Scaling Limits for Internal Aggregation Models with Multiple Sources

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added "least action principle" (Lemma 3.2); small corrections in section 4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version); expanded section 6.

Formation of Heavy Precipitations in Southwestern Part of Iran and its Association with the Madden-Julian Oscillation

The aim of this study is to assess some synoptic characteristics of heavy precipitations in southwestern parts of Iran and evaluate the relationship between them with the Madden-Julian Oscillation (MJO). Research is conducted with regard to distribution of precipitation per month and identifying their steam sources. Daily records of the November-April precipitation data in Abadan, Ahwaz, Bandar-Abbas, Bushehr, Shahr-e-kord and Shiraz stations for the 1975- 2011 period are collected as well as same panel data for Yasuj station from 1990 to 2011. Rainfall data are sorted in descending order and precipitation values that were fallen within the 5% and 10% of highest records are categorized as the heavy precipitation. The most frequent precipitations occurred in January, February and December. The most frequent heavy precipitations in Ahwaz, Bandar-Abbas, Bushehr, Shahr-e-kord and Shiraz stations occurred in phase 8, while in Abadan station occurred in phases 7 and 8. Apparently, due to the short duration precipitations data at Yasuj station, the most frequent heavy precipitation observed in phase 2.Synoptic maps show that harmonized with eastward movement of convective precipitation in Indian or pacific oceans.Heavy precipitation forms in the west region of Iran and moves toward southwest and south Central of Iran and then appears to Afghanistan.Formation of a cyclonic circulation that encompasses the Mediterranean Sea, Red Sea and Persian Gulf plays an important role for moisture supplement of these storm activities. The synoptic maps have indicated that main sources of these heavy rainfalls are moisture produced at the Arab sea and western parts of the Indian Ocean