Calculation of the eigenfunctions of the two-group neutron diffusion equation and application to modal decomposition of BWR instabilities

In this thesis, numerical methods aiming at determining the eigenfunctions, their adjoint and the corresponding eigenvalues of the two-group neutron diffusion equations representing any heterogeneous system are investigated. First, the classical power iteration method is modified so that the calculation of modes higher than the fundamental mode is possible. Thereafter, the Explicitly-Restarted Arnoldi method, belonging to the class of Krylov subspace methods, is touched upon. Although the modified power iteration method is a computationally-expensive algorithm, its main advantage is its robustness, i.e. the method always converges to the desired eigenfunctions without any need from the user to set up any parameter in the algorithm. On the other hand, the Arnoldi method, which requires some parameters to be defined by the user, is a very efficient method for calculating eigenfunctions of large sparse system of equations with a minimum computational effort. These methods are thereafter used for off-line analysis of the stability of Boiling Water Reactors. Since several oscillation modes are usually excited (global and regional oscillations) when unstable conditions are encountered, the characterization of the stability of the reactor using for instance the Decay Ratio as a stability indicator might be difficult if the contribution from each of the modes are not separated from each other. Such a modal decomposition is applied to a stability test performed at the Swedish Ringhals-1 unit in September 2002, after the use of the Arnoldi method for pre-calculating the different eigenmodes of the neutron flux throughout the reactor. The modal decomposition clearly demonstrates the excitation of both the global and regional oscillations. Furthermore, such oscillations are found to be intermittent with a time-varying phase shift between the first and second azimuthal modes.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

A study of air-sea interaction processes on water mass formation and upwelling in the Mediterranean sea

Air-sea interactions are a key process in the forcing of the ocean circulation and the climate. Water Mass Formation is a phenomenon related to extreme air-sea exchanges and heavy heat losses by the water column, being capable to transfer water properties from the surface to great depth and constituting a fundamental component of the thermohaline circulation of the ocean. Wind-driven Coastal Upwelling, on the other hand, is capable to induce intense heat gain in the water column, making this phenomenon important for climate change; further, it can have a noticeable influence on many biological pelagic ecosystems mechanisms. To study some of the fundamental characteristics of Water Mass Formation and Coastal Upwelling phenomena in the Mediterranean Sea, physical reanalysis obtained from the Mediterranean Forecating System model have been used for the period ranging from 1987 to 2012. The first chapter of this dissertation gives the basic description of the Mediterranean Sea circulation, the MFS model implementation, and the air-sea interaction physics. In the second chapter, the problem of Water Mass Formation in the Mediterranean Sea is approached, also performing ad-hoc numerical simulations to study heat balance components. The third chapter considers the study of Mediterranean Coastal Upwelling in some particular areas (Sicily, Gulf of Lion, Aegean Sea) of the Mediterranean Basin, together with the introduction of a new Upwelling Index to characterize and predict upwelling features using only surface estimates of air-sea fluxes. Our conclusions are that latent heat flux is the driving air-sea heat balance component in the Water Mass Formation phenomenon, while sensible heat exchanges are fundamental in Coastal Upwelling process. It is shown that our upwelling index is capable to reproduce the vertical velocity patterns in Coastal Upwelling areas. Nondimensional Marshall numbers evaluations for the open-ocean convection process in the Gulf of Lion show that it is a fully turbulent, three-dimensional phenomenon.