Stochastic modelling of atmospheric gravity waves

Internal gravity waves have an important effect on the large-scale circulation of the middle atmosphere, which is conditioned by the deposition of momentum due to their breaking. The propagation of gravity waves is influenced by the properties of the background wind. This thesis examines this influence: it uses stochastic techniques to study gravity wave propagation through a randomly fluctuating background wind. It begins by describing general features of the atmosphere and gravity wave propagation. The basic equations of fluid flow within the atmosphere are derived. These lead via the WKB approximation to a dispersion relation and to ray equations for gravity wave propagation. Propagation equations, such as the ray equations and the dispersion relation, are derived in a general context. The notion of a Wigner matrix is introduced, and this is used to derive transport equations for a general Hamiltonian system that may contain random components. These results generalise earlier works by Ryzhik et al. and Guo and Wang. Atmospheric gravity waves are described as an application and the equations derived via the WKB approximation are recovered. The major factor influencing the distribution of gravity waves is the spread of their wavenumber as they propagate through a wind. This is described by the Doppler spreading model. A one-dimensional system with a randomly fluctuating background wind, dependent on altitude only, is considered. The model revisits that of Souprayen et al. by using an Ornstein-Uhlenbeck process to describe the wind. Simple equations for the energy spectrum induced by gravity waves are derived. Analytic forms of the energy, spectrum are given and features of the spectrum such as the m 3 (where m is the vertical wavenumber) spectral tail, central wavenumber and scaling with the Brunt-Väisälä frequency are found to be consistent with observations. An equation for the force on the background, induced by gravity wave breaking, is also derived. The analytical results are backed up by numerical simulations. Caustics, that is, singularities caused by the intersection of neighbouring rays, are often overlooked as a possible mechanism for wave breaking. A two-dimensional model is considered in which the background wind is constructed from several Fourier modes, with random phase and amplitude to represent the complexities of a real atmospheric wind. A single parameter controls the amplitude of the random fluctuations of the wind. Numerical techniques are used to detect caustics by computing the ray-tube area. Probability distribution functions for the altitude of caustic events are then obtained. A scaling of the altitude of caustic formation with the amplitude parameter of the wind is proposed.