0.1 μs. Atotal of 4 × 103 samples are displayed corresponding to a time interval of 400 μs. In all three panels, the red dots show the starting location of the particle (theorigin). In addition, the red curve depicts the first 1.4 μs of the simulated trajectory (about one gyroperiod).

not suffer from numerical instability issues when used withsufficiently small update intervals Δt.

IV. ESTIMATION OF THE SINGLE-PARTICLE

ACFS AND GORDEYEV INTEGRALS

In this section, we will describe our processing proceduresof the particle trajectory data for estimating the single-particleACFs and the corresponding Gordeyev integrals.

Consider a long sequence of N simulated particle positionsuniformly sampled in time. For a given radar Bragg vector k,the unbiased estimator for the single-particle ACF at a discretetime delay mΔt is given by⟨

ejk·Δr[m]⟩

=1

N − mR[m], 0 ≤ m ≤ N − 1 (34)

where

R[m] ≡N−m−1∑

n=0

exp (jk · (r[n + m] − r[n])) (35)

is the discrete correlation of a sequence of samples ejk·r[n]. Ingeneral, discrete correlations can be computed efficiently usingthe fast Fourier transform (FFT) technique. In our case, how-ever, we cannot directly apply this procedure because the entiresequence of N particle positions is very large and it cannot befully allocated in the RAM of a computer (for instance, 109

sample positions stored in double-float format would requireabout 24 GB of RAM). Since we are just interested in the firstM samples of this correlation (about 105 samples), let us divide

the full set of positions into L segments of length M , such thatN = LM . We can then reexpress R[m] as

R[m] =L−1∑l=0

2M−1∑n=0

h∗l [(n−m)2M ] gl[n], 0 ≤ m ≤ M−1

(36)

where (a)b denotes the modulo operation (a mod b), and func-tions gl and hl are, respectively

gl[n] ≡{

exp (jk · r[n + lM ]) , 0 ≤ n ≤ M − 10, M ≤ n ≤ 2M − 1

(37)

hl[n] ≡{

gl[n], 0 ≤ n ≤ M − 1gl−1[n − M ], M ≤ n ≤ 2M − 1 (38)

(note that h0 = g0). Using (36), we can compute the correlationR[m] in an iterative way, reducing significantly the amount ofrequired computer memory. The inner summation in (36) is thecircular cross correlation of functions hl and gl, the calculationof which is performed using FFTs.

Estimates of the single-particle ACF 〈ejk·Δr〉 are then ob-tained by dividing R[m] by LM − m. This means that thestatistical variance of our estimates increases with m, since,for larger m, fewer samples are used to estimate the ACF.Assuming that, at every M samples, the simulated positions areindependent random variables, we find that, in the worst casescenario (i.e., when m = M − 1), the variance of our estimateswill be at most inversely proportional to L − 1. Thus, in orderto secure small estimation errors, large values of L have to beconsidered. In our calculations, we used L = 104.


The procedure outlined earlier is used to compute 〈ejk·Δr〉for different values of Bragg vector k. In our calculations, wedefine

k =2π

λB(cos(α)x + sin(α)z) (39)

where λB is the Bragg wavelength (note that λB = λo/2,where λo is the radar wavelength, e.g., λo = 6 m in the caseof Jicamarca, but λB = 3 m). In the previous, the magneticaspect angle α is the complement of the angle between k andthe magnetic field vector B = Boz. Notice that, to estimate〈ejk·Δr〉 for different aspect angles, we do not need to generatea new set of particle positions. We took advantage of this andperformed many of these calculations in parallel on a computer.Our definition of vector k is somewhat arbitrary. Notice that, forthe direction perpendicular to B, we could have chosen k to bey2π/λB instead of x2π/λB . Therefore, for α = 0◦, there aretwo alternative ways of computing the single-particle ACF thatprovide two sets of almost statistically independent estimatesof 〈ejk·Δr〉. Averaging these two sets, we could have reducedthe statistical errors of our estimates. In the future, we will takeadvantage of this in our calculations.

Our next task is the computation of the Gordeyev integralJ(ω) from discrete estimates of 〈ejk·Δr〉. Since our samplesare uniformly distributed in time, we could have simply takenthe FFT of the ACF estimates to perform the Gordeyev inte-gral calculations, but then samples of J(ω) would have beenrestricted to a discrete set of frequencies. We instead used achirp Z-transform algorithm described by Li et al. [26] in ourGordeyev integral computations, which allows the evaluation ofthe transform over any desirable range of frequencies ω’s.

