Quasi-normal mode analysis in BEC acoustic black holes

We perform a quasi-normal mode analysis of black hole configurations in Bose-Einstein condensates (BEC). In this analysis we use the full Bogoliubov dispersion relation, not just the hydrodynamic or geometric approximation. We restrict our attention to one-dimensional flows in BEC with step-like discontinuities. For this case we show that in the hydrodynamic approximation quasi-normal modes do not exist. The full dispersion relation, however, allows the existence of quasi-normal modes. Remarkably, the spectrum of these modes is not discrete but continuous.Comment: 7 pages, 3 figure

The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

Nouvelle carte des isohyètes annuelles et des maxima pluviométriques sur le massif du Piton de la Fournaise (Ile de la Réunion)

L'implantation de douze pluviographes, de 220 à 2490 mètres d'altitude, dans des sites difficilement accessibles des versants sud et est du massif du Piton de la Fournaise à la Réunion, apporte de nouveaux éléments pour le tracé des isohyètes annuelles. Un pluviographe situé à 1600 mètres d'altitude, à l'amont d'un alignement orienté sud-nord dans la zone du Baril sur la planèze sud-est du Volcan, a enregistré une précipitation annuelle supérieure à 18 000 mm en 1993. Dans cette zone, les cartes d'isohyètes moyennes inter-annuelles admises jusqu'alors présentaient des lames d'eau de lordre de 5000 à 6000 mm.an-¹. Les enregistrements obtenus en 1994 puis en 1995 sur le même pluviographe et sur des stations situées à l'est du massif à 1400 et à 1940 mètres d'altitude confirment qu'une très forte pluviométrie affecte cette zone.Une couche d'inversion thermique est fréquemment observée par ballon-sondage entre 2000 et 2500 mètres d'altitude, principalement en hiver austral. Cette inversion est l'une des causes principales de la présence d'une zone de maximum pluviométrique située entre 1400 et 1940 mètres d'altitude sur les planèzes du Volcan. Selon la position de l'inversion, la Réunion est soit dans la situation des îles très hautes, avec des sommets émergeant des nuages, soit dans la situation des îles hautes, pour lesquelles les précipitations sont fortes près des sommets. Les fortes pentes des versants du massif et la présence de grands encaissements contribuent également à expliquer l'emplacement de la zone de maximum pluviométrique et les lames d'eau importantes qui y sont enregistrées.Une nouvelle carte des isohyètes annuelles est proposée pour prendre en compte les nouvelles données. Une zone de maximum pluviométrique définie par l'isohyète 12 000 mm de moyenne annuelle est localisée entre 1300 et 1800 mètres d'altitude, localement jusqu'à 2000 m, sur le versant oriental du massif exposé aux vents dominants. De part et d'autre de cette zone la pluviométrie diminue: les lames d'eau moyennes annuelles décroissent jusqu'à 4000 mm au niveau de la mer, elles représentent 7000 mm au sommet à 2632 mètres d'altitude. L'atténuation rapide des précipitations sur les versants nord-ouest et sud-ouest est liée au changement de l'orientation des versants par rapport aux flux générateurs de précipitations mais sans doute également à l'influence de grandes discontinuités topographiques orientées perpendiculairement aux vents dominants.Reunion Island, located in the south-west Indian Ocean (21° S / 55° E), is composed of 2 volcanoes linked with a hot spot activity: the Piton des Neiges massif (3069 m) in the north-west and the Piton de la Fournaise massif (2632 m) in the south-east (Fig. 1). Climate is intertropical and characterized by two distinct periods: the hot rainfall season, from December to April, and the rather temperate dry season from May to November. South-east to east trade winds prevail during austral winter. However, polar atmospheric disturbances occasionally affect the island. Ascending wet air masses over a rugged topography with high summits and strong slopes release intense orographic rainfalls on the windward eastern slopes of the island. This phenomenon has been reported for other islands where rainfall was shown to be correlated with altitude (Rossignol, 1990; Oki et al., 1991). For trade winds and polar disturbances rainfalls, the maximum zone lies below the thermal inversion layer (Table 1). This layer, present 80% of the time, is found between an altitude of 2000 to 2500 metres in one third of the cases (Fig. 6 and Fig. 7). Cyclonic rainfalls, frequent during the austral summer, increase with altitude. Cyclones generate heavy rains on the eastern slopes of the Piton de la Fournaise massif while they approach the island from the north-east to east sectors (Malick and Mercusot, 1976; Fig. 12).Available isohyet maps (Fig. 3) show a maximum rainfall zone at 860 metres, centred on les Hauts de Sainte-Rose rain gauge, on the north-eastern slope of the Piton de la Fournaise