generation of elf and ulf electromagnetic waves by...

Generation of ELF and ULF electromagnetic waves by modulatedheating of the ionospheric F2 region

B. Eliasson,1,2 C.-L. Chang,3 and K. Papadopoulos1,3

Received 11 May 2012; revised 2 August 2012; accepted 23 August 2012; published 18 October 2012.

[1] We present a theoretical and numerical study of the generation of extremely lowfrequency (ELF) and ultra-low frequency (ULF) waves by the modulation of the electronpressure at the F2-region with an intense high-frequency electromagnetic wave. The studyis based on a cold plasma Hall-MHD model, including electron-neutral and ion-neutralcollisions, which governs the dynamics of magnetostatic waves and their propagationthrough the ionospheric layers. Magnetosonic waves generated in the F2 region arepropagating isotropically and are channeled in the ionospheric waveguide, while shearAlfvén waves are propagating along the magnetic field. To penetrate the ionospherefrom the F2 peak at 300 km to the ground, the magnetostatic waves first propagate asmagnetosonic or shear Alfvén waves that encounter a diffusive layer from about 150 kmto 120 km where the Pedersen conductivity dominates, and then as helicon (whistler-like)mode waves from about 120 km to 80 km where the ions are collisionally glued to theneutrals and the Hall conductivity dominates. By performing numerical simulations andstudying the dispersive properties of the wave modes, we investigate the dynamics andpenetration of ELF/ULF waves through the ionospheric layers to the ground and along thegeomagnetic field lines to the magnetosphere. Realistic profiles of the ionospheric profilesof conductivity and density are used, together with different configurations of thegeomagnetic field, relevant for both the high, mid and equatorial latitudes. Some of theresults are compared with recent HAARP experiments.

Citation: Eliasson, B., C.-L. Chang, and K. Papadopoulos (2012), Generation of ELF and ULF electromagnetic wavesby modulated heating of the ionospheric F2 region, J. Geophys. Res., 117, A10320, doi:10.1029/2012JA017935.

1. Introduction

[2] The propagation of extremely low frequency (ELF)and ultra-low frequency (ULF) electromagnetic waves in theionospheric duct has been the subject of several observationsand theoretical and numerical studies. The guided propaga-tion of MHD waves around the Alfvén speed minimum atthe F2 peak was suggested by Dessler [1958], and was sup-ported by analysis of Argus 3 data by Berthold et al. [1960]and of the naturally occurring pearl pulsations by Tepleyand Landshoff [1966]. These observations indicated that thewaves propagate with the Alfvén speed (700–900 km/s) inthe ionospheric duct. Greifinger and Greifinger [1968] con-ducted a theoretical investigation and derived dispersionrelations of the guided propagation in the magnetic meridian

of low-frequency hydromagnetic waves in the ionosphericduct. They found that there is a low-frequency cutoff ofabout 0.4 Hz for nighttime conditions, and an effective high-frequency cutoff for ground-level signals due to an exponentialdecrease of the transmission coefficient with frequency at highfrequencies. This investigation was later extended to includeoff-meridian propagation [Greifinger and Greifinger, 1973].Numerical studies of Alfvén wave propagation in the iono-sphere have been conducted by Lysak [1997, 1999], whodeveloped a two-dimensional numerical model to study thepropagation of waves in the 1 Hz band in the auroral zone (withvertical geomagnetic field). This model was later extended tothree dimensions and to include the Earth’s dipole magneticfield [Lysak and Song, 2001; Lysak, 2004].[3] Low-frequency waves in the ionosphere, can be excited

by lightning discharges [Greifinger and Greifinger, 1976;Uman, 1987; Berthelier et al., 2008; Milikh et al., 1995],seismic events [Hayakawa et al., 2006], and ionosphericheating by powerful high-frequency (HF) transmitters[Rietveld et al., 1984, 1987, 1989; Barr, 1998; Papadopouloset al., 1990, 2005]. The injection of ELF electromagneticwaves into the equatorial E-region by ground-based antennaswas suggested by Eliasson and Papadopoulos [2009], whofound that induced strong horizontal currents and relatedvertical electric fields can be formed in a few kilometers thick

1Department of Physics and Department of Astronomy, University ofMaryland, College Park, Maryland, USA.

2Institute for Theoretical Physics, Ruhr-University Bochum, Bochum,Germany.

3Technology Solutions, BAE Systems, Arlington, Virginia, USA.

Corresponding author: B. Eliasson, Institute for Theoretical Physics,Ruhr-University Bochum, DE-44780 Bochum, Germany.([email protected])

©2012. American Geophysical Union. All Rights Reserved.0148-0227/12/2012JA017935

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A10320, doi:10.1029/2012JA017935, 2012

A10320 1 of 22

layer, close to the plasma-free space boundary. This effect isreminiscent of the equatorial electrojet [Forbes and Lindzen,1976; Forbes, 1981; Rishbeth, 1997; Rastogi, 1989] where apotential drop due to tidal motion along the equatorial linegives rise to an intense electron current by the Cowlingeffect. The question arises if similar effects can be achievedby magnetosonic waves that are generated at higher altitudesand propagate down to the E-region, and what is the effect ofthe orientation of the geomagnetic field on the penetration ofmagnetic fields through the ionospheric layer. Recent theo-retical and computational results indicate that modulatedF-region HF heating in the polar ionosphere can generateionospheric currents, via the ionospheric current drive (ICD)process, which act as an antenna to inject magnetosonicwaves in the ionospheric plasma [Papadopoulos et al.,2007, 2011a; Papadopoulos, 2009]. The pressure gradientassociated with F-region electron heating drives a localdiamagnetic current, which triggers a magnetosonic wave.When it reaches the E region of the ionosphere, the electricfield of the magnetosonic wave drives Hall currents, whichin turn act as a secondary antenna that injects waves into thebelow Earth-ionosphere waveguide and shear Alfvén wavesupwards to the conjugate regions. To address the problemof wave generation and propagation at different latitudes,we here use a cold plasma Hall-MHD model to studynumerically the generation of low-frequency electromag-netic waves by the heating of the ionospheric layer forrealistic ionospheric parameters and different angles of thegeomagnetic field. In addition, the Hall-MHD model takesinto account wave frequencies both below and above the ioncyclotron frequency and thus supports electromagneticcyclotron and whistler waves. The dynamics and penetrationof ELF waves through the ionospheric layer to the ground isinvestigated for different configurations of the geomagneticfield, relevant for high- and mid-latitude facilities such asHAARP in Alaska and Arecibo in Puerto Rico, and for theequatorial region. A simplified analytic model is used to

estimate the spatial attenuation of the waves propagatingthrough the ionosphere.[4] The paper is organized as follows. In Section 2, we

present an ionospheric plasma model, where the Hall MHDequations are put into a matrix form for numerical conve-nience. Numerical solutions of the model are analyzed inSection 3 for realistic ionospheric profiles and different con-figurations of the geomagnetic field. The propagation of thewaves through the ionospheric layer and down to the plasma-free space boundary to the ground is studied numerically andwith a simplified propagation model based on the dispersionrelation of the waves. In Section 4, we make some compar-isons with recent HAARP experiments where ICD generatedULF/ELF waves were measured on the ground and by theoverflying DEMETER satellite. Finally, some conclusions aredrawn and future perspectives discussed in Section 5.

2. The Ionospheric Model

[5] We are considering a two-dimensional model in the x-zplane, depicted in Figure 1, where the ionospheric layer islocated above z = z0 = 90 km, the atmosphere (free space)below 90 km and the perfectly conducting ground at z = 0.The plasma is magnetized by the geomagnetic field B0,which is tilted at an angle q to the vertical axis. The electronpressure is modulated by a high-frequency radio wave locallyin a heated region near the F2 peak at z = 300 km, x = 0,giving rise to low-frequency electromagnetic waves via theICD process [Papadopoulos et al., 2011a]. Above about150–200 km, the plasma is weakly collisional, and thewave frequency w (corresponding to a few Hz) is larger thanthe ion-neutral collision frequency nin. In this region, thedynamics is dominated by magnetosonic and shear Alfvénwaves, where the ion mass provides the inertia and themagnetic pressure and tension provide the restoring forces.The shear Alfvén waves propagate primarily along the

Figure 1. A schematic picture of the ionospheric model.

ELIASSON ET AL.: GENERATION OF ELF AND ULF EM WAVES A10320A10320

2 of 22

geomagnetic field, while the magnetosonic waves propagateisotropically. The F2 peak at 250–300 km is the region ofminimum Alfvén speed, which leads to the channeling ofmagnetosonic waves in the ionospheric waveguide. Theplasma between about 150 km and 120 km is dominated bycollisions, and both the ions and electrons can be consideredinertialess. Here, the ion and electron E � B0 drifts lead to apartial cancellation of the electric current, which results in adiffusive layer where the Pedersen conductivity sp dominatesthe interaction. In the E-layer, at 90–120 km, the ions arecollisionally glued to the neutrals, and the dynamics isdominated by the Hall conductivity sH due to the electronE � B0 drift. In this region, the whistler branch extends tofrequencies below the ion cyclotron frequency and gives riseto weakly damped helicon waves. The ELF waves are par-tially injected into the neutral atmosphere due to currents inthe bottom-side of the E-region [Papadopoulos et al., 2011a].The propagation of the ELF waves from the source region at300 km to ground is thus an intricate and interesting problemdue to the plasma inhomogeneity and the shifting plasmacharacteristics at different altitudes.