In the previous discussion, we made no distinction betweenelectrons and ions since identical procedures are used to com-pute the Gordeyev integrals for each species. The estimatedelectron and ion Gordeyev integrals are then used to computethe theoretical IS spectra. As a result of our simulation studies,we found out that the ion Gordeyev integrals can be well repre-sented by analytical means. However, closed-form expressionscould not be found for the electron Gordeyev integrals, andtherefore, a numerical electron ACF library had to be con-structed for a wide range of plasma parameters and magneticaspect angles, as needed by Jicamarca applications. The plasmaparameters that this library spans are given in Table III. Theconstruction of the library was computationally demandingand was carried out using a cluster of computers. The TuringCluster, maintained by the Computational Science and Engi-neering Program at the University of Illinois, was used for thispurpose. The Turing system consists of 768 Apple Xserves,each with two 2-GHz G5 processors and 4 GB of RAM. Inthe cluster, hundreds of simulations run simultaneously, whichallowed us to build the library in less than three weeks. Thesame task would have taken many months (probably more thantwo years) using a single desktop computer.

V. STATISTICS OF ION AND ELECTRON TRAJECTORIES

If a plasma is in thermal equilibrium, i.e., if Te = Ti, onecan show analytically that the steady-state solution of the

TABLE IIIPLASMA PARAMETERS SPANNED BY THE LIBRARY

OF SINGLE-ELECTRON ACFs

Fokker–Planck equation is given by the Maxwell–Boltzmannvelocity distribution [27]

f(v) =1

(2πC2)3/2e−

12C2 (v2

x+v2y+v2

z) (40)

where vx, vy , and vz are the components of the particle ve-locity vector, and C =

√KT/m is the corresponding thermal

speed. Notice that this pdf can be written as the product ofthree independent Gaussians, one for each of the velocitycomponents. Since the Fokker–Planck and Langevin equationsare alternative representations of the same Markov process,we expect the distributions of our simulated velocities to beGaussian. Velocity distributions resulting from independent ionand electron simulations in an O+ plasma are shown in Fig. 2.In the simulations, the plasma was considered to be in thermalequilibrium with temperatures Te = Ti = 1000 K (the rest ofthe simulation plasma parameters are given in Table I). Thedistributions were built using more than 109 sampled velocities.In each panel, the expected Gaussian pdf is plotted on top of oursimulation results. As we can see, the computed distributionsmatch exactly the Gaussian curves in these two examples. Werepeated the same test for different plasma parameters andfound that the agreement was excellent in all cases. Becauseof these results, we have the confidence that our simulationprocedure is working properly.

As we mentioned in Section II, the single-particle ACF is acharacteristic function, and therefore, its shape is determinedby the time evolution of the pdf f(Δr, τ) of particle displace-ments Δr. If the plasma is magnetized, particles are forced todiffuse slowly in the plane perpendicular to B, making the pdff(Δr, τ) narrower in that plane than in the parallel direction(for a given τ ). In order to analyze the behavior of f(Δr, τ), wehave computed from the simulated trajectories the distributionsof ion and electron displacements in the directions parallel andperpendicular to B. Since B = Boz, the distributions of theparallel displacement correspond to f(Δz, τ). On the otherhand, the distributions of the perpendicular displacement werecomputed by averaging f(Δx, τ) and f(Δy, τ). In addition, thevariance and covariance of the components of the displacementvector Δr were computed from the simulated trajectories.These mean-square quantities are defined as

〈ΔriΔrj〉 = 〈(ri(t + τ) − ri(t)) (rj(t + τ) − rj(t))〉 (41)

where i and j denote the Cartesian coordinates. These quan-tities were estimated following a procedure similar to the oneused to compute 〈ejk·Δr〉.


Fig. 2. Probability distributions of (a) ion and (b) electron velocities resulting from the simulations. In each plot, the pdfs of the velocity components (displayedin colors) are compared to a Gaussian distribution (black line). Notice that the velocity axes are normalized by the thermal speeds of the corresponding particlespecies. The plasma parameters considered in the simulations are given in Table I.

Fig. 3. Probability distributions of the displacements of a simulated ion in the directions (top panels) perpendicular and (bottom panels) parallel to the magneticfield. On the left, the displacement pdfs are displayed as functions of time delay τ . On the right, sample cuts of the pdfs are compared to a Gaussian distribution.Note that all distributions at all time delays are normalized to unit variance. The displacement axis of each distribution at every delay τ is scaled with thecorresponding standard deviation of the simulated displacements.

Before presenting our statistics of the simulated ion andelectron displacements, notice that expression (8) for the ACF〈ejk·Δr〉 is simply the Fourier transform of f(Δr, τ). Inprinciple, we could have computed 〈ejk·Δr〉 using the Fouriertransforms of the displacement distributions obtained in thesimulations. This would have required, however, a significantincrease in the number of simulated particle positions in orderto reduce the statistical estimation errors to an acceptably small

level. Since we are interested in evaluating 〈ejk·Δr〉 only forsome discrete values of k, we consider the procedure outlined inSection IV to be more accurate, involving fewer computations.

