Models of Convectively Coupled Waves in the Tropical Atmosphere

Clouds are a manifestation of convection in the earth's atmosphere, and the convection is not spatially isotropic but organized on larger scales by waves. In the tropics, clouds and larger scale waves interact as convectively coupled waves (CCWs), and understanding the mechanisms of CCWs is a major unsolved problem. Furthermore, this is not only a theoretical problem but also a practical problem because current atmospheric general circulation models (GCMs) do not adequately represent CCWs. In this thesis, several models for CCWs are developed and studied in order to understand both the mechanisms of CCWs and the representation of CCWs in GCMs. A combination of mathematical theory and numerical simulations is used to understand CCWs in observations and GCMs. In Part I, a simplified tropical climate model is studied to understand the representation (or, in the language of atmospheric science, the parameterization) of CCWs in GCMs. Exact solutions are found for propagating fronts, called precipitation fronts, that mark the boundary between precipitating and non-precipitating regions. The precipitation fronts are shown to be exact solutions for a wide range of parameter values, including both finite values of the relaxation time scale &tgr;c and the limiting case of &tgr;c → 0. Numerical solutions show that the model behaves in a manner similar to parameterizations in actual GCMs: the climatological mean state is remarkably stable, and variability about the mean state is strongly damped. In Part II, another simplified model is studied that includes more complex vertical structures. This model can represent the effects of three cloud types (deep convective clouds, congestus clouds, and stratiform clouds) whereas the model in Part I represents only one cloud type (deep convective clouds). First a system of partial differential equations (PDE) is derived to represent the nonlinear dynamics of gravity waves interacting with background wind shear. These PDE have several interesting mathematical properties: they are a system of nonconservative equations with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of phase space. Theory and numerics are developed to illustrate these properties, and these properties are taken into consideration for developing a simple numerical scheme for these equations. Using these equations, it is shown that a background wind shear causes an asymmetry in eastward- and westward-propagating waves. This might be an important effect for the large-scale organization of convection in CCWs, where convection is often not isotropic but forms in regions of vertically sheared winds. These nonlinear PDE are then used as a nonlinear dynamical core for the multicloud model, which was originally designed with a linear dynamical core for simplicity. The multicloud model includes water vapor as a dynamical variable, and the effects of water vapor and clouds are included as interactive nonlinear source terms. It has been used to represent CCWs, and here it is used in a different parameter regime as a model for squall lines. Finally, the multicloud model is used in yet another parameter regime as a model for an analog of the Madden--Julian oscillation (MJO). This MJO analog captures many features from the observational record, including both the MJO's low-frequency envelope and the intense, chaotic deep convection events propagating within the low-frequency envelope.