Ionospheric Gravity Wave Interactions and Their Representation in Terms of Stochastic Partial Differential Equations

Phenomena in nature that involve diffusion and convection of matter and propagation of waves, e.g., the propagation of waves in geophysical flows, can exhibit randomness properties and thus need to be modeled by stochastic partial differential equations (SPDEs). For example in the ionosphere, the region in the upper atmosphere where there are high concentrations of ions and electrons, wave interactions are influenced by electromagnetic forces that fluctuate randomly in time, and are thus modeled by SPDEs. In this thesis we model interactions between atmospheric waves and the ionosphere induced by upward propagating atmospheric gravity waves (AGWs) starting with the equations of conservation of mass, momentum and energy, and Maxwell's equations. Two important problems are examined: the problem in which the ionosphere is treated as a deterministic medium and the wave interactions are governed by nonlinear partial differential equations (PDEs), and the problem in which the ionosphere is a random medium and the governing equations are nonlinear stochastic partial differential equation (SPDEs) driven by the Brownian motion. In the stochastic case we make use of numerical methods based on Wiener Chaos expansions (WCE) which are effective methods for solving SPDEs driven by Brownian motion. The accuracy of this method is accessed by comparing the results with the exact analytical or semi-analytical solutions for some problems involving stochastic evolution equations comprising the stochastic heat and stochastic advection-diffusion equations, and the stochastic Burgers' equation. In the the deterministic case, we derive analytical solutions for some special simplified configurations and then carry out numerical simulations for time-dependent nonlinear configurations. The results of the simulations of our analytical and numerical models are compared with the conclusions from previous studies which are mainly observations. Our results explain several observed phenomena arising from the interactions of the atmospheric gravity waves with the ionosphere.