Ionospheric modification by powerful HF-waves

Transkript

1 UPTEC F Examensarbete 30 hp December 2008 Ionospheric modification by powerful HF-waves Underdense F-region heating by X-Mode Henrik Löfås

2 Abstract Ionospheric modification by powerful HF-waves Henrik Löfås Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box Uppsala Telefon: Telefax: Observations of modifications of the electron temperature in the F-region produced by powerful high-frequency waves transmitted in X-mode are presented. The experiments were performed during quiet nighttime conditions with low ionospheric densities so no reflections occurred. Nevertheless temperature enhancements of the order of K were obtained. The modifications found can be well described by the theory of Ohmic heating by the pump wave and both temporal and spatial changes are reproduced. A brief overview of several different experimental campaigns at EISCAT facilities in the period from October 2006 to February 2008 are also given pointing out some interesting features from the different experiments. The main focus is then on the campaign during October 2006 and modifications of the electron temperature in the F-region. Hemsida: Handledare: Nickolay Ivchencko Ämnesgranskare: Stephan Buchert Examinator: Tomas Nyberg ISSN: , UPTEC F Tryckt av: Uppsala Universitet

3 Populärvetenskaplig beskrivning Jordens atmosfär består av flera olika lager, närmast jordytan ligger troposfären vilken är den del av atmosfären som är bäst förstådd av människan. Troposfären sträcker sig upp till ca 10 km:s höjd och det är i den här delen av atmosfären som det dagliga vädret formas. Sammansättningen av troposfären är som vanlig luft vilket innebär att den domineras av käve- och syre-molekyler. Ovanför troposfären ligger stratosfären och mesosfären som liknar troposfären när det gäller sammansättning. Ovanför mesosfären på ca 100km:s höjd börjar termosfären eller jonosfären som den också kallas. Vid den här höjden börjar densiteten av atmosfären bli väldigt låg, det beror på att densiteten av atmosfären avtar med avståndet från jorden. När densiteten blir låg innebär det att gas-molekylerna börjar röra sig mer indviduellt och de olika typerna av molekyler blandas inte lika mycket. Detta gör att dels sammansättning men även densiteten av atmosfären på högre höjder ändras väldigt snabbt. På höga höjder är även solinstrålningen mycket starkare eftersom inte så mycket har hunnit bli absorberat än. När dessa högenergetiska solstrålar träffar de neutrala molekylerna joniseras de, vilket innebär att det bildas en positiv jon och en fri elektron. När det bildas fria elektroner i gasen så blir den ledande och ett så kallat plasma har bildats. Det är det här ledande lagret uppe i atmosfären som gjorde det möjligt att med långvågs-radio kommunicera över stora avstånd eftersom dessa vågor reflekteras i jonosfären och markytan och kan därför färdas runt jorden istället för att försvinna ut i rymden. I den här rapporten beskrivs några experiment som har gjorts för att få en bättre förståelse för hur olika processer i jonosfären fungerar. Experimenten har utförts vid EISCAT:s anläggning i Tromsø i norra Norge. EISCAT är en internationell forskningsorganisation som är sponsrad av Sverige, Norge, Finland, Japan, Kina, Storbrittanien och Tyskland. Vid anläggningen i Tromsø finns det bl.a. en radar och en heating -anläggning. Det är dessa två anläggningar som har använts för experimenten som vi har tittat på. Heating anläggningen är ett nät av flera mindre antenner som tillsammans genererar elektromagnetiska vågor i radiofrekvens området (3MHz-10MHz). När de här vågorna kommer upp i jonosfären exiteras olika reaktioner och instabiliteter beroende på hur jonosfären ser ut för dagen. De flesta experimenten vid dessa anläggningar iii

4 Populärvetenskaplig beskrivning iv utförs när jonosfären är i ett sådant tillstånd att vågorna reflekteras eftersom det anses att det är då de intressanta sakerna sker, såsom instabiliteter i plasmat. Dessa instabiliteter driver olika intressanta reaktioner ofta av icke-linjär natur som är speciellt intressanta att studera eftersom icke-linjära reaktioner är svåra att beskriva i teorin. Vid instabiliteter överförs det ofta stora mängder energi från vågen till plasmat inom ett litet område vilket medför stora temperaturförhöjningar i ett väldigt begränsat område. I vårt fall har vi istället undersökt experiment där inte reflektion sker och även en typ av vågor som inte skapar instabiliteterna i plasmat. De här förhållandena har inte undersökts speciellt mycket förut eftersom det inte ska ske speciellt mycket då. Men vi har funnit starka temperaturförhöjningar även i det här fallet. Vi har även funnit att dessa ökningar kan beskrivas med redan existerande teori i form av ohmisk upphettning. Ohmisk upphettning förekommer överallt när en ström går igenom ett medium med resistans viket det jonosfäriska plasmat har. även om det här är en känd effekt är det förvånande att den är så stark i jonosfären och då framförallt för den vågtyp som vi undersökte även kallad extra-ordinära vågor (X-mode). Dessa vågor beskrivs av elektriska fält som oscillerar vinkelrätt mot jordens magnetfält till skillnad från ordninära vågors E-fält som oscillerar längs med. För att beskriva dessa temperaturförhöjningar är det nödvändigt att lösa en partiell differential ekvation, elektron-energi balansekvationen, som beskriver hur elektronerna tar upp energi från foto-elektroner från solen, vågen som vi skickat upp, m.m. samt hur de förlorar energi till omgivande joner och neutrala atomer. Den här ekvationen löses numeriskt och resultaten vi har fått har stämt bra överens med våra observationer. Med hjälp av samma lösare har vi även provat vad som händer om man ändrar frekvensen på den uppskickade vågen eller om densiteten i jonosfären skulle ändras. Dessa resultat är viktiga för att förstå och bestämma hur nya experiment ska utföras för att vidare förstå effekten. Nya experiment har utförts i slutet av oktober 2008 för att validera resultaten som framförts i den här rapporten men de har inte blivit analyserade än.

