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Velocity of decameter electrojet irregularities under strongly driven conditionsGorin, James Donald 22 September 2008
The Earth ionosphere is a highly inhomogeneous medium containing electron density irregularities of various scales, from hundreds of kilometers to tens of centimeters. Understanding the mechanisms responsible for their formation is an important task for various practical applications such as communication, navigation, and safe satellite operation. Of special interest are the decameter irregularities that are abundant at E region heights of ~ 100 120 km. These are excited when enhanced electric field and plasma drifts are setup in the ionosphere. This thesis is aimed at studying the physics of decameter irregularity formation at E region heights with a focus on the extreme conditions of very strong electric fields (plasma flows) of > 50 mV/m (1000 m/s) for which the so called Farley-Buneman (FB) plasma instability is the dominating mechanism of irregularity excitation. The relationship between the irregularity velocity and plasma drift is investigated by considering data of the SuperDARN radar located at Stokkseyri, Iclenad. The radar detects echoes from the irregularities and is thus capable of measuring their velocity. The DMSP satellites measure the plasma drifts in situ at heights of ~ 800 km, but these measurements can be projected onto E region heights at high latitudes. By comparing the radar and satellite data in one direction, we show that irregularity velocity is smaller than the plasma drift by a factor of 2 3 with the stronger difference at faster flows. This contrasts with the theoretical expectation for the velocity to be close to 400 m/s, the nominal ion-acoustic speed at electrojet heights. A two-dimensional comparison is performed by considering a subset of the observations for which the HF echo velocity showed a cosine type variation with the radar look direction. This class of echoes is consistent with predictions of recent theories of the Farley-Buneman instability, but the irregularity velocity magnitude was found to be smaller than the ion-acoustic speed with occasional occurrence of velocities as small as 100 m/s. This implies that either recent theories of the Farley-Buneman instability should be modified or that the typical height of HF echoes is typically below 100 km. Various other properties of decameter irregularities are investigated and discussed in view of the existing theories.
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Velocity of decameter electrojet irregularities under strongly driven conditionsGorin, James Donald 22 September 2008 (has links)
The Earth ionosphere is a highly inhomogeneous medium containing electron density irregularities of various scales, from hundreds of kilometers to tens of centimeters. Understanding the mechanisms responsible for their formation is an important task for various practical applications such as communication, navigation, and safe satellite operation. Of special interest are the decameter irregularities that are abundant at E region heights of ~ 100 120 km. These are excited when enhanced electric field and plasma drifts are setup in the ionosphere. This thesis is aimed at studying the physics of decameter irregularity formation at E region heights with a focus on the extreme conditions of very strong electric fields (plasma flows) of > 50 mV/m (1000 m/s) for which the so called Farley-Buneman (FB) plasma instability is the dominating mechanism of irregularity excitation. The relationship between the irregularity velocity and plasma drift is investigated by considering data of the SuperDARN radar located at Stokkseyri, Iclenad. The radar detects echoes from the irregularities and is thus capable of measuring their velocity. The DMSP satellites measure the plasma drifts in situ at heights of ~ 800 km, but these measurements can be projected onto E region heights at high latitudes. By comparing the radar and satellite data in one direction, we show that irregularity velocity is smaller than the plasma drift by a factor of 2 3 with the stronger difference at faster flows. This contrasts with the theoretical expectation for the velocity to be close to 400 m/s, the nominal ion-acoustic speed at electrojet heights. A two-dimensional comparison is performed by considering a subset of the observations for which the HF echo velocity showed a cosine type variation with the radar look direction. This class of echoes is consistent with predictions of recent theories of the Farley-Buneman instability, but the irregularity velocity magnitude was found to be smaller than the ion-acoustic speed with occasional occurrence of velocities as small as 100 m/s. This implies that either recent theories of the Farley-Buneman instability should be modified or that the typical height of HF echoes is typically below 100 km. Various other properties of decameter irregularities are investigated and discussed in view of the existing theories.
