Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 985

Interaction betweenElectromagnetic Waves andLocalized Plasma Oscillations

BY

JAN-OVE HALL

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2004

Dissertation at Uppsala University to be publicly examined in Haggsalen (Room 10132),AngstromLaboratory, Uppsala University, Lagerhyddsvagen 1, Uppsala, Thursday, June 3, 2004 at 13:15for the Degree of Doctor of Philosophy. The examination will be conducted in English

AbstractHall, J-O. 2004. Interaction Between Electromagnetic Waves and Localized Plasma Oscillations.Acta Universitatis Upsaliensis.Comprehensive Summaries of Uppsala Dissertations from theFaculty of Science and Technology 985. 43 pp. Uppsala. ISBN 91-554-5982-X

This thesis treats interaction between electromagnetic waves and localized plasma oscillations.Two specific physical systems are considered, namely artificially excited magnetic field-alignedirregularities (striations) and naturally excited lower hybrid solitary structures (LHSS). Striationsare mainly density depletions of a few percent that are observed when a powerful electromagneticwave, a pump wave, is launched into the ionosphere. The striations are formed by upper hybrid(UH) oscillations that are localized in the depletion where they are generated by the linearconversion of the pump field on the density gradients. However, the localization is not completeas the UH oscillation can convert to a propagating electromagnetic Z mode wave. This process,termed Z mode leakage, causes damping of the localized UH oscillation. The Z mode leakageis investigated and the theory predicts non-Lorentzian skewed shapes of the resonances for theemitted Z mode radiation. Further, the interaction between individual striations facilitated bythe Z mode leakage is investigated. The LHSS are observed by spacecraft in the ionosphereand magnetosphere as localized waves in the lower hybrid (LH) frequency range that coincideswith density cavities. The localized waves are immersed in non-localized wave activity. Theexcitation of localized waves with frequencies below LH frequency is modelled by scatteringof electromagnetic magnetosonic (MS) waves off a preexisting density cavity. It is shownanalytically that an incident MS wave with frequency less than the minimum LH frequencyinside the cavity is focused to localized waves with left-handed rotating wave front. In addition,the theory is shown to be consistent with observations by the Freja satellite. For frequenciesbetween the minimum LH frequency inside the cavity and the ambient LH frequency, the MSwave is instead mode converted and excites pressure driven LH oscillations. This process isstudied in a simplified geometry.

Keywords: Space physics, plasma physics, radio waves, ionospheric modification, densitydepletions, field-aligned irregularities, lower hybrid solitary structures, lower hybrid cavities,Z mode waves, magnetosonic waves, scattering, mode conversion

Jan-Ove Hall, Department of Astronomy and Space Physics. Uppsala University. Box 515SE-751 20 Uppsala, Sweden

c© Jan-Ove Hall 2004

ISBN 91-554-5982-XISSN 1104-232Xurn:nbn:se:uu:diva-4282 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4282)

To Kristina

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List of Papers

[1] J. O. Hall and T. B. Leyser. Conversion of trapped upper hybrid oscillations andZ mode at a plasma density irregularity.Phys. Plasmas, 10:2509–2518, 2003.

[2] J. O. Hall, Ya. N. Istomin, and T. B. Leyser. Electromagnetic interaction of local-ized upper hybrid oscillations in a system of density depletions. Manuscript.

[3] J. O. Hall, A. I. Eriksson, and T. B. Leyser. Excitation of localized rotating wavesin plasma density cavities by scattering of fast magnetosonic waves.Phys. Rev.Lett., Accepted, 2004.

[4] J. O. Hall. Conversion of localized lower hybrid oscillations and fast magne-tosonic waves at a plasma density cavity.Submitted toPhys. Plasmas, 2004.

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Contents

1 Introduction . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 A Plasma Physics Primer .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Basic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Cold Plasma Approximation . . .. . . . . . . . . . . . . . . . . . . . . . . . 72.3 Finite Larmor Radius Approximation .. . . . . . . . . . . . . . . . . . . . 9

3 Interaction Between Z mode Waves and Localized Upper Hybrid Os-cillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Z Mode Leakage From an Isolated Density Cavity. . . . . . . . . . . 143.1.1 Electrostatic Treatment .. . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Solution by the Scale Separation Method . .. . . . . . . . . . . . 183.1.3 WKB Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.4 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.5 A Quantitative Comparison Between the Methods. . . . . . . 24

3.2 Z mode Interaction Between Density Cavities . . .. . . . . . . . . . . 254 Interaction Between Magnetosonic Waves and Localized Lower Hy-

brid Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Electrostatic treatment . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Electromagnetic treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Excitation of Localized Rotating Left-handed Oscillations . 304.2.2 Excitation of Pressure Driven Lower Hybrid Oscillations . . 32

5 Outlook .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Summary in Swedish. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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List of Figures

3.1 Schematic illustration of ionospheric heating experiments. . . . . 153.2 The local UH frequency as a function ofx . . . . . . . . . . . . . . . . 213.3 The Z mode wave field spectrum. . . . . . . . . . . . . . . . . . . . . . . 25

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x

Introduction

Our understanding of the near-Earth space environment has dramatically in-creased since it became possibility to deploy scientific instruments directlyinto space. The first man-made satellite, Sputnik I, was launched in 1957. Theyear after, the satellite Explorer I, which was equipped with a Geiger counter,was launched. This was the start of a space race during which many fieldsof technology and basic science developed rapidly. Space Physics emergedas a new scientific discipline from the wealth of new and exciting observa-tions. As a spin-off of this space race, today’s society benefits from the use ofsatellites for many purposes, e.g., for communication, weather forecasts, andpositioning. In addition, other scientific disciplines than space physics are us-ing satellites. The Hubble space telescope takes breathtaking images of starsand galaxies, the COBE satellite is measuring the cosmic background radia-tion and gives us a picture of the early universe, just to mention two satellitemissions which are not directly connected to space physics. Modern scien-tific satellites are equipped with an extensive payload of instruments. Theinstruments are measuring physical quantities to accurately describe the spaceenvironment. In a sense, a satellite can be compared with a weather station onEarth measuring the temperature, pressure, and wind speed. However, spaceis filled with plasma instead of neutral gas. A plasma is a mix of charged par-ticles which can generate and respond to electromagnetic fields. It is thereforenecessary to measure a larger set of parameters than in the weather station.Satellite observation has made it possible to determine the large scale structureof the near-Earth space environment. Sounding rockets have been, and stillare, a useful complement to satellite measurements close to Earth, i.e., in theionosphere. Besides thein situ measurements conducted by spacecraft, groundbased radio methods are suitable for studies of the near-Earth plasma.

In addition to the large-scale structures and the boundary layers separatingvarious regions in space, spacecraft measurements have shown that small-scalephenomena are common in space plasmas. An example of such small scaledphenomena is the localized bursts of lower hybrid waves, termed lower hybridsolitary structures (LHSS) or lower hybrid cavities (LHC), first detected in1986 by the Marie sounding rocket [LKYW86]. The wave activity coincideswith density depletions which are elongated along the geomagnetic field. Theperpendicular (to the geomagnetic field) width is observed to be typically a

1

few ion gyro radii and the parallel dimension is estimated to be several order ofmagnitudes larger than the perpendicular. Further, these structures are alwaysobserved to be immersed in nonlocalized wave activity.

The space environment provides many opportunities to make interesting ob-servations of naturally occurring phenomena. From another point of view, thespace plasma is also an excellent plasma laboratory in which we can performcontrolled and designed experiments. The plasma is virtually unlimited in sizein comparison to man-made plasma chambers used in laboratory studies. Forexample, by injecting a powerful electromagnetic wave, a pump wave, into theionosphere we can study the plasma response to electromagnetic waves. Thistype of experiments, which is termed ionospheric heating experiments or iono-spheric modification, can be conducted from a number of facilities around theworld. The pumped plasma can be studied by using radio methods and space-craft. Especially, by using coherent and incoherent radar measurements it ispossible to study the plasma turbulence which is excited by the pump wave. Abroad range of phenomena can be observed in the pumped plasma. One of themost important is the structuring into filamentary plasma irregularities (stria-tions). The irregularities are stretched along the geomagnetic field, similar tothe LHSS, and are generated by localized plasma turbulence which is drivenby the pump wave.

This thesis treats interaction between electromagnetic waves and localizedplasma oscillations. Two specific physical systems are considered, namely theartificially excited density irregularities and the naturally excited lower hybridsolitary structures. Both systems are characterized by field-aligned plasmadensity depletions in which localized wave activity is excited. These local-ized oscillations can interact with long wavelength electromagnetic radiationof various polarizations. For the striations, the localized oscillations have fre-quencies in the upper hybrid (UH) frequency range. The UH oscillations caninteract with electromagnetic Z mode wave. One consequence of this inter-action is that the localized UH oscillation can convert to propagating Z modewaves. The Z mode wave transports energy away from the cavity which causesdamping of the localized UH oscillation. This phenomenon, termed Z modeleakage, is considered in Paper [1]. On the other hand, the Z mode radiationprovides a channel of interaction between individual filaments and can be de-cisive for the global response of the pumped plasma. Paper [2] treats this Zmode interaction. For the LHSS, the localized oscillations have frequencies inthe lower hybrid (LH) frequency range. The localized oscillations can interactwith electromagnetic waves on the R whistler/magnetosonic (MS) dispersionsurface. Paper [3] treats the excitation of localized oscillations with frequen-cies below the LH frequency. The theoretical results are shown to be consistentwith spacecraft observations of LHSS. The treatment in Paper [4] describes theexcitation of pressure driven LH oscillations. Here, the frequency of the inci-

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dent MS wave matches the local LH frequency and mode converts to localizedoscillations.

1.1 OutlineThis thesis consists of two parts, namely the present summary and a collectionof four papers. The summary is organized as follows: In chapter 2 some basicplasma physics is reviewed. Chapter 3 is devoted to wave phenomena relevantfor ionospheric heating experiments. The contents of Paper [1] and [2] aresummarized. Also, some aspects of important previous work are reviewed.Chapter 4 treats the interaction between magnetosonic waves and localizedoscillations in the lower hybrid frequency range. The contents of Paper [3] and[4] are summarized. In chapter 5, some suggestions for future work are given.Chapter 6 contains a short summary in Swedish.