2.1. Plasma Model

[6] We use a cold plasma Hall-MHD model for the iono-sphere, in which the dynamics of the ELF/ULF electro-magnetic fields is governed by Faraday’s and Ampère’s laws

∂B∂t

¼ �r� E ð1Þ

and

r� B ¼ m0en0 vi � veð Þ; ð2Þ

respectively, where m0 is the magnetic vacuum permeability,e is the magnitude of the electron charge, and n0 is theelectron number density. We have neglected the displace-ment current in equation (2) since the wave speed is muchsmaller than the speed of light, and have used the quasi-neutrality condition ni = ne = n0 for the ion and electronnumber densities. We are interested in timescales muchlonger than the lower hybrid oscillation period, so that theelectron inertia can be neglected. The dynamics of the plasmais then governed by the inertia-less electron momentumequation

0 ¼ � e

meEþ ve � B0ð Þ � venve � rPe

men0ð3Þ

and the ion momentum equation

∂vi∂t

¼ e

miEþ vi � B0ð Þ � vinvi; ð4Þ

where me and mi are the electron and ion masses, ven and vinare the electron-neutral and ion-neutral collision frequencies,and Pe is the externally modulated electron pressure, whichprovides the source of ELF/ULF electromagnetic waves. Themodel does not take into account nonlinearities associatedwith HF wave propagation through the D-region, frequentlyused to generate ELF and ULF waves by modulating thePolar Electrojet (PEJ) [Rietveld et al., 1984, 1987, 1989;Papadopoulos et al., 1990, 2005;Moore, 2007; Payne et al.,2007]. We have assumed that collisions with neutrals are

dominant. However, for altitudes larger than about 140 km,electron-ion collisions start to dominate over electron-neutralcollisions [Forbes, 1981]. We neglect this effect here sincethe effect of electron collisions on the wave dynamics isnegligible at higher altitudes.[7] For numerical convenience (see Appendix A), the

system (1)–(4) is written in the form

∂A∂t

¼ �E ð5Þ

and

∂E∂t

¼ �wci Gin þ Genð ÞEþ ɛ�1 r� r� Að Þ½ �

m0� Re r� r� Eð Þ½ �

m0~s

þ wciR i � ∂∂t

� �rPeen0

ð6Þ

where we introduced the vector and scalar potentials A and fvia B =r � A and E =�rf �∂A/∂t, using the gauge f = 0.

The Re and R i matrices (organizing the vectors as columnvectors) are deduced from the electron and ion equations ofmotion (3) and (4) via the definitions (ve � B0 + me venve/e)/

B0 ≡Reve and (vi�B0 +mivin vi/e)/B0 ≡R ivi, respectively, asdiscussed in Appendix A. These matrices are used to con-

struct the inverse of an effective dielectric tensor ɛ�1 ¼� v2A=ɛ0c

2� �

ReR i, where vA ¼ cwci=wpi is the Alfvén speed,and a conductivity tensor s ¼ wci Gin þ Genð Þɛ, were we havedenoted Gen ¼ ven=wce and Gin ¼ vin=wci. Here,wci ¼ e B0=mi

and wce ¼ eB0=me are the ion and electron cyclotron fre-

quencies, wpi ¼ n0 e2=ɛ0 mið Þ1=2 and wpe ¼ n0 e2=ɛ0með Þ1=2are the ion and electron plasma frequencies, and we denoted~s ¼ ɛ0w2

pe=wce. At the lower boundary between the plasmaand free space at z = z0 = 90 km, we have constructedboundary conditions (see Appendix B), by assuming that thex- and y-components of the electric field and vector potential,and their normal derivatives, are continuous. In free space,we have assumed infinite speed of light and that there are noelectric charges or currents, while the ground at z = 0 is per-fectly conducting, so that analytic expressions for the freespace electromagnetic fields can be derived (see Appendix B).

2.2. Model Ionospheric Profiles

[8] We use an ionospheric model with realistic parametersand length scales, shown in Figure 2. For the electron/ionnumber density, we adopt a Chapman profile

n0 zð Þ ¼ nmax exp 1� x � e�x� �; x ¼ z� zmax

H; ð7Þ

where we used the maximum number density nmax =3.4 � 1011m�3 at the F2 peak, located at zmax = 300 km, andwhere H = 120 km is the ionospheric length scale. Theions are assumed to be singly charged oxygen ions(atomic weight = 16), and the geomagnetic field strengthis set to B0 = 4 � 10�5 T, which gives wci = 238 s–1 (38 Hz)and wce = 7.0 � 106 s�1 (1.1 MHz). The maximum of the


3 of 22

plasma density at 300 km in Figure 2a corresponds to a min-imum of the Alfvén speed in Figure 2b.[9] The vertical profiles of the normalized electron and ion

collision frequencies Gen ¼ ven=wce and Gin ¼ vin=wci aregiven by

Gen zð Þ ¼ Gen0 expzen0 � z

Hen0

� �þ exp

zen1 � z

Hen1

� �� ; ð8Þ

and

Gin zð Þ ¼ Gin0 expzin0 � z

Hin0

� �þ exp

zin1 � z

Hin1

� �� ; ð9Þ

respectively, with Gen0 = 10�4, zen0 = 110 km, Hen0 = 20 km,zen1 = 110 km, Hen1 = 5 km, Gin0 = 0.5, zin0 = 120 km, Hin0 =20 km, zin1 = 120 km, and Hin1 = 5 km. For these parameterswe see in Figure 2e that Gen ≪ 1 and Gin ≈ 1 at z = 120 km.The density and collision frequency profiles are used tocalculate the profiles of the parallel, Pedersen, and Hallconductivity, respectively, from sk ¼ ~s G�1

en þ G�1in

� �,

sP ¼ ~s Gen= 1þ G2en

� �þ Gin= 1þ G2in

� �� , and sH ¼

~s 1= 1þ G2en

� �� 1= 1þ G2in

� �� (where ~s ¼ ɛ0w2

pe=wce ¼n0e=B0). As seen in Figure 2f, sP > sH above 120 km, whilesP < sH ≈ ~s in the E-region below 120 km, where the ionsare glued to the neutrals due to collisions.

2.3. Heating Model

[10] We consider a scenario where the F2 peak is heated witha high-frequency, high-intensity radio wave that is amplitudemodulated at a low frequency. This leads to Ohmic heating ofthe electrons and a modulation of the electron pressure Pe =

kBn0Te locally in space, were kB is Boltzmann’s constant andTe the electron temperature. As a model for the modulatedelectron pressure gradient, we use rPe = n0kBrTe, with

rTe ¼ Tmod tanh2 t

Dt

� �cos wtð Þr exp � x2

D2x

� z� zmaxð Þ2D2

z

" #;

ð10Þ

where Tmod is the modulation amplitude of the electron tem-perature, Dt is the rise time, Dx and Dz are the widths of theheated region in the x and z directions, zmax is the altitude ofthe heated region, and w is the modulation frequency. Thegradient of the density at the F2 peak is small and hastherefore been neglected, and we have also neglected arise of the mean electron temperature in the heated region,which will not influence the wave dynamics significantly.The electron pressure gradient enters into the evolutionequation (6) for the electric field.

3. Dynamics of ELF Waves Generatedin the F2 Region

[11] We present here numerical solutions of equations (5)and (6) for different orientations of the geomagnetic field,using the ionospheric profiles plotted in Figure 2, and themodel (10) for the external heating of the electrons.

3.1. Numerical Setup

[12] The simulations are carried out using a computationaldomain of�2240 km ≤ x ≤ 2240 km and 90 km ≤ z ≤ 1020 kmand with Nx � Nz = 320 � 200 intervals, with spatial gridsizes Dx = 14.0 km and Dz = 4.67 km. Periodic boundary

Figure 2. Vertical nighttime profiles of (a) the electron number density h0 (m�3), (b) the Alfvén speed

vA m=sð Þ, (c) the normalized collision frequencies Gin ¼ vin=wci and Gen ¼ ven=wce, and (d) the conductiv-ities sH, sP, sk and ~s S=mð Þ, using the geomagnetic field strength B0 = 4 � 10�5 T. (e and f) Close-ups of

the collision frequencies and conductivities below 160 km.


4 of 22

conditions are used in the x-direction, while a coupling tothe free space field is used at the bottom side at z = 90 kmand first-order outflow boundary conditions are used at thetop boundary at z = 1000 km; see Appendix B for moredetails. Pseudospectral methods are used to calculate thex-derivatives and second-order centered difference schemesare for the z-derivatives. The solution is advanced in timeusing the standard 4th-order Runge-Kutta scheme with atime-step of Dt = 6.25 � 10�5 s. The current density in theplasma is obtained from Ampère’s law as j ¼ r�r� AÞ=m0ð , while j = 0 in free space below z = 90 km. Thefree space electromagnetic fields are reconstructed analyti-cally, as detailed in Appendix B. In the geometry usedhere, the z-component of the magnetic field goes to zero atz = 0 due to the perfectly conducting ground, and By mustbe identically zero in free space due to the y-component ofAmpère’s law, ∂By=∂x ¼ m0 jz, which leads to By = 0 sincejz = 0 in free space.[13] In the heating model (10), we use Tmod = 500 K, Dx =

40 km, Dz = 20 km, zmax = 300 km, Dt = 0.5 s, and w = 4ps�1

(2 Hz).