5 Acknowledgements There are a lot of people I want to thank for the support and help making this thesis. First and the most important is my supervisor at KTH, Nickolay Ivchenko, without him nothing of this would have been possible. He is one of the most encouraging, enthusiastic and inspiring persons I have met, which have helped me a lot when I have got stuck or have had some other problem. I also would like to thank Hanna Dahlgren for help with making things work during my time, and also for involving me in some fun projects building things. A very special thanks also goes to Björn Gustavsson for first the help with the basic code for my analysis. And also for the interesting discussions during the campaign in Tromsø which he arranged in a nice way. I also want to thank for all the help and discussions around the article we have written, which I have learned a lot from. I also would like to thank Mike Rietveld for the discussions about the article and the stay at the EISCAT facilities in Tromsø. I wish to thank Dan Whiter for both the nice company in Stockholm and at our campaign on Svalbard and also for help with the language during the article writing. I want to thank all the people working at the Alfvn laboratory for creating the pleasant working environment. Thanks to Jonas Olsson, Torbjörn Sundberg, Josef Höök, Johanna Warander, Monica Alaniz and the rest of the lab staff for all interesting discussion during lunch and some late nights. At last I want to thank Stephan Buchert and Thomas Leyser at Uppsala University. v

6 Table of Contents 1 Introduction Structure of the Ionosphere Plasma experiments in the Ionosphere Tromsø radar and heating facilities Radar Heating Scope and outline of report Plasma physics Basic formulas Waves in the ionosphere Collision processes Energy balance equation HF Pump Instabilities in F-region Data Overview and Analysis Radar data Density profile The Data Set The October 2006 campaign The December 2006 campaign The February 2008 campaign F-region Heating in underdense conditions Experimental data vi

7 TABLE OF CONTENTS vii 5 Modeling Solution of the energy equation Dependence of T e enhancement on different parameters Discussion and Conclusions 61

8 List of Figures 1.1 Layers of the Atmosphere and the ionosphere Neutral density for typical conditions in Tromsø given by the MSIS-E-90 Atmosphere Model Ion densities for typical conditions in Tromsø according to the International Reference Ionosphere (IRI) Collision frequencies for 17-Oct Dielectric and conductivity coefficients for 17-Oct Profiles for Electric field and energy deposition Overview 17-oct, Critical frequencies from Ionosonde Density profile from radar data E-region modulation 17-Oct, Overview 18-Oct, E-region modulation 18-Oct, Overview 19-Oct, Overview 20-Oct, Overview 11-Dec, E-region modulation 11-Dec, Overview 12-Dec, Overview 13-Dec, Overview 05-Feb, E-region modulation 05-Feb, Overview 06-Feb, Overview 07-Feb, E-region modulation 07-Feb, Overview 08-Feb, viii

9 LIST OF FIGURES ix 4.1 Different time integrations 17-Oct Combined heating cycle 17-oct 2006, average over all heating cycles Combined heating cycle 17-Oct 2006, conditional averaging Height dependence for combined pulse 17-Oct Combined heating cycle 18-Oct Height dependence for combined pulse 18-Oct Combined heating cycle 19-Oct Height dependence for combined pulse 19-Oct Density profile for the three different days, first heating cycle used for simulation Cooling rates in the ionosphere Height dependence of T e enhancement Temporal evolution of the T e enhancement Temperature increase as a function of frequency of heater Rise time as a function of frequency of heater

10 List of Tables 1.1 The Tromsø UHF-system experiments Field strength for the different Tromsø arrays Electron momentum transfer collision frequencies Overview of the experiments Modulation in E-region Modulation in F-region Heat flow used in simulation Observed and simulated parameters x

11 Chapter 1 Introduction The atmosphere of earth is divided into many regions shown in fig The region closest to the earth is the troposphere which extends up to about 10km from the surface. The troposphere is the part best known by mankind and it is in this region our daily weather is formed. Above the troposphere lies the stratosphere which reaches up to about 45 km. Between 45 km and 85 km is the mesosphere which is the outermost layer of the homosphere. The three lowest layers of the terrestrial atmosphere show the same hydrodynamic character which means for example that the mean mass of the molecules does not change. Above 85km lies the thermosphere or the ionosphere which extends several of hundreds of kilometers out to the exosphere which is the outermost region of the atmosphere. In the ionosphere the character of the atmosphere changes, from the molecule dominated homosphere to a region where the ions starts to dominate. There is also a positive temperature gradient leading to temperatures greater than 1000 K due to absorption of solar UV-radiation.[1] The constituents of the atmosphere are mainly molecular nitrogen and oxygen and about one third of the atmosphere s mass is concentrated in the troposphere. At increasing altitude the density of the atmosphere decreases fast and the ratio between the different species stays quite constant until the lower part of the ionosphere. In the ionosphere the density of different species falls of quickly but the rate is dependent on the mass of the constituents therefore the heavier elements fall of more quickly. 1.1 Structure of the Ionosphere The ionosphere is by convention divided into different regions depending on the character of the density and each region has a local maximum in electron density. The lowest region is the D-region about km where the peak density in electrons is around 10 9 m 3, next is the E-region from 85 km to about 130 km where the electron density reaches m 3. Above the E-region is the F-region which consist of two sub regions, where the F 1 is only visible 1

12 1.1. Structure of the Ionosphere 2 Figure 1.1: Layers of the Atmosphere and the ionosphere Density of neutrals O N 2 O 2 altitude (km) number density (m 3 ) Figure 1.2: Neutral density for typical conditions in Tromsø given by the MSIS- E-90 Atmosphere Model

