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This paper studies a 3-D state space representation of Budyko's frameworkdesigned to capture the mutual interdependence among long-term mean actualevapotranspiration (E), potential evapotranspiration (Ep) andprecipitation (P). For this purpose we use three dimensionless anddependent quantities: Ψ  =  E ⁄ P, Φ  =  Ep ⁄ P andΩ  =  E ⁄ Ep. This 3-D space and its 2-D projectionsprovide an interesting setting to test the physical soundness of Budyko'shypothesis. We demonstrate analytically that Budyko-type equations are unableto capture the physical limit of the relation between Ω and Φ inhumid environments, owing to the unfeasibility of Ep ⁄ P  =  0when E ⁄ Ep  →  1. Using data from 146 sub-catchmentsin the Amazon River basin we overcome this inconsistency by proposing aphysically consistent power law: Ψ  =  kΦe, withk  =  0.66, and e  =  0.83 (R2  =  0.93). This power law iscompared with two other Budyko-type equations. Taking into account thegoodness of fits and the ability to comply with the physical limits of the3-D space, our results show that the power law is better suited to model thecoupled water and energy balances within the Amazon River basin. Moreover,k is found to be related to the partitioning of energy viaevapotranspiration in terms of Ω. This suggests that our power lawimplicitly incorporates the complementary relationship of evapotranspirationinto the Budyko curve, which is a consequence of the dependent nature of thestudied variables within our 3-D space. This scaling approach is alsoconsistent with the asymmetrical nature of the complementary relationship ofevapotranspiration. Looking for a physical explanation for the parameters kand e, the inter-annual variability of individual catchments is studied.Evidence of space–time symmetry in Amazonia emerges, since bothbetween-catchment and between-year variability follow the same Budyko curves.Finally, signs of co-evolution of catchments are explored by linking spatialpatterns of the power law parameters with fundamental characteristics of theAmazon River basin. In general, k and e are found to be related tovegetation, topography and water in soils

hydrology and earth system sciences

A scaling approach to Budyko's framework and the complementary relationship of evapotranspiration in humid environments: case study of the Amazon River basin