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 heat flux is introduced 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 introduction 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 implies stable states for an ice-free earth. When the ice-albedo feedback is included, a discontinuous distribution of stable states is found for high response times. Following the thermodynamic analysis of the model, the maximum rates of entropy production at the steady state are obtained. A latitudinally isothermal earth produces the extremum in global entropy production. The material contribution to entropy production (by which we mean the production of entropy by material transport of heat) is a maximum when the latitudinal distribution of temperatures becomes less homogeneous than present values.