A. Statistics of Ion Displacements

In Fig. 3, we show the distributions of ion displacementsin the directions perpendicular and parallel to the magnetic


Fig. 4. Variances of the displacements of a simulated ion in the directions (top panels) perpendicular and (bottom panels) parallel to the magnetic field. Thesimulation results (color lines) are displayed for two time intervals: 5 ms (left panels) and 500 ms (right panels). The dashed lines correspond to the fits of theresults to the theoretical expressions for 〈Δr2

⊥〉 and 〈Δr2‖〉 of the Brownian-motion model.

field. Note that, at every delay τ , the distributions have beennormalized to unit variance by scaling the displacement axisof each distribution with the corresponding standard deviationof the simulated displacements.1 On the left panels, the dis-tributions are displayed as functions of τ , while on the rightpanels, sample cuts of these distributions are compared to aGaussian pdf. For the time interval considered in these plots,we can see that the shapes of the distributions do not changewith time delay and also that they match almost perfectlyto the Gaussian curves. We have observed that, for τ up toand exceeding 10 ms, the distribution of the displacement inthe direction perpendicular to B remains Gaussian in shape.We also found that, at time delays on the order of hundredsof milliseconds, the parallel distribution eventually becomesspikier than a Gaussian. However, by that time, the single-ionACF is negligibly small—typical correlation times of the ionACF for λB = 3 m are on the order of 1 ms (see Fig. 5)—andfor that reason, ion displacements can be regarded as Gaussianrandom variables for all practical purposes. Additionally, weverified that the components of the vector displacement (i.e.,Δrx, Δry , and Δrz) are mutually uncorrelated. The simulationresults presented here were computed for the plasma configu-ration of Table I. The analysis was repeated for other possibleionospheric plasmas, and similar results were observed.

For the case of uncorrelated and jointly Gaussian Δr compo-nents, it is known that (e.g., [1]) the single-particle ACF

〈ejk·Δr〉 = e− 1

2 k2 sin2 α⟨Δr2

‖

⟩× e−

12 k2 cos2 α〈Δr2

⊥〉 (42)

where, assuming a Brownian-motion process with distinctfriction coefficients ν‖ and ν⊥ in the directions parallel

1Note that, in the limit τ → 0, we can write

limτ→0

Δri(τ)

σi(τ)= lim

τ→0

ri(t + τ) − ri(t)

Cτ= vi(t)/C.

Therefore, the distribution of Δri(τ)/σi(τ) at τ = 0 is equal to the distribu-tion of the corresponding velocity component.

and perpendicular to B, the mean-square displacements willvary as⟨

Δr2‖

⟩=

2C2

ν2‖

(ν‖τ − 1 + e−ν‖τ

)(43)

⟨Δr2

⊥⟩

=2C2

ν2⊥ + Ω2

(cos(2γ) + ν⊥τ − e−ν⊥τ cos(Ωτ − 2γ)

)(44)

in which γ ≡ tan−1(ν⊥/Ω), and C ≡√

KT/m and Ω ≡qB/m are the thermal speed and gyrofrequency of the simu-lated particles, respectively. To test the viability of a Brownian-motion model to represent the simulated ion data, we fitted (43)and (44) to the variances of ion displacements obtained in oursimulations and compared the best fit parameters ν‖ and ν⊥ tothe Spitzer collision frequency2 for ion–ion interactions, which,for the case of a single-ion plasma, is given by [24]

νi/i =Nee

4 ln Λi

12π3/2ε2om2i C

3i

(45)

where Λi ≡ 12πNeh2i hD is the ion plasma parameter, and hi

and hD are the ion and plasma Debye lengths defined in (6)and (29), respectively.

An example of our fitting results is shown in Fig. 4. Ingeneral, we found that ν⊥ ≈ νi/i. For instance, for the caseshown in Fig. 4, the best fit collision frequency is ν⊥ = 5.88 Hz,while the Spitzer collision frequency is νi/i = 5.94 Hz. We alsofound that the best fit parallel collision frequency ν‖ is smallerthan the Spitzer frequency by a factor of ∼1.15, i.e., ν‖ < νi/i.

Finally, we note in the same figure that the variances ofΔr‖ and Δr⊥ are very similar for τ < 5 ms. This is expected

2Spitzer collision frequency νs/s′ is the Maxwellian-averaged momentumrelaxation rate of a particle of type s due to its interaction with a background ofparticles of type s′. The inverse of νs/s′ can be interpreted as the time intervalover which the particle velocity vector rotates by about 90◦, an accumulatedeffect due to many small Coulomb deflections known as Spitzer collision.


Fig. 5. Simulated single-ion ACFs at different magnetic aspect angles α’s for two radar Bragg wavelengths: (a) λB = 3 m and (b) λB = 0.3 m. The simulationresults (color lines) are compared to theoretical ACFs computed using expression (47) of the Brownian-motion approximation (dashed lines). Note that there iseffectively no dependence on aspect angle α.

because, for τ in that interval, ν‖τ � 1 and ν⊥τ < Ωiτ � 1,in which case (43) and (44) indicate that⟨

Δr2‖

⟩≈

⟨Δr2

⊥⟩≈ C2

i τ2. (46)

Using this approximation in (42), we find that the single-ionACF simplifies to

〈ejk·Δri〉 ≈ e−12 k2C2

i τ2(47)

which is a well-known result for collisionless and nonmagne-tized plasmas that fits well the simulated single-ion ACFs fordifferent magnetic aspect angles and Bragg wavelengths, asshown in Fig. 5.

