## 1. Introduction

[2] The terrestrial water balance links the partitioning of precipitation (*P*) into evapotranspiration (*E*) and runoff (*R*) with the surface energy balance, and thereby plays a critical role for the climate system over land. It can be expressed as:

where the term *dW*_{s}/*dt* represents the change in soil water storage. At the small scale of a catchment, equation (1) depends on a large number of details, such as soil properties, surface slope and orientation, and vegetation cover. This would seem to imply that a vast amount of small-scale information is required to predict the water balance at larger scales. Optimality principles provide a potential means to simplify the task of scaling up hydrologic fluxes with less detailed information.

[3] *Budyko's* [1974] analysis of the relationship between rainfall and runoff provides a first justification for such principles. On climatic time scales, ≈ 0, thus reducing equation (1) to *P* ≈ *E* + *R*. When *E*/*P* is plotted against the radiative index of dryness *R*_{net}/(*LP*) (net radiation *R*_{net} divided by the energy required to evaporate all precipitation, *LP*, with *L* being the latent heat of vaporization), observations are close to the maximum possible value of *E*/*P*. This upper limit is set by the amount of energy available for evaporation in humid regions and water availability (i.e., precipitation) in arid regions.

[4] Another justification is found in thermodynamics and its foundation in statistical physics. Statistical physics deals with the scaling of microscopic properties to the macroscopic scale and is able to derive simple macroscopic relationships. The common example is the scaling of the nearly chaotic motion of individual molecules to the macroscopic properties of the ideal gas. While the microscopic scale is characterized by the position and velocity of each individual molecule, the macroscopic properties of the gas are described by properties such as temperature, pressure, density. The scaling is achieved by the assumption that the ideal gas is found in the most probable state (i.e., a state of maximum entropy), that is, in the state that represents the vast majority of microscopic states. Statistical mechanics has been extremely successful in deriving relationships such as the ideal gas law and the Boltzmann distribution simply from the assumption of maximum entropy and the constraints of the energy and mass balance.

[5] Thermodynamic arguments have been applied in the past to describe hydrologic fluxes [e.g., *Leopold and Langbein*, 1962; *Rodriguez-Iturbe et al.*, 1992; *Rinaldo et al.*, 1996; *Rodriguez-Iturbe and Rinaldo*, 2001]. *Leopold and Langbein* [1962] compared the flow of water along topographic gradients to temperature conversions of a heat engine, where the heat flow is directed towards lower temperatures. They compared this to the drainage of land where gradients in topography drive the water flux towards lower elevations. However, they used thermodynamics mainly as an analogy.

[6] To apply non-equilibrium thermodynamics to hydrology directly, two difficulties need to be addressed: First, catchment is an open thermodynamic system. It exchanges energy and mass of different entropies with its surroundings, and these exchanges need to be formulated in terms of their chemical potentials [*Kondepudi and Prigogine*, 1998]. Secondly, it is a system far from thermodynamic equilibrium (TE). The corresponding principle to the maximum entropy assumption in equilibrium thermodynamics is still an open issue. The principle of Maximum Entropy Production (MEP) has been proposed to address this deficiency [*Ozawa et al.*, 2003; *Kleidon and Lorenz*, 2005; *Martyushev and Seleznev*, 2006]. Recent theoretical work by *Dewar* [2003, 2005a, 2005b] has attempted to prove the generality of MEP using a similar approach to the one used to derive equilibrium thermodynamics. The MEP principle holds the promise to provide a very general and fundamental understanding of optimality and macroscopic behavior in a wide range of complex systems far from thermodynamic equilibrium and has recently gained interest [*Whitfield*, 2005].

[7] In this review we first provide an overview of non-equilibrium thermodynamics and how it relates to the hydrologic cycle within the Earth system and the water balance on land. The MEP principle is briefly described and poleward heat transport is used to illustrate its application. A simple model is set up to demonstrate possible applications of MEP to land surface hydrology. This example is used to discuss why, how, and at what spatial and temporal scales the MEP principle should apply to land surface hydrology. We close with a discussion, relating potential applications of MEP to previously suggested optimality approaches, potential limitations and implications.