13 1.2. Plasma experiments in the Ionosphere 3 during day time. The peak density in the F-region is around m 3 at the F 2 peak. The ionization is mostly caused by ultraviolet radiation, x-rays from the sun and cosmic radiation and this means that the ionization changes with the solar cycle but also with the time of day. The highest densities in the D-,E- and F 1 often occur at noon local time and during nighttime the electrons disappear from the D-region while in the E- and F 1 -region the electron density falls off to very low values in the order of 10 7 m 3. The behavior of the F 2 -region is more complicated and more correlated to geomagnetic latitude than time of the day. The geomagnetic latitude is derived in the same way as ordinary latitude but using the geomagnetic axis instead of the rotational axis of earth. The simplest way to approximate the geomagnetic field is with a central dipole with an axis inclined 11 to the geographical axis, this field corresponds very well with the observed field. [1] Altitude (km) Density of ions O + NO + + O 2 N + H + He number density (m 3 ) Figure 1.3: Ion densities for typical conditions in Tromsø according to the International Reference Ionosphere (IRI) 1.2 Plasma experiments in the Ionosphere The ionospheric plasma is an example of a naturally occurring plasma laboratory where it is possible to do experiments, and therefore is it of biggest interest both from a plasma physics and a geophysical point of view.[2] A lot of different experiments can be performed in this laboratory, but this report will concentrate on modification of the ionosphere by powerful RF-waves. Motivation to building facilities and performing these experiments comes from:

14 1.3. Tromsø radar and heating facilities 4 First to investigate the response of the natural ionosphere to a well-defined heat source. How the Radio waves interact with the ions are well known but the interesting things to investigate are how the heat conduction, diffusion and chemical reactions in the ionosphere are affected. Second to perform more a kind of laboratory experiments which is possible since the scale length of the ionosphere is much larger than typical scale lengths of interesting plasma phenomena. This means that it is possible to investigate for example plasma instabilities and non-linear interactions in an almost homogeneous plasma much larger than that is possible to attain in laboratories. Third there are those who try to control the ionospheric plasma and in that way get a more secure channel for long distance communication since the ionosphere is used as a reflecting medium for the radio waves. To perform these investigations transmitter sites capable of producing powerful enough radio waves to perturb the ionosphere were constructed in the beginning of the seventies, mostly in the USA and the USSR. These high power facilities are commonly called heaters. Most of the experiments performed on these sites have concerned the F-region modification phenomena where the interesting plasma instabilities occur. But also a number of experiments designed for the D- and E-region have been performed. There are also other ways to perturb and examine the ionosphere, for example have experiments been done where rockets or balloons take some chemicals and release them in the ionosphere to perturb it. 1.3 Tromsø radar and heating facilities At the EISCAT facilities outside Tromsø in Norway basic ionospheric experiments are performed. The aim of the experiments performed is in various ways perturbing the ionosphere to determine geophysical, ionospheric parameters and to investigate fundamental plasma physics. It is possible to divide the experiments into two categories, active geophysical experiments and plasma physics experiments. In the geophysical experiments one perturbs the ionosphere in a controlled way where the start, end time and strength of perturbation are known and from that it is possible to determine aeronomic reaction rate, energy transfer rates and reaction coefficients. In the case of plasma physics experiments the perturbation is used to initialize some nonlinear plasma phenomena and then study how they evolve. So far at the Tromsø site the experiments in the plasma physics category dominate.[3] Radar The EISCAT (European Incoherent Scatter) [4] radar facilities are situated in northern Norway near Ramfjordmoen (69.6, N 19.2 E) outside Tromsø. The

15 1.3. Tromsø radar and heating facilities 5 facility was built between by the Max-Planck-Institut für Aeronomie with support of the University of Tromsø. The facilities contains both a UHF- and a VHF-incoherent radar system. The incoherent scatter radar technology is one of the most powerful techniques for the study of the earth s ionosphere, magnetosphere and the near earth solar wind. The incoherent radar echo results from the interaction between the radiated electro-magnetic waves and the electrons in the ionosphere. This interaction is called Thomson scattering, where the incident wave puts the electrons in a oscillatory motion and they then emit a small fraction of the incident energy as dipole radiation. The same process also occur for the ions, but due to their greater mass the scattered energy is negligible. In the case of monostatic incoherent scatter experiments the radar is transmitting certain modulation patterns at a few different frequencies. The scattered signal is received after the transmission has finished and continues for some time corresponding the altitude range. The next cycle can start when the reception of the previous cycle has finished, a normal cycle lasts for about 5-10ms. The different modulation patterns used are called experiments and at the Tromsø UHF system which is used for the measurements for this report there are four different standard experiments called tau2pl, tau1u, arc1u and mandat. The different experiments give different resolution in altitude and time, and they have also different altitude ranges, the basic parameters are shown in table 1.1. These experiments use a complex form of modulation called alternating codes where the phase is modulated, the phase has either the value 0 or 180. The transmission consist of long pulses divided into bits. After each pulse is transmitted, reception takes place before next pulse is transmitted. The received signal strength is proportional to the pulse length, but long pulses result in poor range resolution, i.e. it becomes ambiguous from where the signal comes. In order to achieve both a strong signal and good range resolution the phase values are changed within pulses according to a so-called code. This code selected in such way, that after reception of a certain number of pulses the contributions from all except a narrow range interval cancel. In this way one can achieve a good range resolution and still operate the radar with a high duty cycle. This alternating codes require that the ionospheric plasma is sufficiently stationary over the sequence of pulses needed to complete one code, typically a few seconds[5] Heating The site in Tromso is quite unique since it is one of the few places in the world with combined incoherent radar and heating facilities. The heater is used for modifying the ionospheric parameters by applying high-frequency electro-magnetic waves. The term heating comes from that the high-power electro-magnetic waves transmitted to the ionosphere raises the temperature of the electrons in ionosphere and thus modifying the plasma state. A detailed description of the heating facilities can be found in for example [6], here a short introduction is given. The heating facility has twelve linear class AB