Evidently, in a magnetized F-region plasma with Coulombcollisions, the oxygen ions diffuse along and across the mag-netic field lines like Brownian-motion particles having isotropicfriction coefficients

ν⊥ ≈ ν‖ ≈ νi/i, (48)

but more significantly, the single-ion F-region ACFs at 50 MHzare essentially the same as in collisionless and nonmagnetizedplasmas, because (a) the ions move by many Bragg wavelengthsλB = 2π/k in-between successive Spitzer collisions and (b)the ions are unable to return to within λB/2π of their startingpositions after a gyroperiod as a consequence of the ion–ioninteractions (giving rise to Spitzer collisions). As an upshot, wewill be able to handle the ion terms analytically in the generalframework equations.

B. Statistics of Electron Displacements

Next, we study the effects of Coulomb collisions on elec-tron trajectories using procedures similar to those applied toions. In Fig. 6, the displacement distributions resulting fromthe simulation of an electron moving in an O+ plasma arepresented. The plasma parameters considered in this simulationare also given in Table I. The top and bottom panels in Fig. 6correspond to the distributions in the directions perpendicularand parallel to B, respectively. On the left, the distributions aredisplayed as functions of τ , while on the right, sample cuts of

the distributions are compared to a Gaussian pdf. As in the ioncase, we note that the normalized distributions for the directionperpendicular to B do not vary with τ and match almost per-fectly to Gaussian curves. However, for displacements parallelto B, the normalized distributions do vary with τ , and theshapes are distinctly non-Gaussian for intermediate values ofτ . More specifically, at very small time delays (τ � 1 μs), thedistributions are Gaussian, but then, in less than a millisecond,the distribution curves become more “spiky” (positive kurtosis)than a Gaussian. Although, at even longer delays τ ’s, thedistributions once again relax to a Gaussian shape, it is clearthat the electron displacement in the direction parallel to Bis not a Gaussian random variable at all time delays τ ’s, forreasons related to electron–electron and electron–ion Coulombcollisions.

Will a Brownian-motion model still prove useful to fit thesimulation data for the electrons even though the displacementprocess seems non-Gaussian? To answer this question, we firstattempted to fit (43) and (44) to the simulated variances of theelectron displacements (as we did for ion displacements earlieron). An example of our fitting results is shown in Fig. 7. Ingeneral, we found that the simulated variance data are wellfitted by the Brownian-motion expressions with the best fitresults of

ν‖ ≈ νe/i (49)

ν⊥ ≈ νe/i + νe/e (50)

where

νe/e =Nee

4 ln Λe

12π3/2ε2om2eC

3e

(51)

νe/i =√

2νe/e =√

2Nee4 ln Λe

12π3/2ε2om2eC

3e

(52)

are the Spitzer electron–electron and electron–ion collisionfrequencies [24], with Λe ≡ 12πNeh

2ehD and he and hD being

defined in (6) and (29), respectively. For instance, in Fig. 7, thebest fit frequencies were ν‖ = 1.469 kHz and ν⊥ = 2.441 kHz,while νe/i = 1.439 kHz and νe/i + νe/e = 2.457 kHz.


Fig. 6. Same as Fig. 3 but for the case of a simulated electron. All distributions at all time delays are normalized to unit variance. Note that the distributions ofthe displacements parallel to B become narrower than a Gaussian distribution in less than a millisecond.

Fig. 7. Same as Fig. 4 but for the case of a simulated electron. The results are displayed for time intervals of 0.02 ms (left panels) and 10 ms (right panels). Notethe different scales for the displacement variances.

Next, we examined whether the Brownian single-particleACF model (42) can be used to fit the simulated electron ACFsusing the best fit friction coefficients identified previously. InFig. 8, we present the single-electron ACFs that were computedfor different magnetic aspect angles and λB = 3 m. The bluecurves correspond to the ACFs obtained from our simulations,while the green curves are the electron ACFs calculated using(42), together with our approximations for ν‖ and ν⊥. Addition-

ally, the electron ACFs for a collisionless magnetized plasmaare also plotted (red curves). We note that the simulated andBrownian ACFs matched almost perfectly at α = 0◦, and also,the agreement is still good at very small magnetic aspect angles[see panels (a), (b), and (c)]. However, substantial differencesbetween the Brownian and simulated ACFs become evidentas the magnetic aspect angle increases [see panels (d), (e),and (f)].


Fig. 8. Simulated electron ACFs for λB = 3 m at different magnetic aspect angles: (a) α = 0◦, (b) α = 0.01◦, (c) α = 0.05◦, (d) α = 0.1◦, (e) α = 0.5◦,and (f) α = 1◦. Note the different time scales used in each plot.

Fig. 9. Simulated electron Gordeyev integrals as functions of Doppler frequency and magnetic aspect angle for two radar Bragg wavelengths: (a) λB = 3 m and(b) λB = 0.3 m. An O+ plasma with electron density Ne = 1012 m−3, temperatures Te = Ti = 1000 K, and magnetic field Bo = 25 μT is considered.

