A thermodynamic interpretation of Budyko and L'vovich formulations of annual water balance: Proportionality Hypothesis and maximum entropy production

The paper forms part of the search for a thermodynamic explanation for the empirical Budyko Curve, addressing a long-standing research question in hydrology. Here this issue is pursued by invoking the Proportionality Hypothesis underpinning the Soil Conservation Service (SCS) curve number method widely used for estimating direct runoff at the event scale. In this case, the Proportionality Hypothesis posits that the ratio of continuing abstraction to its potential value is equal to the ratio of direct runoff to its potential value. Recently, the validity of the Proportionality Hypothesis has been extended to the partitioning of precipitation into runoff and evaporation at the annual time scale as well. In this case, the Proportionality Hypothesis dictates that the ratio of continuing evaporation to its potential value is equal to the ratio of runoff to its potential value. The Budyko Curve could then be seen as the straightforward outcome of the application of the Proportionality Hypothesis to estimate mean annual water balance. In this paper, we go further and demonstrate that the Proportionality Hypothesis itself can be seen as a result of the application of the thermodynamic principle of Maximum Entropy Production (MEP). In this way, we demonstrate a possible thermodynamic basis for the Proportionality Hypothesis, and consequently for the Budyko Curve. As a further extension, the L'vovich formulation for the two-stage partitioning of annual precipitation is also demonstrated to be a result of MEP: one for the competition between soil wetting and fast flow during the first stage; another for the competition between evaporation and base flow during the second stage.