3.2. Simulation Results

[14] We present below simulations for the cases of vertical,oblique and horizontal magnetic field. The vertical magneticfield case corresponds to experiments performed using theHAARP ionospheric heater in Gakona Alaska [Papadopouloset al., 2011b] and allows us to compare the results withexperimental measurements on the ground and in the iono-sphere. The oblique case corresponds to ionospheric heatingexperiments performed or planned at Arecibo, Puerto Rico andSura, Russia. The final case corresponds to predictions of aheater located in the dip equator, in places such as Jicamarca,Peru or Tuba, India.

3.2.1. Vertical Magnetic Field: High Latitude Heater[15] Snapshots of the electromagnetic wave fields for a

vertical geomagnetic field, q = 180�, are shown in Figures 3and 4, at t = 5 s after the transmitter has been turned on. Themain features, seen in Figure 3, are magnetosonic waves that

Figure 3. Magnetosonic wave propagation at t = 5 s for vertical geomagnetic field (0 = 180�) and the iono-spheric profiles in Figure 2, using a source modulation frequency of 2 Hz. (a and b) The x- and z-componentsof the magnetic field (pT) and (c and d) the associated y-components of the electric field mV=mð Þ and currentdensity nA=m2ð Þ. (e–g) Close-ups of Bx, Bz and jy below 150 km and in free space below 90 km.

Figure 4. Shear Alfvén wave propagation at t = 5 s for ver-tical geomagnetic field (q = 180�) and ionospheric profilesgiven by Figure 2, using a modulation frequency of 2 Hz.The shear Alfvén wave is associated with (a) the y-componentof the magnetic field, (b) the x-component of the electric field,and (c and d) the x- and z-components of the current density.


5 of 22

are propagating horizontally in each direction away from theheated region at x = 0, at the Alfvén speed minimum at aboutz = 300 km. For the ionospheric profiles in Figure 2, thetheoretical Alfvén speed at z = 300 km is approximately 4�105 m=s (see Figure 2b), which gives a wavelength of about200 km for 2 Hz, in agreement with Figure 3. The magne-tosonic waves are primarily associated with the compressionand rarefaction of the total magnetic field, and hence withthe z-component of the wave magnetic field and the associ-ated y-components of the electric field and current density.As seen in Figures 3a and 3b, the magnetosonic wave isassociated with a wave magnetic field Bz ≈ 10 pT and elec-tric field Ey≈5 mV=m. Below 150 km, shown in Figure 3d,the structure of the wave magnetic field changes rapidly, andthe amplitude decreases by a factor of a few. These changesin the magnetic field are associated with localized currents inthe y-direction, visible in Figure 3g.[16] Smaller amplitude shear Alfvén waves that are prop-

agating vertically along the magnetic field lines are seen inFigure 4. The shear Alfvén waves are associated with they-component of the magnetic field, the x-component of theelectric field and the x- and z-components of the currentdensity. The generation of shear Alfvén waves is due tomode conversion of magnetosonic waves via Hall currents inthe E-region [Papadopoulos et al., 2011a], and due to the

presence of the Hall term (proportional to Re) in the right-hand side of equation (6). Shear Alfvén waves would not begenerated by the modulation of the electron pressure in anideal MHD model.3.2.2. Oblique Magnetic Field: Mid-Latitude Heaters[17] The case of oblique magnetic fields, q = 135� is shown

in Figures 5 and 6. Figure 5 shows signatures of magnetosonicwaves propagating along the ionospheric duct at 300 km and

Figure 5. Magnetosonic wave propagation at t = 5 s for oblique geomagnetic field (q = 135�) and theionospheric profiles in Figure 2, using a source modulation frequency of 2 Hz. (a and b) Thex- and z-components of the magnetic field (pT) and (c and d) the associated y-components of the electricfield mV=mð Þ and current density nA=m2ð Þ. (e–g) Close-ups of Bx, Bz and jy below 150 km and in free spacebelow 90 km.

Figure 6. Shear Alfvén wave propagation at t = 5 s foroblique geomagnetic field (q = 135�) and ionospheric pro-files given by Figure 2, using a modulation frequency of2 Hz. The shear Alfvén wave is associated with (a) they-component of the magnetic field, (b) the x-componentof the electric field, and (c and d) the x- and z-componentsof the current density.


6 of 22

carried by the x- and z-components of the magnetic field andthe y-components of the electric field and current density. Therapidly changing wave magnetic fields below 150 km inFigure 5e are associated with localized currents in they-direction, seen in Figure 5g. The shear Alfvén mode, seen inFigure 6, propagates primarily along the geomagnetic field,and is associated with the y-component of the magnetic fieldand the x- and z-components of the electric field and currentdensity. (Since the electric field parallel to the geomagneticfield is almost zero, we have Ez ≈ Ex for q = 135�, and wetherefore show only Ex in Figure 6.)3.2.3. Horizontal Magnetic Field: Dip-Equator Heaters[18] Finally, we discuss simulation results with almost

horizontal, curved geomagnetic field lines, shown inFigures 7 and 8, which is relevant for projected dip equatorheaters. The most striking feature, seen in Figure 7, are theclear signatures of shear Alfvén waves, carried by By, Ez, jxand jz and propagating in a narrow band along the geo-magnetic field lines. The shear Alfvén waves reach theplasma-free space boundary about 1000 km away from theheated region, where they are partially injected into the earth-ionosphere waveguide.[19] The magnetosonic wave, seen in Figure 8, is associ-

ated with a strong compressional x-component of the mag-netic field and associated y-components of the electric fieldand current density. Also visible is the appearance of a thinhorizontal current sheath jy, seen in Figure 8h, associatedwith a strong vertical electric field Ez, seen in Figure 8g. Thiseffect is similar to the equatorial electrojet, where a weakhorizontal electric field Ey perpendicular to the geomagneticfields tries to drive a Hall current in the vertical direction.Since current continuity close to the plasma-free spaceboundary prevents large currents to propagate vertically,

Figure 7. Shear Alfvén wave propagation at t = 5 s foroblique geomagnetic field (q ¼ 90� þ x=Lð Þ � 180�=p andL = 3000 km) and ionospheric profiles given by Figure 2,using a modulation frequency of 2 Hz. The shear Alfvénwave is associated with (a) the y-component of the magneticfield, (b) the z-component of the electric field, and (c and d)the x- and z-components of the current density.

Figure 8. Snapshot of magnetosonic wave propagation at t = 5 s for curved geomagnetic field lines(q ¼ 90� þ x=Lð Þ � 180�=p and L = 3000 km), using the ionospheric profiles in Figure 2 and a sourcemodulation frequency of 2 Hz. (a and b) The x- and z-components of the magnetic field (pT)and (c and d) the associated y-components of the electric field mV=mð Þ and current density nA=m2ð Þ .(e–h) Close-ups of Bx, Ey, Ez and jy below 150 km.


7 of 22

a large vertical electric field is set up, which in turn drives theintense electron current in the y direction. The resultingCowling conductivity sC ¼ s2

H=s2P þ 1

� �sP, is a few orders

of magnitude larger than the Pedersen conductivity in the Eregion. For the case when magnetic fields are injected intothe ionosphere by a ground-based antenna from below[Eliasson and Papadopoulos, 2009], it was found that theCowling effect can lead to intense horizontal currents andvertical electric fields in the E-region. For the case consideredhere, we see in Figure 8h that a sharp drop of the amplitude ofBx at z ≈ 100 km is associated with the intense current com-ponent jy via Ampère’s law ∂Bx=∂z ¼ m0 jy, and a large ver-tical electric field Ez in Figure 8g. The oscillatory currentsheet in the E-region could inject electromagnetic waves intothe Earth-ionosphere waveguide. Figure 8h shows that they-component of the current sheet extends about 10 kmvertically and 1000 km horizontally in the x-direction withan amplitude of 0:3 nA=m2. If we assume that the currentsheet also extends 1000 km in the y-direction, we have adipole moment of about 107 Am, which can give significantinjection of electromagnetic waves into the earth-ionospherewaveguide.

3.3. Ground Magnetic Field Signatures

[20] For comparison with experiments, where the mag-netic field is measured by ground-based magnetometers, theamplitude of the magnetic field leaking down to the groundis calculated from the simulation data. The results are sum-marized in Figure 9, which shows the amplitude of Bx on theground and at the plasmas-free space boundary, for the dif-ferent orientations of the geomagnetic field discussed inFigures 3–8. For the vertical geomagnetic field (q = 180�),

shown in Figure 9a, there is a minimum at x = 0 where thewave magnetic field has a node, while the wave field issignificantly larger 30–40 km away from the origin. Thisskip distance of ICD generated waves has been observed inthe high-latitude experiments at HAARP, as discussed inSection 4. The amplitude maxima at x ≈ 100 km are asso-ciated with the shear Alfvén waves that propagate verticallyalong the magnetic field lines, while the smaller maxima at x� 800 km are associated with magnetosonic waves that havebeen guided in the ionospheric duct. For the oblique geo-magnetic field (q = 135�), we see in Figure 9b that there areamplitude maxima somewhat to the left of x = 0 and at x =800 km. The wave pattern of Bx and Bz in Figures 5a and 5bindicates that the amplitude maximum to the left of x = 0 inFigure 9b corresponds to magnetosonic waves injectedalmost vertically from the heating region, while the maxi-mum at x = 800 km corresponds primarily to magnetosonicwaves that have been injected obliquely and reflected fromthe upper ionospheric layer with higher Alfvén speed abovethe F2 peak. Finally, for the case of almost horizontal,curved magnetic field, shown in Figure 9c, the amplitudemaxima at x = � 100 km are due to large amplitude mag-netosonic waves excited in the heated region, as seen inFigure 8, while the amplitude is somewhat smaller at x = 0.The amplitude maxima at x = � 800 km correspond pri-marily to magnetosonic waves that have been injectedobliquely and reflected from the upper ionospheric layerwith higher Alfvén speed, seen in Figures 8a and 8b.