In summary, our study revealed that the single-electron ACFsneeded for ISR spectral models cannot be accurately modeledas a Brownian-motion process over all aspect angles. Thefundamental reason for this is the deviation of the electrondisplacements parallel to B from a Gaussian statistics, despitethe fact that displacement variances are well fitted by theBrownian model. Certainly, a non-Gaussian process cannot befully characterized by a model that specifies its first and secondmoments only; this is particularly true for the estimation of the

characteristic function of the process 〈ejk·Δre〉 that depends onall the moments of the process distribution.

What is the physical reason for the electron displacements tobe non-Gaussian in the direction parallel to B at intermediatetime delays? We believe that the answer is related to the factthat, at low and high electron speeds (in comparison with Ce),the Fokker–Planck collision coefficients (which are functions ofthe particle speed) are dominated by different types of physicalprocesses that correspond to electron–ion and electron–electron


interactions, respectively. While the electron–electron colli-sions that are dominant at high speeds give rise to randomchanges in the electron speed, the same quantity is conserved inelectron–ion collisions (due to the large mass difference of elec-tron and ions) dominating at low speeds. This asymmetry leadsto the formation of a low-speed population of electrons (roughlyafter one electron–electron collision time) staying close totheir starting locations for longer periods (during which manyelectron–ion collisions may take place) than would have beenexpected in a (speed-independent) Brownian-motion model forCoulomb collisions. As a result, the electron displacements atintermediate delays τ ’s have distributions that are sharper than aGaussian, a shape which is consistent with an enhanced popula-tion at small electron displacements. This population eventuallydisperses, and the distributions relax back to a Gaussian format large time delays corresponding to many electron–electroncollision times as would be expected in view of the centrallimit theorem. This description of what we suspect is goingon is also consistent with ν‖ fitting best νe/i (as opposed toνe/i + νe/e) at the intermediate time scales of importance to ourfitting procedure. Note that, despite our best fit result ν‖ ≈ νe/i,electron–electron Coulomb collisions still play a crucial rolein shaping the single-electron ACF by determining the timescale over which the displacements parallel to B return backto normal (Gaussian) distribution.

As for the absence of similar effects in the direction perpen-dicular to B, this can be explained by the fact that the mainrole that the collisions play in the dynamics of the electrons inthe perpendicular plane is to knock them off their gyrocenters,a process that is not sensitive to the distinctions betweenelectron–electron and electron–ion collisions (and, hence, ν⊥ ≈νe/i + νe/e, as we have found out).

Returning back to Fig. 8, the fact that the correlation time ofthe simulated electron signal is longer than what is predictedby the other models is a signature effect of Coulomb collisionsencountered at small (but nonzero) aspect angles. As discussedin the next section, this feature of the electron ACF determinesthe width of the electron Gordeyev integral, and therefore,it has an impact on the shape of the IS spectrum averagedover small magnetic aspect angles. Since the Brownian-motionmodel cannot be used for the calculation of electron ACFsand no other simplified model was found, a numerical libraryof electron ACFs had to be built, as already described in theprevious section. The collisional IS spectra to be presented inthe next section were produced using the numerical library.

VI. COLLISIONAL IS SPECTRUM:RESULTS AND COMPARISONS

Fig. 9 shows the plots of Re{Je(ω)}—the real part of theelectron Gordeyev integral proportional to 〈|nte(k, ω)|2〉—computed using our new numerical library as a function ofDoppler frequency and magnetic aspect angle for two differ-ent Bragg wavelengths, namely, λB = 3 m and λB = 0.3 m.Notice the difference between the frequency axes used in theseplots—while, in the first plot, we are considering a range ofω/2π from −1 to 1 kHz, in the second one, the frequency rangeis ten times wider. A few degrees away from perpendicular to

Fig. 10. Simulated IS spectra as a function of magnetic aspect angle andDoppler frequency for λB = 3 m (e.g., Jicamarca radar Bragg wavelength).An O+ plasma with physical parameters given in Table I is considered.

B, where the effect of collisions can be neglected, the band-width of the electron Gordeyev integral Je(ω) is proportional tothe product of Bragg wavenumber k and electron thermal speedCe. Thus, if the radar wavelength is reduced by a factor of ten,the bandwidth of Je(ω) will increase by the same factor. If thesame were true for viewing directions close to perpendicularto B, the plots displayed above would have looked identical.However, that is not the case, and our results illustrate that thedependence of Je(ω) on k and Ce deviates from kCe at smallaspect angles.

We observe that both Gordeyev integrals become narroweras the magnetic aspect angle decreases. However, in the caseof λB = 0.3 m, Re{Je(ω)} stops shrinking at an angle around0.02◦, while in the case of λB = 3 m, the same happens ata smaller angle (∼ 0.01◦). This is another effect of Coulombcollisions, namely, the saturation of electron Gordeyev integralwidth at very small α. Without having considered collisions,Re{Je(ω)} for α = 0◦ would have approached a Dirac deltafunction at ω = 0 (accompanied by a series of smaller deltaslocated at multiples of the electron gyrofrequency Ωe).

