Abstract
Abstract. This paper studies a 3-D state space representation of Budyko's framework designed to capture the mutual interdependence among long-term mean actual evapotranspiration (E), potential evapotranspiration (Ep) and precipitation (P). For this purpose we use three dimensionless and dependent quantities: Ψ = E ⁄ P, Φ = Ep ⁄ P and Ω = E ⁄ Ep. This 3-D space and its 2-D projections provide an interesting setting to test the physical soundness of Budyko's hypothesis. We demonstrate analytically that Budyko-type equations are unable to capture the physical limit of the relation between Ω and Φ in humid environments, owing to the unfeasibility of Ep ⁄ P = 0 when E ⁄ Ep → 1. Using data from 146 sub-catchments in the Amazon River basin we overcome this inconsistency by proposing a physically consistent power law: Ψ = kΦe, with k = 0.66, and e = 0.83 (R2 = 0.93). This power law is compared with two other Budyko-type equations. Taking into account the goodness of fits and the ability to comply with the physical limits of the 3-D space, our results show that the power law is better suited to model the coupled water and energy balances within the Amazon River basin. Moreover, k is found to be related to the partitioning of energy via evapotranspiration in terms of Ω. This suggests that our power law implicitly incorporates the complementary relationship of evapotranspiration into the Budyko curve, which is a consequence of the dependent nature of the studied variables within our 3-D space. This scaling approach is also consistent with the asymmetrical nature of the complementary relationship of evapotranspiration. Looking for a physical explanation for the parameters k and e, the inter-annual variability of individual catchments is studied. Evidence of space–time symmetry in Amazonia emerges, since both between-catchment and between-year variability follow the same Budyko curves. Finally, signs of co-evolution of catchments are explored by linking spatial patterns of the power law parameters with fundamental characteristics of the Amazon River basin. In general, k and e are found to be related to vegetation, topography and water in soils.
Highlights
The pioneering work of Budyko (1974) introduced a theoretical framework to link the long-term average water and energy balances in a river basin considering the dominant controls on actual evapotranspiration (E), assuming that the water balance is mostly governed by water availability and energy availability
We have introduced a physically consistent scaling approach towards a 3-D generalization of the Budyko framework in humid environments, which opens up a new research avenue to understand the coupling between the long-term mean annual water and energy balances catchments, and the hydrological effects brought about by climate change
Results for the FAO agro-climatic stations and catchments in the US, China and the Amazon indicate that the well-known Budyko function that relates vs. corresponds to a particular bi-dimensional cross section of a broader coupling existing between, and and in turn of the mutual interdependence between precipitation (P ), potential evapotranspiration (Ep) and actual evapotranspiration (E)
Summary
The pioneering work of Budyko (1974) introduced a theoretical framework to link the long-term average water and energy balances in a river basin considering the dominant controls on actual evapotranspiration (E), assuming that the water balance is mostly governed by water availability (precipitation, P ) and energy availability (represented for convenience by potential evapotranspiration, Ep). According to Budyko, mean annual E approaches mean annual P as the climate becomes drier, provided that water storage change in the catchment is negligible. Such water and energy coupling is represented in a bi-dimensional space relating two nondimensional variables, the evapotranspiration ratio (E/P ), Published by Copernicus Publications on behalf of the European Geosciences Union. Carmona et al.: Scaling approach to Budyko’s framework and the aridity index (Ep/P ), such that
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