Stability of equilibrium in a one-dimensional hydrostatic model of the dry atmosphere

Abstract

Experts in the field of mathematical modeling of the climate system have different views about which class of models should be employed to analyze and predict climate for time scales corresponding to climatic processes. In this paper, we investigate the properties of a model constructed using the hydrostatic hypothesis. A one-dimensional (horizontally homogeneous) hydrostatic model of a dry atmosphere is considered. Air is considered an ideal gas. The source of heat is the external short-wave radiation flux entering the upper boundary of the atmosphere. This energy is partly absorbed by the atmospheric layers and the underlying surface, partly returned to space. The atmospheric layers and the underlying surface radiate in the long-wave range. In general, the absorption coefficient and heat capacity are specific for the atmospheric layers and are everywhere positive. In the model, the radiation balance of a segment of the atmospheric column above a unit area of the underlying surface determines the change in the internal energy and the volume occupied by the segment.The pressure value always remains equal to the weight of a part of the atmospheric column above the segment (hydrostatic hypothesis). The underlying surface is always in the state of radiation equilibrium. Under these assumptions: a) there is a single equilibrium vertical temperature distribution in the column and corresponding air pressure and density distributions (they are calculated using the hydrostatic assumption and the equation of a state of the ideal gas); b) the temperature distribution is asymptotically stable, i.e. any other initial distribution of non-negative temperature values tends with time to equilibrium uniformly on the vertical. Thus, one can expect that the numerical analogs of the model considered in this work will also be stable, which is important for the computational implementation of both the one-dimensional model and its three-dimensional version.