3.4. Transition and Propagation of ELF/ULF WavesThrough the Ionospheric Layers

[21] In order to understand the propagation of ELF andULF waves through the ionospheric layers down to the

Figure 9. The amplitude of the horizontal magnetic field Bx at the plasma-free space boundary at z = 90 km(dotted line) and on ground (solid line) for (a) vertical geomagnetic field q = 180�, (b) oblique geomagneticfield q = 135�, and (c) curved geomagnetic field q ¼ 90� þ x=Lð Þ � 180�=p with L = 3000 km.


8 of 22

neutral atmosphere, we use a simplified model that estimatesthe spatial attenuation of the wave amplitude along thepropagation path. In this model, a plane wave is launched at300 km and is propagating down to the plasma-free spaceboundary at 90 km. The relative amplitude for waves prop-

agating vertically is estimated as exp � R zz1kI zð Þdz

h i, where

the spatial attenuation rate kI is obtained from the dispersionrelation of the wave with the assumption that the iono-spheric profile is slowly varying in comparison with thewave, and z1 is the altitude of the source of the wave (z1 =300 km in our case). The dispersion relation is obtained byFourier analyzing the system (5) and (6) in space and time,by assuming, for example, that the geomagnetic field isdirected along the x-axis and that A and E are proportional toexp(ikkx + ik? z�iw t). This yields the dispersion relation

X 2 � Xv2A k2 þ k2k� �

a22 þ k2?a11h i

þv4A k2k2k a222 þ a223

� �þ k2k2?a11a22h i

¼ 0;ð11Þ

where we have denoted X = w[w + iwci (Gin + Gen)]and k2 = kk

2 + k?2 , and aij are given by a11 ¼

GinGen � i w=wcið ÞGen; a22 ¼ 1þ GinGen � i w=wcið ÞGen; anda23 ¼ Gin � Gen � iw=wci. In addition to the roots ofequation (11), we also have solutions given by X = 0,which are non-propagating and hence do not contributeto the wave dynamics. We next define k? = k sin (q) and kk =k cos (q), where q = 0� corresponds to parallel propagationand q = 90� corresponds to perpendicular propagation to themagnetic field lines. Inserted into equation (11), this gives

X 2 � Xv2Ak2 1þ cos2 qð Þ� �

a22 þ sin2 qð Þa11�

þ v4Ak4½ cos2 qð Þ a222 þ a223

� �þ sin2 qð Þa11a22� ¼ 0: ð12Þ

The spatial attenuation is now obtained by assuming that thewavenumber is complex, k = kR + i kI, where kR is the realwavenumber and kI the spatial attenuation rate, while the wavefrequency w is real, and by solving equation (12) for k.[22] It is illustrative to discuss some limiting cases of

equation (11). In the collisionless limit Gin = Gen = 0, wehave X = w2, a11 = 0, a22 = 1, and a23 ¼ �iw=wci , whichinserted into equation (11) gives the dispersion relation forthe collisionless shear Alfvén and magnetosonic waves,

w2 � v2Ak2

� �w2 � v2Ak

2k

� �� w2

w2ci

v4Ak2k2k ¼ 0: ð13Þ

[23] For parallel propagation k = kk, we have from (13)

w2∓v2Ak2kwwci

� v2Ak2k ¼ 0 ð14Þ

for the whistler (upper sign) and ion cyclotron (lower sign)branches of the shear Alfvén wave. For parallel propagationalong the magnetic field lines, k? = 0 and k = kk,equation (11) yields X � vA

2kk2(a22 ∓ ia23) = 0, or

w2 þ iGen∓1ð Þv2Ak2k þ iw2ci Gin þ Genð Þ

h i wwci

� 1þ GenGin � i Gin � Genð Þ½ �v2Ak2k ¼ 0: ð15Þ

for the right-hand circularly polarized helicon/whistler (uppersign) and left-hand circularly polarized ion cyclotron (lowersign) branches of the shear Alfvén wave. On the other hand,assuming perpendicular propagation to the magnetic fieldlines, kk = 0 and k = k?, equation (11) yields the two solutionsX–vA

2 k?2 a22 = 0 and X–vA

2 k?2 a11 = 0. Setting X–vA

2 k?2 a22 = 0

yields the dispersion relation for dampedmagnetosonic waves,

w2 þ i Gin þ Genð Þw2ci þ Genv

2Ak

2?

� wwci

� 1þ GenGinð Þv2Ak2? ¼ 0;

ð16Þ

which recovers the magnetosonic mode w2 = vA2 k?

2 in the limitGen = Gin = 0. Setting instead X–vA

2 k?2 a11 = 0 gives strongly

damped modes.[24] The right-hand circularly polarized shear Alfvén

waves lie on the same dispersion curve as the helicon waves,and connects smoothly to the collisionless shear Alfvén wavefor li kk≫ Gin, where li ¼ c=wpi is the ion inertial length. Forthe strongly collisional cases li kk < Gin/2, the frequencyalmost exactly obeys the helicon wave dispersion relationw = vA

2 kk2/wci. Hence, by studying the dispersion relation (15)

numerically, we found the following rule of thumb

likk <Gin

2→helicon waves

likk ≫ Gin;→shear Alfv�en=whistler waves:

[25] The shear Alfvén waves become more damped forincreasing ion-neutral collision frequency, while heliconwaves become less damped for increasing ion-neutral colli-sion frequency. The helicon waves are weakly damped forGin ≫ 1, since then the Hall conductivity dominates over thePedersen conductivity, but become strongly damped forGin ⪅ 1, when the Pedersen conductivity dominates. Thesefeatures are clearly seen in Figure 10a, where we have solvedequation (15) for complex wavenumbers kk = kR + ikI fordifferent values of the wave frequency w = 2pf, and have usedthe model ionospheric profiles in Figure 2 to produce profilesof the spatial attenuation rate kI as a function of altitude. Theattenuation rate has a strongly peaked maximum at around120 km, which is the Pedersen conductivity dominated layer,and decreases rapidly both above and below this altitude.Below 110 km, we have weakly damped helicon waves, andabove 150 km, we have weakly damped shear Alfvén waves.The decrease of the amplitude in the propagation from z1 =

300 km is estimated as exp � R zz1kI z;ð Þ; d; z

h i, and is plotted

in Figure 10b. Above the altitude where w = Gin wci = vin, thedynamics is dominated by the polarization current whichgives rise to the Alfvén waves. For the ionospheric profiles inFigure 2, this altitude is 165 km for 2 Hz, 130 km for 10 Hzand 115 km for 50 Hz. Hence, as seen in Figure 10b, waveswith higher frequencies can penetrate the plasma to loweraltitudes as weakly damped Alfvén waves, and waves withlarge enough frequencies can connect directly to the weaklydamped helicon waves below 120 km, where the dynamics isdominated by the Hall current. This leads to an increase of therelative amplitude at 90 km for frequencies larger than about


9 of 22

10 Hz. The increase in amplitude for low frequencies below10 Hz is due to that the penetration through the Pedersenlayer is diffusive with the diffusion coefficient D ≈ 1=ðm0sPÞ.Hence the skin depth for wave penetration can be estimatedas l � 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim0 w sPp

, which increases for decreasing w.[26] In Figure 11, we investigate the propagation of left-

hand polarized waves along the geomagnetic field linesthrough the ionospheric layer. For large wavenumbers, theyconverge to the ion cyclotron resonance and at small wave-numbers to the shear Alfvén waves. However, in contrast tothe right-hand polarized wave, they do not have a weaklydamped “helicon” mode at large ion-neutral collision fre-quencies. Using the ionospheric profiles in Figure 2, thespatial attenuation rate and integrated spatial attenuation areplotted in Figures 11a and 11b, and predict a higher degreeof damping for the left-hand polarized waves compared tothe right-hand polarized waves in Figure 10. Left-handpolarized waves with frequencies above the ion cyclotronfrequency 38 Hz are non-propagating, and are therefore notshown in Figure 11.[27] For applications to the equatorial ionosphere, it is also

interesting to investigate the propagation of magnetosonicwaves perpendicular to the geomagnetic field lines. Solu-tions of equation (16) are plotted in Figure 12. The spatialattenuation rate (as well as the real wavenumber) increaseswith increasing values of Gin. As seen in Figure 12, themagnetosonic waves become strongly attenuated once theyenter the E-region at z ≈ 120 km, where Gin ≳ 1, and theirrelative amplitudes quickly become vanishingly small beforereaching the plasma-free space boundary.