Trying out the Brownian-motion expression for the electronACF as a guide, we noticed that the bandwidth of Re{Je(ω)}in the limit α = 0◦ varies according to

k2C2e

ν⊥ν2⊥ + Ω2

e

. (53)

This expression is obtained by placing (44) into (42) and con-sidering ν⊥τ � 1, a limit in which the electron ACF becomesan exponential function. Notice that the bandwidth dependenceon kCe has changed from linear (at relatively large α) toquadratic (at very small α), a change that is related to thetransition between the collisionless and collisional regimes ofelectron diffusion. Furthermore, using ν⊥ ≈ νe/i + νe/e fromthe last section and taking Ωe � ν⊥, we can verify that band-width dependence (53) is proportional to

Ne√Te

. (54)


Fig. 11. Sample cuts of the simulated spectra of Fig. 10 taken at different magnetic aspect angles: (a) α = 0◦, (b) α = 0.01◦, (c) α = 0.05◦, (d) α = 0.1◦,(e) α = 0.5◦, and (f) α = 1◦. Our (blue lines) simulation results are compared to (green lines) the IS spectra computed using the Brownian-motion model, (redlines) the collisionless electron model, (light-blue lines) the simulation results of [5], and (magenta lines) the aspect-angle-dependent collision frequency modelof [28]. Note the different frequency scales used in each plot.

Since, at very small aspect angles, the electron Gordeyev inte-gral dominates the shape of the overall IS spectrum, we expectthe IS spectral width to exhibit the same dependence (54) ondensity and temperature. We will demonstrate that to be the caseby specific examples shown later in this section.

Fig. 10 shows a surface plot constructed from full IS spec-trum calculations for λB = 3 m (e.g., Jicamarca radar) as afunction of aspect angle α and Doppler frequency ω/2π. In thisself-normalized surface plot, we observe how the IS spectrumsharpens significantly at small aspect angles. For instance, justin the range between 0.1◦ and 0◦, the amplitude of the spectrumbecomes ten times larger, while its bandwidth is reduced by thesame factor.

Sample cuts of the surface plot taken at a number of magneticaspect angles are shown in Fig. 11. In these plots, the simulatedspectra (blue curves) are compared to other spectral models.The green curves correspond to the spectra computed usingthe Brownian-motion model discussed in the previous section,while the red curves correspond to the collisionless electroncase (i.e., the case when Coulomb collisions for the electronsare neglected but collisions for the ions are still considered).We note that the agreement with the Brownian-motion spectrais excellent at very small aspect angles (α ≤ 0.01◦). However,as α increases, our collisional spectra become narrower than theBrownian-motion spectra. We also observe that the collisionlessspectra are, in general, wider than our simulation results. Addi-tionally, note that, at very small aspect angles, the collisionlessspectrum has a humped shape (similar to the one expected atlarge aspect angles but with a much narrower spectral width). Inthis regime, electrons and ions exchange their roles in defining

the shape of the collisionless spectrum so that the spectral widthbecomes proportional to the electron thermal speed Ce andsin α, as previously discussed by Kudeki et al. [2].

In the limit of α → 0◦, the collisionless spectrum develops adelta function at ω = 0, implying an infinite signal correlationtime in the collisionless limit. The reason for this is that, inthe absence of collisions, the magnetic field restricts the motionof the electrons in the plane perpendicular to B, forcing themto gyrate always around the same magnetic field lines. Withcollisions, the electrons manage to diffuse across the field lines,and consequently, the correlation time of the IS signal becomesfinite, and its spectrum broadens out of its delta function limit.However, collisions do not cause spectral broadening at allaspect angles. Notice that, in Fig. 11, the collisional spectrumfor α ≥ 0.01◦ is narrower than the collisionless spectrum, im-plying that collisions in this range of aspect angles have causedan increase of the IS signal correlation time. The explanationfor this is that, at aspect angles larger (and also somewhatsmaller) than 0.01◦, the shape of the IS spectrum is dominatedby electron diffusion in the direction parallel to B. As collisionsimpede the free motion of particles, the electrons diffuse slowerin a collisional plasma than in a collisionless one, which impliesthat the electrons stay closer to their original locations forlonger periods. As a result, the correlation time of the signalscattered by the electrons also becomes longer, causing thebroadening of the IS signal ACF and the associated narrowingof the signal spectrum in this aspect angle regime, as firstexplained by Sulzer and González [5].

Also in Fig. 11, we compare our simulated spectra to theresults of Sulzer and González [5] for α ≥ 0.5◦ [see panels (e)


Fig. 12. Electron density dependence of the simulated IS spectra for λB = 3 m at aspect angles α = 0◦ (left panel), α = 0.1◦ (central panel), and α = 1◦

(right panel). An O+ plasma in thermal equilibrium with Te = Ti = 1000 K and Bo = 25 μT is considered. Note the different frequency scales used ineach plot.

