Higher order approximation of ray theory applied to orographic gravity waves

The modeling of the propagation of orographic waves using ray techniques in realistic background wind conditions usually encounters places, called caustics, where the ray approximation breaks down. In this work we examine a higher order approximation than usual ray theory applied to orographic waves in realistic background wind conditions. Our analytical results show that this formulation is free of the singularities that arise in ray theory. We extend previous results to a background wind that depends on vertical and horizontal coordinates. We apply the higher-order approximation to waves that are propagating in a background wind that is turning with height, same conditions as in Shutts' work. We show that the evolution of the amplitude is well defined in this approximation for all times and locations including the initial point and the converging point (caustic) where a maximum in wave amplitude is found. We compare our results with a linear numerical model, there is good agreement of the amplitude and wavenumber evolution for both short and long times in other words this formulation may represent both the near and far field as long as the slowly varying wave train assumption remains valid.