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4

A Plasma Physics Primer

Plasma physics is the theoretical foundation of space physics and describesthe collective behavior of quasi-neutral gases of charged and neutral particles,i.e., a gas consisting of an equal amount of positive and negative charges. Thetheory can be applied to describe various phenomena in space and laboratoryplasma. Other interesting and important applications of plasma physics arefound in the research in thermo nuclear fusion as a future source of energy. Toquantitatively describe the dynamics of the plasma it is necessary to formulatethe equations of motion. In the majority of applications of plasma theory inspace physics it is sufficient to use classical mechanics to describe the parti-cle motion and classical electrodynamics to describe the electromagnetic field.Relativistic effects can be ignored in most cases, but need to be consideredin certain astrophysical systems. Instead of tracking each individual particlein the plasma it is often convenient to use statistical mechanics to describethe gas of electrons and ions. Statistical mechanics describes the plasma interms of the distribution functionf . The distribution function describes, in aprobabilistic sense, how many particles there are in a certain volume elementhaving a certain velocity at a certain time. Mathematically speaking, the dis-tribution function depends on seven variables. If short range interactions, suchas binary collisions, are neglected the time evolution off is described by theVlasov equation. As the particles in the plasma are electrically charged, anymotions inside the plasma will result in currents and local space charge. TheVlasov equation must therefore be solved together with Maxwell’s equationsto describe the electromagnetic fields excited by the currents and space charge.This set of equations describes self-consistently the motion of the plasma andthe electromagnetic fields. In addition to the high dimensionality of the Vlasovequation, the governing equations are nonlinear which makes the mathemati-cal analysis difficult. Even with today’s supercomputers it is only possible tosolve relatively simple model problems. However, the complexity of the gov-erning equations is only reflecting the great richness of physical phenomena inthe dynamics of plasma.

Wave phenomena in plasma have been studied extensively both experimen-tally in laboratories and space plasma and by theoretical considerations. Inaddition to the transversal electromagnetic wave mode found in vacuum, aplasma can support many different modes of wave propagation. The theory for

5

small amplitude waves in a homogeneous plasma is well developed. It is pos-sible to solve the linearized equation of motions exactly for small fluctuationsaround a stationary Maxwell distribution in the velocity space. The solutiondescribes the dispersive properties of the wave, i.e., the relation between wavefrequency and wavelength, and the polarization of the electric and magneticwave fields. Many interesting phenomena, such as collisionless damping ofplasma waves, are described by the linear theory for homogeneous plasma.However, when plasma are encountered in nature they are always inhomoge-neous to some extent. There is no general method in existence for analyzingwaves in inhomogeneous plasma. Each physical system must be analyzed in-stead with specialized models and methods. When allowing for finite ampli-tude the various wave modes are coupled by nonlinear terms and energy canflow between the wave modes. This type of wave interaction is termed para-metric process. As an example, a large amplitude transverse electromagneticwave can decay into an electrostatic Langmuir wave and another electromag-netic wave. Also, a few forms of strong turbulence, e.g. Langmuir turbulence,are fairly well understood [Rob97].

The mathematical formulation of the Vlasov equation and Maxwell’s equa-tions are given in section 2.1. In the present thesis the Vlasov equation is notused directly, instead two types of approximations are used. The cold plasmaapproximation is used as a starting point for the analytical treatment presentedin Paper [3] and is considered in section 2.2. The finite Larmor radius (FLR)approximation is important in the treatments in Papers [1,2,4]. This approxi-mation is briefly discussed in section 2.3.

2.1 Basic EquationsThe evolution of the distribution functionfα for speciesα is described by theVlasov equation. If all other forces but the electromagnetic are neglected theequation is [Che83]

∂ fα

∂t+v · ∂ fα

∂x+

qα

mα(E+v×B) · ∂ fα

∂v= 0 (2.1)

whereqα (mα) is the charge (mass) of the speciesα, E is the electric field,andB is the magnetic field. The distribution functionfα is a function of thespatial coordinatesx, the velocityv, and timet, i.e., fα = fα (x,v, t). Thedistribution function contains a statistical description of the microscopic stateand all macroscopic quantities can be calculated as statistical moments offα.The number of particles per unit volumenα, i.e., the number density, can becalculated by integratingfα over the velocity space, we have that

nα (x, t) =Z

fα (x,v, t) dv3 . (2.2)

6

The macroscopic mean velocity is given by the integral

vα (x, t) =1

nα (x, t)

Zv fα (x,v, t) dv3 . (2.3)

The electrical charge densityρ and current densityJ can be expressed in termsof the distribution function. The charge densityρ is

ρ(x, t) = ∑α

qα

Zfα (x,v, t) dv3 = ∑

αqαnα , (2.4)

and the current densityJ is

J(x, t) = ∑α

qαnα

Zv fα (x,v, t) dv3

= ∑α

qαnαvα = ∑α

Jα , (2.5)

where the sums are over all species in the plasma.The fundamental equations in classical electrodynamics are the well known

Maxwell equations which describeE andB. The equations are [Jac75]

∇ ·E = ρ/ε0 , (2.6)

∇ ·B = 0, (2.7)

∇×E = −∂B∂t

, (2.8)

∇×B = ε0µ0∂E∂t

+µ0 J , (2.9)

whereJ andρ are given by Eqs. (2.4) and (2.5), respectively. The above set ofequations describes the dynamics of the plasma and the electromagnetic fieldself-consistently.

2.2 Cold Plasma ApproximationIt is often useful to consider various approximations of the Vlasov equation.A common and important one is the cold plasma approximation in which thethermal motion of the particles in the plasma is neglected. All particles in avolumedx3 aroundx move with the same velocityvα. By averaging Eq. (2.1)over the velocity space one obtains the continuity equation for each species,giving

∂nα

∂t+∇ · (nαvα) = 0. (2.10)

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By calculating the first statistical moment of Eq. (2.1) one obtains a momentumbalance equation, which is

∂vα

∂t+(vα ·∇)vα =

qα

mα(E+vα ×B) . (2.11)

When thermal motion is included a term due to the pressure will appear inEq. (2.11).

The conservation laws Eqs (2.10) and (2.11) are, despite their simple struc-ture, difficult to solve because of the nonlinear terms. However, in many situ-ations it is sufficient to consider small amplitude deviations from a stationarystate which can be characterized by the magnetic fieldB0, electric fieldE0,densityn0, and velocityv0. The equations of motions can be linearized aroundthe background quantities and all fluctuations can be assumed to have a har-monic time dependence, e.g.,E(x, t) = Re[E(x)exp(−iω t)] whereω is thewave (angular) frequency. For a magnetized plasma withB0 = B0z, E0 = 0,v0 = 0, andn0 = n0(x), Eq. (2.11) can be linearized and solved with respect tovα in terms ofE. The current of speciesα is in the linear approximation givenby

Jα = iωε0ω2

pα

ω2−Ω2α

(E⊥− i

Ωα

ωz×E⊥

)+ iωε0

ω2pα

ω2 Ezz ≡ σαE , (2.12)

whereE⊥ is the electric field component perpendicular toB0, Ωα = qαB0/mαis the gyro frequency, andω2

pα = q2αn0/(ε0mα) is the plasma frequency. For

our purposes it is useful to define the susceptibility tensorχα in terms of theconductivity tensorσα asχα = i/(ε0ω)σα. Further, the dielectric tensorε isdefined by

ε = I +∑α

χα , (2.13)

whereI is the identity matrix. These definitions allow us to write Eq. (2.9) as∇×B =−iω/c2 εE , wherec is the vacuum light speed. The electric displace-ment,εE, can be written as

εE = SE⊥ + iDz×E⊥ +PEzz . (2.14)

The elements areS = (R+L)/2, D = (R−L)/2, and

P = 1−∑α

ω2pα

ω2 , (2.15)

where

R = 1−∑α

ω2pα

ω2

ωω+Ωα

and L = 1−∑α

ω2pα

ω2

ωω−Ωα

. (2.16)

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In a rectangular coordinate systemε can be represented as

ε =

S −iD 0

iD S 0

0 0 P

. (2.17)

Note that the cold plasma is temporal dispersive asε depends onω. Further, inan inhomogeneous plasma the elements ofε are functions ofx because of thevariations ofωpα = ωpα(x) ∝

√n0(x).

A wave equation governingE can be obtained by combining the Maxwellequations. We have that

∇× (∇×E)− k20 εE = 0, (2.18)

wherek0 = ω/c. The wave magnetic field can be calculated from Eq. (2.8),i.e.,B =−i/ω∇×E. A homogeneous plasma with constantn0 supports planewaves of the formE = exp(ik ·x)e wherek is the wave vector ande describesthe polarization ofE. For plane waves Eq. (2.18) is reduced to a set of algebraicequations. In order to have a non trivial solution of these equationsk andωmust satisfy the dispersion relation

det|N2(kik j −δi j)+ εi j(ω)| = 0, (2.19)

whereN = k/k0 is the refractive index,k = k/k, andδi j is the Kronecker delta.The dispersion relation can be solved with respect toω or any componentsof k, depending on the situation. The polarization vector for each solutioncan be calculated from the wave equation. The plane wave solutions describeall wave properties in a homogeneous plasma. However, when some of thebackground quantities are spatially inhomogeneous the wave equation (2.18)must be solved by some other method. In the above discussion we used a waveequation forE, but it is also possible to derive a wave equation forB. We havethat

∇× (ε−1∇×B)− k2

0 B = 0 (2.20)

andE = iω/k20 ε−1B. This formulation is applicable whenε is non singular,

i.e., det|ε| = RLP = 0.

2.3 Finite Larmor Radius ApproximationThe cold plasma approximation describes many plasma wave phenomena witha good accuracy, even when the plasma is rather hot. However, the cold plasmaapproximation fails completely near certain resonance frequencies where the

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approximation predicts infinitely small wavelengths, or infinitely largek. How-ever, a kinetic treatment based on the Vlasov equation shows that these reso-nances are artificial. The spatial dispersion described by the Vlasov equationbecomes important for small length scales and allows us to describe the physicson the cold plasma resonances. Many important properties of the spatial dis-persion can be described by the finite Larmor radius (FLR) approximation.

