Chaotic systems are characterised by exponential separation between close-by trajectories, which in particular leads to deterministic unpredictability over an infinite time-window. It is now believed, that such butterfly effect is not fully relevant to account for the type of randomness observed in turbulence. For example, tracers in homogeneous isotropic flows are observed to separate algebraically, following a universal cubic growth, independent from the initial separation. This regime, known as Richardon’s regime, suggests that at the level of trajectories, and unlike in chaos theory, randomness may in fact emerge in finite-time. This phenomenon called ‘spontaneous stochasticity’ originates from the singular nature of the underlying dynamics, and provides a candidate framework for turbulent randomness and transport. While spontaneous stochasticity has been mathematically formalised in simplified turbulence models , a precise and systematic tool for quantifying the various facets of this phenomenon is to this day missing.
We introduce in this thesis a 3d rough flow inspired by the Weierstrass function, entitled ‘the WABC model’. We show that Lagrangian trajectories in this model have a finite-time dispersion, even in the limit of infinitesimal initial dispersion. This direct observation of spontaneous stochasticity is impossible to perform in real flows due to numerical or experimental constraints. To circumvent this technical issue, we adapt the definition of spontaneous stochasticity in our model to create a criterion based on transition probabilities. We show that this criterion is more suited for the analysis of real flows. We verify its sensibility to spontaneous stochasticity in the WABC model. This criterion is then applied on experimental data, where preliminary results tend to highlight traces of spontaneous stochasticity.