Second differential of the entropy as a criterion for the stability in low-dimensional climate models

The second differential of the entropy is used for analysing the stability of a thermodynamic climatic model. A delay time for the heat flux is introd uced whereby it becomes an independent variable. Two different expressions for the second differential of the entropy are used: one follows classical irreversible thermodynamics theory; the second is related to the introducti on of response time and is due to the extended irreversible thermodynamics theory. The second differential of the classical entropy leads to unstable solutions for high values of delay times. The extended expression always im plies stable states for an ice-free earth. When the ice-albedo feedback is included, a discontinuous distribution of stable states is found for high r esponse times. Following the thermodynamic analysis of the model, the maxim um rates of entropy production at the steady state are obtained. A latitudi nally isothermal earth produces the extremum in global entropy production. The material contribution to entropy production (by which we mean the produ ction of entropy by material transport of heat) is a maximum when the latit udinal distribution of temperatures becomes less homogeneous than present v alues.