Entropy Production In Thermodynamic Climate Models

We present a non-equilibrium theory in a system with heat and radiative fluxes. The obtained expression for the entropy production is applied to a simple one-dimensional climate model based on the first law of thermodynamics. In the model, the dissipative fluxes are assumed to be independent variables, following the criteria of the Extended Irreversible Thermodynamics (BIT) that enlarges, in reference to the classical expression, the applicability of a macroscopic thermodynamic theory for systems far from equilibrium. We analyze the second differential of the classical and the generalized entropy as a criteria of stability of the steady states. Finally, the extreme state is obtained using variational techniques and observing that the system is close to the maximum dissipation rate. Introduction In the late 60's and in the 70's the development of one-dimensional climate models based on the first law of thermodynamics had its apogee. The importance of these simple models was centered in the possibility to understand the weight of the parameters that intervene in the climate system [1-5]. These previous works led scientists to create more sophisticated climate models with two and three dimensional variables [6-10] until arriving to the present General Circulation Models (GCM), that suffer of a high level of complexity with the intention to simulate all the physical processes present in the atmosphere. Thus, different types of GCMs have been used to diagnose the laws that govern the interactions between the elements of the climate system and to predict possible climate evolution in different possible scenarios [11-12]. However, due to their complexity these models do not permit a direct application of new concepts in the study of climate. On the other hand, the one-dimensional climate models do not give quantitative solution for the climate but are very useful tools for introducing new parameters and for observing how the new terms affect and modify the state of the system. Following this way, some authors have recently studied the role of entropy in climate. Li and Chylek [13] have used a one-dimensional latitudinaldependent climate model developed by North [3] for obtaining the expression of the J. Non-Equilib. Thermodyn. · 1998 · Vol. 23 · No. 1 © Copyright 1998 Walter de Gruyter · Berlin · New York Brought to you by | provisional account Unauthenticated | 84.88.138.106 Download Date | 10/21/13 1:51 PM Entropy production in thermodynamic climate models 63 rate of entropy production. Furthermore, Li et al. [14] consider a one-dimensional radiative-convective model to observe the vertical distribution of the rate of entropy production. Looking for the possibility of an extremal of the dissipation in the clipiate system, formerly claimed by Paltridge [15-18], Stephens and O'Brien [19] have calculated the contribution of the radiative flux of entropy using satellite data and comparing them with analytical results. Later, O'Brien and Stephens [20] use a Paltridge box model pointing out that it follows the principle of maximum dissipation proposed by Ziegler [21]. Paltridge developed a box-model assuming that the climate system is close to the maximum dissipation rate. With this constraint, the Paltridge model gives values of temperature and cloud cover extraordinarily close to the real ones as has been shown by Grassl [22], introducing the ice-albedo feedback. Therefore, our purpose is to formulate a non-equilibrium climate model in order to obtain a general expression of the rate of entropy production outside equilibrium. We will follow Nicolis and Nicolis [23] to explore wether or not the climate system is governed by an extremal principle related with the entropy production. Moreover, with the aim of doing a rigorous deduction according to the most recent macroscopic theories, we will assume the dissipative fluxes of the system to be independent variables. Thus, a macroscopic expression will be obtained in the framework of the Extended Irreversible Thermodynamics (EIT). This theory has been successfully used in a wide range of macroscopic systems but it has never been used in climatic systems. In Section 2, a general expression of the rate of the generalized entropy production outside equilibrium is obtained in a system with heat and radiative fluxes. In Section 3, a one-dimensional climate model is developed following the one proposed by North [3-4] with the intention of evaluating a qualitative behaviour of the rate of entropy production. In the horizontal diffusive model used the horizontal heat flux is approximated using a Fourier's law type equation. As this model is vertically averaged, it is not necessary to know the radiation flux, only its divergence. We will introduce, then, an expression for the radiative flux in order to apply the development proposed by EIT. When its divergence is averaged vertically the parameterization of North's model is obtained. Then we will be able to define a generalized entropy with an expected larger range of validity than the expressions used before. Both, the rate of the classical and extended entropy production, are compared in Section 4 where we also study their time evolution for different cases. They present different behaviours when different initial conditions are chosen. Thus, at the steady state a maximum or a minimum in the rate of entropy production is obtained in function of the values assumed for the initial conditions. Even, the attainment of this extremal at the steady state in reference to the time evolution is not always fulfilled. In Section 5, we discuss the possibility of using the second differential of the