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The Mid-Latitude Ionosphere: Modeling and Analysis of Plasma Wave Irregularities and the Potential Impact on GPS SignalsEltrass, Ahmed Said Hassan Ahmed 26 March 2015 (has links)
The mid-latitude ionosphere is more complicated than previously thought, as it includes many different scales of wave-like structures. Recent studies reveal that the mid-latitude ionospheric irregularities are less understood due to lack of models and observations that can explain the characteristics of the observed wave structures. Since temperature and density gradients are a persistent feature in the mid-latitude ionosphere near the plasmapause, the drift mode growth rate at short wavelengths may explain the mid-latitude decameter-scale ionospheric irregularities observed by the Super Dual Auroral Radar Network (SuperDARN). In the context of this dissertation, we focus on investigating the plasma waves responsible for the mid-latitude ionospheric irregularities and studying their influence on Global Positioning System (GPS) scintillations.
First, the physical mechanism of the Temperature Gradient Instability (TGI), which is a strong candidate for producing mid-latitude irregularities, is proposed. The electro- static dispersion relation for TGI is extended into the kinetic regime appropriate for High- Frequency (HF) radars by including Landau damping, finite gyro-radius effects, and tem- perature anisotropy. The kinetic dispersion relation of the Gradient Drift Instability (GDI) including finite ion gyro-radius effects is also solved to consider decameter-scale waves gen- eration. The TGI and GDI calculations are obtained over a broad set of parameter regimes to underscore limitations in fluid theory for short wavelengths and to provide perspective on the experimental observations.
Joint measurements by the Millstone Hill Incoherent Scatter Radar (ISR) and the Su- perDARN HF radar located at Wallops Island, Virginia have identified the presence of decameter-scale electron density irregularities that have been proposed to be responsible for low-velocity Sub-Auroral Ionospheric Scatter (SAIS) observed by SuperDARN radars. In order to investigate the mechanism responsible for the growth of these irregularities, a time series for the growth rate of both TGI and GDI is developed. The time series is computed for both perpendicular and meridional density and temperature gradients. The growth rate comparison shows that the TGI is the most likely generation mechanism for the observed quiet-time irregularities and the GDI is expected to play a relatively minor role in irregular- ity generation. This is the first experimental confirmation that mid-latitude decameter-scale ionospheric irregularities are produced by the TGI or by turbulent cascade from primary irregularity structures produced from this instability. The quiet- and disturbed-times plasma wave irregularities are compared by investigating co-located experimental observations by the Blackstone SuperDARN radar and the Millstone Hill ISR under various sets of geomagnetic conditions. The radar observations in conjunction with growth rate calculations suggest that the TGI in association with the GDI or a cascade product from them may cause the observations of disturbed-time sub-auroral ionospheric irregularities.
Following this, the nonlinear evolution of the TGI is investigated utilizing gyro-kinetic Particle-In-Cell (PIC) simulation techniques with Monte Carlo collisions for the first time. The purpose of this investigation is to identify the mechanism responsible for the nonlinear saturation as well as the associated anomalous transport. The simulation results indicate that the nonlinear E x B convection (trapping) of the electrons is the dominant TGI sat- uration mechanism. The spatial power spectra of the electrostatic potential and density fluctuations associated with the TGI are also computed and the results show wave cascad- ing of TGI from kilometer scales into the decameter-scale regime of the radar observations. This suggests that the observed mid-latitude decameter-scale ionospheric irregularities may be produced directly by the TGI or by turbulent cascade from primary longer-wavelength irregularity structures produced from this instability.
Finally, the potential impact of the mid-latitude ionospheric irregularities on GPS signals is investigated utilizing modeling and observations. The recorded GPS data at mid-latitude stations are analyzed to study the amplitude and phase fluctuations of the GPS signals and to investigate the spectral index variations due to ionospheric irregularities. The GPS measurements show weak to moderate scintillations of GPS L1 signals in the presence of ionospheric irregularities during disturbed geomagnetic conditions. The GPS spectral indices are calculated and found to be in the same range of the numerical simulations of TGI and GDI. Both simulation results and GPS spectral analysis are consistent with previous in-situ satellite measurements during disturbed periods, showing that the spectral index of mid- latitude density irregularities are of the order 2. The scintillation results along with radar observations suggest that the observed decameter-scale irregularities that cause SuperDARN backscatter, co-exist with kilometer-scale irregularities that cause L-band scintillations. The alignment between the experimental, theoretical, and computational results of this study suggests that turbulent cascade processes of TGI and GDI may cause the observations of GPS scintillations that occur under disturbed conditions of the mid-latitude F-region ionosphere. The TGI and GDI wave cascading lends further support to the belief that the E-region may be responsible for shorting out the F-region TGI and GDI electric fields before and around sunset and ultimately leading to irregularity suppression. / Ph. D.
