Fields near a caustic and cusp in an isotropic ionosphere

Physics

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Earth Ionosphere, Electric Field Strength, Magnetic Flux, Numerical Integration, Airy Function, Magnetic Dipoles, Saddle Points, Steepest Descent Method

Scientific paper

A vertical Hertzian magnetic dipole is assumed to be transmitting into an isotropic and horizontally stratified ionosphere, and the field strength is calculated near a caustic or cusp. The field can be expressed by an integral representing an angular spectrum of plane waves produced by reflection in the ionosphere. Integration for a region far from a cusp or caustic is done by the method of steepest descent or the saddle point method, where the saddle points are points where the complex phase is stationary. The contribution from each saddle point depends on the higher derivatives of the phase. At a point near a caustic the saddle points are close, and the third derivative must be taken into account. The field can then be expressed as an Airy integral function. For a point near a cusp, three points are close together and the fourth derivative becomes appreciable. In this case, the field is represented by the Pearcey (1946) function. For a lossy atmosphere, the fields are used to calculate the power absorbed per unit volume.

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