In the general framework of the atmosphere, the in-depth study of extreme events of complex nature such as tropical cyclones remains an open problem. Nowadays, there are different useful studies aiming to enlarge the current knowledge of these events. Under this perspective, the search for mechanisms and geographical areas of formation and dynamics of tropical cyclones at different latitudes needs better constraints on their preliminary results. This work focuses on a diagnostic analysis of a tropical cyclone, with the aim of identifying key points that characterize its evolution, from a dynamic and observational point of view. The study is applied to Hurricane Faraji, the most powerful tropical cyclone of the 2021 Indian Ocean season, classified as fifth-category on the Saffir–Simpson intensity scale. The study develops in three main sections, all related to each other. The starting point is a large set of satellite products from both polar and geostationary platforms. From the data acquired by the polar instruments, an accurate study of the evolution of the hurricane is carried out, focused on the extraction of physical information related to the system (temperature and altitude of the associated cloudy system, temperature gradient, pressure in the different regions) at moderate resolutions. From the data acquired by the geostationary instruments, it is possible to obtain very high temporal resolution pictures of the temperature field of the cyclone, from which a study of the turbulent dynamics is carried out. In particular, to investigate the maximum energy content in the different regions of the cyclone, the Proper Orthogonal Decomposition (POD) technique is used to extract the associated spectra of both the spatial and temporal components, studied separately on three different ranges of scales.