• beta plane
• beta shift
• enhanced diffusion
• Euler equations
• homogenisation
• Ito-Stratonovich diffusion limit
• point vortices
• vortex aggregation
• vortex-wave
• zonal flows

The aim of this work is to take a step towards an explanation, at the Lagrangian level, of the formation of large-scale structures in two-dimensional fluids.
The first part of this work is devoted to the classical existence and uniqueness theorems: the result of Yudovich and the weak formulation of Schochet. The case of atomic measures gives us the point-vortex system. We will review Marchioro and Pulvirenti theorem and use the point-vortices as a convenient approximation to perform numerical simulations of the true Euler equations.
In the second part we consider the enhanced diffusion problem: a passive scalar which is advected by a strongly oscillating vector field, diffuses approximately according to the heat equation. We
present two different ways of formalising the concept of strongly oscillating vector fields, periodic homogenisation and the Ito-Stratonovich diffusion limit.
In the last part of the thesis, inspired by the framework of the vortex-wave system of Marchioro and Pulvirenti, we prove that highly concentrated vortices immersed in a large-scale vorticity field move towards the local maxima of the underlying field. We also explain the so-called beta-shift, i.e. the northward motion of positive vortices in a rotating planet. Finally, we write a simple one-dimensional model of zonal-flows displaying enhanced diffusion, vortex aggregation and beta-shift, and compare it with the Jupyter wind profile.