[28] Since the magnetosonic wave and the right-handpolarized whistler mode wave lie on the same dispersionsurface with different propagation directions to the geo-magnetic field lines, it is interesting to investigate the spatialattenuation rate at oblique angles to the geomagnetic fieldlines. In Figure 13a, we have plotted the integrated spatial

attenuation exp � R z0z1kI z;ð Þ; d; z

h i, of a plane wave gener-

ated at z1 = 300 km, which has propagated vertically down toz0 = 90 km. The main result is that there is a strong attenu-ation of (magnetosonic) waves propagating exactly perpen-dicularly to the magnetic field (corresponding to q = 90�).This can be seen in Figure 13a where the waves are mod-erately attenuated for q = 0� (helicon/whistler waves) and q =45�, while for q = 90�, the wave is strongly attenuated exceptat frequencies below 1 Hz. As seen in Figure 13b, theangular dependence of the integrated spatial attenuation isvery different for the low frequency 2 Hz and the higherfrequencies 10 Hz and 50 Hz. The 2 Hz signal is stronglyattenuated only in an angular interval of a few degreesaround the purely perpendicular propagation, which indi-cates that the reason for the amplitude maxima at x = � 100km in Figure 9c, is that the magnetosonic wave has propa-gated obliquely (not exactly perpendicularly) to the geo-magnetic field lines through the ionosphere down to theplasma-free space boundary. Waves with frequencies 10 Hzand 50 Hz are strongly attenuated in an interval of about �20� from perpendicular propagation. At q = 90� the inte-grated spatial attenuation is �10�2, 10�3 and 10�8, respec-tively, for frequency 2 Hz, 10 Hz and 50 Hz. Hence, one

Figure 10. (a) The vertical profiles of the spatial attenuation rate kI(z) and (b) the integrated spatial atten-

uation exp � R zz1kI z;ð Þ; d; z

h iof a right-hand polarized shear Alfvén wave excited at z1 = 300 km, and

propagating parallel to the geomagnetic field lines, for f = 2 Hz (solid lines), f = 10 Hz (dashedlines) and f = 50 Hz (dotted lines).


10 of 22

Figure 11. (a) The vertical profiles of the spatial attenuation rate kI(z) and (b) the integrated spatial attenu-

ation exp � R zz1kI z;ð Þ; d; z

h iof a left-hand polarized shear Alfvén/ion cyclotron wave excited at z1 = 300 km,

and propagating parallel to the geomagnetic field lines, for f = 2 Hz (solid lines) and f = 10 Hz (dashed lines).For frequencies above the ion cyclotron frequency fci = 38 Hz, the wave mode becomes non-propagating andstrongly damped.

Figure 12. (a) The vertical profiles of the spatial attenuation rate kI(z) and (b) the integrated spatial atten-

uation exp � R zz1kI z;ð Þ; d; z

h iof a magnetosonic wave excited at z1 = 300 km, and propagating perpendic-

ular to the geomagnetic field lines, for f = 2 Hz (solid line), f = 10 Hz (dashed line) and f = 50 Hz (dotted line).


11 of 22

would expect that there is an effective high-frequency cutoffof the signal around approximately 5–10 Hz measured onground in the equatorial region for ULF waves generated atthe vertically overhead F2 peak.[29] Finally, it is worth noting that a significant amount of

absorption of the HF electromagnetic wave also can occur inthe D-region ionosphere below 90 km, where the product ofthe electron density and the electron-neutral collision fre-quency has a maximum. This effect is not accounted for bythe present model, where the modulated electron pressure isassumed to be an external source.

4. Some Experimental Results

[30] While the emphasis on the paper was the theoreticalmodeling of ICD in general and its properties as a functionof the geomagnetic latitude of the heater, we present in thissection specific results of ICD experiments that relate tofeatures demonstrated in the modeling. The most compre-hensive set of experiments were conducted using theHAARP ionospheric heater in Gakona, Alaska and some ofthe ground measurements were reported by Papadopouloset al. [2011b]. There are also two experimental results frommid-latitude heaters, one using the Arecibo heater [Gangulyet al., 1986] that was subsequently destroyed by floods and arecent one using the Sura facility [Kotik and Ryabov, 2011].

We restrict our discussion here to HAARP experiments thatobserved ICD generated ULF/ELF waves measured on theground and simultaneously by DEMETER overflying at670 km altitude and close to the HAARP Magnetic Zenith(MZ). In all cases the electrojet current, as determined bythe magnetometer measurements and by using the strengthof 2 kHz signals as proxy for the strength of electrojet[Papadopoulos et al., 2011b], was absent. In all the exam-ples shown here the ground signature is the magnetic sig-nature of the generated signal while the satellite measuresthe electric field signal. Noise prevents magnetic field mea-surements at ULF/ELF frequencies in the noisy trough ion-osphere above HAARP.[31] A typical ICD experiment was performed on

October 30, 2010 from 06:00:00–06:19:30 UT. During theexperiment, HAARP transmitted at 2.8 MHz, O-mode, atpeak power (3.6 MW), an amplitude modulated squarewaveform at 2.5 Hz, with the heater beam pointing along themagnetic zenith direction. The local VHF Riometer showedlow absorption at �0.2 dB at 30 MHz. The onsite fluxgatemagnetometer showed flat HDZ traces with no fluctuation,indicating a very quiet ionosphere. The onsite digisondeshowed foF2 at 1.45 MHz and F-peak at 260 km altitudewith extremely weak E-layer. Figure 14 shows the wavespectrum measured simultaneously on the ground approxi-mately 20 km away from the HAARP heater and by the

Figure 13. The integrated spatial attenuation exp � R z0z1kI z;ð Þ; d; z

h iat the bottom-side ionosphere

z0 = 90 km for a plane wave excited at z1 = 300 km, (a) as a function of frequency for q = 180� (solidline), q = 135� (dashed line) and q = 90� (dotted line), and (b) as a function of q for wave frequency2 Hz (solid line), 10 Hz (dashed line) and 50 Hz (dotted line). The low frequency wave 2 Hz is stronglyattenuated in a narrow region around q = 90�, corresponding to perpendicular propagation to the geomag-netic field lines.


12 of 22

overflying DEMETER. Figure 14 (top left) is a projection ofthe DEMETER orbit moving with respect to the HAARPMZ. The red marking on the orbit represents the orbit partduring which there was a strong signal at the injected fre-quency. The time duration was 20–25 s that corresponds to adistance of 100–150 km. The most important aspect of themeasurement is the observed strong confinement of the ELFwaves to the injection field line as expected by shear Alfvénwaves and as seen in the simulation in Figure 4.[32] This should be compared with Figure 15a that shows

the electric field spectrum measured by DEMETER duringa 0.1 Hz modulation along three color-coded parts of thetrajectory. Notice that the 0.1 Hz signal was detectedbetween 06:51:19 and 06:53:19 UT, approximately six timeslonger than the signal measured in the previous cases. In factthe low frequency and strong signal in this case allowed forthe measurement of the waveform of the electric field shownin Figure 15b. We identify this wave with a magnetosonicwave that propagates isotropically, as seen in the simulationresults of Figure 5.[33] A most unexpected result revealed by the simulations

is the distribution of the ELF magnetic amplitude on theground signatures as a function of distance from the heater,shown in Figure 9. Previous experience with ELF and ULFwaves generated by modulating the Polar Electrojet (PEJ)[Rietveld et al., 1984, 1987, 1989; Papadopoulos et al.,1990, 2005; Moore, 2007; Payne et al., 2007] indicatedthat the wave amplitude had a maximum in the vicinity ofthe heater while monotonically decreasing with distance in afashion consistent with guided wave propagation. However,as noted in Section 3.3, ICD driven waves have a minimumat the ground location defined by its interception with themagnetic field line that passes through the heated volume.This is seen clearly in the vertical magnetic field case(Figure 9a) where the magnetic field has a null at thetransmitter location and a maximum approximately 200 kmaway. Figure 16 shows the results of a HAARP testconducted under daytime conditions during the period17–25 August, 2009. The magnetic signals were measuredsimultaneously in Gakona and in Homer 300 km away. Anumber of frequencies between 12 Hz and 44 Hz were gen-erated and simultaneously measured at the two sites. Thesignals are numbered from 1–40 and their frequency is color-coded. Signals 18–27 were generated by PEJ modulationwhile the remaining by ICD. It is clear that signals 18–27 arelarger in Gakona than Homer consistent with previous PEJobservations. However, the situation is reversed for theremaining ICD signals many of which were detected only inHomer and not in Gakona. This, as explained previously andseen in the simulations, is a result of the two-step ELF gen-eration by ICD. Namely the Hall current that acts as a sec-ondary antenna that generates the ground signals occurs atthe intersection of the magnetosonic wave with the Hallregion. As a result the minimum occurs at the intersection ofthe magnetic field line with ground and the maximum at adistance that depends on the heating altitude and the geo-magnetic latitude.[34] We finally present some results indicative of far

propagation of magnetosonic waves in the Alfvénic duct andmeasurements on the ground that indicate the presence of

skip distance such as seen in the simulations of Figure 9.Magnetometers were located in Gakona, Alaska, as well asin Lake Ozette, WA (2100 km), Makua Valley, Hawaii(4700 km) and Guam (7700 km) marked in Figure 17 (bot-tom right). The three diagrams in Figure 17 show magne-tometer measurements taken during three different campaignsin 2009. It is important to mention that all the measurementsshown in the figures correspond to ICD generation. The farsites did not record any signals 0.1–40 Hz range of ourexperiments during PEJ generation.[35] A more comprehensive analysis of the far site results

along with detailed modeling will be presented elsewhere.As an example we present here only the analysis and spectrameasured by the ground ELF sites during the ICD experi-ment conducted May 6, 2009 between 08:00:15 and08:20:15 UT, at a modulation frequency 3.8 Hz and detectedby all the available ground sites in Figure 17. Figure 18shows the ground magnetometer measurement at Gakonaindicating that there was no electrojet over the site during theexperiment time. This was also verified by the absence of2 kHz signals when it was tested between 08:00:00 and08:00:15. Figure 19 shows the ionosonde measurementindicating the absence of D/E regions and a strong F-layerwith FoF2 close to 3.00 MHz. The HAARP heater operatedat full power (3.6 MW), O-mode and frequency 3.25 MHz,consistent with under-dense heating. Figure 20 shows theELF spectra measured at the four sites during the 20 minutesexperiment time. In all cases there is a modest 3.8 Hz peak,and the S/N ratio is approximately 3 dB. The measuredamplitudes vary between 30–40 fT.[36] Our tentative conclusion of the results shown in

Figure 17 is that the signals received in the far sites propa-gated laterally in the Alfvénic waveguide as magnetosonicwaves generated by ICD. Their ground signatures werelocalized in regions where the magnetosonic wave reachedthe E-region after reflection from the top of the Alfvénicguide. We attribute the spottiness of the ground measure-ments in the fact that the skip distance in the Alfvénic guidedepends on the ionospheric parameters that are highlydynamic and variable with latitude and time and to the valueof the Hall conductivity at the E-region reflection points ofthe magnetosonic waves. We should also caution the readerthat a comparison of the simulations with the far field mea-surements beyond a qualitative or notional level is not pos-sible, since the simulations assumed a laterally homogeneousionosphere and a highly unrealistic constant direction of theambient magnetic field with latitude.

