  The global structure of the atmosphere and the oceans is a continuous source
of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets
are determinant in the air and water circulation around the Earth. In the last
fifty years, thanks to the development of more and more precise and extensive
observations, it has been possible to study in detail the atmospheric
formations of the giant-gas planets in the solar system. For those planets,
jets are the dominant large scale structure. Starting from the 70s, various
theories combining observations and mathematical models have been proposed in
order to describe their formation and stability. In this paper, we propose a
purely algebraic approach to describe the spontaneous formation of jets on a
spherical domain. Analysing the algebraic properties of the 2D Euler equations,
we give a characterization of the different jets' structures. The calculations
are performed starting from the discrete Zeitlin model of the Euler equations.
For this model, the classification of the jets' structures can be precisely
described in terms of reductive Lie algebras decomposition. The discrete
framework provides a simple tool for analysing both from a theoretical and and
a numerical perspective the jets' formation. Furthermore, it allows to extend
the results to the original Euler equations.