16 1.3. Tromsø radar and heating facilities 6 Experiment Pulses (µs) Sampling Resolution (km) Ranges (km) Time name rate (µs) resolution (s) tau2pl two tau1 two manda arc Table 1.1: The Tromsø UHF-system experiments (from EISCAT a ) a tetrode amplifiers each capable of producing 100kW continuously radiating power. In total the facility is therefore capable of generating 1.2MW (12 100kW) of continuous power in the frequency range from 3.85MHz to 8MHz. The amplifiers deliver the power to an antenna array and depending on the array the effective power radiated (ERP) can be estimated. The ERP is the power that would need to be radiated to achieve the same power density if one used instead of a directional antenna a simple half-wave dipole antenna. For the high gain antennas that are typically used at the world s heating facilities the ERP is only a small fraction of the actually radiated power and can be calculated as the product of the power from the amplifiers times the antenna gain. The antenna gain is often given in db which means it is not possible to just multiply. Instead the ERP is given as 10log (ERP) = 10log np + G (1.1) where n is the number of amplifiers, P is the power of one amplifier and G is the antenna gain in db. There are three different antenna arrays, array 2 and 3 consist of six rows of six crossed dipoles and are capable of radiating in the frequency ranges MHz respectively 5.5-8MHz. Both these arrays have a antenna gain of 24dB (± 1 depending on frequency). This means that the maximum effective radiated power (ERP) is 300MW for these arrays. Array 1 consist of 12 rows of 12 antennas with a total gain of 30dB which gives an ERP of 1200MW in the frequency region 5.5-8MHz. Each amplifier can also be connected to any of the antenna arrays and it is possible to use subsets of the antenna arrays simultaneously and in that way use different frequencies at the same time. The transmitters can be tuned in a continuous way in the frequency range, but they are only allowed to use a discrete set of frequencies decided by the government.the transmitted waves are either circular or linear polarized. There are also two different senses of circular polarization, left- or right-handed. Since the waves are deflected when they propagate through the ionosphere, in the reflection region they propagate perpendicular to the magnetic field instead of parallel as when they were transmitted. This means that the left- and right-handed waves have turned into either O- or X-mode waves. Therefore it is said that the transmitter is either in O- or X-mode when circular polarized waves are transmitted. The

17 1.4. Scope and outline of report 7 main wave parameters such as frequency, polarization, beam direction and maximum power is chosen during the tuning stage before experiment. Then during the experiment it is possible to modulate the transmitted wave in various ways, for example in complicated on-off schemes. It is also possible to use power stepping to use different power, change the polarization or the beam direction on each cycle. For wave propagation in free space it is possible to calculate the electric field at a distance R as E = 0.25 (ERP)/R (1.2) Typical values for the Tromsø facilities for two different altitudes are given in table 1.2 One also needs to have in mind that depending on the density in the Array 1 Array 2 Height(km) E (V/m) E (V/m) Table 1.2: Field strength for the different Tromsø arrays (from [6]) D- and E- region there will be absorption in these layers as well. During times with high densities in these layers the waves sometimes can not penetrate through them but instead they will be reflected before they reach the F-region. 1.4 Scope and outline of report A lot of both theoretical and experimental work have been done concerning modulation in the E- and F-region by O-mode.(see for example [2],[3],[7],[6] and [8]) Most of these experiments have been done when the ionosphere have been over dense so reflection occur, since the common thought is that the interesting things happens when there are reflection. In this report instead an underdense ionosphere modulated in the F-region with X-mode waves is the main concern. Even if it is possible to conclude that no instabilities occur there is still clear modulation which will be analysed in more detail. The report will start by introducing the necessary plasma and wave relations for the analyse in chapter 2. Then in chapter 3 an overview over all the experiments performed will be given with a short description of what is visible each day. In chapter 4 the more deep analysis of the F-region modulations will be performed. Chapter 5 will continue with making use of the theoretical relations obtained earlier to model the results obtained in the previous chapter. In the last chapter, chapter 6 there will be a discussion of the results from the study.

18 Chapter 2 Plasma physics 2.1 Basic formulas There are three fundamental parameters characterizing a plasma, these are the particle density n measured in particles per cubic meter, the temperature T measured in electron-volt or Kelvin where 1eV = 11605K and finally the magnetic field B measured in Tesla. From these fundamental parameters is it the possible to deduce some other parameters that describe the plasma. Some of the important ones are first the Debye length which is given by λ 2 D = ǫ 0κT e n 0 e 2 (2.1) where ǫ 0 is the permitivity of free space, κ the Boltzmann constant and e is the electron charge. The Debye length is the characteristic length scale of the plasma and it is a measure of the relaxation distance i.e. how far into the plasma a charge e will affect before it is shielded by surrounding charges. The second important parameter is the plasma frequency given by ω 2 ps = n se 2 s ǫ 0 m s (2.2) The plasma frequency is possible to define for each species if the plasma contains more than one so e s,m s and n s denote the charge, mass and number density for each species. The plasma frequency is the characteristic time scale of the plasma, and the electron plasma frequency is by far the most important. As seen from eq. 2.2 the plasma frequency is independent of the magnetic field and describes the oscillation resulting from charge and density perturbations. The third important parameter is the cyclotron frequency ω cs = e sb m s (2.3) which is the frequency describing the gyration for charged particles around the field lines of the background magnetic field.[9] 8

