2.3 Budyko framework
The Budyko hypothesis assumes that the long-term (annual scale) partitioning of P into ET and runoff can be determined from available water measured as precipitation and available energy measured as potential evapotranspiration (\(E_{P}\)). Based on the Budyko hypothesis, the ratio between actual evapotranspiration (\(E_{A}\)) and P is related to the aridity index (the ratio between EP and P , \(\frac{E_{P}}{P}\)), or the climate dryness index (Budyko, 1974):
\(\frac{\mathbf{E}_{\mathbf{A}}}{\mathbf{P}}\mathbf{=}\left\{\frac{\mathbf{E}_{\mathbf{P}}}{\mathbf{P}}\mathbf{\tanh}\left(\frac{\mathbf{E}_{\mathbf{P}}}{\mathbf{P}}\right)\left[\mathbf{1-cosh}\left(\frac{\mathbf{P}}{\mathbf{E}_{\mathbf{P}}}\right)\mathbf{+sinh}\left(\frac{\mathbf{P}}{\mathbf{E}_{\mathbf{P}}}\right)\right]\right\}^{\mathbf{0.5}}\)(4)