To see how these resonances arise in the cold plasma theory and how theyare effected by FLR effects we consider wave propagation strictly perpendicu-lar to the background magnetic field. The cold plasma theory admits two wavemodes for a givenω. These modes are the O (ordinary) and X (extraordinary)mode. Withk = kx ⊥ B andB0 = B0z the O mode electric field is linearly po-larized withE ‖ B0 and the X mode electric field is elliptically polarized withE ⊥ B0. For the X mode withE ⊥ z the wave equation (2.18) reduces to(

εxx εxy

εyx εyy −N2

)(Ex

Ey

)= 0. (2.21)

In order to have a non trivial solution withE = 0 the determinant of the matrixon the left-hand side must equal zero. We obtain the dispersion relation

N2 =εxxεyy − εxyεyx

εxx, (2.22)

which in combination with Eq. (2.21) gives the polarization vector

e = εxyx− εxxy . (2.23)

Note thatk = k0N tends to infinity, or the wavelength tends to zero, whenεxx =0. Further,E becomes parallel tok and describes a purely longitudinal wave.In addition, according to the Maxwell equation (2.8), the waveB vanishes andthe wave is purely electrostatic. For a two component plasma with electronsand one ion species the resonances occur whenω is equal to the UH frequency,

ω2UH = ω2

pe+Ω2e (2.24)

or equal to the LH frequency,

ω2LH =

ω2pi

1+(ωpe/Ωe

)2 , (2.25)

whereΩe is the electron gyro frequency andωpe (ωpi) is the electron (ion)plasma frequency. To resolve the wave properties atωUH andωLH it is neces-sary to include corrections due to FLR. For a hot plasma in a magnetic field thesusceptibility tensorχ depends both onω andk. Thek dependence describes

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the spatial dispersion due to thermal motion in the plasma. In our case withk ⊥ B0, χα = χα(ω,λα) whereλα = (ρLαk)2/2 andρLα is the Larmor radius.If we regardλα as a small parameter we can expandχα in a Taylor series. Tothe lowest non-vanishing order,

χα ≈ χ(0)α +λαχ(1)

α . (2.26)

χ(0)α can be identified as the susceptibility tensor for a cold plasma. In the UH

frequency range the ion dynamics can be neglected and the corrected dielectrictensor is

ε(1) ≈ ε+ρLeχ(1)k2/2, (2.27)

whereε = I +χ(0) is the cold plasma dielectric tensor. In view of the dispersionrelation Eq. (2.22), the most important correction is toεxx which is vanishingat ωUH in the cold plasma approximation. With the corrected dielectric tensorε(1) we obtain the dispersion relation

ΛxxN4− εxxN2 + εxxεyy − εxyεyx = 0, (2.28)

whereΛxx = −(ρLek0)2χ(1)xx /2. Note that Eq. (2.28) is a quadratic equation

in N2 and describes two modes. The new mode, which is not described byEq. (2.22), is a pressure driven oscillation which propagates forω > ωUH. BothsolutionsN2 are now finite, even forω = ωUH (see Figs. 2 and 3 in Paper [1]).The FLR expansion method is not limited to the considered case withkz = 0.It is also possible to retain the full kinetic parallel dispersion [Bra98].

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Interaction Between Z mode Waves andLocalized Upper Hybrid Oscillations

One of the most important effects of injecting large amplitude electromagneticwaves from the ground into the ionosphere is the structuring of the plasma intofilamentary plasma irregularities. The structures are stretched along the ambi-ent geomagnetic field because of the much larger thermal conductivity alongthe magnetic field than across. The perpendicular (to the magnetic field) lengthscale of the small scale irregularities is smaller than the ion Larmor radius butmuch larger than the electron Larmor radius. The density depletions are a fewpercent deep. These striations are central to our understanding of a number ofphenomena, including anomalous absorption of radio waves [SKJR82], stimu-lated electromagnetic emissions (SEE) [TKS82, Ley01], Langmuir turbulenceevolution [ND90], and field-aligned scattering of radio waves [MKW74]. Thestructures are formed by UH oscillations that are trapped in the depletion wherethey are generated by linear conversion of the pump field on the density gradi-ents [GZL95, IL97]. As a consequence, the striations are only observed whenthe electromagnetic pump wave is in O mode which is reflected above the layerwhere the wave frequencyω matches the UH frequency. The X mode is re-flected at a lower altitude and does not reach the UH layer. Figure 3.1 shows aschematic picture of the experiment.

The dynamics of the plasma density and the plasma temperature when ex-posed to a powerful O mode pump wave can be described by [IL97](

∂∂t

−D1,2∇2⊥

)Ψ1,2 = Q , (3.1)

whereΨ1 andΨ2 are linear combinations of the relative density profileη andthe plasma temperature. The right-hand side describes the plasma heating andis proportional to the squared amplitude of the high frequency electric fieldEh, i.e., Q ∝ |Eh|2. The O mode pump waveEO is scattered off the densityirregularity and excites electrostatic UH oscillations with electric fieldE1. Thisscattering process is described by

∇ · [(H−η)E1] = ∇η ·EO , (3.2)

whereH is a differential operator describing UH oscillations in a homoge-neous plasma. The above system of equations describes self-consistently the

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evolution of the electrostatic fieldE1, the plasma density, and the temperatureof the plasma. The large amplitude pump wave excites UH waves by scatteringoff a density perturbationη which initially is small. This scattering process isdescribed by Eq. (3.2). The excited UH wave is damped by weak collisionaldamping. In the collisional process the energy in the UH wave is dissipatedinto heat. As a result of the heating, the electron temperature will increase inregions where|Eh|2 = |EO+E1|2 is large. If the amplitude of the pump wave issufficiently large, an initially shallow density cavity will start to grow. This isthe so called thermal parametric instability (TPI). The TPI [GT76] is generallyaccepted in the community as the initial state of the striations.

The filamentary density irregularities with their trapped UH oscillationsconstitute a strongly inhomogeneous form of plasma turbulence. However,the short wave UH oscillations are not completely trapped, but are partiallytransmitted through the trapping depletion into the long wave electromagneticZ mode, a phenomenon which has been termed Z mode leakage [DMPR82,Mjø83]. The Z mode wave transports energy away from the cavity and de-creases the amplitude of the localized oscillation that sustain the cavity. The Zmode radiation can therefore directly affect the dynamics of the density cav-ity by decreasing the locally dissipated energy. The Z mode leakage from anisolated density cavity is considered in section 3.1.

The Z mode leakage from the individual filaments also facilitates an inter-action mechanism [Mjø83] between different filaments. The energy lost fromone filament is gained by its neighbors. This mechanism reduces the total out-flux of energy from the collective of striations. Hence, the Z mode radiationacts as an agent for long range interaction between regions of localized UHturbulence. With this interaction the small scaled turbulence is coupled to thelarge scale behavior of the turbulent plasma. This Z mode interaction betweendensity cavities is considered in section 3.2.

3.1 Z Mode Leakage From an Isolated Density CavityTo investigate the Z mode leakage from one isolated density cavity we use thesimplest possible model. The O mode pump wave is assumed to propagateparallel toB0 = B0z so that the circular polarized fieldEO is perpendicular toB0. The leakage process is considered in the stationary state and a preexistingstationary density cavity is assumed. The cavity is modeled by a one dimen-sional structure so that the density is varying in one direction perpendicularto B0, n(x) = n∞

0 [1+η(x)] where|η| 1 is the density variation associatedwith the cavity andn∞

0 is the density of the ambient plasma. The depletion isassumed to be small scale in the sense that the characteristic widthL⊥ perpen-dicular toB0 is much smaller than the wavelength of the freely propagatingZ mode wave, i.e.,L⊥ LZ ≡ 2π/kZ. This condition is realized whenω is

14

upper hybrid waves

Radio wave

X mode cutoff

O mode cutoff

Upper hybrid level

Strong excitation of

3~10 density cavities

Magnetic fieldA

ltitu

de

Radio transmitter

~200 km

Figure 3.1: Schematic illustration of ionospheric heating experiments.

below and not too close toω∞UH (UH frequency of the ambient homogeneous

plasma). For simplicity, the cavity is considered to be symmetric aroundx = 0,i.e., η(−x) = η(x). Further, the wave properties of the O mode pump alongB0 are neglected and the pump is scattered off the cavity to excite the electricfield E which propagates in the direction of the density gradient, i.e.,E = E(x).The model must include FLR effects in order to describe UH waves since thecold plasma theory predicts a resonance at the UH frequency, as discussed insection 2.3. The dielectric properties can be approximately described by thedielectric tensor operator

E = ε∞ +ηχ(0)−1/2ρ2Lχ(1) d2

dx2 , (3.3)

which was derived using the lowest order FLR approximation in Paper [1]. Thefirst termε∞ = I + χ(0) is the dielectric tensor for a cold magnetized plasmaevaluated for the plasma parameters outside the cavity. The second term isa correction due to the density variation across the magnetic field. Thus,ε∞ + ηχ(0) is the local cold plasma dielectric tensor. The FLR correction isdescribed by the third term. By combining Maxwell’s equations and the di-electric tensor operator, Eq. (3.3), we obtain a vector wave equation governingE, giving [

k−20 Λ

d2

dx2 + ε∞ +ηχ(0)]

E = −ηχ(0)E0 , (3.4)

where the tensorΛ = yy+ zz− (ρLk0)2χ(1)/2 contains terms originating fromthe Maxwell equation∇×E = iωB and the FLR correction. The right-hand

15

side of Eq. (3.4) describes the scattering of the O mode pump field off thedensity cavity. TheEz component is, due to the structure of the tensorsΛ, ε∞,andχ(0), not coupled toEx or Ey and describes O mode waves propagatingperpendicular toB0. This component is not excited by the pump and will notbe considered further. The two perpendicular components are coupled anddescribe two wave modes.

