VHF propagation modes within ionospheric waveguides

A three dimensional waveguide model of VHF ducting has been developed. The waveguide is aligned along the Earth's magnetic field and has an elliptical cross-section which, based on observations of ionospheric bubble irregularities, is taken to have axes of 10–100 km. Rays propagating to the conjugate hemisphere via the duct undergo relatively few reflections at the duct walls (typically 3–20). Consequently the waveguides are not fully excited and significant power is transmitted by modes which are not fully trapped. A complete analysis of the propagation characteristics of such ducts requires the consideration of trapped and leaky modes, focusing and tunnelling. The cross-section of the guide is defined by specifying the vertical and East-West horizontal axes at the apex of the guide (magnetic equator). Comparison of cases with the same vertical axis shows that, when the vertical axis is larger than the East-West axis, the curvature on the upper surface of the guide is consequently increased and tunnelling becomes more important. On the other hand, when the vertical axis is the smaller, focusing is greater and leaky rays become increasingly important. In some models the region of highest power in the conjugate hemisphere is due almost entirely to rays which have suffered some leakage. Obviously non-trapped modes need to be considered when attempting to explain phenomena such as trans-equatorial propagation at VHF.     

VHF transequatorial propagation via three dimensional waveguides

The three dimensional model of VHF propagation along ionospheric waveguides or ducts developed by Platt I. G. and Dyson P. L., 1989 [J. atmos. terr. Phys. 51, 897–910] has been used to study transequatorial propagation (TEP) along equatorial bubble irregularities aligned along the earth's magnetic field. Bubbles with circular and elliptical cross-sections and different apex heights have been used to determine the region illuminated in the conjugate hemisphere by TEP and the power of the signals. The results show that the three dimensional waveguide model can explain many features of TEP, including those not well explained by the simpler waveguide models used previously. The calculations therefore confirm that TEP can be caused by guided propagation along equatorial bubble irregularities aligned along the magnetic field. The results show that a bubble located on the transmitter's longitude can illuminate a wide area of the conjugate hemisphere. A well defined maximum central power region occurs which in most cases is centred at a latitude slightly lower than the conjugate of the transmitter. Bubbles with circular cross-section can illuminate a region more than ten degrees in latitudinal extent and five degrees in longitude. Bubbles which are vertically elongated at their apex illuminate a much narrower region in longitude. TEP between points with relatively large separation in longitude can result from either leaky rays or ducts tilted out of the magnetic meridian plane. This explains some observations of TEP previously thought to be inconsistent with waveguide propagation. The variation of power loss with frequency depends on a number of factors such as bubble shape and height. Co-ordinated experiments involving a transmitter and several receivers in the conjugate hemisphere should enable properties of bubbles to be determined.     

The effect of multi-duct structure on whistler-mode wave propagation

Ducting of whistler-mode waves is investigated by ray-tracing in a magnetospheric model in which multiple and complex duct structures are superimposed on a smooth magnetospheric plasma distribution. When two or more ducts are present, propagation through a duct in which the ray does not become trapped is found to result in little deviation of the ray path, showing that upgoing waves can traverse several ducts before becoming trapped in one that is suitably positioned.The presence of two ducts situated in the same meridian plane and close together in L-value (ΔL ~ 0.07) is found to enable a double-duct trapping mode. This has the special property of guiding waves with frequency above half the local electron gyrofrequency across the equatorial plane in such a way that they become retrapped in the duct from which they had previously escaped at its local detrapping frequency. This may explain the observation of whistlers with a particularly high ratio of cut-off frequency to nose frequency (Bernhardt, 1979).Ducting is also investigated for a more complex duct structure in which fine structure is superimposed on a broader larger enhancement main duct. Here, it is found that rays which are first trapped to propagate in the main duct at low altitude can be further trapped to be ducted inside fine structure enhancements at higher altitude. This can result in certain components of multipath whistlers always being excited together and also having a common exit-point in the lower ionosphere. This is shown to be consistent with experimental observations.     

