The Role of Mathematics in the Understanding of the Dynamics of Meteorogical Situations that Produce Heavy Rain over the Spanish Mediterranean Zone C. Ramis, R. Romero, V. Homar and S. Alonso Group of Meteorology. Department of Physics. Universitat de les Illes Balears. 07122 Palma de Mallorca. Spain Monograf´ ıas de la Real Academia de Ciencias de Zaragoza 31, 175–198, (2009). Abstract Globally, floods and flash floods are the natural hazards that annually produce the greatest number of fatalities as well as economic losses. Mainly in autumn, the Spanish Mediterranean zone is affected by these meteorological phenomena. The most common meteorological situation that produces such episodes can be characterized at surface by an European anticyclone and a weak cyclone located on the Algerian coast that induce humid easterly winds impinging on the Spanish Mediterranean coast. At the middle and high troposphere a cold cyclonic centre is usually located to the south or southwest of the Iberian peninsula, producing southwesterly flow over the western Mediterranean. The conceptual model that can be devised from the synoptic situation attributes an important role to the orography, being the factor that provides enough lifting to the humid low level parcels to trigger convection. Evaporation from the sea is responsible for the high amount of water vapor that carries the flow associated with the pressure gradient produced by the European anticyclone and the Algerian low. We have verified the validity of this conceptual model by means of numerical simulations of several cases of torrential rain combined with a factor separation technique. In order to analyze the role played by the high-level disturbance, we apply numerical techniques aimed at isolating the ingredients that contribute to the development of convection and intense rain. In particular, we investigate the role of Potential Vorticity nuclei, associated with the upper-level cold low, through numerical experiments. Initial conditions in the model are thus soundly perturbed by modifying the Potential Vorticity field after applying an inversion technique. The results confirm that the high-level Potential Vorticity is responsible for the development of an easterly low-level jet. The interaction of such circulation and the orography determines where the heavy rain focuses. 175