Accurate coupling between the resolved scale dynamics and sub-grid scale physics is essential for accurate modelling of the atmosphere. Previous emphasis has been towards the temporal aspects of this so called physics-dynamics coupling problem, with little attention towards the spatial aspects. When designing a model for numerical weather prediction there is a choice for how to vertically arrange the required variables, namely the Lorenz and Charney-Phillips grids, and there is ongoing debate as to which is the optimal. The Charney-Phillips grid is considered good for capturing the large scale dynamics and wave propagation whereas the Lorenz grid is more suitable for conservation. However the Lorenz grid supports a computational mode. In the first half of this thesis it is argued that the Lorenz grid is preferred for modelling the stably stratified boundary layer. This presents the question: which grid will produce most accurate results when coupling the large scale dynamics to the stably stratified planetary boundary layer? The second half of this thesis addresses this question. The normal mode analysis approach, as used in previous work of a similar nature, is employed. This is an attractive methodology since it allows one to pin down exactly why a particular configuration performs well. In order to apply this method a one dimensional column model is set up, where horizontally wavelike solutions with a given wavenumber are assumed. Applying this method encounters issues when the problem is non normal, as it will be when including boundary layer terms. It is shown that when addressing the coupled problem the lack of orthogonality between eigenvectors can cause mode analysis to break down. Dynamical modes could still be interpreted and compared using the eigenvectors but boundary layer modes could not. It is argued that one can recover some of the usefulness of the methodology by examining singular vectors and singular values; these retain the appropriate physical interpretation and allow for valid comparison due to orthogonality between singular vectors. Despite the problems in using the desirable methodology some interesting results have been gained. It is shown that the Lorenz grid is favoured when the boundary layer is considered on its own; it captures the structures of the steady states and transient singular vectors more accurately than the Charney-Phillips grid. For the coupled boundary layer and dynamics the Charney-Phillips grid is found to be most accurate in terms of capturing the steady state. Dispersion properties of dynamical modes in the coupled problem depend on the choice of horizontal wavenumber. For smaller horizontal wavenumber there is little to distinguish between Lorenz and Charney-Phillips grids, both the frequency and structure of dynamical modes is captured accurately. Dynamical mode structures are found to be harder to interpret when using larger horizontal wavenumbers; for those that are examined the Charney-Phillips grid produces the most sensible and accurate results. It is found that boundary layer modes in the coupled problem cannot be concisely compared between the Lorenz and Charney-Phillips grids due to the issues that arise with the methodology. The Lorenz grid computational mode is found to be suppressed by the boundary layer, but only in the boundary layer region.

Over the years, global climate modelling has advanced, aiming for realistic and precise models by increasing their complexity. An integral component of climate models, the physics parameterizations, are a major limitation, but are required due to limited computational power. Grid adaptivity is an avenue that is being explored to mitigate these challenges, but comes with its own difficulties. For example, the question of whether the physics should be ``scale-aware’’, by adjusting according to the resolution and the fact that parameterizations are optimized for specific grid ranges. To research these challenges, test cases that work in both the adaptive and non-adaptive cases are required. This thesis concentrates on physics parameterizations of Atmospheric Global Climate Models (AGCMs) presenting the current hierarchy of idealized physics parameterizations found in the literature. It focuses on and provides a comprehensive explanation of a simplified dry physics model for AGCMs, exploring where it is situated in the current hierarchy and its steady states in the uncoupled case. A coupling of the physics model to the adaptive dynamical core wavetrisk is explained and explored. This includes characterizing the results in the non-adaptive case for time convergence, grid convergence, and the effects of the soil, while also benchmarking the climatology of the coupling. The simplified dry physics model introduces another level of complexity in the current dry physics hierarchy and is stable in the coupled and uncoupled cases. A decreasing temperature trend with height is observed, however warmer surface temperatures and cooler upper atmosphere temperatures, than that of Earth, are produced in the steady states. Additionally a linear rate of convergence in space is noted and an improvement in parallel efficiency with resolution is required. Overall these results can be used as a benchmark for future coupling in the adaptive case. / Thesis / Master of Science (MSc)