MF and HF ducting within equatorial bubbles

MF and HF conjugate ducting is often observed while satellite sounders are within equatorial bubbles. This paper examines two possible forms of this propagation. The first is guiding by the bubble itself, the second is ducting along irregularities, of small cross-section, embedded in the bubble. One bubble model, based on observations by Dyson and Benson [Geophys. Res. Lett. 9, 795 (1978)], gives some results at variance with observation. Nevertheless it is considered that slight changes to the model, such as asymmetries between the conjugate ionospheres, should remove the discrepancies. The results show that bubbles themselves will definitely produce conjugate ducting on occasions. The alternative explanation requires ducts of small cross-section, distributed throughout the bubble in order to produce conjugate ducting on successive ionograms. Good matches between calculated and observed echo traces for conjugate echoes were obtained using this model. It is likely that both forms of propagation occur.     

Error sources and travel time residuals in plasmaspheric whistler interpretation

In the interpretation of observed whistlers by curve fitting, systematic travel time residuals appeared which were studied by extensive simulations using ray-tracing, numerical integration and curve fitting. The residuals were found to originate from the commonly used approximations in the refractive index and ray path of whistler mode waves, which result in travel time increments or decrements, not accounted for in whistler interpretation. These approximations and the assumed form of the electron density distribution also lead to systematic errors in the diagnostics of plasmaspheric electron density by whistlers. In addition, the effects of other error sources, including random measurement errors, are also reviewed briefly.It is shown that the fine structure of residual curves is connected to propagation conditions. Thus, their study may yield a new research tool for studying whistler trapping, ducting structures and other features of whistler propagation. The application of residual analysis in conjunction with digital matched filtering of whistlers seems to be especially promising for further whistler studies.     

Cross-meridian refraction of ducted whistler-mode waves

The propagation of a ducted whistler-mode wave is considered for the case in which the wave normal has an azimuthal component. The meridional gradient in electron density associated with the duct is taken to be sufficient to keep the meridional component of the wave normal close to the direction of the ambient magnetic field. Because of the strong anisotropy effects at whistler-mode frequencies, this may lead to a considerable difference, even in cross-meridian propagation, from the unducted case, where there is no such constraint. When there is axial symmetry, the deflection in longitude may be found by a series of elementary steps followed by quadrature, and results are presented for a dipole magnetic field and realistic plasmaspheric density models. For the case of an azimuthal density gradient, a first-order theory is given in which the variation in longitude is given by quadrature, and results are shown exhibiting the dependence on frequency and on the density model used. The case of a duct limited in longitude is also discussed. Here., the ray path oscillates in longitude, and a simple analysis enables the `wavelength' and the relative amplitude of the oscillation to be estimated in terms of the density enhancement and the azimuthal width of the duct. Finally, in an Appendix, the effects of the oscillation in the meridional component of the wave normal. which are second-order, are analysed.     

The effects of ionospheric horizontal electron density gradients on whistler mode signals

Whistler mode group delays observed at Faraday, Antarctica (65° S, 64° W) and Dunedin, New Zealand (46° S, 171° E) show sudden increases of the order of hundreds of milliseconds within 15 minutes. These events (‘discontinuities’) are observed during sunrise or sunset at the duct entry regions, close to the receiver's conjugate point. The sudden increase in group delay can be explained as a tilting of the up-going wave towards the sun by horizontal electron density gradients associated with the passage of the dawn/dusk terminator. The waves become trapped into higher L-shell ducts. The majority of the events are seen during June-August and can be understood in terms of the orientation of the terminator with respect to the field aligned ducts. The position of the source VLF transmitter relative to the duct entry region is found to be important in determining the contribution of ionospheric electron density gradients to the L-shell distribution of the whistler mode signals.     

Investigation by ray-tracing of the effect of a summer-winter asymmetry on whistler ducting

A ray-tracing model of the inner magnetosphere (L < 6) is constructed for a plasma distribution asymmetric about the equatorial plane, thus representing summer and winter conditions in the two hemispheres. At the reference height of 900 km, the oxygen ion concentration and electron density are taken to vary by factors of ten and two respectively between the hemispheres. The concentrations of hydrogen and helium ions at the reference level are chosen to ensure electron density continuity across the equatorial plane. The altitude at which ducts terminate is modelled to differ between the two hemispheres in accordance with the numerical simulations of Bernhardt and Park (1977).It is shown that the different plasma distributions in the two hemispheres affect the paths of ducted rays and consequently the likelihood of the reception of one-hop whistlers in the conjugate hemisphere. The difference in final latitude between propagation in the symmetric and asymmetric models for the same initial latitude is largest when ducts extend down to 300 km altitude in the conjugate hemisphere. When ducts terminate at greater altitude, the effect of a difference in termination heights between the two hemispheres generally has a larger effect than that of the plasma asymmetry. Both these effects may play a role in determining the seasonal variation of whistler occurrence.     

