The investigator and colleagues study the impact of using simplified models as well as the effect of noise on uncertainty propagation in terms of statistical solutions in several fluid systems. The physical problems under investigation are large Prandtl number or small Ekman number regimes in Rayleigh-Benard convection, large Prandtl-Darcy number or small Darcy number regimes in convection in fluid saturated porous media, and stochastically forced quasi-geostrophic systems with random noise. Model reduction in the parameter regimes for the convection problems leads to singular limit problems involving multiple time and spatial scales. One of the goals here is to investigate what kind of statistical properties can be approximated using the statistical properties of the simplified models derived via singular limits. In the case of systems with random noise, which are modeled by stochastic partial differential equations, the investigator and his colleagues investigate possible noise-induced statistical stabilities, i.e., stability of statistical properties with respect to initial condition and parameters.
There are abundant uncertainties in nature. For instance, prediction of the path of a hurricane is influenced by (imperfect) information about the initial distribution of atmospheric and sea surface temperatures, wind, and the atmospheric jet stream, as well as by the imperfect model itself. The climate (long-time behavior) of the earth may be influenced by the ozone layer, concentrations of carbon dioxide, and other factors. Transport of contaminants in ground water is influenced by porosity and permeability of the media, which are random. Studying the influence of various uncertainties on the geophysical flows considered here helps us better understand the behavior of the flows.
