<p> The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar conservation laws, many of the same types of waves and a similar (quasi-) balanced state. It is desirable that numerical models posses similar properties, and the prototypical example of such a scheme is the 1981 Arakawa and Lamb (AL81) staggered (C-grid) total energy and potential enstrophy conserving scheme, based on the vector invariant form of the continuous equations. However, this scheme is restricted to a subset of logically square, orthogonal grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others). </p><p> It is also possible to obtain these properties (along with arguably superior wave dispersion properties) through the use of a collocated (Z-grid) scheme based on the vorticity-divergence form of the continuous equations. Unfortunately, existing examples of these schemes in the literature for general, spherical grids either contain computational modes; or do not conserve total energy and potential enstrophy. This dissertation extends an existing scheme for planar grids to spherical grids, through the use of Nambu brackets (as pioneered by Rick Salmon). </p><p> To compare these two schemes, the linear modes (balanced states, stationary modes and propagating modes; with and without dissipation) are examined on both uniform planar grids (square, hexagonal) and quasi-uniform spherical grids (geodesic, cubed-sphere). In addition to evaluating the linear modes, the results of the two schemes applied to a set of standard shallow water test cases and a recently developed forced-dissipative turbulence test case from John Thuburn (intended to evaluate the ability the suitability of schemes as the basis for a climate model) on both hexagonal-pentagonal icosahedral grids and cubed-sphere grids are presented. Finally, some remarks and thoughts about the suitability of these two schemes as the basis for atmospheric dynamical development are given.</p>

<p> Scintillometer measurements of the turbulence inner-scale length lo and refractive index structure function C²n allow for the retrieval of large-scale area-averaged turbulent fluxes in the atmospheric surface layer. This retrieval involves the solution of the non-linear set of equations defined by the Monin-Obukhov similarity hypothesis. A new method that uses an analytic solution to the set of equations is presented, which leads to a stable and efficient numerical method of computation that has the potential of eliminating computational error. Mathematical expressions are derived that map out the sensitivity of the turbulent flux measurements to uncertainties in source measurements such as l_o. These sensitivity functions differ from results in the previous literature; the reasons for the differences are explored. </p>

<p>Well known dynamic models for air and stormwater quality typically involve the application of deterministic differential equations (DDEs) or random differential equations (RDEs) that apply Monte Carlo simulation. An alternative to RDEs are stochastic differential equations (SDEs), which are DDEs that incorporate random noise. In this thesis, we develop air and stormwater quality models that employ DDEs, RDEs and SDEs numerically solved by finite difference methods. The numerical results of the model variants are compared to each other and empirical data. The outcome demonstrates the utility of the SDE approach. The stormwater model is based on a one-dimensional advection-diffusion partial differential equation (PDE) that simulates the stream transport of copper in a small area within Los Angeles. Two air models are implemented, an ordinary differential equation model based on the continuity equation and a two-dimensional advection-diffusion PDE. The models approximate carbon monoxide levels in Costa Mesa and the Coachella Valley in California. The numerical PDEs are solved with the Strang splitting method, where the Lax-Wendroff and Crank-Nicolson methods are employed for the advection and diffusion subproblems respectively. For the SDE case the Euler-Maruyama method is applied to the source term subproblem. </p>