19 2.2. Waves in the ionosphere 9 There are several ways of describing a plasma. One commonly used is the model of two-fluid approximation. In this description the plasma is treated as one electron fluid and one ion fluid interacting with each other via electromagnetic fields. For this model two sets of equations are necessary, one set describing the fluid. n s t + (n sv s ) = 0 (2.4) ( ) v m s n s t + (v s )v s = e s n s (E + v s B) P s (2.5) which is the continuity equation and the momentum equation where v s is the mean velocity of different species, E the steady state electric field and P s is the pressure. In eq. (2.5) collisions between the different species are neglected. To connect the different fluids, Maxwell s equations are needed ǫ 0 E = n s e s (2.6) E = B (2.7) t B = 0 (2.8) 1 B = E e s n s v s + ǫ 0 (2.9) µ 0 t Equations forms a set of equations describing the plasma. For completeness one would also need an equation of state describing the relation between the pressure and the density which is omitted here and in the following the thermodynamical pressure will be neglected, i.e. the pressure is assumed to be zero. This set of equations is not unique in that way that it is the only way describing the plasma but just one of many models. To describe more complicated instabilities and non-linear phenomena one for example has to consider the kinetic model instead which will be described briefly in later chapters. 2.2 Waves in the ionosphere All plasma phenomena are possible to describe with use of the Maxwell s equations combined with the Lorentz force equation here represented by the fluid momentum equation. One natural way to start is with the description of linear waves since many important phenomena can be described by them and the theory of linear waves is well understood. The developing of the linear waves will follow the treatment in chapter 6 in [10]. If the plasma is assumed to be a cold magnetized plasma that is no collisions and the background magnetic field along the z-axis. The waves are then described by Maxwell s equations ( ) and the momentum equation, (2.5), derived in the previous section. To describe the evolution of the waves it is necessary to examine Ampére s law (2.9) and for that first an expression for the velocity is needed.

20 2.2. Waves in the ionosphere 10 Assume that the waves can be described as a small amplitude electromagnetic field then E = E 1 (x,t) (2.10) B = B 0 + B 1 (x,t) (2.11) where B 0 = B 0 ẑ is the background magnetic field and E 1,B 1 describe the oscillating field. Since the amplitude of the terms describing the oscillating field are small it is possible to linearize the momentum equation using this assumption and dropping non-linear terms. m dv dt = q (E 1(x,t) + v B 0 ) (2.12) The waves can be described by oscillatory electric fields with time dependence exp( iωt) and since (2.12) is linear the particle responses are just a superposition of different Fourier modes of the applied field. Therefore it is possible to just consider one Fourier mode. E 1 = Ẽ(x,ω)exp( iωt) (2.13) If it is also assumed that the particles follow the wave motion then the velocity has the same time dependence exp( iωt) hence (2.12) simplifies to ) iωmṽ = q (Ẽ(x) + ṽ B0 (2.14) where the time factor has been left out. The vector equation (2.14) can be solved for v with some algebraic manipulation resulting in ( ) ṽ = iq Ẽ (x)ẑ + Ẽ (x) ωm 1 ωc/ω 2 2 iω c ẑ Ẽ(x) ω 1 ωc/ω 2 2 exp( iωt) (2.15) In this expression it is possible to identify three different velocities. The first term on the right hand side describes the parallel quiver velocity, this motion does not involve the background magnetic field since the motion is parallel to the magnetic field. The second term describes the polarization drift velocity and the last term describes the E B-drift velocity. The velocity (2.15) can now be used in Ampére s law (2.9) to get an expression for the electromagnetic wave 1 µ B = 0 j = iǫ 0 q j n j v j + ǫ 0 Ẽ t j ω 2 pj ω ( ) Ẽ Ẽ z ẑ + 1 ωcj 2 iω cj ẑ Ẽ /ω2 ω 1 ωcj 2 iωǫ 0 Ẽ /ω2 (2.16)

21 2.2. Waves in the ionosphere 11 This equation can be written in a more compact way introducing the dielectric tensor K 1 µ B ( K = ǫ 0 Ẽ) (2.17) 0 t where and K Ẽ = S id 0 id S P Ẽ (2.18) S = 1 j ω 2 pj ω 2 ω 2 cj (2.19) D = j ω cj ω ω 2 pj ω 2 ω 2 cj (2.20) P = 1 j ω 2 pj ω 2 (2.21) where the summation is performed over electrons and the different ion species. It is now possible to derive the dispersion relation for the waves using the dielectric tensor. First one can write the Maxwell s equations as two coupled equations B = 1 ( K ) E c 2 t E = B t (2.22) (2.23) where the tilde has been left out but E,B are still small amplitude oscillating fields. The equation describing the waves is obtained by taking the curl of (2.23), substituting it into (2.22) and assuming a phase dependence as exp(ik x iωt) resulting in the algebraic equation k (k E) = ω2 c 2 K E (2.24) This is the dispersion relation for the electromagnetic waves in the plasma describing the relation between the wave-vector k and the frequency ω. Introducing the refractive index n = ck ω (2.25) which is a renormalization of the wave-vector so that light waves has unity, results in nn E n 2 E + K E = 0 (2.26)

