Large-Scale Turbulence in Geophysical Flows: Confinement, Topography, and Rotation

Large-scale flows in geophysical and astrophysical systems often display features associated with two-dimensional turbulence, despite evolving in three-dimensional domains. Their behaviour reflects a competition between nonlinear advection, rotation, geometric confinement, and spatial variations in fluid depth. This thesis investigates how these mechanisms shape energy transfer and coherent structure formation beyond standard planar and quasi-geostrophic descriptions: we consider the case of rotating turbulence in finite-depth spherical shells, and model non-rotating shallow flows over topography. First, direct numerical simulations of the three-dimensional incompressible Navier-Stokes equations are used to study turbulent flows in spherical shells varying the shell depth and the rotation rate. A spectral analysis combining spherical harmonics with radial coarse-graining is developed to resolve the full three-dimensional distribution of kinetic energy. The results reveal a scale-dependent separation between the predominantly tangential large-scale motion and radial fluctuations dominating the smaller scales. Rotation organizes the flow into zonal structures throughout the investigated parameter range, but these structures do not always imply an inverse cascade. In deeper shells, Taylor columns efficiently collect energy into the largest-scale zonal modes, leaving a spectral depression at intermediate scales, around the equator. Tighter radial confinement, instead, geometrically hinders the formation of Taylor columns and permits a more continuous inverse energy transfer, resembling 2D turbulence. This mechanism is explained in terms of alignment between the radial confinement and the vertical rotation: the two constraints cooperate at the poles but compete in the equatorial band. In this light, jet formation is not an indicator of the transition towards two-dimensional dynamics, which, in spherical geometry, strongly depends on radial confinement. Second, a non-rotating shallow-flow model is derived from the three-dimensional Navier-Stokes equations for a fluid of variable depth beneath a rigid lid. Formulated through the mass-transport stream function, the model retains the nonlinear coupling between the flow and bottom topography. A structure-preserving numerical method is developed and validated to investigate its long-time dynamics. The topographic modulation precludes non-trivial self-similar stationary states in freely decaying viscous flows. When energy is artificially preserved, stationary configurations arise from a Reynolds-dependent balance between nonlinear advection and diffusion, rather than from conventional minimum-enstrophy selection. In the absence of rotation, numerical simulations, with both deterministic and stochastic forcing, display the formation of domain-scale vortex dipoles whose vortices systematically avoid topographic hills, regardless of the vorticity sign. At larger Reynolds numbers, the dynamics can remain trapped in metastable excited states. Taken together, these results expose the limitations of standard reduced models for large-scale turbulence. They highlight the importance of intermediate regimes, where nonlinear interactions and non-planar geometries can produce dynamics beyond the reach of conventional single-layer and rapidly rotating approximations.