Explicit analytical inversion of the parametric Budyko equations

Authors: Reaver, Nathan; KAPLAN, DAVID - University Of Florida; KLAMMLER, HARALD - Federal University Of Bahia (UFBA); JAWITZ, JAMES - University Of Florida

Submitted to: SSRN:First look
Publication Type: Pre-print Publication
Publication Acceptance Date: 9/7/2024
Publication Date: 9/7/2024
Citation: Reaver, N.G., Kaplan, D.A., Klammler, H., Jawitz, J.W. 2024. Explicit analytical inversion of the parametric Budyko equations. SSRN:First look. https://dx.doi.org/10.2139/ssrn.4949309.
DOI: https://doi.org/10.2139/ssrn.4949309

Interpretive Summary: The hypothesis named for the Soviet climate scientist Mikhail Budyko posits that long-term average climate is the main factor in determining a region’s water availability. Observed regional water resource availability across a wide range of climates from wet to dry generally follows a distinctive pattern that was described by Budyko and is referred to as the Budyko curve. With the Budyko curve, one can make rough predictions of available water resources from a region’s long-term average climate. Attempts to improve the predictive ability of the Budyko framework introduced a region-specific parameter to represent the impacts of biologic and physical characteristics (other than average climate) on water resource availability. This parametric Budyko framework suggests that a region’s water resource availability should follow a very specific pattern (determined by the region-specific parameter) as its climate gets wetter or drier. However, these patterns have not been observed in nature, meaning the parametric Budyko framework lacks predictive power. In this study, we show mathematically that the region-specific parameter can change due to changes in average climate alone, meaning that it cannot only represent regional biological and physical characteristics. Additionally, we use our results to develop a new expression for an important constant in mathematics.

Technical Abstract: A distinctive global clustering pattern emerges (i.e., the “Budyko curve”) when the evaporative behavior of multiple catchments, described by the evaporative index (E/P), is plotted against their average climate, described by the aridity index (E0/P). The Budyko framework describes the central tendency of this emergent pattern with semi-empirical, non-parametric relationships, enabling the probabilistic estimation of E/P from E0/P. Further extensions to this framework with parametric Budyko equations have deterministically extrapolated the global emergent curve behavior to individual catchments (i.e., “the catchment trajectory conjecture”) by introducing a catchment-specific parameter (n or w), intended to describe differences in climate and landscape features not represented by E0/P. However, the catchment trajectory conjecture is not empirically supported and therefore the inclusion of n or w renders the parametric Budyko equations underdetermined. Consequently, values of n and w must always be obtained from known values of E/P and E0/P, precluding the use of the parametric Budyko equations in predicting E/P. As such, n and w do not necessarily represent catchment-specific characteristics, but rather can be interpreted as alternative parameterizations of Budyko space, with coordinates (E0/P, n) or (E0/P, w) instead of (E0/P, E/P). We illustrate this by analytically inverting both forms of the parametric Budyko equations, producing explicit expressions for n and w only in terms of E/P and E0/P. Subsequently, we use these expressions to show that n or w can vary due to changes in E0/P alone. Additionally, we use the expression for w to develop a novel infinite family of series representations for Euler’s number.