5. Conclusions

[37] We have here developed a cold plasma model forthe propagation of ELF and ULF electromagnetic waves inthe Earth’s ionosphere, which we have used to study thedynamics of waves that are generated by the modulation ofthe electron pressure via external high-frequency heating ofthe F2 peak. Of particular interest is the penetration of theULF waves through the ionospheric layer down to the free-space atmosphere and to the ground for different configura-tions of the geomagnetic field lines. At high altitudes, whereion-neutral and electron-neutral collisions are relatively rare,


13 of 22

Figure 14. (a) Electric field spectrum measured by DEMETER during 2.5 Hz modulation of the HAARPheater. The heater operated at full power (3.6 MW), 2.8 MHz frequency and O-mode polarization. The topleft diagram indicates the DEMETER location with respect to the HAARP MZ. (b) Magnetic field spec-trum measured at Gakona simultaneously with DEMETER.


14 of 22

Figure 15


15 of 22

the dynamics is governed by the polarization current, whichgives rise to the magnetosonic and shear Alfvén waves.When the ion-neutral collision frequency is larger than thewave frequency, but smaller than the ion cyclotron fre-quency, the ions and electrons experience E � B0 drift, andthe ion current partially cancels the electron Hall current.This gives rise to a diffusive behavior of the plasma wherethe Pedersen conductivity dominates. In the Hall regionbelow 120 km, the ions are collisionally glued to the neutralswhile the electrons are mobile, which gives rise to weaklydamped helicon wave dynamics governed by the Hall con-ductivity of the electrons. To understand the propagation ofELF and ULF waves through these layers down to theground, we have performed simulations of the governingtime-dependent equations for typical ionospheric parametersand analyzed the dispersive properties of the waves. We findthat in the auroral region, where the geomagnetic field is

almost vertical, the right-hand circularly polarized shearAlfvén wave can connect smoothly to the helicon wave via atransition through the Pedersen layer where it loses aboutone order of magnitude wave energy (a factor of a few waveamplitude). A similar behavior is seen for oblique geomag-netic fields, typical for mid-latitude heating facilities, whereshear Alfvén waves propagate along the geomagnetic fieldlines and are injected into the earth-ionosphere waveguide.For horizontal geomagnetic fields the dynamics is interest-ing in that current continuity demands that no large verticalcurrents can exist close to the plasma-free space boundary.The physics is here very similar to the equatorial electrojet,where the Cowling effect gives rise to intense horizontalcurrents perpendicular to the geomagnetic field. Our simu-lation has shown that modest amplitudes of the heating in theexperiment can give rise to thin ELF current sheets in theE-region with dipole moments of about 107 Am, which may

Figure 15. (a) Electric field spectrum measured during overfly of HAARP when the heater was modulated at 0.1 Hz.The heater was operating at full power (3.6 MW), and O-mode, 3.2 MHz frequency. The insert indicates by color theDEMETER locations when the data shown were plotted. Notice that the 0.1 Hz frequency appears as dominant only duringthe DEMETER crossings near the MZ position while it disappears before and after. Ground data were too noisy at 0.1 Hz toresolve the 0.1 Hz line during the crossing. Notice that the detection time was close to 2 minutes, vs. only 20–30 seconds forthe cases shown in Figure 14. (b) Waveform of the electric field measured by DEMETER during the part of the orbit markedby red, showing clear 10-second oscillations.

Figure 16. ELF wave amplitudes simultaneously measured at Gakona, AK, site of HAARP and Homer,AK, 330 km away for different frequencies 12–44 Hz (color coded). The different ELF signals are enu-merated 1–40 (horizontal axis). The blue circle indicates ELF waves (signals 18–27) generated by PEJcurrent modulation, while the remaining by ICD. Contrary to the PEJ generated waves the amplitudesof the ICD waves were larger at the further site consistent with the simulation results shown in Figure 9a.


16 of 22

work as an antenna to inject electromagnetic waves into theearth-ionosphere waveguide. Some features of the modelingresults are relevant for recent measurements of ICD gener-ated ULF/ELF waves at HAARP, including a skip distance,where the ICD generated signal amplitude at the ground has a

minimum near the heater, in contrast to PEJ generated sig-nals. Some far field observations are also consistent with ICDgenerated magnetosonic waves guided in the ionosphericwaveguide, not observed for PEJ generated waves. These

Figure 17. ICD generated ELF waves measured at 1. Gakona, Alaska, 2. Lake Ozette, WA (2100 km),3. Makua Valley, Hawaii (4700 km) and 4. Guam (7700 km) during three campaigns conducted in2009. Notice that in many cases the waves were detected only at the far sites and not in Gakona. A numberof times the waves were measured at two or three sites. In only one case waves at 3.8 Hz with amplitude 30 fTwere detected in all four sites (May 6, 2009, 08:00:15 UT to 08:20:15 UT). We attribute the unpredictabledetection to the fact that the skip distance seen in the simulations of Figure 9 depends critically in the iono-spheric conditions and to the noise at the detection site.


17 of 22

Figure 18. Gakona magnetometer measurement indicating the absence of the electrojet during the testingperiod (marked by red arrow).

Figure 19. Gakona ionosonde measurement showing very week D/E layers and a well- developed F-layerwith FoF2 close to 3 MHz.


18 of 22

issues along with injection of ICD generated waves in theradiation belts will be discussed in a future publication.

Appendix A: Derivation of the GoverningEquations Using Dielectric and ConductivityTensors

[38] We here derive a cold plasma model for ELF andULF wave propagation in the ionosphere, describing theinteraction between the magnetic and electric fields in termsof dielectric and conductivity tensors. In order to reduce theanalytic work, it is convenient to introduce matrix formalismand organize the vectors as column vectors to describe theelectron and ion dynamics. By denoting

ve � B0

B0þ meven

eB0ve ≡ Reve ðA1Þ

and

vi � B0

B0� mivin

eB0vi ≡ R ivi; ðA2Þ

where Re and R i are tensors/matrices operating on the vec-tors, the electron and ion momentum equations (3) and (4)can be written

E ¼ �B0Reve �rPe

en0ðA3Þ

and

∂vi∂t

¼ e

miEþ B0R ivi

� �; ðA4Þ

respectively. From equation (A3) we have, for example,

ve ¼ �R�1

e EþrPe=ðen0Þ½ �=B0 , where R�1

e is the inverse

of the tensor/matrix Re, given in equation (A10) below. Anevolution equation for the electric field can be found bytaking the time derivative of equation (A3) and expressingthe right-hand side in terms of E and B. The result is

∂E∂t

¼ e

miB0 R i � Re

� �E� B2

0ReR i r� Bð Þm0en0

" #

� B0

m0en0Re r� r� Eð Þ½ � � 1

en0r ∂Pe

∂tþ eB0

miR i

rPe

en0ðA5Þ

In deriving equation (A5), we have used that R i and Re

commutate, i.e. R iRe ¼ ReR i. Equation (A5) can be writtenin the form

∂E∂t

¼ ɛ�1 �sEþr� B

m0

� �� Re r� r� Eð Þ½ �

m0~s

� 1

en0r ∂Pe

∂tþ wciR i

rPe

en0; ðA6Þ

Figure 20. ELF spectra measured at the four sites shown in Figure 17 taken during the 20-minute trans-mission of 3.8 Hz modulated HF waves by the HAARP heater. The amplitude of the detected signal(marked by red arrows) was between 30–40 fT while the signal-to-noise ratio was close to 3 dB.


19 of 22

where ɛ ¼ � ɛ0c2=v2A� �

ReR i

� ��1is the effective dielectric

tensor, vA ¼ cwci=wpi is the Alfvén speed,s ¼ ~s R�1

e � R�1

i

� �is the conductivity tensor, and we denoted ~s ¼ ɛ0w2

pe=wce .