A model for the electron temperature variation along geomagnetic field lines and its effect on electron density profiles and VLF paths

A realistic model for the temperature variation along geomagnetic field lines is described. For high altitudes (>1500 km) the temperature is taken to increase as the nth power of radial distance (n−2), giving temperatures consistent with those measured in situ by high altitude satellites. For realistic temperatures at low altitude an extra term is included. The temperature gradient along the field line is then 0.9–1.6° km⁻¹ during the day and 0.5–0.7° km⁻¹ during the night at 1000 km, reducing to about half these values at 2000 km, for the latitude range 35–50°. This is consistent with calculations made from nearly simultaneous satellite measurements at 1000 and 2500 km. It is shown that assuming diffusive equilibrium, including the new temperature model, more realistic equatorial electron density profiles result than for isothermal field lines.The temperature gradient model is also purposely formulated to be of a form that enables the temperature modified geopotential height to be obtained without numerical integration. This renders the model particularly suitable for ray-tracing calculations. A ray-tracing model is developed and it is shown that unducted ray paths are significantly altered from the corresponding paths in an equivalent isothermal model; there is greater refraction and magnetospheric reflection takes place at lower altitudes. For summer day conditions, an inter-hemispheric unducted ray path becomes possible from 26° latitude that can reach the ground at the conjugate.     

A note on ray curvature

Although formulae have long existed for ray curvature in an inhomogeneous cold collisionless magnetoplasma, they have not always been applied consistently. General expressions applicable to propagation in a meridional plane in an axially symmetric plasma in the presence of a poloidal potential magnetic field, such as a dipole field, are re-presented, and some previous results re-examined.     

Observations of the high latitude trough using EISCAT

Data taken by EISCAT are presented as contours of electron density, ion and electron temperature and plasma velocity versus invariant latitude and local magnetic time.Three nights near midsummer were studied and in each case a trough in electron density occurred north of invariant latitude 64° shortly after local midnight (MLT 0200) and remained a prominent feature for about 3 h before moving poleward. The minimum in electron density was associated with a marked increase in ion temperature, but the electron temperature showed litttle change. In this respect the high latitude trough is clearly different from the mid-latitude trough.Full velocity measurements were not available for all three nights, but it seems that the appearance of the trough followed the start of a strong eastward plasma velocity combined with a strong upward velocity along the magnetic field line. The sudden change in plasma velocity causes frictional heating, which explains the increase in ion temperature. Upward plasma velocity is a major factor in the formation of the trough, with enhanced recombination making a smaller contribution.     

Reflection of electromagnetic waves by density gradients in a magnetised plasma

The reflection of electromagnetic waves by density gradients in a plasma with a magnetic field derived from a scalar potential is studied. Conditions are found for reflection at grazing incidence at a surface S containing field lines. No restriction is placed on the angle between the wave-normal and the field, and account is taken of the curvature of the surface and the anisotropy of the medium. Particular reference is made to whistler-mode waves in the magnetosphere. The minimum electron density enhancement needed for reflection is calculated. Guidance by small-scale irregularities and ‘super whistlers’ (whistlers with upper cut-off above one-half the equatorial electron cyclotron frequency) are briefly discussed.     

Analytic properties of the whistler dispersion function

The analytic properties of the dispersion function of a whistler are investigated in the complex frequency plane. It possesses a pole and a branch point at a frequency equal to the minimum value of the electron gyrofrequency along the path of propagation. An integral equation relates the dispersion function to the distribution of magnetospheric electrons along the path and the solution of this equation is obtained. It is found that the electron density in the equatorial plane is very simply related to the dispersion function. A discussion of approximate formulae to represent the dispersion shows how particular terms can be related to attributes of the electron density distribution, and a new approximate formula is proposed.     

Experimental studies of daytime LF radio propagation and their implications for D-region electron densities

Measurements are presented of interference phenomena in amplitude and phase of VLF and LF signals along propagation paths from central England to the Norwegian Sea. The data are interpreted by means of the ‘wave-hop’ propagation theory, incorporating full wave evaluation of ionospheric reflection coefficients with realistic D-region models. No published electron density profiles are found which completely satisfy the experimental data, but modified profiles are presented which provide a better fit to the observations.     

