## 1. Introduction

The Earth receives a certain amount of energy from the Sun, in the form of visible light, which it has to radiate back to space, in the form of infrared light, to maintain a steady state. Most Earth System processes, including weather and climate, can be regarded as little more than steps in this process of energy conversion from one form to the other, going through various other forms of energy (potential energy, kinetic energy, heat...). Although in each of these steps the overall quantity of energy has to be conserved, its quality can vary (Peixoto *et al.*, 1991; Peixoto and Oort, 1992). The field of physics that deals with such energy conversions in different forms and the corresponding evolution of its quality is called thermodynamics, and the quality of energy is measured by a state function called *entropy*. The constraint of conservation of the exchanged quantity of energy is well-known and commonly used in climate sciences. Less attention is paid to the additional information one can gain by monitoring creation and exchanges of entropy, either as a diagnostic tool or as a prediction principle relying on an extremum property. This information can be discarded with little consequences for some processes for which physical laws naturally allow for a macroscopic description (and for the subsequent computation of the exchanged energy due to the process), but in some other cases it might bring in a fundamental constraint that should not be ignored.

A typical example is atmospheric heat transport: considering the observational fact that the net top-of-atmosphere radiative budget is positive in the Tropics and negative near the Poles, one has to invoke a poleward atmospheric transport to ensure global energy balance in the steady state (see e.g. Trenberth and Solomon (1994) for a quantitative discussion based on observations and Lucarini and Ragone (2010) for a modelling counterpart). An important difficulty comes from the turbulent nature of the laws of atmospheric motion and the lack of a satisfactory theory of turbulence. We are thus forced to integrate numerically the equations of motion at a very high computational cost. On the other hand, the laws of radiation allow for a macroscopic description more easily. To cope with this difficulty, the options up to now have been either to deal with the overdetailed (from the minimalist standpoint we are considering) microscopic equations of motion of the atmosphere, or to use empirical parameterizations.

An alternative was suggested by Paltridge (1975, 1978) (see O'Brien and Stephens (1995) for a nice reformulation). He was looking for a large-scale description of heat transport and suggested a variational principle stating that atmospheric heat transport adjusts so as to maximize the production of entropy. Several climate studies based on similar models have been published since this seminal work: Gerard *et al.* (1990), Grassl (1981), Lorenz *et al.* (2001), Wyant *et al.* (1988) for zonally-averaged climate, Ozawa and Ohmura (1997), Pujol and Fort (2002) for vertical convection. The Maximum Entropy Production Principle can be interpreted as a maximum energy transport efficiency requirement. The general idea was first expressed by Lorenz (1960). See for instance Lin (1982), Lucarini (2009), Lucarini *et al.* (2010), Peixoto and Oort (1992), Vos and Wel (1993) and references therein for discussions in terms of available potential energy and Carnot engines, and Pauluis and Held (2002b, 2002a) for the competing effect of water vapour removal by the atmosphere.

The purpose of this paper is not to discuss the justification of the Maximum Entropy Production Principle. For a rather comprehensive review of different approaches and applications, the reader is refered to Martyushev and Seleznev (2006) and Kleidon and Lorenz (2005). In this paper, we present a new formulation of Paltridge's model: we have developed a different approach for the treatment of radiation, which we believe to be more rigorous and physically sound. In particular we provide a derivation of the analytical expression of the radiative coefficients which are introduced as *ad hoc* in Paltridge's model. We also avoid the questionable maximum convection hypothesis, critical in Paltridge's model (O'Brien and Stephens, 1995; Pujol and Llebot, 2000), by applying the MEP principle on the vertical dimension as well, and discard Paltridge's cloud cover variable and oceanic transport for the sake of cogency. Formulated this way, the model is as free of tunable coefficient, empirical parameterization and spurious assumption as possible, therefore constituting a clean basis to evaluate the validity of the MEP hypothesis as applied to climate. The only information needed *a priori* to compute the coefficients are the surface albedo, the vertically-integrated water vapour and ozone density and the *CO*_{2} concentration. In particular, we are now able to run a sensitivity experiment with respect to the surface albedo parameter. Here, we investigate the effect of the surface albedo distribution corresponding to the presence of large ice-sheets in the Northern Hemisphere during the Last Glacial Maximum. The results for pre-industrial and LGM climates are compared with standard simulations with the IPSL_CM4 atmosphere-ocean general circulation model using similar boundary conditions.