The Existence and Uniqueness of Weak Solutions for Precipitation Fronts: a Novel Hyperbolic Free Boundary Problem in Several Space Variables

The determination of the large scale boundaries between moist and dry regions is an important problem in contemporary meteorology. These phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families (drying, slow moistening, and fast moistening) of precipitation fronts. The last two wave types violate Lax’s shock inequalities yet are robustly realized. This formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit. A new L-contraction estimate is also established at positive relaxation values. 0. Introduction The goal here is to prove the existence and uniqueness of weak solutions to a novel hyperbolic free boundary value problem in several space variables that has emerged recently in the analysis of precipitation fronts in the large scale tropical atmosphere ([FMP], [SM], [PFM], [KM1], [KM2]). Precipitation fronts are the boundaries between the zones of extremely moist air (with constant precipitation) such as over the Indonesian marine continent, the Indian ocean, and Western Pacific, and the zones of extremely dry air in the tropics and subtropics that occur over areas such as the Galapagos islands at the equator or the Arabian peninsula in the subtropics. An important practical question in contemporary meteorology for long range weather prediction and climate change projections is what determines the boundaries of the precipitating fronts as well as their evolution in time. Such assessments are performed, for example, by the Intergovermental Panel for Climate Change (IPCC) by running extremely complex general computer models called GCM’s. An important practical issue with the GCM’s is how they treat moisture and what type of moisture waves do they (1) Department of Mathematics and Climate Atmosphere Ocean Science, Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA email: [email protected]. (2) Partially supported by the National Science Foundation and the Office of Naval Research. (3) Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email: [email protected]. (4) Partially supported by the National Science Foundation.