In the homogeneous plasma outside the cavity the wave modes are uncou-pled and the general solution to Eq. (3.4) can be written as a superposition ofplane waves. We have that

E(x) = (A+X eikXx +A−

X e−ikXx)eX +(A+Z eikZx +A−

Z e−ikZx)eZ , (3.5)

whereA±I are some constants. The wave numberskX andkZ can be obtained

from the dispersion relation

ΛxxN4− ε∞xxN2 + ε∞

xxε∞yy − ε∞

xyε∞yx = 0, (3.6)

whereN = k/k0 is the refractive index. The polarization vectoreX andeZ canbe calculated from the wave equation. Forω < ω∞

UH, we haveNX ≈NUH whereN2

UH = ε∞xx/Λxx < 0 is the refractive index for an UH wave in the electrostatic

FLR approximation. The other solution isN2Z ≈ (ε∞

xxε∞yy − ε∞

xyε∞yx)/ε∞

xx whichis the refractive index for a Z mode wave in the cold plasma approximation.For ω > ω∞

UH, we haveNZ ≈ NUH. For our case withω < ω∞UH, the X mode

polarization vectoreX is essentially parallel tox, i.e., in the direction of prop-agation, and describes a longitudinal electrostatic wave. With an appropriatenormalization,eX ≈ x/

√Λxx. The Z mode polarization vector forω < ω∞

UH iseZ ≈ −ε∞

xy/ε∞xx x + y. In the inhomogeneous plasma inside the cavity the two

modes are coupled and mode conversion can occur.

3.1.1 Electrostatic TreatmentBefore considering the conversion process described by Eq. (3.4) it is instruc-tive to consider the electrostatic approximation in the inhomogeneous plasmain some detail. By assuming that the excited field is purely electrostatic, i.e.,E =−∇φ = Ex(x)x whereφ is the electrostatic potential, the vector wave equa-tion (3.4) is reduced to a single equation, we have that

LUHEx = −QOηEO , (3.7)

whereLUH = k−2UHd2/dx2 + 1+ QUη, QU = χ(0)

xx /ε∞xx, andQO = xχ(0)eO/ε∞

xx.For localized waves withω < ω∞

UH the UH oscillation is evanescent outsidethe cavity andEx must go to zero far away from the cavity. Ifω > ωmin

UH(the minimum UH frequency inside the cavity) the UH oscillations can prop-agate inside the cavity and are excited by the O mode pump. Equation (3.7)

16

can be solved using the WKB approximation [DMPR82]. However, for ourpurposes it is more convenient to use a Green’s function approach to “in-vert” the operatorLUH. The Green’s function is determined by the equationLUHGUH(x,x1) = δ(x− x1) and the boundary conditionGUH → 0 as|x| → ∞,whereδ(x) is the Dirac delta function.GUH describe the excitation of UHwaves by a point source located atx = x1. An attractive representation ofGUH

can be constructed as a superposition of all possible eigenmodes which canbe excited. In a homogeneous plasma the eigenmodes are plane waves of theform ψk = exp(ik · x) and any wave field can be written as a superposition ofsuch waves. In our case with an inhomogeneous plasma, the harmonic waveexp(ik ·x) is no longer an eigenmode and we have to find a set of eigenmodesappropriate for the considered density profile. There are two types of eigen-modes, namely nonlocalized propagating waves with a continuum of possiblefrequencies and localized oscillations with a discrete set of eigen frequencies.The Green’s function can be written as a superposition of these modes [Fri48]

GUH(x,x1) =M

∑n=0

ψn(x)ψ∗n(x1)

1−∆n/∆+

1iπ

Z ∞

−∞

ψ+k (x)ψ−

k (x1)1− k2/∆

k dk

W [ψ+k ,ψ−

k ], (3.8)

where the first term derives from the discrete spectrum ofLUH and the secondfrom the continuous spectrum. The parameter∆ ≡ −Q−1

U ∝ ω−ω∞UH quan-

tifies the deviation ofω from the ambient UH frequency. The normalizedψn which are associated with the discrete spectrum and the discrete eigenval-ues∆n (minη < ∆0 < ∆1 < · · · < ∆M < 0) are determined by the equationLUH ψn = (1− ∆n/∆)ψn and the boundary conditionsψn → 0 as |x| → ∞.Due to the symmetry ofη(x) aroundx = 0, the eigenfunctionsψn are evenfor evenn and odd for oddn. The eigenfunctionsψ±

k satisfy the equationLUHψ±

k = (1−k2/∆)ψ±k and the boundary conditionψ±

k ∝ exp(±i k x) asx →±∞. W [ψ+

k ,ψ−k ] is the Wronskian.

For localized oscillations withω < ω∞UH, or equivalently∆ < 0, only the dis-

crete part of the spectrum can be excited resonantly. The integrand in Eq. (3.8)has poles atk = ±i

√|∆| and the integral term represents excitation of quasimodes. If the contribution from the continuous spectrum is neglected the solu-tion to Eq. (3.7) can be written as

Ex(x)/EO ∝M

∑n=0

η∗n

∆−∆nψn(x) , (3.9)

whereηn =R ∞−∞ ηψn dx1. Even though we have not specifiedψn and∆n, the

solution Eq. (3.9) gives a clear picture of the excitation of localized electro-static UH oscillations. Whenω is such that∆ ≈ ∆n the amplitude of the UHoscillation will be large. When∆ = ∆n the pump wave is resonantly excitingthe eigen modeψn and the stationary solution Eq. (3.9) predicts an infinite am-plitude at the resonance. Physically, the amplitude cannot be infinite and the

17

oscillation must be damped by some linear or nonlinear mechanism. Further,the pump wave, which is assumed to be homogeneous, cannot excite the oddresonances. The UH amplitude is proportional to the overlapηn which is van-ishing whenψn is odd. The spectral representation ofEx has previously beenapplied to describe parametric decay of localized UH oscillations into lowerhybrid waves [Mjø97].

3.1.2 Solution by the Scale Separation MethodIn the electrostatic approximation discussed above it is possible to interpret theexcitation of localized UH oscillations in terms of the cavities eigen modes.This interpretation is attractive, but in order to model the Z mode leakage wemust include electromagnetic terms. Due to coupling between the componentsin Eq. (3.4) we are faced with a fourth order differential equation and the eigenfunction expansion method can not be applied directly. This is the topic ofPaper [1] where we present an electromagnetic generalization of the electro-static treatment above. Analytic expressions for the Z mode amplitude and thelocalized wave field are given in terms of the electrostatic eigenfunctions. Themethod developed in Paper [1] is similar to the source approximation first in-troduced to obtain analytic expressions for mode conversion of Langmuir andelectromagnetic waves in unmagnetized plasma [HLFM89].

The componentsEx andEy are coupled in the wave equation (3.4) both in-side and outside the cavity. The coupling is due to the magnetization of theplasma which implies thatε∞ and Λ have off-diagonal elements. We knowfrom the plane wave solution Eq. (3.5) that the two modes are independent ofeach other outside the cavity. Thus, by expressingE in terms of the polariza-tion vectors the equations are uncoupled in the ambient homogeneous plasma.As eX and eZ are not parallel it is possible to describe any polarization as asuperposition of these vectors. Without loss of generality we can write thesolution to Eq. (3.4) as

E(x) = EX(x)eX +EZ(x)eZ . (3.10)

In this representation ofE, Eq. (3.4) is transformed to the component equations

LX EX +QXZηEZ = −QXOηEO , (3.11)

LZ EZ +QZXηEX = −QZOηEO , (3.12)

which are uncoupled outside the cavity whereη = 0. The operators areLX ≡k−2

X d2/dx2 + 1+ QXX η andLZ ≡ k−2Z d2/dx2 + 1+ QZZη wherekX ≡ k0 NX

andkZ ≡ k0 NZ. The coupling constants areQIJ = (e†I χ(0) eJ)N−2

I whereI,J =X,Z,O. It is important to note that the transformed equations (3.11) and (3.12)are equivalent to the original Eq. (3.4). The operatorLX is approximately equal

18

to LUH which describes localized UH oscillations. We can therefore identifyEq. (3.11) as a generalization of the electrostatic wave equation (3.7).

The above choice of basis vectors is rather natural and it decouples theequations outside the cavity. However, it is important to have a physical in-terpretation ofEX(x) andEZ(x) for all x. Outside the cavity,EX is evanescentsincek2

X < 0 and tends to zero far away from the density cavity. In this limitthe wave is a pure Z mode wave sinceE(x) = EZ(x)eZ ∝ exp(±ikZx)eZ. Thus,in the region outside the cavityEZ can be interpreted as the amplitude of thepropagating Z mode wave. When approaching the cavity from the outsideωwill be equal to or greater than the localωUH in some region inside the cavity.When passing through the transition region whereω = ωUH the long wave-length Z mode wave is mode converted andE can be characterized as an UHoscillation. The UH oscillation is essentially an electrostatic wave. The localwave number forEX is real inside the cavity, i.e.,k2

X [1+QXX η(x)] > 0, andEX

is a spatially oscillating quantity. The local wave number forEZ remains realthroughout the cavity, i.e.,k2

Z[1+ QZZη(x)] > 0. BothEX andEZ are nonzeroinside the cavity and both quantities are necessary to describeE inside thecavity. The polarization vectoreX is approximately parallel to the directionof propagation so thatEX(x)eX describes a localized electrostatic oscillation.However,E is not purely electrostatic and the contributionEZ(x)eZ to E canbe interpreted as an electromagnetic correction inside the cavity. In summary,we can interpretEX as the electrostatic part of the localized UH oscillation.EZ describes the Z mode wave field outside the cavity and an electromagneticcorrection to the electrostatic UH field inside the cavity.

With the interpretations ofEX andEZ given above we can regard Eq. (3.11)as the UH wave equation. It describes the interaction between the localizedUH oscillation and the electromagnetic Z mode as well as the excitation by theO mode pump wave. Equation (3.12) describes excitation of Z mode waves bythe localized source of UH oscillations and by scattering the pump wave di-rectly into a Z mode wave. The interaction between the modes is not localizedto any specific point. Instead, the wave interaction is effective in the wholecavity. This is an important conceptual difference between the present methodand the previously used WKB method where the interaction is assumed to takeplace at some coupling points. A more detailed discussion of the WKB solu-tion is given in section (3.1.3).

