Matching the Budyko functions with the complementary evaporation relationship: consequences for the drying power of the air and the Priestley–Taylor coefficient

Abstract. The Budyko functions B₁(Φ_p) are dimensionless relationships relating the ratio E / P (actual evaporation over precipitation) to the aridity index Φ_p = E_p / P (potential evaporation over precipitation). They are valid at catchment scale with E_p generally defined by Penman's equation. The complementary evaporation (CE) relationship stipulates that a decreasing actual evaporation enhances potential evaporation through the drying power of the air which becomes higher. The Turc–Mezentsev function with its shape parameter λ, chosen as example among various Budyko functions, is matched with the CE relationship, implemented through a generalised form of the advection–aridity model. First, we show that there is a functional dependence between the Budyko curve and the drying power of the air. Then, we examine the case where potential evaporation is calculated by means of a Priestley–Taylor type equation (E₀) with a varying coefficient α₀. Matching the CE relationship with the Budyko function leads to a new transcendental form of the Budyko function B₁′(Φ₀) linking E / P to Φ₀ = E₀ / P. For the two functions B₁(Φ_p) and B₁′(Φ₀) to be equivalent, the Priestley–Taylor coefficient α₀ should have a specified value as a function of the Turc–Mezentsev shape parameter and the aridity index. This functional relationship is specified and analysed.