massif. There, the average rainfall is higher than 10 000 mm.year-1, and decreases in the west and south directions: at the same altitude in Baril, on the south side of the massif, annual rainfall totals were estimated between 5000 and 6000 mm (Anonyme, 1975; Bargeas et al., 1984; Robert, 1986).In 1993, twelve tipping bucket rain gauges connected to data loggers or remote transmitters were installed between 220 and 2490 metres to precisely determine isohyets and define the presence of an altitude-dependent maximum rainfall zone (Fig. 2). On the south side of the massif, four rain gauges (altitude 650, 900, 1200 and 1600 m) form the north to south Baril transect along the main slope of the mountain.During the first sampling year at Baril 1600, where water totals are maximum, the annual rainfall was higher than 18 000 millimetres (Table 2; Fig. 4). This constitutes the highest amount ever recorded on Reunion Island. A previous record was set at les Hauts de Sainte-Rose rain gauge (15 381 mm, from 1 August 1979 to 31 July 1980). Return periods over the 27 February 1993 - 26 February 1994 time interval calculated from Météo France reference rain gauge data show that the studied year was not exceptionally wet (Table 3 and Fig. 5). Since 1993, other measurements from the Baril transect, as well as from Bois Blanc and Enclos transects on the east side of the massif, suggest an under-evaluation of isohyet estimations, principally on the east and the south sides of la Fournaise volcano (Fig. 8, Fig. 9 , Fig. 10 and Fig. 11; Table 4).A comparison between the climatologic and topographic environment of Reunion Island and other similar islands such as Hawaii, Maui, Kauai or Oahu in the Hawaiian archipelago (Bean et al., 1994; Ekern et al., 1971; Giambelluca et al., 1986; Giambelluca and Nullet, 1992; Giambelluca and Sanderson, 1993; Juvik and Nullet, 1994; Nullet and Juvik, 1994; Nullet et al., 1995; Peterson, 1972; Schroeder, 1993), Tenerife in the Canary islands (Custodio et al., 1991), Fogo in the Cape Verde islands (Juvik et al., 1995), or Guadeloupe in the Caribbean Sea (Robin and Rossignol, 1988) is presented (Table 5).In Hawaii a distinction has been made between high and very high islands. High islands (such as Oahu, Kauai, east Maui, Guadeloupe or Gran Canaria) have an elevation lower or equal to the thermal inversion layer mean altitude. Water totals are very high and located near the summits. In very high islands (such as Hawaii, west Maui, Tenerife or Fogo), with summits often higher than the thermal inversion layer, rainfalls are more moderate and located lower on windward slopes.La Fournaise volcano experiences these two situations alternately, depending on the presence, altitude and strength of the thermal inversion layer. When it is strong and low, during austral winter, rainfalls are restricted to low elevation zones. The frequent lack of thermal inversion during the hot season, or its high position when present, and the occurrence of tropical depressions cause heavy rains near the summits. These elements and the presence of great embankments and steep slopes, which are increasing relief effects, contribute to give to Reunion Island an original climatic and hydrologic environment. The maximum rainfall zone has a large extension in altitude, along the upper part of steep windward slopes. Recorded water totals and rainfall gradients count among the highest recorded values.Subsequently, atmospheric and topographic elements defining spatial rainfall variations are discussed, and a new isohyet map is proposed (Fig. 13). A maximum rainfall zone over 12 000 mm.year-¹ is defined on the whole eastern side of the massif between 1300 and 1800 metres, and locally up to 2000 metres. Above and below this zone, rainfall decreases to 7000 mm.year-¹ at the summit of the Volcano, and to 4000 mm.year-¹ at sea level respectively. Isohyets are nearly parallel to contour lines on all sides of the Volcano except along the north-western and south-western slopes. On the north-western part of the massif, rainfall decreases to 4000 - 5000 mm.year-¹ since slope exposure to prevailing trade winds diminishes. South-western slopes rapidly become leeward, and rainfall amounts to less than 2000 mm.year-¹

On the robustness of acoustic black hole spectra

We study the robustness of the spectrum emitted by an acoustic black hole by considering series of stationary flows that become either subsonic or supersonic, i.e. when the horizon disappears. We work with the superluminal Bogoliubov dispersion of Bose--Einstein condensates. We find that the spectrum remains remarkably Planckian until the horizon disappears. When the flow is everywhere supersonic, new pair creation channels open. This will be the subject of a forthcoming work.Comment: 4 pages, 2 figure, jpconf.cls; to appear in the proceedings of the Spanish Relativity Meeting ERE201