Cascate turbolente nel modello Quasi-Geostrofico di Superficie

This thesis explores non-equilibrium corrections in direct cascades of Surface Quasi-Geostrophic (SQG) model. The Surface Quasi-Geostrophic (SQG) model serves as a simplified yet effective tool for studying turbulence in geophysical fluid dynamics, specifically near the Earth's surface. By providing a computationally manageable framework, SQG equations capture essential features of turbulent flows, making them valuable for analyzing boundary layer turbulence, coastal oceanography, and the interaction between ocean currents and atmospheric processes. Widely applied in weather prediction, climate modeling, and the understanding of oceanic and atmospheric phenomena, the SQG model finds practical use among meteorologists, oceanographers, and climate scientists, enhancing insights into complex turbulent processes and improving predictions in various environmental contexts. The SQG model belongs to a broader class of 2D turbulence models known as 'alpha-turbulence'. These models are composed of a transport and diffusion equation for a scalar field in a two-dimensional, incompressible velocity field. The velocity field is expressed in terms of a stream function, which is, in turn, determined by the transported field. The functional relationship between the transported field and the stream function depends on a parameter (alpha), the value of which is specific to the model, for example SQG for alpha=1, as well as Navier-Stokes equations for alpha=2 or Charney-Hasegawa-Mima for alpha=-2. Previous theoretical studies showed that in 3D turbulence the instantaneous energy spectra reveal deviations from the classical prediction (the so-called Kolmogorov spectrum). These deviations are attributed to temporal fluctuations in the energy flux. This implies that, at certain instants, there may be an energy imbalance between large and small scales. In these investigations was observed a power-law for the difference between the mean spectrum and the instantaneous spectrum characterized by an exponent of "-7/3". The original contribution of this thesis was to extend this formalism to calculate the spectral corrections for the generic 'alpha-turbulence' model. Specifically, we computed the non-stationary generic correction for 'alpha-turbulence' and subsequently derived the particular correction for the SQG model from this general framework. For the case alpha=2 corresponding to the Navier-Stokes model, we observed that the corrections are not subdominant, indicating that the assumptions are not self-consistent. In contrast, for the SQG model, the correction exhibits the same exponent as observed in 3D turbulence, justifying its validity. We validated these results through numerical simulations on square periodic domains with periodic boundary conditions, enabling the use of pseudospectral methods to solve the equations. We computed instantaneous and averaged spectra, extracting statistics of spectral fluctuations. As these temporal fluctuations are linked to potential enstrophy fluctuations, reflecting the intermittency properties of this field, the final part of the study analyzed intermittency through both structure functions and probability density functions of increments at a specific scale. The results highlighted strong intermittency in these potential vorticity increments.