## 1. Introduction

[2] Gravity waves generated by tropospheric convection can propagate high into the atmosphere, reaching altitudes of 200 kilometers or more. While high resolution simulations using fully nonlinear mesoscale atmospheric models can capture the processes near the tropospheric source, extending these models with sufficient resolution to such high altitudes is computationally impractical. Thus, alternative approaches are required to model the deep penetration of convectively generated gravity waves into the middle and upper atmosphere.

[3] One such approach for convectively generated gravity waves has been developed by *Vadas and Fritts* [2001, 2004, 2005, 2009] and has been used to interpret a number of observations, including airglow data from the mesopause region near 85 km altitude [*Vadas et al.*, 2009] and ionospheric soundings of the F-layer near 250 km altitude [*Vadas and Crowley*, 2010].

[4] The Vadas-Fritts model consists of a Fourier-Laplace integral representation for the near-field around the convective forcing, and a ray-tracing for the propagation of the gravity waves into the far-field. The initial conditions for the far-field ray-tracing are deduced from the near-field integral representation.

[5] In this paper, we examine the Vadas-Fritts ray-tracing model. Our main analysis tool is the stationary phase approximation for the near-field integral representation, which we present based on the theory of *Lighthill* [1996]. The stationary phase approximation, like the Vadas-Fritts theory, takes spectral wave amplitudes from the near-field integral representation and produces a ray solution for the gravity waves emerging from the forcing. It thus provides all the information necessary to initialize a far-field ray-tracing. Mostly, the stationary phase theory supports the Vadas-Fritts ray-tracing initialization, but there are some small differences that suggest potential refinements to the specification of the initial conditions.

[6] We also examine the Vadas-Fritts method of computing the spatial wave amplitudes in the ray-tracing. Their method is new, at least in the context of gravity waves, in the way that it accounts for the three dimensional geometrical spreading of the rays. In principle, the effects of geometrical spreading on wave amplitudes can be obtained by advecting a ray-tube volume element along each ray, but this is notoriously difficult in practice, as discussed by *Hasha et al.* [2008]. The Vadas-Fritts method is a practical alternative to the ray-tube method and is potentially useful for other gravity wave applications. We analyze the Vadas-Fritts method in a different but equivalent way to their formulation, making explicit use of a ray Jacobian and relating the ray Jacobian to the density of the rays. The advantages of this approach are summarized at the conclusion of the paper.

[7] The paper is organized as follows. In section 2 the Vadas-Fritts model equations are presented, along with an integral representation for the solution. This is followed in section 3 by a presentation of the stationary phase approximation for the integral representation. In section 4 the ray Jacobian and density of rays are discussed in relation to the Vadas-Fritts method of wave amplitude calculation. In section 5 the specific forms for the convective forcing terms of the Vadas-Fritts model are introduced, and examples of the solution are presented based on the stationary phase analysis. In section 6 the stationary phase and Vadas-Fritts ray initializations are compared. In section 7, the extension to a non-uniform background is discussed. In section 8 our main results are summarized, with suggestions for further analysis.

[8] We assume a uniform background throughout this paper, except to indicate in section 7 how the method can be extended to a non-uniform background. We also make the Boussinesq approximation and ignore dissipation. While dissipative effects are an important component of the Vadas-Fritts ray-tracing model, they are not needed for our purposes and so are omitted. The presentation is fairly general until section 5, where the specific Vadas-Fritts forcing terms are introduced.

[9] We give a modified derivation of the Vadas-Fritts integral representation using a Fourier transform in time, rather than the Laplace transform of the Vadas-Fritts theory. The modified derivation is based on work by *Lighthill* [1996], and its equivalence to the Vadas-Fritts solution is shown in Appendix A.