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 $\nu \sim {H/L} \sim L/\pi R$ where ${H}$ is a characteristic height of the ionospheric layer, ${L}$ is the scale of its horizontal non-uniformity, ${R}$ is the Earth radius.

中文翻译：

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用光滑摄动法模拟3D非均匀电离层中HF无线电波的传播

射线方程的渐近解用于电磁波在光滑非均匀地球电离层管道中传播的数值模拟。作为绝热不变方法的一种扩展，我们的方法使人们可以轻松地计算和可视化全球HF无线电波的所有描述性和高能特性，包括多跳和弹跳传播模式之间的过渡，对立聚焦，几何发散和欧姆吸收。在数值示例中使用了F2层的准抛物线模型，其参数取自国际模型IRI-2012。从数学的角度来看，这是一个小参数的两尺度渐近展开的例子 $ \ nu \ sim {H / L} \ sim L / \ pi R $ 哪里 $ {H} $ 是电离层的特征高度， $ {L} $ 是其水平不均匀的比例， $ {R} $ 是地球半径。