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This thesis addresses the statistical modeling of turbulence, focusing on three main aspects: the critical transition from laminarity to turbulence, the effects of the so-called intermittency and the energy dynamics of a turbulent flow. The central part of the thesis consists of six papers, divided into two parts. In Part I we develop two new indices to quantify the proximity to critical transitions in stochastic dynamical systems, with particular attention to the transition from laminarity to turbulence in fluids (Paper A). The two indices are tested on two toy models and then applied to the detection of critical events in a magnetised fluid and in financial time series. We define a third index Y, which quantifies the effects of intermittency and does not require very long time series. This index turns out to be effective in recovering the structure of the turbulent flow (Papers B, C). In Paper D we show that Y is also sensitive to the turbulent behavior of financial markets, providing a possible early warning indicator of the proximity to critical events. In Part II we introduce a new local observable as the arrival times of tracer particles at a particular point in the fluid as a proxy of the turbulent velocity field. We model the universal self-organising structure of this observable in an effective and parsimonious way. In the second paper of Part II, we model the continuous-time dynamics of the energy budget of the turbulent field. We show that this observable can be characterised as the exponential of a stochastic integral on a Lévy basis, under the assumption that the energy transmission across time scales is a multiplicative cascade process. |
