Self‐focusing of radio waves in an underdense ionosphere

The theory of self‐focusing instabilities in the ionosphere is developed emphasizing the critical parameters required to obtain sufficiently fast temporal and spatial growth rates so that the instability may be observed. It is shown that self‐focusing will not occur unless 2 cf /π f p ² l <1, where f is the radio wave frequency, f p a typical ionospheric plasma frequency, and l the spatial growth length. (In the F region, l ≈ 25 km is used, while in the E region l ≈ 1 km.) In the F region, the threshold power flux P c is P c,F ≈ (1.5 µW/m²) ( f /15 MHz)³ ( T e /1000 °K) 4 × (10 6 cm −3 / n e )³ C F , where n e and T e are typical electron densities and temperatures, and C F ∼1 depends on spatial and temporal growth rates. In the E region, the result is P c,E ≈ (1 mW/m²) ( f /15 MHz)³ (10 5 cm −3 / n e ) C E , where C E ≈ 1 again depends on growth rates. Dimensional analysis indicates nonlinear saturation will set in when variations of order unity occur in the radio wave intensity. The corresponding relative electron density fluctuations are given by δ n / n ∼π cf / f p ² l . Applications to planned ionospheric heating experiments and ionospheric modification by the microwave beam from a satellite power station are discussed.