On the Mean Residence Time in Stochastic Lattice-Gas Models

A heuristic law widely used in fluid dynamics for steady flows states that the amount of a fluid in a control volume is the product of the fluid influx and the mean time that the particles of the fluid spend in the volume, or mean residence time. We rigorously prove that if the mean residence time is introduced in terms of sample-path averages, then stochastic lattice-gas models with general injection, diffusion, and extraction dynamics verify this law. Only mild assumptions are needed in order to make the particles distinguishable so that their residence time can be unambiguously defined. We use our general result to obtain explicit expressions of the mean residence time for the Ising model on a ring with Glauber + Kawasaki dynamics and for the totally asymmetric simple exclusion process with open boundaries

Processi di propagazione delle piene e di infiltrazione lungo i conoidi del Cellina e Meduna

The thesis deals with the problem of modeling flood propagation along large and permeable alluvial fans in which non negligible infiltration and sub-channel flow is present. In particular, the study considers the large alluvial fans of Cellina and Meduna rivers, Livenza River Basin, District of the Eastern Alps. These alluvial fans, located at the mouth of their mountain basins and manned by hydroelectric reservoirs, represent the connection between the mountain part and the plain part of the two streams, which then merge into the Livenza River. In the early chapters of the thesis, the natural environment is described and a review is made of some of the previous studies about the propagation and infiltration phenomena along these two fans. In chapter three the proposed mathematical model for simulating runoff and infiltration phenomena, coupled together, is described. Starting from an available finite element model, which integrates the two-dimensional shallow water equations, a new model was developed. The proposed model couples the two-dimensional shallow water flow model with an original sub-surface module which accounts for the presence of infiltration and sub-channel flow. In the proposed model the interaction between surface runoff and sub-surface flow, linked through the vertical infiltration, is based on mass conservation (i.e., no dynamic interaction is included in the model). The sub-surface model uses a mixed "Horton-Dunne"representation of the infiltration processes and a loss of water for infiltration toward the deep layers of the ground is also included. The saturation of the hypodermic layer is assumed to occur when the amount of water incoming into the computational element (from adjacent cells and / or by precipitation) exceeds the infiltration capacity towards the deep layers (Hortonian mechanism). Moreover, we assume that surface runoff starts when the groundwater surface intersects or exceeds the land surface, that is to say, when the hypodermic layer is completely saturated (Dunne mechanism). The fourth chapter describes the monitoring networks and the available data used to validate the model. In the fifth chapter a sensitivity analysis, with respect to the main parameters used in the model, is described. The results of these simulations highlighted that the sub-channel flow is generally small when compared to free surface flow. However, the analysis also showed that flood peak and volume are largely reduced during the propagation both because of the water loss by infiltration and because of water storage in the ground surface layer. The sixth chapter describes the application of the model to three recent real events, provided with measured data. These are the flood events that occurred in November 2000, 2002 and 2010. The results demonstrate the effectiveness of the proposed model to simulate the flow over large, permeable alluvial fan