  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. Since 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. Analyzing the algebraic properties of the 2D Euler equations, we give a
complete characterization of the jet formations. In view of the applications
and the numeric integration of the Euler equations, we show how the jets
formation also appear in the discrete Zeitlin model of the Euler equations. For
this model, the classification of the jet formations can be precisely described
in terms of reductive Lie algebra decomposition. Our results provide a simple
mechanism for the jets formation, which can be directly applied in the
numerical integration of their dynamics.