Fig. 13. Te/Ti dependence of the simulated IS spectra for λ = 3 m at aspect angles (left panels) α = 0◦, (central panels) α = 0.02◦, and (right panels)α = 0.1◦. The same spectra are plotted on the top and bottom panels; however, on the bottom, every spectrum has been normalized to its peak value. An O+

plasma with Ne = 1012 m−3, Ti = 1000 K, and Bo = 25 μT is considered. Each curve corresponds to different values of electron temperatures Te’s (1000,1500, 2000, 2500, and 3000 K). Note the different frequency scales used in each column.

and (f)]. The comparison is good, except for a minor offset thatwas traced to a minor coding error in the simulation programused by Sulzer and González [5]. At smaller aspect angles,comparisons are made with the spectrum model of Woodman[28], which is effectively an ad hoc extrapolation of the Sulzerand González [5] model to very small magnetic aspect an-gles (including the α → 0◦ limit). Woodman [28] fitted theBrownian-motion spectral model to the results of Sulzer andGonzález [5] and developed an empirical formula for the elec-tron collision frequency that depends on the magnetic aspectangle. As can be observed in panels (a), (b), (c), and (d), oursimulated spectra are not quite as sharp as Woodman’s spectra(although the differences are small). These differences can beexpected since Woodman [28] did not really have the spectra

to fit at very small aspect angles, and therefore, his collisionfrequency formula may require a little adjustment for α < 0.1◦.

Fig. 12 shows the dependence of the simulated IS spectraon the mean electron density Ne that controls the Coulombcollision rates. As discussed earlier in this section, we expectthe spectrum at α = 0◦ to become wider as Ne grows [seerelation (54)]. This behavior is clearly shown in the left panelof Fig. 12. As Ne increases, collisions become more frequent,and consequently, the electrons diffuse out to longer distancesacross the magnetic field lines. Hence, the spectrum bandwidthincreases with Ne, so long as the collision frequency remainssmaller than the gyrofrequency (otherwise, the collisions startimpeding the electron motion rather than just perturbing thegyromotions to facilitate diffusion). However, a few hundredths


of a degree away from perpendicular to B, the shape of theIS spectrum starts being controlled by diffusion along themagnetic field lines. In this regime, the spectrum becomesnarrower as Ne grows because the increasing number of col-lisions impedes the movement of the electrons (as alreadydiscussed before). This behavior is observed in the central panelof Fig. 12. Also note that the collisional spectrum approachesthe collisionless spectrum as Ne is reduced, in consistencywith the dependence of collision frequencies on Ne. Finally, acomparison of the central and rightmost panels of Fig. 12 showsthat collision effects become less significant at even largeraspect angles where the spectrum is shaped by ion dynamics.In that regime, the spectral shapes become independent of Ne

as long as khe � 1.The temperature dependence of the simulated IS spectra

at different aspect angles will be examined with the help ofFig. 13. Considering an O+ plasma with Ne = 1012 m−3 andTi = 1000 K, we have computed the IS spectra for a Braggwavelength of λB = 3 m and electron temperatures of Te =1000, 1500, . . . , 3000 K. Fig. 13 shows the results for threedifferent aspect angles, namely, α = 0◦ (left panels), α = 0.02◦

(center panels), and α = 0.1◦ (right panels). Notice that thesame spectra are plotted in the top and bottom panels, butin the bottom panels, we have normalized each spectrum byits peak value in order to emphasize the changes in spectralwidths.

Fig. 13 includes a wealth of information pertinent to theultimate application of our theory in estimating the ionospherictemperatures from Jicamarca radar data collected with beamsperpendicular to B. First, we note that the left panels of Fig. 13illustrate how the spectrum bandwidth at α = 0◦ is inverselyproportional to the square root of Te, as discussed earlier.Second, we note that this dependence of spectral width on Te

changes at larger aspect angles, as shown in the remaining pan-els of the figure—for aspect angles larger than about α = 0.01◦,spectral width increases with increasing Te. Third, the spectralamplitudes change rapidly with aspect angle. For instance, inthe case of Te = Ti = 1000 K, the peak of the spectrum atα = 0◦ is about ten times larger than the one at α = 0.1◦.Finally, the amplitude differences for different α’s are enhancedat higher values of Te (in comparison with Ti), which is relatedto the aspect angle dependence of the volumetric radar crosssection (e.g., [29] and [30])

σ ≡ 4πr2eNeη(k) (55)

of ISR echoes. In (55), re is the classical electron radius, and η(k)is an electron scattering efficiency factor (see [30]) defined as

η(k) ≡∫

dω

2π

⟨|ne(k, ω)|2

⟩Ne

(56)

which is primarily dependent on temperature ratio Te/Ti andmagnetic aspect angle α. A plot of this factor obtained fromour collisional simulation results is shown in Fig. 14. Note that,for α ≈ 1◦, the factor is independent of Te/Ti, but it increasesand decreases with the temperature ratio at small and largeaspect angles, respectively. The efficiency factor η(k) plays an

Fig. 14. Electron scattering efficiency factor η(k) resulting from the fre-quency integration of the simulated IS spectra as a function of electron-to-iontemperature ratio Te/Ti and magnetic aspect angle α. An O+ plasma withNe = 1012 m−3, Ti = 1000 K, and Bo = 25 μT is considered.

important role in Ne estimation from ISR power data, and itsaccurate determination in the collisional regime at small aspectangles is one of the main contributions of this paper.

