The thesis reports on the development of a new quasi-elastic nonhydrostatic model, cast in a terrain-following coordinate based on the full pressure field. The equations used are the ƒã coordinate by White (1989). The equations are filtered of vertically propagating acoustic waves. However, since Lamb waves are present, the equations may be termed quasi-elastic. In contrast to similar quasi-elastic pressure-based models, the equations and the numerical solution procedure presented here are formulated independent of the use of a reference state thermodynamic profile. Thus, it is possible that the equations may be used to simulate atmospheric motion at spatial scales larger than the meso-scale. A novel split semi-Lagrangian procedure is formulated to solve the quasi-elastic equations on a grid that is nonstaggered in both the horizontal and vertical. A nonstaggered grid is appealing to use in semi-Lagrangian discretizations of the atmospheric equations, since only one set of trajectories needs to be calculated during each advection time step. However, it is well known that the nonstaggered grid has poor gravity wave dispersion properties. In this study, this problem is alleviated by using high-order centered spatial differencing, and by applying a spatial filter to remove two-grid-interval waves from the grid. It is shown that large time steps (large Courant numbers) are allowed during the semi-Lagrangian advection step. This makes the method computationally attractive compared to explicit or split-explicit procedures that use an Eulerian approach to treat the advection terms. For situations where the fast moving gravity waves carry a non-negligible amount of the energy, the split semi-Lagrangian approach may even be computationally more efficient than the widely used semi-implicit semi-Lagrangian solution procedures. The thesis reports on a large set of bubble convection tests performed with the new kernel. It is concluded that the new model is worth developing further.