Diffraction of acoustic-gravity waves in the presence of a turning point

Acoustic-gravity waves (AGWs) in an inhomogeneous atmosphere often have caustics, where the ray theory predicts unphysical, divergent values of the wave amplitude and needs to be modified. Unlike acoustic waves and gravity waves in incompressible fluids, AGW fields in the vicinity of a caustic have never been systematically studied. Here, asymptotic expansions of acoustic gravity waves are derived in the presence of a turning point in a horizontally stratified, moving fluid such as the atmosphere. Sound speed and the background flow (wind) velocity are assumed to vary gradually with height, and slowness of these variations determines the large parameter of the problem. It is found that uniform asymptotic expansions of the wave field in the presence of a turning point can be expressed in terms of the Airy function and its derivative. The geometrical, or Berry, phase, which arises in the consistent Wentzel–Kramers–Brillouin approximation for AGWs, plays an important role in the caustic asymptotics. In the dominant term of the uniform asymptotic solution, the terms with the Airy function and its derivative are weighted by the cosine and sine of the Berry phase, respectively. The physical meaning and corollaries of the asymptotic solutions are discussed.