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Investigação da usabilidade do GBAS no Brasil / Investigation of GBAS usability in BrazilPereira, Vinícius Amadeu Stuani [UNESP] 13 September 2018 (has links)
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Previous issue date: 2018-09-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Dentre os métodos de posicionamento GNSS (Global Navigation Satellite System) utilizados pela aviação no suporte das fases de aproximação e pouso preciso de aeronaves, destacam-se o SBAS (Satellite-Based Augmentation System) e o GBAS (Ground-Based Augmentation System). O GBAS tem a capacidade de corrigir a maioria dos erros envolvidos na pseudodistância a partir do DGNSS (Differential GNSS), desde que a camada ionosférica apresente um comportamento não perturbado na região do aeroporto. Entretanto, dependendo do fluxo de ionização solar, da atividade geomagnética, do ciclo de manchas solares, do ângulo zenital do Sol e da localização geográfica, a ionosfera pode sofrer fortes perturbações, proporcionando uma ameaça à integridade do GBAS, uma vez que podem ser diferentes os efeitos ionosféricos em pequenas distâncias. Assim, investigações dos erros sistemáticos devido à camada ionosférica no GBAS tem sido objeto de estudos há alguns anos. Nesse sentido, modelos de risco ionosférico, que visam determinar a máxima decorrelação ionosférica espacial existente entre a estação GBAS e a aeronave que se aproxima num aeroporto, foram desenvolvidos ou avaliados, principalmente para o hemisfério norte, mais precisamente para o território norte-americano, onde se destaca o CONUS (Conterminous United States) Threat Model. Nessa área o comportamento da ionosfera é mais estável em comparação com o observado sobre o Brasil, localizado na região ionosférica equatorial e de baixas latitudes, que apresenta a ocorrência da Anomalia de Ionização Equatorial (AIE), bolhas ionosféricas, irregularidades ionosféricas, cintilação ionosférica e Anomalia Magnética do Atlântico Sul (AMAS). A implantação de um GBAS no Brasil, por meio do Departamento de Controle do Espaço Aéreo (DECEA), despertou o interesse de seu uso com segurança. Sendo assim, a pesquisa propôs investigar a aplicabilidade do modelo CONUS de risco ionosférico para GBAS no território brasileiro, utilizando o método dos pares de estações (station-pair method), além de estimar os parâmetros para os principais aeroportos internacionais do Brasil, considerando a variação sazonal, bem como investigar os benefícios quanto ao uso dos sinais GLONASS (Global’naya Navigatsionnaya Sputnikovaya Sistema), Galileo e da portadora L5 do GPS (Global Positioning System) no modelo. Para isso, foram utilizados dados GNSS de várias redes ativas entre os anos de 2000 e 2016, bem como dados do GBAS instalado no aeroporto internacional do Rio de Janeiro/RJ (Galeão). Para a determinação dos parâmetros do modelo de risco e do parâmetro de integridade σvig (vertical ionospheric gradient sigma), esse último utilizado para estimar os níveis de proteção horizontal e vertical da aeronave, foi implementado um sistema denominado MoR_Ion. Os parâmetros do modelo CONUS estimados para o Brasil, utilizando sinais GPS para a combinação de portadoras L1/L2, mostraram que é inviável o uso de um GBAS considerando todo o território nacional. Uma alternativa foi estimativa local e temporal para os aeroportos de interesse. Valores obtidos indicaram que o GBAS pode, provavelmente, ser utilizado nos aeroportos internacionais de São Paulo/SP (Cumbica), Rio de Janeiro/RJ (Galeão), Brasília/DF (Presidente Juscelino Kubitschek) e Recife/PE (Gilberto Freyre) com algumas restrições quanto à estação do ano, hora do dia e elevação dos satélites. Já para o aeroporto internacional de Porto Alegre/RS (Salgado Filho) é o único, entre os analisados, em que nenhuma restrição à instalação do GBAS no local foi identificada a partir do conjunto de dados processados. Resultados empregando os sinais GPS e Galileo, para a combinação L1/L5, apresentaram ser melhores que os da combinação L1/L2. Já em relação ao GLONASS, verificou-se que há uma semelhança com os resultados do GPS. A determinação do σvig em tempo real para cada satélite disponível se apresentou como uma alternativa interessante, uma vez que transmite para a aeronave a real condição ionosférica no momento da aproximação e pouso, ao contrário da atual configuração do GBAS de transmitir um valor fixo de σvig que, teoricamente, contempla todas as possíveis perturbações ionosféricas. Estimativas de níveis de proteção para aproximação no Galeão indicaram que há a possibilidade de se realizar um procedimento CAT-I, utilizando satélites GPS ou GLONASS (combinação L1/L2), desde que sejam aplicadas restrições local-temporais previamente estabelecidas. Verificou-se, também, que a utilização dos satélites GLONASS em concomitância com o GPS possibilita a obtenção de valores que atendem aos limiares para um pouso CAT-III, uma vez que uma maior