In analogy to the electrostatic treatment above, the operatorsLX andLZ

can be inverted by using an appropriate Green’s function. By invertingLZ wecan calculateEZ in terms ofEX andEO from Eq. (3.12). Similarly,EX canbe calculated from Eq. (3.11). By combining the results one obtains a singleintegral equation governingEX. We have that

EX(x)−Z ∞

−∞KX(x,x1)EX(x1)dx1 = g(x) , (3.13)

19

where the kernelKX(x,x1) and the sourceg(x) are given in terms of the Green’sfunctions. The kernel is

KX(x,x1) ∝Z ∞

−∞GX(x,x2)η(x2)eikZ|x2−x1|η(x1)dx2 , (3.14)

where the exponential factor originates from the Z mode Green’s function. Byusing the length scale separationkZL⊥ 1, we can approximate

η(x2)eikZ|x2−x1|η(x1) ≈ η(x2)η(x1) , (3.15)

which makes it possible to write the kernel as a product of one function ofxand one function ofx1, i.e., KX(x,x1) ≈ K1(x)K2(x1), which thus constitutea degenerate kernel approximation [Kre89]. With the degenerate kernel it isstraightforward to solve the integral equation (3.13). When using the eigenfunction expansion ofGX the solution can be written as

EX(x)/EO ∝∑M

n=0η∗

n∆−∆n

1+ i ∑Mn=0

νn/2∆−∆n

ψn(x) , (3.16)

whereνn ∝ |ηn|2 andηn =R ∞−∞ ηψn dx1. Equation (3.16) is the desired elec-

tromagnetic generalization of the solution Eq. (3.9), which was obtained in theelectrostatic approximation. There is a clear correspondence between the elec-tromagnetic and electrostatic results. The most important difference is that theelectromagnetic solution is “normalized” such that the amplitude at the reso-nance is finite. The width of the resonance is determined by the real part ofνn. The imaginary part ofνn describes a shift of the resonance frequencies.The resonances are now determined by the condition∆ = ∆n + Im(νn)/2. Thefrequency shift is due to interaction with the electromagnetic wave and can becompared to the frequency shift arising when coupling two harmonic oscilla-tors. However, it can be shown that the shift is small in comparison to theresonance width and can therefore be neglected. Having solved forEX it ispossible to calculate the Z mode amplitude using Eq. (3.12), giving

EZ(x)/EO = −QXO

QXZ

[a+ i ∑M

n=0νn/2∆−∆n

1+ i ∑Mn=0

νn/2∆−∆n

]exp(i kZ |x|) , (3.17)

wherea = 1 is a constant. The Z mode power spectrum|EZ|2 is not just a sumof Lorentzian resonances. Instead, the resonances are skewed and distorted asseveral terms in the sums can be important. However, if the spacing betweenthe resonances∆n+1−∆n is large in comparison to the resonance widthνn onlyone term contributes and|EZ|2 is Lorentzian shaped around the resonances.Another interesting feature of the solution Eq. (3.17) is that it reduces to

|EZ/EO| = |QXO/QXZ | (3.18)

20

−4 −2 0 2 40

1

x/L⊥

ωUH/ω∞UH

ω/ω∞UH ω > ωUH ω < ωUH

ω = ωUH

1 532 4

Figure 3.2: The local UH frequency as a function ofx (solid line) andω(dashed line).

at the resonances∆ = ∆n. The ratioQXO/QXZ does not depend on the specificshape of the cavity, it depends only onω and the plasma parameters whichcharacterize the ambient plasma. The resonantly excited localized UH oscilla-tions facilitates a shape independent channel for the conversion from O modeto Z mode.

3.1.3 WKB Solution

The analytical treatment of the conversion between Z mode and UH wave dis-cussed above is specially designed for localized UH oscillations. Previoustreatments [DMPR82, Mjø83] are based on the WKB approximation. In orderto compare the two methods, the WKB treatment is briefly described in thissection.

The general idea is to formulate a set of simpler wave equations which arevalid in limited regions. These approximate equations are solved in the re-gions where they are valid. The regions are constructed in such a way thateach piecewise solution can be continued into neighboring regions. The so-lution is obtained by matching the asymptotical solutions. The method has awide range of applications in plasma physics and other areas of physics. Inthe present mode conversion problem we can identify three different types ofregions. Figure 3.2 shows the local UH frequency as a function ofx (solid line)for an arbitrary density profile. It is assumed thatω (dashed line) is betweenthe ambient UH frequency and the cavity’s minimum UH frequency. The firsttype of region is characterized byω < ωUH , regions 1 and 5 in Fig. 3.2, wherethe wave field can be regarded as a pure Z mode wave. Asω is not equal to theUH resonance frequency an approximate wave equation can be obtained from

21

the cold plasma theory. We have that

d2Edx2 + k2

Z(x)E = 0, (3.19)

wherek2Z(x) is the local Z mode wavenumber. The second type of region is

characterized byω > ωUH which corresponds to region 3 in Fig. 3.2. Guidedby the dispersion relation for a homogeneous plasma the wave field is assumedto be a purely electrostatic UH oscillation. By using the FLR approximationwe can derive

d2Edx2 + k2

UH(x)E = 0, (3.20)

wherek2UH(x) is the local UH wavenumber. The interaction with the O mode

pump wave can be described by an additional term in the right-hand side ofEq. (3.20). The third type of region describes the transition between the twoprevious types andω is allowed to be equalωUH. These transition regionscorrespond to region 2 and 4 in Fig. 3.2. It is not possible to describe thewave field as a single wave mode governed by a second order equation. Wemust use instead the full system of equations Eq. (3.4), or equivalently a singlefourth order differential equation. However, the transition region can be madearbitrarily narrow around the pointx0 whereω = ωUH and the coefficientsin the governing fourth order equations can be Taylor expanded around thecoupling pointx = x0. We obtain an approximate equation of the form

d4Edx4 +κ2(x− x0)

d2Edx2 + γE = 0, (3.21)

whereκ andγ are constants. The coefficient in front of the second derivative isvanishing at the coupling point whereω = ωUH. Equation (3.21) is sometimesreferred to as the “Standard Equation” [Sti92]. The standard equation is auseful model in many mode conversion problems.

Equations (3.19) and (3.20) can be solved by using the WKB approxima-tion. This is straightforward sincekUH(x) and kZ(x) are finite and nonzeroin the regions where Eqs. (3.19) and (3.20) are defined. The WKB solutionscan be continued into the transition region, but not across the coupling point.Equation (3.21) can be solved by contour integration [Sti92], giving

E(x) ∝Z

Cdu exp

[1κ2

(− 1

3u3 −κ2xu

+ γu

)]. (3.22)

The integration contourC in the complexu plane can be chosen in differentways to describe four linearly independent solutions to Eq. (3.21). The onlyrestriction onC is that the integrand is vanishing at the endpoints or thatC isclosed. By choosingC appropriately the solution can be matched asymptoti-cally to the WKB solutions outside and inside the cavity to obtain the solution.

22

Besides the technical difficulties of connecting the solutions across theω =ωUH layer, the method is easy to understand. The wave motion in the regioninside and outside can be interpreted in terms of the WKB solutions. In thisapproximation the interaction between the modes is confined to the couplingpoints at the density slopes. In the mode conversion theory developed in Pa-per [1] the governing wave equations (3.11) and (3.12) and their solutions aredefined everywhere. In addition, electromagnetic effects are retained inside thecavity, in contrast to the WKB method which considers the UH oscillations tobe purely electrostatic. There are no distinct coupling points in the scale sep-aration method. Instead the wave modes interact throughout the cavity. Thisinteraction gives rise to the coupling between the electrostatic cavity modesand is responsible for the skewing of the resonances. The WKB method can-not describe this effect as the interaction is confined to the coupling points. Asseen, the philosophy behind the two methods is completely different.

3.1.4 Numerical Solution

Even though the basic wave equation (3.4), which describes the mode con-version between localized UH oscillations and Z mode waves, has a simplemathematical structure it is difficult to solve analytically. Two different ap-proximate solutions have been discussed above. In both the scale separationmethod and the WKB method it is necessary to introduce several simplifi-cations and approximations. The scale separation method predicts that theresonances are skewed. As this feature of the conversion might be useful forexperimental detection of localized UH turbulence by measuring the emittedZ mode it is valuable to compare the analytical result with a numerical solu-tion. We do this in order to verify that the skewness effect is real and not anartifact of the approximations. A numerical solution can be obtained withoutintroducing any further approximations or simplifications. Eq. (3.4) and theboundary conditions was solved in Paper [1] using a standard finite differencemethod [GO92]. The numerical procedure will be briefly described below.

In the finite difference method the spatial coordinate is discretized and thesolution is calculated inN equally spaced pointsxi. The differential operatorsin Eqs. (3.11) and (3.12) are approximated by finite differences. The boundaryconditionsEX → 0 andEZ ∝ exp(±ikZ) as |x| → ∞ give information aboutthe asymptotic behavior at infinity. In a numerical calculation it is necessaryto truncate the spatial interval and consider the solution in a finite intervalbetween some pointsx− andx+. By choosingx− (x+) sufficiently far to theleft (right) of the cavity the plasma density will be constant at these artificialboundaries. As the plasma is homogeneous nearx− and x+, the solutionsare independent propagating plane waves. The emitted Z mode satisfies the

23

equationdEZ

dx∓ ikZEZ = 0, (3.23)

which can be used as a boundary condition atx = x±. Similarly, a boundarycondition for the evanescent quantityEX is found to be

dEX

dx±|kX |EZ = 0 (3.24)

atx = x±. The boundary conditions (3.23) and (3.24) can be discretized and aresuitable for the numerical calculation. When discretizing the wave equationsand the boundary conditions one obtains a set of 2N linear algebraic equations.The whole set of equations can be arranged in a matrix equation. We have that(

AXX AXZ

AZX AZZ

)(aX

aZ

)=

(bX

bZ

), (3.25)

where(aX)i = EX(xi) and(aZ)i = EZ(xi) are column vectors,AIJ areN ×Nblock matrixes, andbI areN components column vectors describing the exci-tation. The matrix equation can be solved numerically by using some iterativemethod. The solution gives an approximation of the wave field betweenx− andx+. To test the numerical solution it is useful to plot|EZ| outside the cavity.As EZ describes a propagating plane wave the amplitude|EZ| should be con-stant. Any artificial reflections on the boundariesx− andx+ due to numericalinaccuracies or due to the truncation of the infinite interval will be visible asmodulations of|EZ|.

