| Summary: | An asymptotic solution of ray equations is used for numerical simulation of the electromagnetic wave propagation in a smoothly non-uniform Earth-ionosphere duct. Being an extension of the adiabatic invariant approach, our method allows one to easily calculate and visualize all descriptive and energetic characteristics of global HF radio waves, including transition between multi-hop and ricochet propagation modes, antipodal focusing, geometric divergence and ohmic absorption. A quasi-parabolic model of F2 layer with the parameters taken from the international model IRI-2012 is used in numerical examples. From the mathematical point of view, it is an example of two-scale asymptotic expansion in a small parameter ν ~ H/L ~ L/πR where H is a characteristic height of the ionospheric layer, L is the scale of its horizontal non-uniformity, R is the Earth radius.
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