Using the definitions (A1) and (A2) of Re and R i , we find

that ɛ�1sE ¼ �wci R i � Re

� �E ¼ wci Gin þ Genð ÞE, were

we have denoted Gen ¼ ven=wce and Gin ¼ vin=wci. Hence,equation (A6) can be written

∂E∂t

¼ �wci Gin þ Genð ÞEþ ɛ�1r� B

m0� Re r� r� Eð Þ½ �

m0~s

� 1

en0r ∂Pe

∂tþ wciRi

rPe

en0ðA7Þ

The last step is to introduce the scalar and vector potentials fand A via B = r � A and E ¼ �rf� ∂A=∂t using thegauge f = 0. Faraday’s law (1) is then fulfilled trivially andwe obtain the system (5)–(6).[39] It is instructive to discuss some limiting cases of the

governing equations without the driving terms proportionalto Pe. We note that the left-hand side of equation (A6) isassociated with the polarization current, i.e. jp ¼ ɛ ∂E=∂t. Inthe collisionless limit wci Gin þ Genð Þ≪ ∂=∂tj j, the polariza-tion current dominates the physics and we have shear Alfvénand magnetosonic waves. In the limit Ge = Gi = 0, equation

(A7), using the expressions of ɛ�1, Re and R i , reduces to

the collisionless Hall-MHD equation

∂E∂t

¼ � v2AB20

r� Bð Þ � B0½ � � B0 � c2wce

w2peB0

r� r� Eð Þ½ � � B0:

ðA8Þ

[40] In the frequency range much below the ion cyclotronfrequency, equation (A8) reduces further to the ideal MHDequation

∂E∂t

¼ � v2AB20

r� Bð Þ � B0½ � � B0; ðA9Þ

which, together with equation (1), supports the shear Alfvénand magnetosonic wave modes. In the opposite limit, whenwci Gin þ Genð Þ≫ j∂=∂tj , the ion inertial effects (and thusthe polarization current) can be neglected, and the currentis given by the plasma conductivity as j ¼ sE . We thenhave from equation (A6) the quasi-steady state limit (omit-ting the externally modulated electron pressure) �sEþr� B=m0 ¼ 0, which, coupled with equation (1), supportshelicon/whistler wave dynamics in the frequency rangebelow the ion cyclotron frequency when sH ≫ sP, and dif-fusive behavior for sP ≫ sH.[41] There are also time-independent and non-propagating,

damped solutions of the system (5) and (6). We see thatE = 0 and A = rU, where U(r) depends only on space,is a time-independent solution (conserved quantity) ofequations (5) and (6). For the case Gin + Gen = constant, wehave with E = �rf into (5) and (6), ∂A=∂t ¼ �E and∂E=∂t ¼ �wci Gin þ Genð ÞE , which have solutions propor-tional to exp[�wci (Gin + Gen)t] that decay in time but do notpropagate in space. This mode does not involve the excita-tion of magnetic fields since r � E = �r � rf = 0 in

equation (1). In our model, since the term proportional tor∂Pe=∂t in the right-hand side of equation (6) is the gradientof a scalar, it primarily excites the non-propagating modesand not the electromagnetic modes.[42] We next give the expressions of the dielectric and

conductivities tensors, and their inverses, in matrix form. Forthe case where B0 is directed along the x-axis, we have

Re ¼Gen 0 0

0 Gen 1

0 �1 Gen

264

375; R

�1

e ¼

G�1en 0 0

0Gen

1þ G2en

�1

1þ G2en

01

1þ G2en

Gen

1þ G2en

2666664

3777775

ðA10Þand

R i ¼�Gin 0 0

0 �Gin 1

0 �1 �Gin

264

375; R

�1

i ¼

�G�1in 0 0

0�Gin

1þ G2in

�1

1þ G2in

01

1þ G2in

�Gin

1þ G2in

2666664

3777775:

ðA11Þ

[43] The conductivity tensor is

s ¼ ~s R�1

e � R�1

i

� �¼

sk 0 00 sP �sH

0 sH sP

24

35; ðA12Þ

where the parallel, Pedersen, and Hall conductivity,respectively, are given by sk ¼ ~s G�1

en þ G�1in

� �,

sP ¼ ~s Gen= 1þ G2en

� �þ Gin= 1þ G2in

� �� , and sH ¼

~s 1= 1þ G2en

� �� 1= 1þ G2in

� �� . The dielectric tensor ɛ is

proportional to the conductivity tensor s, as ɛ ¼s= wci Gin þ Genð Þ½ �, and the inverse of the dielectric tensoris given by

ɛ�1 ¼ ɛ�1

0

v2Ac2

GinGen 0 00 1þ GinGen Gin � Gen

0 Gen � Gin 1þ GinGen

24

35: ðA13Þ

[44] The general case of oblique and r-dependent geo-magnetic fields are obtained by multiplying the above

matrices by rotation matrices, as A ¼ S AxS�1

, where Ax

represents any of the matrices for the case when the geo-magnetic field is parallel to the x-axis. The rotation matrixand its inverse are given by

S ¼sin qð Þ 0 � cos qð Þ0 1 0

cos qð Þ 0 sin qð Þ

24

35;

S�1 ¼

sin qð Þ 0 cos qð Þ0 1 0

� cos qð Þ 0 sin qð Þ

24

35; ðA14Þ

where q is the angle between the vertical z-axis and themagnetic field lines, defined via the relations sin qð Þ ¼B0x=B0 and cos qð Þ ¼ B0z=B0, and which may vary in space,


20 of 22

q = q(r). The matrices Re, R i, ɛ, and s, and their inverses,are all of the general form

Ax ¼ak 0 00 aP aH0 �aH aP

24

35 ðA15Þ

for =90�, from which we obtain the general case of arbitraryq as

A ¼a11 a12 a13�a12 a22 a23a13 �a23 a33

24

35 ðA16Þ

where a11 = ak sin2 (q) + aP cos2 (q), a12 = aH cos (q), a13 =

(ak–aP) cos (q) sin (q), a22 = aP, a23 = aH sin (q), and a33 =ak cos2 (q) + aP sin2 (q).

Appendix B: Boundary Conditions and Free SpaceElectromagnetic Fields

[45] In our model, free space is characterized by infinitespeed of light, while the ground is perfectly conducting. Atthe plasma-free space boundary, the x- and y-components ofthe electric field and vector potential, as well as their normalderivatives, are continuous over the boundary, while at theground the x- and y-components of the electric field andvector potential are zero. Hence, the boundary conditionsand free space fields are analogous for E and A, and it issufficient to discuss the electric field.[46] The free space electric field Efree is governed by

r2Efree = 0. Fourier transforming in the x-direction byassuming that Efree is proportional to exp(ikxx), gives

∂2Efree

∂z2� k2xEfree ¼ 0; ðB1Þ

with the general solution

Efree ¼ C1 exp kxj jzð Þ þ C2 exp � kxj jzð Þ; ðB2Þ

whereC1 andC2 are vector-valued integration constants withcomponents along the x- and y-directions. Since Ex and Ey

and their normal derivatives are continuous at z = z0, we havefor the x- and y-components of the electric field,

Eplasma z ¼ z0ð Þ ¼ Efree z ¼ z0ð Þ¼ C1 exp kxj jz0ð Þ þ C2 exp � kxj jz0ð Þ ðB3Þ

and

∂Eplasma

∂zz ¼ z0ð Þ ¼ ∂Efree

∂zz ¼ z0ð Þ

¼ kxj j C1 exp kxj jz0ð Þ � C2 exp � kxj jz0ð Þ½ �: ðB4Þ

[47] At the perfectly conducting ground we have Efree(z = 0)= 0, hence C2 = �C1, and from equation (B3) we obtain(dropping the subscripts “plasma” and “free” on E)

C1 ¼ �C2 ¼ E z ¼ z0ð Þexp kxj jz0ð Þ � exp � kxj jz0ð Þ : ðB5Þ

[48] Eliminating C1 and C2 from equation (B4), we obtainthe plasma-free space boundary condition at z = z0,

∂E∂z

¼ a kxj jð ÞE; ðB6Þ

where

a kxj jð Þ ¼ exp kxj jz0ð Þ þ exp � kxj jz0ð Þexp kxj jz0ð Þ � exp � kxj jz0ð Þ kxj j ðB7Þ

for |kx| > 0, and a ¼ 1=z0 for kx = 0. The inverse Fouriertransform of E with respect to kx gives the boundary con-dition in real space.[49] The free space electric field in the region 0 < z < z0 is

obtained from (B2) by using (B5) as

Efree ¼ exp kxj jzð Þ � exp � kxj jzð Þexp kxj jz0ð Þ � exp � kxj jz0ð ÞE z ¼ z0ð Þ: ðB8Þ

[50] The same boundary conditions are used for the x- andy-components of the vector potential A, so that

∂A∂z

¼ a kxj jð ÞA; ðB9Þ

at z = z0, and

Afree ¼ exp kxj jzð Þ � exp � kxj jzð Þexp kxj jz0ð Þ � exp � kxj jz0ð ÞA z ¼ z0ð Þ: ðB10Þ

[51] The z-component of the vector potential is obtainedfrom the assumption that By ¼ ∂Ax=∂z� ∂Az=∂x ¼ 0 at theboundary, or with ∂=∂x ¼ ikx,

Az ¼ 1

ikx

∂Ax

∂zðB11Þ

at z = z0. Similarly, since A and E are related by a timederivative, we have

Ez ¼ 1

ikx

∂Ex

∂zðB12Þ

at z = z0. The vertical components of the free space vectorpotential and electric field are

Az;free ¼ 1

ikx

∂Ax

∂z¼ kxj j

ikx

exp kxj jzð Þ þ exp � kxj jzð Þexp kxj jz0ð Þ � exp � kxj jz0ð ÞAx z ¼ z0ð Þ

ðB13Þ

and

Ez;free ¼ 1

ikx

∂Ex

∂z¼ kxj j

ikx

exp kxj jzð Þ þ exp � kxj jzð Þexp kxj jz0ð Þ � exp � kxj jz0ð ÞEx z ¼ z0ð Þ

ðB14Þ

respectively, for 0 < z < z0. The free space magnetic field isobtained as Bfree = r � Afree, while the current in free spaceis zero, jfree = 0.[52] The numerical approximation of the boundary con-

ditions (B6) and (B9) are done with one-sided difference


21 of 22

approximations ðE1 � E0Þ=Dz ¼ aE0 and ðA1 � A0Þ=Dz ¼aA0, where the subscript 0 denotes the unknown value at theboundary point and subscript 1 the known value at theneighboring point inside the computational domain. Solvingfor E0 and A0, we obtain the numerical boundary conditions

E0 ¼ E1

1þ aDz; ðB15Þ

and

A0 ¼ A1

1þ aDz; ðB16Þ

respectively. The boundary conditions (B11) and (B12) areapproximated as

Az;0 ¼ 1

ikx

Ax;1 � Ax;0

DzðB17Þ

and

Ez;0 ¼ 1

ikx

Ex;1 � Ex;0

Dz; ðB18Þ

where Ex,0 and Ax,0 are obtained from equations (B15) and(B17), respectively.At the top boundary at z = 1020 km, we usefirst-order outflow boundary conditions ∂A=∂t þ vA∂A=∂z ¼0 and ∂E=∂t þ vA∂E=∂z ¼ 0 for E and A, where vA is theAlfvén speed at the top boundary, and where the z derivativesare calculated with one-sided difference approximations.