3.1.5 A Quantitative Comparison Between the MethodsTo quantitatively study the resonance features described by the scale sepa-ration method a specific density profile is considered. We considerη(x) =−ηsech2(x/L⊥), whereη > 0 quantifies the amplitude of the depletion andL⊥ parameterizes the cavities perpendicular width. For this profile it is pos-sible to calculate the eigenfunctionsψn and the eigenvalues∆n analytically(see Paper [1] for details). The Z mode wave field can be calculated by us-ing Eq. (3.17). Figure 3.3 shows|EZ|2 as a function of∆ (solid line) for aset of plasma parameters appropriate for the F region ionosphere. The evenresonances at∆ = ∆0, ∆ = ∆2, and∆ = ∆4 are clearly seen. In addition, theresonances are slightly skewed as Eq. (3.17) indicates. To validate the accu-racy of the scale separation method the basic wave equation is solved using thenumerical method described in section 3.1.4. As seen from the figure, thereis a good agreement between the scale separation method and the numericalsolution (drawn with-signs in Fig. 3.3). The numerical solution also exhibitsthe characteristic skew shaped resonances. The resonance amplitude is in good

24

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.50

0.1

0.2

0.3

∆ [%]

|EZ /

EO

|2

∆=∆0

∆=∆2

∆=∆4

Figure 3.3: The Z mode wave field spectrum outside the cavity as a func-tion of ∆ from the length scale separation method (solid line) andfrom the WKB approximation (dashed line) and from the numer-ical solution () for η = 4%, L⊥ = 0.5 m, ωp/2π = 4.5 MHz,Ωe/2π = 1.35 MHz, andTe = 1500 K (electron temperature).

agreement with the simple formula|EZ/EO|= |QXO/QXZ |. The WKB solution(dashed line) reproduces the position of the resonances but does not describethe skewness.

3.2 Z mode Interaction Between Density Cavities

The Z mode leakage from an isolated cavity is strong and determines the am-plitude of the localized UH oscillation. Another important implication of thisleakage is that the individual filaments can interact through the Z mode radia-tion. The Z mode generated in one cavity can propagate to a neighbor cavityand mode convert to localized UH oscillations. The Z mode interaction be-tween individual filaments is investigated in a one dimensional geometry inPaper [2]. The basic equations are the same as in Paper [1], i.e., Eqs. (3.11)and (3.12). This treatment is briefly discussed in this section.

To illustrate the physics it is convenient to first consider the simpler prob-lem with the scattering of Z mode waves off a density cavity. This problem isalmost the same as the O mode driven problem considered in Paper [1]. How-ever, the incident Z mode has a long but finite wavelength so that it can exciteodd modes as well as even modes. This is in contrast to the O mode case whereonly even modes are excited. It is shown in Paper [2] that the odd and evenmodes behave very different. In order to describe the excitation of odd modesand to obtain a finite amplitude the approximation Eq. (3.15) is not sufficientlyaccurate. Instead, it is necessary to expand the exponential factor to the second

25

order inkZL⊥ 1,

η(x2)eikZ|x2−x1|η(x1) ≈ η(x2)[1+ ikZ|x2− x1|− k2

Z|x2− x1|2/2]

η(x1) .(3.26)

The zeroth order term does not give any contribution to the odd modes. Thefirst order term gives rise to a shift of the system’s resonance frequencies. Thewidth of the resonance is described by the second order term. Physically, thefrequency shift and resonance width is due to the interaction between the elec-tromagnetic Z mode wave and the electrostatic UH oscillation. For the evenmodes the frequency shift is small in comparison to the width of the resonanceand can therefore be neglected. For the odd modes the relation between shiftand width is the opposite. The frequency shift is much larger than the widthof the resonance. It is therefore necessary to include the shift in the calcula-tions. The shift is small but the physics of the shifted resonance is differentfrom the unshifted electrostatic resonances. The incident Z mode wave is to-tally reflected off the cavity at both even and odd shifted resonances. Due tothe reflection, the total field forx < 0 will be a standing wave. However, thestructure of the standing wave field is different for odd and even resonances.For even resonances the scattered field forx < 0 is out of phase with the inci-dent wave and there will be a node on the location of the cavity. For the oddresonances the scattered field is in phase with the incident wave and there willbe a maximum on the location of the cavity. The solution of the scatteringproblem shows that the Z mode field and/or its derivative can be discontinuouson a length scale larger than the width of the cavity.

Instead of describingEZ in detail inside the cavity where it represents anelectromagnetic correction to the localized UH oscillation, a set of condi-tions for the jump inEZ anddEZ/dx across the cavity can be derived. It canbe shown that at an even resonance the mean value ofEZ near the cavity is−QXO/QXZEO, which is in agreement with results obtained in Paper [1]. Atan odd resonance, the derivative ofEZ has opposite sign on the left and rightside of the cavity. We have that

dEZ

dx |x=x j+δ+

dEZ

dx |x=x j−δ= 0, (3.27)

whereδ > 0 andx j is the center of cavityj. These conditions can be usedto calculate the Z mode wave field between the cavities and the amplitude ofthe localized UH oscillations. In addition to the one dimensional treatment,an extension to cylindrical geometry is indicated. However, the actual modeconversion mechanism is still considered as one dimensional.

26

Interaction Between Magnetosonic Waves andLocalized Lower Hybrid Oscillations

An interesting example of naturally occurring small scale phenomena in spaceplasma is the localized bursts of waves in the lower hybrid (LH) frequencyrange. This phenomena was first observed by the Marie sounding rocket in theupper ionosphere [LKYW86]. The waves appeared as spikes in the measuredspectrogram and the phenomena were therefore initially termed “spikelets”. Itwas later confirmed that the wave activity coincides with density depletionsthat are elongated along the geomagnetic field. The relative depth of the cav-ities is typically a few percent. The perpendicular width is observed to betypically a few ion gyro radii and the parallel dimension is estimated to be sev-eral order of magnitudes larger than the perpendicular. The localized waves areobserved to have frequencies both above and below the LH frequency in theambient plasma. However, the localized waves are not only observed wherethe frequency is larger than the local LH frequency. A large fraction of thewave power withω < ω∞

LH is located at frequencies well below the cavitiesminimum LH frequency. These elongated density structures with localizedwaves in the LH frequency range, now termed LH cavities (LHC) or LH soli-tary structures (LHSS) are frequently observed by spacecraft in the ionosphere[KVC+92, EHD+94, BSP+98] and magnetosphere [TEA03] (see also a recentreview by Schuck et al. [SBK03]). Further, LHSS are always observed to beimmersed in nonlocalized wave activity nearωLH. The nonlocalized wavesare collectively referred to as the “hiss” and are believed to be generated byprecipitating electrons. The electric field associated with the hiss is strongestaboveωLH but wave activity belowωLH is also observed. The wave activityaboveωLH is mainly attributed to LH resonance cone waves. The magneticwave field has no sharp decrease in amplitude belowωLH which is consis-tent with electromagnetic waves on the whistler/MS wave dispersion surface.It has been observed, by using interferometricin situ measurements, that thewave front of the localized waves is rotating around the ambient geomagneticfield [PKSB97, BSP+98, TEA03]. The wave front is rotating in a left-handed(right-handed) sense for frequenciesω below (above) the LH frequencyωLH.The characteristic rotational properties are consistent with theoretical resultsfrom electrostatic cold plasma theory [Sey94, SSP+98].

The excitation of right-handed waves withω > ωLH has been explained in

27

terms of scattering LH resonance cone waves off the density cavities [SSP+98].The interaction between the nonlocalized LH resonance cone waves and thedensity cavity occurs through the Hall current due to the plasma inhomogene-ity. This Hall current is decisive for the wave properties inside the cavity, asit accounts for the sense of rotation of the radially localized waves. This scat-tering process is not able to explain the excitation of left-handed waves forω < ωLH where the resonance cone waves are evanescent. In Paper [3] it isshown that the left-handed localized waves can be excited by MS waves whichare incident on the cavity. The MS wave is focused by the Hall current mech-anism to left-handed rotating oscillations. The treatment is valid forω < ωmin

LH(the minimum LH frequency in the cavity). The theoretical results are shownto be consistent with spacecraft observations of LHSS.

The theories presented in [SSP+98] and Paper [3] describe the excitationof waves above the ambient LH frequency and below the cavity’s minimumfrequency, respectively. The transition regime between left-handed and right-handed withωmin

LH < ω < ω∞LH is investigated in Paper [4] in a simplified geom-

etry. In this regime the frequency of the incident MS wave matches the localωLH somewhere inside the cavity and the incident wave is mode converted tolocalized pressure driven LH oscillations.

This chapter is organized as follows: In section 4.1 the electrostatic descrip-tion of rotating oscillations is discussed. Section 4.2 describes the electromag-netic treatment presented in Paper [3] and [4]. The excitation of rotating wavesby focusing of MS waves is considered in section 4.2.1. In section 4.2.2 thetransition regimeωmin

LH < ω < ω∞LH is considered where mode conversion oc-

curs instead.

4.1 Electrostatic treatmentThe first analytical treatment of rotating LH oscillations in a preexisting den-sity cavity was based on the cold plasma approximation and the assumptionthat the electric field is purely electrostatic [Sey94]. Further, only wave prop-agation strictly perpendicular to the ambient magnetic fieldB0 = B0z wasconsidered. Using the same approximations in a homogeneous plasma givesthe dispersion relationω = ωLH which describes non propagating LH oscil-lations. However, in a inhomogeneous plasma the dispersive properties aresignificantly altered by the Hall current. We assume a cylindrically symmet-ric plasma density variation,n0(r⊥) = n∞

0 [1+η(r⊥)] wheren∞0 is the density

of the ambient plasma andη = η(r⊥) is the density variation associated withthe preexisting density cavity. In the electrostatic approximationE = −∇⊥φ,whereφ is the electrostatic potential. The LH frequency domain is character-ized byΩi ω Ωe which allows us to evaluate the electron velocity in the

28

drift approximation, giving the electron current

Je ≈ e2n0(r⊥)/me

Ωe

(i

ωΩe

∇⊥φ+ z×∇⊥φ)

, (4.1)

where the first term is the polarization current and the second term the Hallcurrent due to theE×B drift. The ions can be regarded as unmagnetized asω Ωi , giving Ji ≈ −ie2n0(r⊥)/(miω)∇⊥φ. The fluctuations in the electroncharge density can be calculated from the linearized continuity equation, giv-ing

ρe ≈ ε0ω2

pe(r⊥)ωΩe

[ωΩe

(∇2⊥φ+∇⊥φ ·∇⊥η

)− i(z×∇⊥φ) ·∇⊥η], (4.2)

where we have used(∇⊥n0)/n0 ≈ ∇⊥η. The polarization current describes acompressional motion of the electron fluid and give rises to the first two termsin Eq. (4.2). The second term is a correction due to the density gradient whichis small in comparison to the first term and can therefore be neglected. In ahomogeneous plasma the Hall current describes a non-compressional motionand the only contribution is the correction due to the density gradient. Simi-larly, the ion charge density has a negligible small correction due to the densitygradient andρi ≈−ε0ω2

pi(r⊥)/ω2∇2⊥φ. A wave equation forφ can be obtained

by using the Poisson equation,∇2⊥φ = −ρ/ε0. We have that[

1+ω2

pi(r⊥)

ω2 − ω2pe(r⊥)Ω2

e

]∇2⊥φ+ i

ω2pe(r⊥)ωΩe

[z×∇⊥φ] ·∇⊥η = 0. (4.3)

In a homogeneous plasma the second term, which is due to the Hall current inthe inhomogeneous plasma, is vanishing and the wave equation is reduced tothe well-known dispersion relation for LH oscillations, i.e.,ω = ωLH.