quantidade de satélites e, consequentemente, uma melhor configuração geométrica, é disponibilizada. Um estudo de caso utilizando o time-step method para a região do aeroporto de São José dos Campos/SP, onde se encontram cinco estações em um raio de 10 km, indicou que gradientes desse método podem ser empregados na estimativa dos valores dos parâmetros. Entretanto, tal método tem pouca semelhança com a arquitetura de uma estação GBAS e uma aeronave que se aproxima e, adicionalmente, não soluciona a decorrelação temporal. Por fim, um método alternativo que pode indicar a realização ou não do pouso consiste no monitoramento das irregularidades ionosféricas em tempo real na região circundante de um determinado aeroporto. Experimento realizado em tempo real, mas utilizando dados GPS e GLONASS de março de 2014 (próximo ao pico do ciclo solar 24), mostrou fortes irregularidades para a região do Galeão, com a frente ionosférica se deslocando de sudoeste a nordeste. Assim, uma medida que pode ser empregada para estimar os níveis de proteção consiste em não utilizar os sinais dos satélites que atravessam tais irregularidades. / Among the methods of GNSS (Global Navigation Satellite System) positioning used by the aviation in the support of the phases of approach and precise landing of aircraft, stand out the SBAS (Satellite-Based Augmentation System) and the GBAS (Ground-Based Augmentation System). GBAS has the ability to correct most of the errors involved in pseudorange from DGNSS (Differential GNSS), provided that the ionospheric layer exhibits undisturbed behavior in the airport region. However, depending on the flow of solar ionization, geomagnetic activity, sunspot cycle, zenith angle of the sun and geographic location, the ionosphere can suffer severe disturbances, posing a threat to the integrity of the GBAS, since the ionospheric effects may be different at small distances. Thus, investigations of systematic errors due to the ionospheric layer in GBAS have been the subject of studies for some years. In this sense, ionospheric threat models, which seek to determine the maximum existing spatial ionospheric decorrelation between the GBAS station and the aircraft approaching an airport, have been developed or evaluated, especially for the northern hemisphere, more precisely to the US territory, which highlights the CONUS (Conterminous United States) Threat Model. In this area, the ionosphere behavior is more stable compared to that observed in Brazil, located in the equatorial and low latitude ionospheric region, which presents the occurrence of Equatorial Ionization Anomaly (EIA), ionospheric bubbles, ionospheric irregularities, ionospheric scintillation and South Atlantic Magnetic Anomaly (SAMA). The implementation of a GBAS in Brazil, through the Department of Airspace Control (DECEA), aroused the interest of its use with safety. Therefore, the research proposed to investigate the applicability of the CONUS Threat Model to GBAS in the Brazilian territory, using the station-pair method, besides estimating the parameters for the main international airports of Brazil, considering the seasonal variation, as well as investigating the benefits of using the GLONASS (Global’naya Navigatsionnaya Sputnikovaya System), Galileo and GPS (Global Positioning System) L5 carrier in the model. For this purpose, GNSS data from several active networks were used between 2000 and 2016, as well as data from GBAS installed at Rio de Janeiro International Airport (Galeão). For the determination of the parameters of the threat model and the σvig (vertical ionospheric gradient sigma) integrity parameter, the latter used to estimate the aircraft horizontal and vertical protection levels, a system called MoR_Ion was implemented. The parameters of the CONUS model estimated for Brazil, using GPS signals for the combination of L1/L2 carriers, showed that it is impracticable to use a GBAS considering the entire national territory. An alternative was a local and temporal estimate for the airports of interest. The values obtained indicate that the GBAS can probably be used in the international airports of São Paulo/SP (Cumbica), Rio de Janeiro/RJ (Galeão), Brasília/DF (President Juscelino Kubitschek) and Recife/PE (Gilberto Freyre) with some restrictions on the season, time of day and satellite elevation. At the international airport of Porto Alegre/RS (Salgado Filho) it is the only one, among the analyzed ones, in which no restriction to the installation of the GBAS in the place was identified from the data set processed. Results using the GPS and Galileo signals for the L1/L5 combination were better than the L1/L2 combination. Regarding GLONASS, it was found that there is a similarity with the GPS results. The determination of the real time σvig for each available satellite was presented as an interesting alternative, since it transmits to the aircraft the actual ionospheric condition at the time of approach and landing, unlike the current