数学大讲堂

Small is Beautiful: Understanding Complex Meteorological Phenomena in Tropics with Simple Mathematical Models

Abstract: Nonlinear partial differential equations describing dynamics of the atmosphere, the so-called primitive equations, are complex and contain a plethora of sophisticated parameterizations of various physical processes, the most important of them related to phase transitions of water. In spite of undeniable progress in numerical modeling of weather and climate with the help of the primitive equations, predictions of climate models strongly diverge in what concerns the evolution of the water vapor, and the accurate forecasts of precipitations are still problematic in many cases. Inversely, the influence of the processes including moisture upon dynamics, primarily the moist convection which is of special importance in tropics, is not sufficiently understood.

I will show how to construct a simple conceptual model of the moist atmosphere, the moist-convective shallow water model, by applying the procedure of vertical averaging to the primitive equations and using the basic conservation laws to include thermodynamics of phase transitions. The model, in its simplest one-layer version, is quasi-linear hyperbolic, and allows for application of the state-of-the-art well-balanced finite-volume numerical methods. In spite of its simplicity, the model captures the essential features of large-scale dynamics of the moist atmosphere, while requiring incomparably less computational resources than “big” weather and climate models. An important advantage of the model is that it organically incorporates an arbitrary topography. I will illustrate the capabilities of the model on the examples of evolution of tropical cyclones, interactions of equatorial waves with the Maritime Continent and Indo-Pacific oceanic warmpool, and the so-called Madden-Julian events, the organized periodically arising patterns of enhanced moist convection which are slowly moving eastwards in Indo-Pacific region strongly influencing its weather. I will also discuss numerical challenges in the context of natural generalizations of the model.