Determination of the turbulent spectrum in the ionosphere by a tomographic method

A theory of tomographic reconstruction of the statistical properties of the random turbulent ionospheric plasma is presented. Derived integral equations for the coherence functions of the measured fields allow the determination of inhomogeneous layer coordinates and the reconstruction of cross-sections of the electron density correlation functions. For statistically homogeneous layers and a transmitter on board a moving satellite with a linear receiving array on the ground, we have the possibility of determining the three-dimensional correlation function structure or its spectrum using a set of two-dimensional cross-sections. One receiver allows the reconstruction of the spectrum of the two-dimensional cross-section of the correlation function. We also consider the solution of the inverse problem for non-homogeneous fluctuations. In this case the distribution of the electron density fluctuations, its variance and the correlation coefficient, characterizing the spatial structure of fluctuations may be reconstructed by a tomographic technique. Experimental results on the identification of the layer height of the irregularities and on the spectrum of the two-dimensional cross-sections of the correlation function measurements are presented.     

Recent progress in transequatorial propagation—review

Transequatorial propagation of HF and VHF radio waves is placed into three categories according to the physical mechanisms. Specular reflection off the underside of the anomalously dense equatorial F-layer is predictable by ray tracing and limited to frequencies less than about 60 MHz. Multipoint reflection from bottomside irregularities applies for the same radio frequencies but is associated with travelling ionospheric disturbances and spread-F traces on ionograms. This type of TEP may be used as a technique for studying some of the properties of bottomside irregularities. Ducted propagation of VHF waves depends upon high plasma density gradients and occurs along equatorial plasma bubbles during the evening hours. Observations on the ducted VHF mode relate to the behaviour of plasma bubbles.     

A quantitative study of the high latitude ionospheric trough using EISCAT's common programmes

Eighteen days of EISCAT data were used in a systematic study of the high latitude trough. Apart from a few days at midwinter, the pattern was the same in all cases. Near midnight the reversal of plasma flow from westward to eastward caused significant frictional heating of the ion population. At the same time a strong plasma velocity was observed upwards along the magnetic field line. This was the result of

• 1.(i) a southward neutral wind
• 2.(ii) a vertical wind driven by Joule heating
• 3.(iii) diffusion. Both enhanced recombination—associated with the increase in ion temperature—and the escape of plasma along the field line contribute to the drop in electron density.

     

Frequency dependence of anomalous absorption caused by high power radio waves

High power radio waves can modify the ionospheric electron density distribution to produce field aligned plasma irregularities which give rise to the anomalous absorption of HF radio waves. The coefficient of anomalous absorption of a vertically propagated radio wave due to scattering from field aligned irregularities has been calculated, taking into account the effects of the geomagnetic field on electron motions. These results are compared with those of other theoretical models. Furthermore, the scale lengths of field aligned irregularities produced by a high power radio wave during recent high latitude modification experiments have been determined from measurements of the anomalous absorption by means of this theory.     

Gyro line observations with the EISCAT VHF radar

This paper reports the first successful gyro line experiment with the EISCAT VHF (224 MHz) radar. The incoherent scatter gyro line (also known as ‘resonance line’ and ‘whistler line’ in the literature) corresponds to the electrostatic wave mode ω ≈ Ω cos α known to be present in a weakly magnetized plasma (Ω is the electron gyro frequency and α is the angle between the scattering wave vector and the magnetic field). The line is very weak, but has the great advantage from an observational point of view that its position in the scattered spectrum is only marginally dependent on the electron density and temperature. This means that filter offsets can be easily predicted and that a long pulse and long integration times can be used in the experiment. Measurements were made at angles of 55 and 69° with the geomagnetic field where the gyro line frequencies are approximately 800 and 500 kHz, respectively. The line was seen in the altitude region 100–220 km, being most intense at 160–170 km. The strong dependence of the gyro line on the magnetic field may be used to study variations in the field. Other interesting aspects of the line to be investigated in future experiments are the effects of suprathermal particles, the possible effects of stimulated scattering, and the heating effects in an ionospheric modification experiment.