GBAS configuration of transmitting a fixed value of σvig which theoretically covers all possible ionospheric disturbances. Estimates of protection levels for approach in Galeão indicated that there is the possibility of performing a CAT-I procedure, using GPS or GLONASS satellites (L1/L2 combination), provided that previously established local-temporal restrictions are applied. It was also verified that the use of GLONASS satellites in concomitance with GPS allows obtaining values that meet the thresholds for a CAT-III landing, since a larger number of satellites and, consequently, a better geometric configuration, is made available. A case study using the time-step method for the airport region of São José dos Campos/SP, where five stations are located within a 10 km radius, indicated that gradients of this method can be used to estimate the parameter values. However, this method has little resemblance to the architecture of a GBAS station and an aircraft approaching and additionally does not solve the temporal decorrelation. Finally, an alternative method that may indicate whether or not the landing is carried out is to monitor ionospheric irregularities in real time in the surrounding region of a given airport. A real-time experiment using GPS and GLONASS data from March 2014 (near the peak of the solar cycle 24) showed strong irregularities for the Galeão region, with the ionospheric front moving from southwest to northeast. Thus, one way that can be used to estimate protection levels is to not use satellite signals that cross such irregularities. / FAPESP: 2015/20522-7
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Modeling of Plasma Irregularities Associated with Artificially Created Dusty Plasmas in the Near-Earth Space EnvironmentFu, Haiyang 22 January 2013 (has links)
Plasma turbulence associated with the creation of an artificial dust layer in the earth's ionosphere is investigated. The Charged Aerosol Release Experiment (CARE) aims to understand the mechanisms for enhanced radar scatter from plasma irregularities embedded in dusty plasmas in space. Plasma irregularities embedded in a artificial dusty plasma in space may shed light on understanding the mechanism for enhanced radar scatter in Noctilucent Clouds (NLCs) and Polar Mesospheric Summer Echoes (PMSEs) in the earth's mesosphere. Artificially created, charged-particulate layers also have strong impact on radar scatter as well as radio signal propagation in communication and surveillance systems. The sounding rocket experiment was designed to develop theories of radar scatter from artificially created plasma turbulence in charged dust particle environment. Understanding plasma irregularities embedded in a artificial dusty plasma in space will also contribute to addressing possible effects of combustion products in rocket/space shuttle exhaust in the ionosphere.
In dusty space plasmas, plasma irregularities and instabilities can be generated during active dust aerosol release experiments. Small scale irregularities (several tens of centimeter to meters) and low frequency waves (in the ion/dust scale time in the order of second) are studied in this work, which can be measured by High Frequency (HF), Very High Frequency (VHF) and Ultra High Frequency (UHF) radars. The existence of dust aerosol particles makes computational modeling of plasma irregularities extremely challenging not only because of multiple spatial and temporal scale issue but also due to complexity of dust aerosol particles.
This work will provide theoretical and computational models to study plasma irregularities driven by dust aerosol release for the purpose of designing future experiments with combined ground radar, optical and in-situ measurement. In accordance with linear analysis, feasible hybrid computational models are developed to study nonlinear evolution of plasma instabilities in artificially created dusty space plasmas. First of all, the ion acoustic (IA) instability and dust acoustic (DA) instability in homogenous unmagnetized plasmas are investigated by a computational model using a Boltzmann electron assumption. Such acoustic-type instabilities are attributed to the charged dust and ion streaming along the geomagnetic field. Secondly, in a homogenous magnetized dusty plasma, lower-hybrid (LH) streaming instability will be generated by dust streaming perpendicular to the background geomagnetic field. The magnetic field effect on lower-hybrid streaming instabilities is investigated by including the ratio of electron plasma frequency and electron gyro frequency in this model. The instability in weakly magnetized circumstances agree well with that for the ion acoustic (IA) instability by a Boltzmann model. Finally, in an inhomogeneous unmagnetized/magnetized dust boundary layer, possible instabilities will be addressed, including dust acoustic (DA) wave due to flow along the boundary and lower-hybrid (LH) sheared instability due to flow cross the boundary.