22 2.2. Waves in the ionosphere 12 which is a set of three homogeneous equations in the three components of E. If it is also assumed that the plasma is spatially uniform in the x-y plane which means that the density only changes along the z-axis (parallel to B) and the x-axis (perpendicular to B). The dispersion relation can now be written on matrix form S n 2 z id n x n z id S n 2 0 n x n z 0 P n 2 x E x E y E z = 0 (2.27) To simplify it is possible to introduce polar coordinates in k-space (or n-space) with B defining the axis and θ as the polar angle between B and k. Hence the relation between spherical and Cartesian components for the refractive index looks like and equation (2.27) becomes S n 2 cos 2 θ id n 2 sin θ cos θ id S n 2 0 n 2 sinθ cos θ 0 P n 2 sin 2 θ n x = nsin θ (2.28) n z = ncos θ (2.29) n 2 = n 2 x + n 2 z (2.30) E x E y E z (2.31) = 0 (2.32) In the situation when n 2 there is a so called resonance. The resonance corresponds to the wavelength going to zero (λ 1/ k ) and the dissipative effect will become very large since the fractional attenuation per wavelength is constant and there will be a near-infinite amount of wavelengths were the wave amplitude is decreased by the same fraction for each of these. One other case is when n 2 0 then there is a cut-off. The cut-off corresponds to a reflection of the wave since the refraction index changes from pure real to pure imaginary. To find these two cases it is necessary to calculate the determinant of the matrix in (2.32) which after some algebra can be written as where An 2 Bn 2 + C = 0 (2.33) A = S sin 2 θ + P cos 2 θ (2.34) B = (S 2 D 2 )sin 2 θ + PS(1 + cos 2 θ) (2.35) C = P(S 2 D 2 ) (2.36) Since it is only high frequency waves that are considered here where ω ω pi,ω ci the terms S,D,P can be simplified since the ion terms does not

23 2.2. Waves in the ionosphere 13 contribute to the sum. Hence the elements in the dielectric tensor can be simplified to S = 1 ω2 pe ω 2 ω 2 ce (2.37) n 2 = 1 P = 1 ω2 pe ω 2 (2.38) ω ce ωpe 2 D = ω(ω 2 ωce) 2 (2.39) Using the high-frequency coefficients derived above in (2.33) an expression for the refractive index can be obtained ( ) 2 ω2 pe ω 1 ω2 pe 2 ω 2 where Γ = ( ) 2 1 ω2 pe ω ω2 2 ce (2.40) ω sin 2 θ ± Γ 2 ( ) ωce 4 2 ω 4 sin4 θ + 4 ω2 ce ω 2 1 ω2 pe ω 2 cos 2 θ (2.41) This is the Altar-Appleton-Hartree dispersion relation first derived in 1932 to investigate how the long-distance short wave radio transmissions bounced off the ionosphere and it also has the property that it shows the deviation from the vacuum dispersion n 2 = 1. Now it is possible to get the cut-off frequencies, these frequencies determines the critical frequencies in the ionosphere when the waves are reflected. If it is assumed that the ionosphere is only non-uniform along the z-axis, the x-y plane is assumed uniform in a small region around the wave, hence it is only the z-component of the wave-vector that changes, the x-component will stay constant. This means that k k z and when the wave travels and n 2 0 also k z 0. When the magnitude of the k z decreases but not k x the wave will curve and at the reflection point the wave-vector will be perpendicular to the background magnetic field, that is θ = π/2. First consider the solutions for waves propagating along the magnetic field (θ = 0) without the high frequency approximation, then the determinant of 2.40 looks like ( (S n 2 ) 2 D 2) P = 0 (2.42) with the roots P = 0 (2.43) n 2 S = ±D (2.44) The second equation (2.44) can be written as n 2 = R (2.45) n 2 = L (2.46)

24 2.2. Waves in the ionosphere 14 where R = S + D (2.47) L = S D (2.48) where R and L stands for Right and Left and the origin of the terminology will be discussed further down. The solutions to these equations are R = 1 s L = 1 s ω 2 ps ω(ω + ω cs ) ω 2 ps ω(ω ω cs ) (2.49) (2.50) Now it is possible to see that R diverges for ω = ω cs and L for ω = ω cs. Since the ion cyclotron frequency is positive and the electron cyclotron frequency is negative it is possible to say that the R resonance is due to the electrons and the L resonance to ions. Since the electron gyrate around the magnetic field in the right-hand sense the wave corresponding to R resonance is called right hand polarized. The ions gyrate in the opposite direction and therefore the wave corresponding to the L resonance is called a left hand polarized wave. As mentioned before the wave curves when it travels through the ionosphere and at the reflection point it has curved so it travels perpendicular to the magnetic field. In this case there will be two other types of wave mode. One mode where the wave oscillates along the magnetic field called Ordinary mode (O-mode). And one mode where it oscillates transverse to the magnetic field called extra-ordinary mode (X-mode). For these two modes the roots of the determinant of eq. (2.40) looks like n 2 + = P (2.51) n 2 = S2 D 2 S (2.52) where n + and n are the refractive index for O- and X-mode waves. Now it is easy to set n 2 = 0 and solve to get the critical frequencies. For O-mode it is the same equation as for waves traveling in a unmagnetized plasma with solution ω 2 = ω 2 pe (2.53) and for X-mode the solution is ω 2 pe = ω(ω ± ω ce ) (2.54) (2.53) and (2.54) are the critical frequencies for reflection for O- and X-mode waves.