generalized entropy as a stability criteria for the climate system. The classical expression has been largely used in thermodynamic fluid systems for obtaining the stability at stationary conditions. This hypothesis applied to the climate has been initially proposed by Nicolis and Nicolis [23]. The second differential of the generalized entropy has been studied by Jou et al [24] where the convexity requirement implies a maximum permissible value for the dissipative fluxes, directly related with the range of applicability of the EIT theory. Here, the classical and the generalized expression of the second J. Non-Equilib. Thermodyn. -1998 · Vol. 23 · No. 1 Brought to you by | provisional account Unauthenticated | 84.88.138.106 Download Date | 10/21/13 1:51 PM 64 T. Pujol, J.E. Llebot differential of the entropy are studied and their obviously 4iff behaviour is emphasized. On the other hand, a maximum entropy production at the steady state can be obtained which is a minimum in reference to the time evolution. It is due to the existence of an extremal entropy production for fixed conditions, which value has been calculated at the steady state using variational techniques (Sect. 6). We have analyzed three expressions related to the rate of entropy production: the global, the thermal and the difference between the radiative flux and the radiative energy contribution of entropy. Furthermore, the time evolution of the rate of entropy production is shown in comparison with the extremal value obtained using the variational principle. This calculation is only acceptable for a time evolution close to the steady state presuming that the extremal conditions are also applicable not very far from the equilibrium. However, a variational principle outside the steady state is applied with some additional restrictions, obtaining a good approach of the extremal values to the observed ones along the time evolution. In Section 7, finally, we conclude the work with the discussion of the results obtained. 2. Generalized entropy production In the last years the common acceptance of studying the climate through complex 2-dimensional and 3-dimensional models, to take into account the great quantity of physical phenomena present in the atmospheric system, has not permitted to introduce new concepts. There have been some attempts to investigate the role of the rate of entropy production in the climate system using simple 1-dimensional models [13-14, 23] or using it as a fundamental part for the calculation of the variables in a box model [15-20,22]. In a closed thermodynamic system the study of the behaviour of the rate of entropy production has been extensively used [25]. However, in an open system with a radiation field (e.g., the climate) it is still unclear how the entropy production will evolve and wether it is of importance for the states reached by the system. Thus, it is not clear wether or not the system follows an extreme principle at the steady state [23, 26, 27]; also it is unclear wether the rate of entropy production can be expressed'as a bilinear form, though there have been attempts to do so [28, 29]. Even in the treatment of the entropy on the climate, some authors have considered only the thermal part [23] pointing out that the radiative contribution has no relevance on the dynamics of the system and, in consequence, on the state that it is led to [20]. The assumption of a thermodynamic 1-dimensional model permits a great technical simplification, with only three thermodynamics variables: the temperature, the heat and the radiative fluxes. However, there have been attempts to find an extremal behaviour of the rate of entropy production in simple dynamic models [30, 31] with the intention to obtain a principle equivalent to the principle of minimum entropy production proposed by Prigogine [32], not necessarily restricted to closed systems [25]. Nevertheless, the existence of the radiation field in the climate implies a great difficulty and raises doubts in the success of this task. Here, we will obtain a general expression of entropy production for non-equilibrium situations, assuming an open system with two perpendicular fluxes, the heat flux and J. Non-Equilib. Thermodyn. · 1998 · Vol. 23 · No. 1 Brought to you by | provisional account Unauthenticated | 84.88.138.106 Download Date | 10/21/13 1:51 PM Entropy production in thcrmodynamic climate models 65 the radiative flux. When the system is outside the steady state, the classical phenomenological laws are not satisfied and we consider the fluxes as independent variables. The thermodynamic theory that assumes the introduction of the fluxes as independent variables has been called Extended Irreversible Thermodynamics (BIT) because, in principle, it is an extension of the classical non-equilibrium thermodynamics based on the local equilibrium hypothesis. HIT mainly deals with a generalized entropy, obtained from a generalized Gibbs equation. This generalized thermodynamics has enlarged the applicability of a macroscopic theory in the study of multiple physical systems (e.g., fluids, rheology, cosmology,...) and has obtained significant agreements with mesoscopic theories (kinetic theory, statistical mechanics and information theory) [24]. Now, we try to use it in a climatic system. We assume the generalized expression of the rate of entropy production to be not only a function on the extensive variables as the classical expression but also of the dissipative fluxes. Then, s* = s* + s* where the subscripts r and m indicate the radiation and the matter part respectively and * denotes the generalized entropy. With the assumption that 5* = s*(u, Q, R), the generalized Gibbs equation has the form