The first bracket term in Eq. (4.3) is equal toεxx = S and depends onr⊥through the local plasma frequencies. As a first approximation, the variationof S can be neglected in comparison to the Hall current term. Assuming aparabolic density profile it is possible to obtain a spectrum of localized waves,both below and aboveω∞

LH. In addition, the obtained solutions are rotating inthe same sense as in the observed LHSS. Thus, the Hall current interactionaccounts for the sense of rotation and admits localized waves withω rangingfrom well below to well aboveω∞

LH.The analysis outlined above describes localized waves with a rotating wave

front. It is natural to ask if it is possible to excite these waves by a linear mech-anism. The ambient hiss of nonlocalized propagating waves provides a sourceof energy. An extension of the above theory is presented in Ref. [SSP+98]where the restriction of only perpendicular propagation is relaxed. With this

29

generalization, the theory describes LH resonance cone waves which are prop-agating in the ambient plasma whenω > ωLH. The theory describes the ex-citation of right-handed waves by the scattering of LH resonance cone wavesoff the preexisting density cavity. In a homogeneous plasma the extension tokz = 0 allows for propagating waves on the LH resonance cone, which is de-scribed by the dispersion relationSk2

⊥ +Pk2z = 0. Taking the Hall current term

into account the following generalization of Eq. (4.3) is obtained

∇2⊥φ+ k2

⊥φ+ iDS

[z×∇⊥φ] ·∇⊥η = 0, (4.4)

wherek2⊥ = −Pk2

z /S. This theory describes the excitation of right-handedwaves aboveωLH by scattering LH resonance cone waves off the preexistingdensity cavity. The treatment in [SSP+98] is purely electrostatic and cannotdescribe the excitation of left-handed waves since the resonance cone wavesare evanescent forω < ωLH.

4.2 Electromagnetic treatmentIn the following two subsections the work in Paper [3] and [4] is summarized.In Paper [3] we present the first analytic electromagnetic theory for the linearexcitation of radially localized left-handed waves. It is shown that an inci-dent MS wave is focused inside the cavity to excite left-handed localized os-cillations. The interaction between the nonlocalized MS wave and the densitycavity occurs through the Hall current mechanism discussed above. The devel-oped theory is valid for frequencies below the cavity’s minimum LH frequency.The results are shown to be consistent with observations of LHSS by the Frejasatellite [EHD+94]. In Paper [4] the frequency range between the ambientLH frequency and the cavity’s minimum LH frequency is considered. In thisnarrow frequency range an incident MS wave can mode convert to pressuredriven LH oscillations and LH resonance cone waves. The mode conversionprocess between MS waves and pressure driven oscillations is investigated in asimplified geometry. It is shown that the MS wave excite localized LH wavesfor a set of resonance frequencies.

4.2.1 Excitation of Localized Rotating Left-handed Oscillations

To model the excitation of left-handed rotating oscillations it is necessary toinclude electromagnetic effects. For frequencies belowωLH, the cold plasmatheory describes an electromagnetic mode which is termed whistler wave orfast MS wave. In the following discussion the wave mode is referred to asthe MS mode. In a homogeneous plasma the MS waves are described by the

30

approximate dispersion relation

N2⊥ =

(R−N2z )(L−N2

z )S−N2

z. (4.5)

The present treatment is restricted toω < ωminLH for which R > 0, L < 0, and

S < 0 so thatS−N2z = 0 andN⊥ is finite. Contrary to the previously considered

LH resonance cone waves the MS waves can propagate outside the cavity whenkz is sufficiently small, i.e.,N2

z < R.It is convenient to start with the local dielectric tensorε = ε(x) as all dielec-

tric properties are local in a cold plasma. The elements ofε are functions ofxdue to the density variation associated with the cavity. By using the Maxwellequations and the cold plasmaε one can derive a wave equation forB, givenby [Eps56],

∇× (ε−1∇×B)− k2

0B = 0, (4.6)

whereε−1 is the tensor inverse ofε. Equation (4.6) describes both the prop-agating MS mode and an evanescent mode. The evanescent wave mode canapproximately be eliminated to obtain one single wave equation forBz. Theresulting wave equation depends on the parallel wave number but there is noqualitative difference between the oblique propagation and strictly perpendic-ular propagation. Therefore, the casekz = 0 is studied in detail. The waveequation forBz is

∇2⊥Bz + k2

⊥Bz + iDS

[z×∇⊥Bz] ·∇⊥η = 0 (4.7)

wherek⊥ = k0N⊥, S, andD are calculated for the plasma parameters outsidethe cavity. The third term in Eq. (4.7) is due to the Hall current in the inhomo-geneous plasma. The electric field can be calculated in terms ofBz by usingMaxwell’s equations, givingEz = 0 and

E⊥ = − iωk2

0RL(iD∇⊥Bz +Sz×∇⊥Bz) . (4.8)

To quantitatively study the coupling of localized and propagating MS waveswe consider a cylindrically symmetric density cavity, withη = η0(1− r2

⊥/a2)for r⊥ < a andη = 0 otherwise, whereη0 < 0 characterizes the depletion am-plitude anda is the perpendicular half-width of the cavity. The total wavefield is a superposition of the incident wave and an outward propagating scat-tered wave. Equation (4.7) can be solved by using the method of separation ofvariables. With unit amplitude of the incident wave the solution is

Bz(r⊥,ϕ, t) =∞

∑m=−∞

imei(mϕ−ωt)Bm(r⊥) , (4.9)

31

whereϕ is the azimuthal angle. Each term in Eq. (4.9) can be interpreted as awave with a wave front that rotates around the center of the cavity. The wavefront rotates in a right-handed sense form > 0 and in a left-handed sense form < 0. The radial partial waves are continuous but defined piecewise insideand outside the cavity. They are

Bm(r⊥) =

αmJm(κmr⊥), r⊥ < a

βmH(1)m (k⊥r⊥)+ Jm(k⊥r⊥), r⊥ > a ,

(4.10)

whereκ2m ≈ −2Dη0m/(Sa2) describes the wave properties inside the cavity

andJm (H(1)m ) is the Bessel (Hankel) function of the first kind and orderm. It

can be shown that|κm| k⊥ for m = 0 so that the length scale of the wave ismuch shorter inside than outside the cavity. The right-handed waves (m > 0)are evanescent inside the cavity sinceκ2

m < 0. These partial waves are defo-cused by the cavity so that the right-handed component will always be smallinside the cavity. For left-handed wavesκm is real andκm k⊥ so that thephase velocity is much smaller inside the cavity than outside. The amplitudeinside the cavity is

αm =k⊥J′m(k⊥a)H(1)

m (k⊥a)− k⊥Jm(k⊥a)H(1)′m (k⊥a)

κmJ′m(κma)H(1)m (k⊥a)− k⊥Jm(κma)H(1)′

m (k⊥a). (4.11)

It is important to note thatαm = α−m. The amplitudeαm is large when the de-nominator in Eq. (4.11) is small. The denominator does not have any roots forrealω and the amplitude will always be finite. However, the denominator hasimaginary rootsωmn. The amplitude will be maximum at the resonance fre-quencyω = Re(ωmn) and the width of the resonance is described by Im(ωmn).Due to the defocusing of the right-handed waves only the left-handed waveswill be resonantly excited andBz will have a left-handed rotating wave front.The sense of rotation is consistent with observations of LHSS. In addition, thetheoretical predictions of|E| are demonstrated to be consistent with LHSS ob-servations by the Freja satellite from a previously published event [EHD+94].The theoretically calculated and the measured averaged|E| are compared andfound to be in good agreement (see Fig. 2 in Paper [3]).

4.2.2 Excitation of Pressure Driven Lower Hybrid Oscillations

The focusing of MS waves considered in Paper [3] is valid for frequenciesbelow the minimum LH frequency inside the cavity. The caseω > ω∞

LH hasbeen considered previously by other authors. This leaves us with a transitionregime between the left-handed and right-handed waves withωmin

LH < ω < ω∞LH.

In this regimeω will be equal to the local LH frequency at some points on the

32

density slopes of the cavity and the local variation of the plasma density mustbe considered.

The treatment in Paper [4] is a first attempt to study mode conversion insideLHSS in the transition regimeωmin

LH < ω < ω∞LH. It is shown that an incident

MS wave can effectively excite localized LH oscillations. The theory is fullyelectromagnetic and includes finite Larmor radius effects. To have an analyt-ically tractable model of the conversion processes, only density variations inone direction perpendicular to the ambient magnetic fieldB0 = B0z are con-sidered. Further, the waves are considered to propagate parallel to the densitygradient and hence perpendicular toB0. The basic equations are derived from aFLR approximation [CK83]. These equations have the same form as Eq (3.4).By using the same analytical method as in Paper [1] and [2] it is shown thatthe MS wave excites localized pressure driven LH waves for a set of resonancefrequencies.