25 2.3. Collision processes Collision processes To get a measure of the energy transfer between the heated electrons and the surrounding gas the collisions between both electrons and ions and electrons and neutrals have to be examined. Theoretically this can be done in some different ways, the following approach will follow chapter 9 in [1] and make use of the kinetic approach where the particles in the gas are described by the Maxwellian distribution function f(v s ) = n i (m i /2πκT s ) 3/2 e msv 2 s 2κTs (2.55) The normalization for f(v s ) is chosen so that f(v)dv = 4π v 2 f(v)dv = n s (2.56) 0 where the integral is taken over all particle velocities in a spherical coordinate system. The distribution function is a common way of evaluating different properties of a gas, for example to find the kinetic energy of a particle in the gas it is necessary to average the 1 2 mv2 s over the velocity distribution 1 ǫ s = 2 m svsf(v 2 s )dv s (2.57) or by using (2.55) ǫ s = 3 2 κt s (2.58) one ends up with the commonly know expression for the kinetic energy. To investigate the collisions between electrons and neutrals it is first possible to assume that the interaction only depends on the relative velocity between the particles g = v 1 v 2 (2.59) and it can be represented by the cross-section σ(g). Then the number of collisions per unit time made by a particle of type 1 with type 2 is ν 12 (g) = n 2 gσ(g) (2.60) For the present study it is the momentum transfer cross section that is of biggest interest since it relates to both energy and momentum exchange. σ D (g) = 2π q(θ,g)(1 cos θ)sin θdθ (2.61) Now it is possible to use (2.61) in (2.60) to get the collision frequency for electrons with particles in a real gas ν en = n n gσ D (g) (2.62)

26 2.3. Collision processes 16 Element Collision frequency (sec 1 ) N n N2 ( T e )T e O n O2 ( Te 1/2 )Te 1/2 O n O Te 1/2 H n H ( T e )Te 1/2 He n He Te 1/2 Table 2.1: Electron momentum transfer collision frequencies ν en (from [1]) To get the average momentum transfer cross section it is necessary to average (2.62) over all relative speeds consistent with two independent Maxwellian distributions and taking T e /m e T n /m n which gives the collisional frequency ν en = 4 3 (8κT e/πm e ) 1/2 QD (2.63) where Q D is the average momentum transfer cross section given by Q D = (1/c 6 e) 0 veσ 5 D (v e )e v 2 e c 2 e dv e (2.64) c 2 e = 2κT e m e (2.65) From (2.63) it is possible to calculate theoretical values for the collision frequencies between electrons and neutrals but since the theoretical methods usually does not give accurate values for the momentum cross section since it is too many parameters to take into account. Therefore it is necessary to calculate the collisions frequencies from experimental data instead. From [1] the experimental collision frequencies are given in table 2.1 When it comes to the collisions between charged particles for example electrons and ions a theoretical treatment is easier since the force between the particles is the ordinary Coulomb force F 12 = Z 1Z 2 e 2 4πǫ 0 r 2 (2.66) where Z is the charge state of each particle and r is the radius of separation. The differential scattering cross-section for elastic collisions was derived by Rutherford q(θ,g) = (Z 1 Z 2 e 2 /4πǫ 0 µg 2 ) 2 sin 4 (θ/4) (2.67) where µ = m 1 m 2 /(m 1 + m 2 ) is the reduced mass and θ is the center of mass scattering angle. Using the mom. cross section given in (2.61) it is possible to remove the angular dependence of (2.67) by integration with the limits (θ min,π/2) which yields σ D (g) = 2π(Z 1 Z 2 e 2 /4πǫ 0 µg 2 ) 2 (ln(1 cos θ m ) 1 ) (2.68)

27 2.4. Energy balance equation 17 where θ m is the minimum scattering angle. With the definition (2.68) takes the form 1 cos θ m = Λ (2.69) σ D = 2π(Z 1 Z 2 e 2 /4πǫ 0 µg 2 ) 2 ln Λ (2.70) Now the hard part is to calculate the Λ factor. From [11] an expression is obtained ( ) 16πǫ0 κt e ln Λ = ln γ 2 z i e 2 k2 e + k 2 ( i (k 2 k e ki 2 ln i + ke) 2 1/2 ) (2.71) k e where k 2 i = n i Z 2 i e 2 /ǫ 0 κt i (2.72) k 2 e = n e e 2 /ǫ 0 κt e (2.73) and γ = is the Euler-Mascheroni constant. To get the collision frequencies for electrons moving in a gas of ions, it is now possible as in the previous case to use (2.70) in (2.62) and average over all relative speeds and taking T e /m e T i /m i yields Q D = (π/2)(e 2 /4πǫ 0 κt e ) 2 lnλ (2.74) The average collision frequency between electrons and ions is then obtained by using (2.74) in (2.63) which results in ν ei = 1 3ǫ 2 0 Zi 2e4 ln Λ m 1/2 e (2πκT e ) n 3/2 i (2.75) As example the collisions frequencies for 17th of October 2006 are shown in fig Energy balance equation To determine how electromagnetic waves transfer energy and affect the ionospheric plasma the transport equation for the plasma has to be examined. There the motions of the different species are determined by pressure gradients, gravity and the external electric and magnetic fields. The transport equations here are derived for the ionosphere which is a partly ionized, collision dominated plasma moving in a neutral background gas and affected by the forces of gravity, electric and magnetic fields. The equations describing the energy transfers in the plasma are the energy balance equations for the different species. These equations can be derived from the Boltzmann or Vlasov equations which are the general equations when the kinetical description of the plasma is used. The energy equation is the third moment from these equations and looks as[12]

28 2.4. Energy balance equation Electron colision (ν en, ν ei, ν e ) e n e i e i+e n 400 altitude (km) Collision frequency (Hz) Figure 2.1: Collision frequencies for 17-Oct n sκ T s t n sκv s T s + n s κt s V s + φ s = M s A s V s (2.76) where φ s = K e T s is the s:th species heat flux vector where only the parallel component is considered and K e is the parallel coefficient of thermal conductivity. M s is the kinetic energy moment and A s is the momentum moment both derived from the collision term in the Boltzmann equation. According to [12] the thermal conductivity can be described as K e = T 5/2 e (T 2 e /n e ) s n s Q D ev cm 1 s 1 K 1 (2.77) The equation (2.76) is possible to simplify for the scope of this report. Due to the greater inertia of the ions the effect of the HF-waves can be neglected compared to the effect on the electrons. For electrons the right hand side of (2.76) describes the gain in energy from photo ionization, other background sources and external sources like electromagnetic waves and also the loss in energy due to both elastic and inelastic collisions with ions and neutrals. In this report the flow of electrons are also neglected and the ionosphere is only considered to be inhomogeneous in the z-direction.