[53] Acknowledgments. This work was partially supported by ONRMURI N00014-07-1-0789 and DARPA-DSO HR0011-09C-0099 contractto BAE-Systems. The authors are grateful to Nail Gumerov, GennadyMilikh, Xi Shao, Aram Vartanyan and A. S. Sharma at the University ofMaryland and to T. Wallace, John Labenski and I. Doxas at BAE Systems- AT for useful discussions. B. E. acknowledges the hospitality of Univer-sity of Maryland, as well as partial support by the DFG FOR1048 (Bonn,Germany).[54] Robert Lysak thanks Jean-Andre Sauvaud and another reviewer

for their assistance in evaluating this paper.

ReferencesBarr, R. (1998), The generation of ELF and VLF radio waves in the iono-sphere using powerful HF transmitters, Adv. Space Res., 21(5), 677–687,doi:10.1016/S0273-1177(97)01003-X.

Berthelier, J.-J., M. Malingre, R. Pfaff, E. Seran, R. Pottelette, J. Jasperse,J.-P. Lebreton, and M. Parrot (2008), Lightning-induced plasma turbu-lence and ion heating in equatorial ionospheric depletions, Nat. Geosci.,1, 101–105, doi:10.1038/ngeo109.

Berthold, W. K., A. K. Harris, and H. J. Hope (1960), World-wide effects ofhydromagnetic waves due to Argus, J. Geophys. Res., 65(8), 2233–2239,doi:10.1029/JZ065i008p02233.

Dessler, A. J. (1958), The propagation velocity of world-wide sudden com-mencements of magnetic storms, J. Geophys. Res., 63(2), 405–408,doi:10.1029/JZ063i002p00405.

Eliasson, B., and K. Papadopoulos (2009), Penetration of ELF currents andmagnetic fields into the Earth’s equatorial ionosphere, J. Geophys. Res.,114, A10301, doi:10.1029/2009JA014213.

Forbes, J. M. (1981), The equatorial electrojet, Rev. Geophys., 19, 469–504,doi:10.1029/RG019i003p00469.

Forbes, J. M., and R. S. Lindzen (1976), Atmospheric solar tides and theirelectrodynamic effects—II. The equatorial electrojet, J. Atmos. Terr.Phys., 38(9–10), 911–920, doi:10.1016/0021-9169(76)90074-X.

Ganguly, S., W. Gordon, and K. Papadopoulos (1986), Active nonlinearultralow-frequency generation in the ionosphere, Phys. Rev. Lett., 57,641–644, doi:10.1103/PhysRevLett.57.641.

Greifinger, C., and P. S. Greifinger (1968), Theory of hydromagneticpropagation in the ionospheric waveguide, J. Geophys. Res., 73(23),7473–7490, doi:10.1029/JA073i023p07473.

Greifinger, C., and P. Greifinger (1973), Wave guide propagation of micro-pulsations out of the plane of the geomagnetic meridian, J. Geophys. Res.,78(22), 4611–4618, doi:10.1029/JA078i022p04611.

Greifinger, C., and P. Greifinger (1976), Transient ULF electric and mag-netic fields following a lightning discharge, J. Geophys. Res., 81(13),2237–2247, doi:10.1029/JA081i013p02237.

Hayakawa, M., S. Pulinets, M. Parrot, and O. A. Molchanov (2006), Recentprogress in seismo electromagnetics and related phenomena, Phys. Chem.Earth, Parts A/B/C, 31(4–9), 129–131.

Kotik, D., and A. Ryabov (2011), New results of experiments on generationULF/VLF waves with SURA facility, Abstract SA42A-06 presented at2011 Fall Meeting, AGU, San Francisco, Calif.

Lysak, R. L. (1997), Propagation of Alfvén waves through the ionosphere,Phys. Chem. Earth, 22, 757–766, doi:10.1016/S0079-1946(97)00208-5.

Lysak, R. L. (1999), Propagation of Alfvén waves through the iono-sphere: Dependence on ionospheric parameters, J. Geophys. Res., 104,10,017–10,030, doi:10.1029/1999JA900024.

Lysak, R. L. (2004), Magnetosphere-ionosphere coupling by Alfvén wavesat midlatitudes, J. Geophys. Res., 109, A07201, doi:10.1029/2004JA010454.

Lysak, R. L., and Y. Song (2001), A three-dimensional model of the prop-agation of Alfvén waves through the auroral ionosphere: First results,Adv. Space Res., 28(5), 813–822, doi:10.1016/S0273-1177(01)00508-7.

Milikh, G. M., K. Papadopoulos, and C. L. Chang (1995), On the physics ofhigh altitude lightning, Geophys. Res. Lett., 22(2), 85–88, doi:10.1029/94GL02733.

Moore, R. C. (2007), ELF/VLF wave generation by modulated heating ofthe auroral electrojet, PhD thesis, Stanford Univ., Stanford, Calif.

Papadopoulos, K. (2009), Ionospheric current drive at low frequencies, EosTrans. AGU, 90(52), Fall Meet. Suppl., Abstract SA11B-01.

Papadopoulos, K., C. L. Chang, P. Vitello, and A. Drobot (1990), Onthe efficiency of ionospheric ELF generation, Radio Sci., 25(6),1311–1320, doi:10.1029/RS025i006p01311.

Papadopoulos, K., T. Wallace, G. M. Milikh, W. Peter, and M. McCarrick(2005), The magnetic response of the ionosphere to pulsed HF heating,Geophys. Res. Lett., 32, L13101, doi:10.1029/2005GL023185.

Papadopoulos, K., B. Tesfaye, H. Shroff, X. Shao, G. Milikh, C. Chang,T. Wallace, U. Inan, and D. Piddyachiy (2007), F-region magnetosphericULF generation by modulated ionospheric heating, Eos Trans. AGU,88(52), Fall Meet. Suppl., Abstract SM53D-04.

Papadopoulos, K., N. A. Gumerov, X. Shao, I. Doxas, and C. L. Chang(2011a), HF-driven currents in the polar ionosphere, Geophys. Res. Lett.,38, L12103, doi:10.1029/2011GL047368.

Papadopoulos, K., C.-L. Chang, J. Labenski, and T. Wallace (2011b), Firstdemonstration of HF-driven ionospheric currents, Geophys. Res. Lett., 38,L20107, doi:10.1029/2011GL049263.

Payne, J. A., U. S. Inan, F. R. Foust, T. W. Chevalier, and T. F. Bell (2007),HF modulated ionospheric currents, Geophys. Res. Lett., 34, L23101,doi:10.1029/2007GL031724.

Rastogi, R. G. (1989), The equatorial electrojet, in Geomagnetism, vol. 3,edited by J. Jacobs, pp. 461–525, Elsevier, New York.

Rietveld, M. T., R. Barr, H. Kopka, E. Nielson, P. Stubbe, and R. L. Dow-den (1984), Ionospheric heater beam scanning: A new technique for ELFstudies of the auroral ionosphere, Radio Sci., 19(4), 1069–1077,doi:10.1029/RS019i004p01069.

Rietveld, M. T., H.-P. Mauelshagen, P. Stubbe, H. Kopka, and E. Nielsen(1987), The characteristics of ionospheric heating-produced ELF/VLFwaves over 32 hours, J. Geophys. Res., 92, 8707–8722, doi:10.1029/JA092iA08p08707.

Rietveld, M. T., P. Stubbe, and H. Kopka (1989), On the frequency depen-dence of ELF/VLF waves produced by modulated ionospheric heating,Radio Sci., 24(3), 270–278, doi:10.1029/RS024i003p00270.

Rishbeth, H. (1997), The ionospheric E-layer and F-layer dynamos—Atutorial review, J. Atmos. Sol. Terr. Phys., 59(15), 1873–1880,doi:10.1016/S1364-6826(97)00005-9.

Tepley, L., and R. K. Landshoff (1966), Waveguide theory for iono-spheric propagation of hydromagnetic emissions, J. Geophys. Res., 71(5),1499–1504, doi:10.1029/JZ071i005p01499.

Uman, M. A. (1987), The Lightning Discharge, Int. Geophys. Ser., vol. 39,edited by W. L. Donn, Academic, New York.


22 of 22

generation of elf and ulf electromagnetic waves by...

Documents