33

34

Outlook

In future work it is essential to extend the theory in Paper [1] to a cylindrical ge-ometry. This would allow the localized wave to have an azimuthal componentof the wave vector in addition to the radial component. Unfortunately, it is notclear how to generalize the scale separation method to cylindrical geometry. Acrucial point in the scale separation method is that the wave field is expressedin terms of the polarization vectors. This is important since the equations areuncoupled outside the cavity in this representation. The polarization vectorsfor plane waves are constant in the homogeneous plasma outside the cavity.In a cylindrical geometry, with cylindrical waves, the polarization vectors arenot constants but are a function of the distance from the center of the cavity.They will only agree with the plane wave polarization in the far field. As aconsequence, the governing wave equations will be coupled not only insidethe cavity but in the whole near field region. However, a deeper understandingof the coupling described by the coefficientsQXZ andQZX might give a hintabout how to proceed.

Another important generalization is to allow for a nonzero wave vector com-ponent along the ambient magnetic field and a large scale density variation tomodel the plasma stratification in the ionosphere. With these generalizations itis possible to describe the absorption and scattering of the O mode pump wave.The present treatment of Z mode interaction should also be generalized. Thetreatment is confined to the resonance frequencies for a single density cavityat which total reflection occurs. However, the system as a whole has otherresonance frequencies than those of each individual cavity. This means thatthe localized UH oscillations can be excited to even larger amplitudes at someother frequencies. It is also important to generalize the treatment to describethe mode conversion in a cylindrical geometry. This will give a more realisticmodel of the scattering processes.

Similar to the case of UH–Z mode interaction, the MS wave facilitates in-teraction between neighboring LHSS. The theory in Paper [3] should thereforebe extended to describe the MS wave field in the presence of several cavities.This extension will describe multiple scattering of a MS wave incident on thesystem of cavities. The amplitude of the total MS wave field should, due to themultiple scattering, be larger in the region between the cavities than far awayfrom the system. This global focusing of the MS wave field should possibly be

35

observable by spacecraft measurements. The present treatment of excitationof pressure driven LH waves by scattering MS waves off a cavity is limitedto a one dimensional density structure and wave vectors strictly perpendicularto ambient magnetic field. In future work it is essential to extend the treat-ment to a cylindrically symmetric density cavity. This is essential since theHall current interaction becomes important and influences the local dispersiveproperties inside the cavity. Further, the formulation in Paper [4] should be ex-tended to allow a nonzero wave vector component along the ambient magneticfield. This would allow the incident MS wave to mode convert to inertia drivenresonance cone waves as well as to the considered pressure driven waves. Withthese two extensions of the theory it is possible to study the transition from theleft-handed waves withω < ωmin

LH to the right-handed waves withω > ω∞LH.

36

Acknowledgments

First of all I would like to thank my supervisor Thomas Leyser for all supportand encouragement through my doctoral education.

The present work was supported by the Advanced Instrumentation and Mea-surements graduate school (AIM) and the Swedish Foundation for StrategicResearch (SSF). The research work has been carried out at the Department ofAstronomy and Space Physics at Uppsala University and in the “physics inspace” program, which is a part of the Swedish Institute of Space Physics. The“physics in space” research group is lead by Bo Thide and I am grateful to himfor giving me the opportunity to work there. Many scientists has visited andworked in our research group during the years. It has always been interestingand educating to meet these people. A thanks to all of you. In particular, Iwant to express my gratitude to Yakov Istomin for collaboration and enlight-ening discussions during his visits in Uppsala.

I am thankful to Anders Eriksson for fruitful collaboration on the lowerhybrid cavity problem. I thank present and former graduate students in spaceand plasma physics, including Jan Bergman, Tobia Carozzi, Bengt Eliasson,Roger Karlsson, David Sundkvist, and Anders Tjulin, for being good friendsand colleagues. Also, I am grateful to Christer Wahlberg for giving me theopportunity to assist in the education of undergraduate students.

Thanks to a generous traveling fund provided by the AIM graduate school,I have had the possibility to attend international conferences. In addition, thetraveling fund made it possible for me to stay an extended period at UCLA.During that stay, I had the privilege to discuss my work with George Morales.I am grateful to George Morales for helping me to find the right track to asolution of the Z mode leakages problem. Also, I thank Alfred Wong for hishospitality.

I thank Roger Karlsson, Harley Thomas, Thomas Leyser, and Kristina An-dersson for their valuable comments on the manuscript to the present summary.

My family has always been there for me and I thank my parents Maudand Borje and my brother Thomas for their encouragement. Finally, I thankmy fiancee Kristina Andersson for her unlimited support and encouragementduring these years.

Uppsala, April 2004

37

38

Summary in Swedish

Vaxelverkan mellan elektromagnetiska vagor och lokalis-erade plasmaoscillationer

I denna avhandling studeras vaxelverkan mellan elektromagnetiska vagor ochlokaliserade plasmavagor. Arbetetar i huvudsak teoretiskt men matdata fransatellitobservationer har ocksa anvands i viss utstrackning. Tva olika fysik-aliska system har betraktats. Artikel 1 och 2 beskriver vaxelverkan mellanlokaliseradeovrehybridvagor och Z-vagor. Speciellt har Z-vag emissioner franovrehybridvagor lokaliserade i plasma kaviteter studerats. Dessa emissionerar starka och bestammerovrehybridvagornas amplitud. Dessutom, Z-vagornakan propagera mellan kaviteterna och darmed koppla samman enskilda kavitet-er. Denna vaxelverkanar viktig for var forstaelse och tolkning av radiovagsex-periment i jonosfaren. Artikel 3 och 4 beskriver vagfenomen somar relevantafor solitara lagrehybridstrukturer som observerats i jonosfaren och i magne-tosfaren. Exitation av lokaliserade vagor genom spridning av magnetosoniskavagor har studerats. Nedan foljer en kort sammanfattning av de fyra artiklarsomar inkluderade i denna avhandling.

Artikel 1: Konversion av infangadeovrehybridoscillationer och Z-vagor vid en plasmakavitet

I denna artikel presenteras analytiska uttryck som beskriver konversion mellankvasi-infangadeovrehybridoscillationer och Z-vagor. Ovrehybridoscillation-ernaar lokaliserade i en plasmakavitet dar de exiteras av en extern vag medordinar polarisation.Ovrehybridoscillationernaar inte fullstandigt infangadei kaviteten, istallet emitteras Z-vagor fran kaviteten. Detta problemar viktigtfor en vidare utveckling av teorier som beskriver generering av plasmakaviteterunder radiovagexperiment i jonosfaren. Ett system av ekvationer som beskriv-er konversionen har harletts. Dessa ekvationer har losts i en endimensionellgeometri genom att nyttja att systemet har tva olika langdskalor. Den elektro-magnetiska Z-vagen har en mycket langre vaglangdan kavitetens utstrackningvinkelratt det geomagnetiskafaltet. Resultatetar giltigt for godtyckligt lagaresonanser av det kvasi-infangade elektriska faltet. Teorin forutsager att res-onanserna for Z-vagornas amplitudar skeva och inte Lorentzianska.

39

Artikel 2: Elektromagnetisk vaxelverkan mellan lokaliseradeovre-hybridoscillationer i ett system av plasmakaviteterI denna artikel studeras elektromagnetisk vaxelverkan mellan plasmakavitetersom exiteras under radiovagexperiment i jonosfaren. Dessa kaviteter skapas avlokaliseradovrehybridturbulens som drivs av en extern vag med ordinar polar-isation. Radarmatningar och observationer fran sondraketer visar att ett stortantal kaviteter genereras i den del av plasma som den externa vagen belyser.Det ar valkant att den lokaliserade turbulensen genererar Z-vagor som kanpropagera fritt utanfor kaviteterna, se artikel 1. Vagorna som emitteras fran enkavitet faller in mot de narliggande kaviteterna. Denna vaxelverkan kopplardarmed samman turbulensen i de olika kaviteterna.

Artikel 3: Exitation av roterande lokaliserade vagor i plasmakavi-teter genom spridning av magnetosoniska vagorEn analytisk beskrivning av elektromagnetiska vagor i ett icke-homogent plas-ma har anvants for att undersoka exitation av lokaliserade vagor med roterandevagfront. Exitationar undersokt for frekvenser under lagrehybridfrekvensen.Pa grund av asymmetri i vaxelverkan mellan vagen och plasmat fokuseras deninfallande magnetosoniska vagen till en lokaliserad vag vars vagfront roterarrunt en magnetfaltslinje genom kavitetens centrum. Resonansfrekvenser ochresonansernas vidd beraknas. Teorinar viktig for forstaelsen av solitara lagre-hybridstrukturer somar vanligt forekommande i rymdplasmor. Genom att jam-fora de teoretiska resultaten med observationer av lagrehybridstrukturer franFreja satelliten har vi visat att dessaar konsistenta med varandra.

Artikel 4: Konversion av lokaliserade lagrehybridoscillationer ochmagnetosoniska vagor vid en plasmakavitetI denna artikel presenteras analytiska uttryck som beskriver konversion mellanlokaliserade tryckdrivna lagrehybridoscillationer och magnetosoniska vagor.De ekvationer som beskriver konversionen har losts i en endimensionell ge-ometri. Detta genom att anvanda samma skallangdseparationsmetod som tidi-gare anvants for att beskriva konversion mellanovrehybridoscillationer ochZ-vagor (se artikel 1). Analysen visar att en infallande magnetosonisk vag kanexitera tryckdrivna lagrehybridoscillationer i ett frekvensband mellan den om-givande lagrehybridfrekvensen och kavitetens lagsta lagrehybridfrekvens. Exi-tationenar speciellt stark da den infallande vagens frekvens sammanfaller mednagra resonanta frekvenser. Den teori som presenterasar viktig for forstaelsenoch tolkningen av observationer av solitara lagrehybridstrukturer i rymdplas-mor.

40

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Acta Universitatis UpsaliensisComprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and TechnologyEditor: The Dean of the Faculty of Science and Technology

Distribution:Uppsala University Library

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ISSN 1104-232XISBN 91-554-5982-X

A doctoral dissertation from the Faculty of Science and Technology, UppsalaUniversity, is usually a summary of a number of papers. A few copies of thecomplete dissertation are kept at major Swedish research libraries, while thesummary alone is distributed internationally through the series ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology.(Prior to October, 1993, the series was published under the title “ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science”.)