A time-dependent climatic feedback system involving sea-ice extent, ocean temperature, and CO2

Abstract

An idealized, time and latitude dependent, nonlinear, basically thermodynamic model is developed for an equinoctial, all-ocean planet to illustrate the possible roles of a number of positive and negative feedbacks that are present in the earth's climatic system. The mean atmospheric properties (e.g., temperature and cloud cover) and the sea-surface temperature are treated as fast response parts of the total system that are coherent with the sea ice extent measured by the sine of the ice edge latitude, ?. The mean depth- and latitude-average ocean temperature, ?, is the other prognostic variable that can vary along with ? over long time periods. The model is based on a set of parameterizations for all the modes of heat transfer across the top of the sea water, some of which represent considerable improvements over those incorporated in previous statistical dynamical climate models. In particular, the sensible heat flux now includes the “rectifier” effects associated with synoptic air mass exchanges that arise as a consequence of the baroclinicity of the atmosphere. This introduces a strong positive feedback leading to instability of the system, that can be viewed as the physical manifestation of the more familiar “ice-albedo” feedback. Another positive feedback included is due to longwave emissivity changes associated with CO₂ changes that, in turn, are postulated to arise in response to the variations of mean ocean temperature ?. Among the negative feedbacks is one due to the insulating effect of sea ice on ? (which, in turn, influences the ice edge extent through an upward subsurface heat flux at the ice edge), and another due to the changes of the amount and path length of solar radiation at the ice edge. The model is expressed mathematically as a coupled autonomous polynomial system of 6th degree governing the two state variables ? and ?. Analyses are made of the equilibria, sensitivity of the equilibria to changes in all the parameters, linear stability and structural stability of the equilibrium paths for changes in selected parameters of special interest (e.g., the solar constant and CO₂ level) constituting climatic prediction of the “second kind”. Finally, some phase plane portraits for specified values of parameters are presented showing the evolution of the system from arbitrary initial conditions (prediction of the “first kind”) and illustrating the time dependent properties of the system. For one reasonable set of parameters, damped oscillations of a period on the order of a thousand years are obtained, while for other possible parameter values unstable behavior is manifest. Because of the instabilities for large departures from equilibrium, this system cannot exhibit stable limit cycles (i.e., auto-oscillatory behavior), but realistic modifications are suggested that offer the possibilities of such behavior. DOI: 10.1111/j.2153-3490.1980.tb00938.x