Soluções W.K.B. para o cálculo da intensidade de campo na baixa ionosfera

Abstract

This work presents methods to roughly estimate the field of a wave spreading in the ionosphere. The ionosphere is one of the most complex means when one wants to calculate the field intensity of an electromagnetic wave, because it possesses an electronic density that varies with the seasons, and mainly with the hour of the day. Another dificulty for solutions of direrential equations is that the ionosphere characteristic is of an anisotropic material that produces two propagation modes due to its two refraction indexes. This anisotropy is originating from the existence of the Earth magnetic field. This work had as principal objective the calculation of the wave field in the ionosphere, but initially ignoring any magnetic field, but considering only the in°uence of the electronic density. When the Earth magnetic field is neglected, which is not the real case, the ionosphere behaves as an isotropic material and the solution of the direrential equations that govern the fields can be given by functions known as: Airy functions when the lineal density model is considered, Bessel functions when the exponential model is considered, and Weber functions when the density model is parabolic. When the electronic density does not follow any known function to represent the real ionosphere, the wave field can be calculated using approximate techniques, or numerical solutions. The last ones can produce rounding mistakes for descending wave. The more well-known approximated are the W.K.B. solutions which are presented in the form of an exponential of the integral of the refraction index becomes by the same refraction index. Close to the re°ection point, the refraction index reduces which makes the W.K.B. solutions fail in certain situations. For a real ionosphere, the electron collision frequency cannot be ignored, mainly in the low ionosphere, where it varies with the local atmospheric pressure. When the collision frequency is high, the ionosphere refraction index become a complex number and cannot be zero for a real height. In this situation, the W.K.B. solutions are therefore valid because the refraction index in the denominator of the field calculation formula does not assume too small values that are too small. When longitudinal or traverse propagation is considered, the direction of the Earth magnetic field has two propagation modes, that is, the ordinary (O) and extraordinary (X) waves which spread in an independent way and the field solutions can be given by two direrential equations, each one representing a propagation mode. When the longitudinal or traverse propagation is not the case, the equations that govern the ionosphere fields present in a coupled form. This means that to calculate a propagation mode there is a need to know the solution of the other mode. There is a factor that relates the two modes that is called coupling factor. Fortunately, for most cases, this coupling factor is small and since it is much smaller than the refraction index, the homogeneous equations can be used to determine the fields, together with the W.K.B. solutions. For the lower ionosphere and low frequencies (<3 MHz), the field attenuation is large and the reflection coeficient is small. This facilitates to calculate the field intensity inside of the ionosphere because the re°ected or descending wave can be neglected. This work analyzes just the case of vertical incidence, but considering the magnetic field making any angle with the propagation direction. The W.K.B. solutions present a great advantage over the numerical solutions because of mathematical formulas of easy understanding and simple behavior.