With applications to active rocket experiments, plasma irregularity features in a linear/nonlinear saturated stage are characterized and predicted. Important parameters of the dust aerosol clouds that impact the evolution of waves will be also discussed for upcoming dust payload generator design. These computational models, with the advantage of following nonlinear wave-particle interaction, could be used for space dusty plasmas as well as laboratory dusty plasmas. / Ph. D.
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Secondary Electromagnetic Radiation Generated by HF Pumping of the IonosphereNorin, Lars January 2008 (has links)
Electromagnetic waves can be used to transmit information over long distances and are therefore often employed for communication purposes. The electromagnetic waves are reflected off material objects on their paths and interact with the medium through which they propagate. For instance, the plasma in the ionosphere can refract and even reflect radio waves propagating through it. By increasing the power of radio waves injected into the ionosphere, the waves start to modify the plasma, resulting in the generation of a wide range of nonlinear processes, including turbulence, in particular near the reflection region. By systematically varying the injected radio waves in terms of frequency, power, polarisation, duty cycle, inclination, etc. the ionosphere can be used as an outdoor laboratory for investigating fundamental properties of the near-Earth space environment as well as of plasma turbulence. In such ionospheric modification experiments, it has been discovered that the irradiation of the ionosphere by powerful radio waves leads to the formation of plasma density structures and to the emission of secondary electromagnetic radiation, a phenomenon known as stimulated electromagnetic emission. These processes are highly repeatable and have enabled systematic investigations of the nonlinear properties of the ionospheric plasma. In this thesis we investigate features of the plasma density structures and the secondary electromagnetic radiation. In a theoretical study we analyse a certain aspect of the formation of the plasma structures. The transient dynamics of the secondary radiation is investigated experimentally in a series of papers, focussing on the initial stage as well as on the decay. In one of the papers we use the transient dynamics of the secondary radiation to reveal the intimate relation between certain features of the radiation and structures of certain scales. Further, we present measurements of unprecedentedly strong secondary radiation, attributed to stimulated Brillouin scattering, and report measurements of the secondary radiation using a novel technique imposed on the transmitted radio waves.
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Opening New Radio Windows and Bending Twisted BeamsNordblad, Erik January 2011 (has links)
In ground based high frequency (HF) radio pumping experiments, absorption of ordinary (O) mode pump waves energises the ionospheric plasma, producing optical emissions and other effects. Pump-induced or natural kilometre-scale field-aligned density depletions are believed to play a role in self-focussing phenomena such as the magnetic zenith (MZ) effect, i.e., the increased plasma response observed in the direction of Earth's magnetic field. Using ray tracing, we study the propagation of ordinary (O) mode HF radio waves in an ionosphere modified by density depletions, with special attention to transmission through the radio window (RW), where O mode waves convert into the extraordinary (X, or Z) mode. The depletions are shown to shift the position of the RW, or to introduce RWs at new locations. In a simplified model neglecting absorption, we estimate the wave electric field strength perpendicular to the magnetic field at altitudes normally inaccessible. This field could excite upper hybrid waves on small scale density perturbations. We also show how transmission and focussing combine to give stronger fields in some directions, notably at angles close to the MZ, with possible implications for the MZ effect. In a separate study, we consider electromagnetic (e-m) beams with helical wavefronts (i.e., twisted beams), which are associated with orbital angular momentum (OAM). By applying geometrical optics to each plane wave component of a twisted nonparaxial e-m Bessel beam, we calculate analytically the shift of the beam's centre of gravity during propagation perpendicularly and obliquely to a weak refractive index gradient in an isotropic medium. In addition to the so-called Hall shifts expected from paraxial theory, the nonparaxial treatment reveals new shifts in both the transverse and lateral directions. In some situations, the new shifts should be significant also for nearly paraxial beams.
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