29 2.5. HF Pump 19 Using these assumptions the electron energy balance can be described as 3 2 n eκ T e(t,z) = ( K e (T e ) T ) e(t,z) + Q(t,z) + Q 0 (t,z) L e (T e ) (2.78) t z z where Q is the energy deposited from the external HF-source, Q 0 is the background sources and L e is the rate of energy loss due to both elastic and inelastic collisions with ions and neutrals. 2.5 HF Pump To calculate the term Q appearing in eq. (2.78) it is necessary to calculate the energy deposited by the wave in the plasma. This is done using the ordinary ohmic heating relation Q =< J E >= 1 2 Re ( E σ E ) (2.79) where σ is the conductivity and E is the total electric field given by E (z,t) = 1 2 ( E(z)e iωt + c.c ) (2.80) where ω is the angular frequency of the wave. In general the total electric field can be described by one right hand circularly polarized component E +, one left hand circularly polarized component E and one parallel component E z. But in the case for this consideration only the E + and E are relevant since there is no E z when the propagation is along the magnetic field. The spatial dependence of the amplitude of the field can be expressed as ( ) E(z) = E ± (z)exp i dzk ± (z) ê ± (2.81) where ê ± = 1 2 (ˆx ± iŷ) (2.82) and the wave number k is a complex quantity in general, in the case of this study it is always positive since there is no reflection. In the coordinate system defined by eq. (2.82) and eˆ z = ẑ the complex index of refraction defined in eq (2.25) can be written as a sum of the diagonal elements in the real dielectric tensor and HF electric conductivity tensor as where n 2 ±(z) = ǫ ± (z) + i ǫ 0 ω σ ±(z) (2.83) ǫ ± = 1 (ω ± ω c)ω 2 p(z) ω ((ω ± ω c ) 2 + ν 2 e) σ ± = ν e ω 2 p(z) ((ω ± ω c ) 2 + ν 2 e) (2.84) (2.85)

30 2.5. HF Pump 20 and the parallel components are given by ǫ z (z) = 1 ω2 p(z) ω 2 + νe 2 σ z (z) = ν eωp(z) 2 ω 2 + νe 2 (2.86) (2.87) These elements are related to the elements S,D,P given earlier for K by the transformation in eq.(2.82) and in this case collisions are also accounted for. Calculated values for the dielectric and conductivity coefficients for the 17th of ε (4.04MHz) + ε (4.04MHz) ε (4.04MHz) z Dielectric coefficient Conductivity coeff σ + /(ε 0 ω) σ /(ε 0 ω) σ z /(ε 0 ω) altitude (km) Altitude (km) Figure 2.2: Dielectric and conductivity coefficients for 17-Oct 2006 October are shown in fig As seen from the figures the conductivity coefficient is several orders of magnitude smaller than the dielectric coefficient and it only affects the refractive index near a reflection layer where ǫ ± 0. As the case is here there is no reflection and n ± (z) ǫ ± (z) (2.88) In the case of no reflection the expression for the electric field given in eq. (2.81) only need to be corrected for the beam divergence, this can be done according to [13] by using the expression E ± (z) = E(0) ( ǫ± (0) ǫ ± (z) ) 1/4 ( (A(0) A(z) ) 1/2 exp ( i ω c z 0 dz n ± (z )) ) (2.89) where the zero refers to the bottom of the ionosphere so E(0) is calculated from eq A(z) is the area of the beam and since the Tromsø heater has a beam divergence of 14.5 [6], A is given by ( A(z) = π z tan φ ) 2 φ = 14.5 π (2.90)

31 2.6. Instabilities in F-region 21 Figure 2.3 shows a calculation of the E-field corresponding to October 17 with an estimated ERP of 180MW and an absorption of about 5dB in the D- and E-region. When the E-field and the conductivity are known it is easy to calculate the energy deposited by the wave in the ionosphere, results are shown in figure Pump wave X mode(6.2 MHz) X mode(4.04 MHz) O mode(4.04mhz) HF heating X mode(4.04 MHz) O mode(4.04mhz) altitude (km) altitude (km) E field (V/m) Energy deposition (evm 3 s 1 ) x 10 8 Figure 2.3: Profiles for the Electric field and the energy deposition used for solution of the energy equation for October 17. As seen from the figure the energy deposition is strongly dependent on both frequency and mode while the field strength is not. 2.6 Instabilities in F-region A powerful High Frequency (HF) radio wave can perturb the ionospheric plasma significantly and induce a great number of interesting plasma-physical phenomena near the reflection altitude. This includes instabilities and mode conversions in Langmuir and upper hybrid turbulence leading to large electron temperature enhancements [14, 15, 16, 17], creation of field aligned density depletions [18, 19], acceleration of electrons [20], radio induced emissions at HF and optical frequencies [21, 22, 23, 24]. To this much theoretical and experimental work has been done and for a review of the nonlinear phenomena in the ionosphere see [25]. Along the entire wave propagation path there is also an everpresent Ohmic heating due to non-zero HF-conductivity and associated collisional damping of the HF-wave. The Ohmic effect has been studied for overdense conditions by [13], [26, 27]. For such conditions most of the energy deposition occurs just below the reflection altitude, from there heat is then convected up and down along the magnetic field [28, 29]. Due to the multitude of wave-plasma processes acting in the region close to the reflection altitude it is difficult to accurately separate the effects of Ohmic heating from the effects of nonlinear and resonant processes. Even though the instabilities do not affect the results for this report, a short introduction of them will be given in this section.