VII. SUMMARY

We have reported in this paper on the modeling of F-regionISR spectrum, including the effects of Coulomb collisions formodes propagating nearly perpendicular to the geomagneticfield B. Our procedure to model the IS spectrum is based onthe simulation of the trajectories of charge carriers moving ina magnetized plasma with suppressed collective interactions.The effects of Coulomb collisions are considered using theFokker–Planck model of Rosenbluth et al. [6] within a particledynamics formalism in which particle motion is describedby a stochastic Langevin equation. The results of our studiesshow that the electron displacements cannot be reconciled withsimplified Brownian-motion models of Coulomb collisions. Anumerical library of single-electron ACFs was developed inorder to include the collisional electron dynamics properly inIS spectral models.

In the next stage of our studies, we will make use of our newspectral model to fit the beam-averaged ISR spectra measuredin Jicamarca radar experiments with antenna beams pointedperpendicular to B. For this purpose, the measured spectrumhas to be carefully modeled because, within the range of smallmagnetic aspect angles illuminated by the Jicamarca antenna,the theoretical IS spectra vary quite rapidly. In addition, notethat magnetoionic propagation effects that are significant at50 MHz should also be included in the model. This furthercomplicates the description of Jicamarca measurements, sincedifferent normal modes of propagation are excited within thewidth of the antenna beam, modes that vary from linearly po-larized at α = 0◦ to circularly polarized at aspect angles largerthan a degree, with a transition between these two regimeshappening at an angle of approximately 0.5◦. A model of beam-weighted ISR spectrum that takes into account magnetoionicpropagation and collisional effects is under development. Theanalysis of the sensitivity of this model to electron and iontemperatures is also the subject of our current studies.


Most of the perpendicular-to-B radar experiments conductedat Jicamarca are aimed at collecting measurements from theF-region. For this reason, we focused our attention in thepresent work on O+ plasmas dominant at F-region heights.To model the perpendicular-to-B radar observations in thetopside ionosphere, we will need to extend our spectral modelto consider the effects of collisions on plasmas composed ofhydrogen and helium ions. This will be the subject of futureresearch.

ACKNOWLEDGMENT

The authors would like to thank the University of Illinois forthe use of the Turing Cluster, which is maintained and operatedby the Computational Science and Engineering Program ofthe university. Turing is a 1536-processor Apple G5 Xservecluster devoted to high-performance computing in engineeringand science.

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measurements using incoherent scatter power data collected at ALTAIR,”Ann. Geophys., vol. 24, no. 5, pp. 1333–1342, Jul. 2006.

Marco A. Milla (M’07) received the Bachelor’sdegree in electrical engineering from the PontificiaUniversidad Católica del Perú, Lima, Peru, in 1997and the M.S. and Ph.D. degrees from the Universityof Illinois, Urbana, in 2006 and 2010, respectively.

He joined the Jicamarca Radio Observatory asa Signal-Processing Engineer in 1998 and workedthere for about five years in different research andengineering projects. Since 2004, he has been withthe Remote Sensing and Space Sciences Group, De-partment of Electrical and Computer Engineering,

University of Illinois. His doctoral research involves the development ofincoherent scatter radar techniques for the estimation of ionospheric stateparameters. In particular, he has studied Coulomb collisions and magnetoionicpropagation effects on the incoherent scatter spectrum measured with antennabeams pointed perpendicular to the Earth’s magnetic field.

Erhan Kudeki (M’05) received the Ph.D. degreefrom the School of Electrical Engineering, CornellUniversity, Ithaca, NY, in 1983. His dissertationwas on the plasma instabilities of the equatorialionosphere.

After spending about two years as a PostdoctoralResearcher with Cornell University and with theJicamarca Radio Observatory near Lima, Peru, hejoined the faculty of the Department of Electricaland Computer Engineering, University of Illinois,Urbana, in 1985, where he teaches courses on cir-

cuits and linear systems, electromagnetics, and communication and remotesensing applications of radio-wave propagation. He is a coauthor (with DavidC. Munson, Jr.) of an undergraduate textbook on circuits and linear systemsentitled Analog Signals and Systems (Pearson/Prentice-Hall, 2009) published inthe Illinois ECE Series. His research is mainly focused on radar remote sensingof the upper atmosphere and ionosphere, with special emphasis on equatorialplasma electrodynamics and incoherent scatter radar techniques.

Prof. Kudeki was the recipient of the National Science Foundation CEDARPrize Lecture in 2006 for his work on incoherent scatter observations at smallmagnetic aspect angles.

Documents

Incoherent Scatter Spectral Theories—Part II: Modeling the Spectrum for Modes Propagating Perpendicular to ${\bf B}$