Small time Chung-type LIL for Lévy processes

We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

Processes controlling surface, bottom and lateral melt of Arctic sea ice in a state of the art sea ice model

We present a modelling study of processes controlling the summer melt of the Arctic sea ice cover. We perform a sensitivity study and focus our interest on the thermodynamics at the ice–atmosphere and ice–ocean interfaces. We use the Los Alamos community sea ice model CICE, and additionally implement and test three new parametrization schemes: (i) a prognostic mixed layer; (ii) a three equation boundary condition for the salt and heat flux at the ice–ocean interface; and (iii) a new lateral melt parametrization. Recent additions to the CICE model are also tested, including explicit melt ponds, a form drag parametrization and a halodynamic brine drainage scheme. The various sea ice parametrizations tested in this sensitivity study introduce a wide spread in the simulated sea ice characteristics. For each simulation, the total melt is decomposed into its surface, bottom and lateral melt components to assess the processes driving melt and how this varies regionally and temporally. Because this study quantifies the relative importance of several processes in driving the summer melt of sea ice, this work can serve as a guide for future research priorities.