Civil, Structural & Geotechnical Engineering
Turc (1954) & Budyko Curve Watershed Water-Balance Formula: Annual AET & Runoff Yield
Partition long-term annual catchment precipitation P into Actual Evapotranspiration (AET) and Streamflow Runoff Q = P - AET using Mikhail I. Budyko’s (1974) Aridity Curve (P, PET) and Lucien Turc’s (1954) Temperature-Precipitation Law.
Enforce simultaneous energy-limit (AET <= PET) and water-limit (AET <= P) catchment bounds!
At long-term mean annual catchment scales (where soil-water storage change Delta_S ~= 0), Actual Evapotranspiration AET cannot exceed available Precipitation P in arid deserts (water limit: AET -> P as Aridity Index phi = PET / P -> infinity) and cannot exceed Potential Evapotranspiration PET in wet rainforests (energy limit: AET -> PET as phi = PET / P -> 0)!
Taking the geometric mean of Schreiber’s (1904, 1 - EXP(-phi)) and Ol’dekop’s (1911, phi * TANH(1 / phi)) asymptotic curves, Soviet climatologist Mikhail I. Budyko (1974, Climate and Life) formulated the celebrated universal Budyko Curve (expressing hyperbolic tangent identically via exponentials (1 - EXP(-2 / phi)) / (1 + EXP(-2 / phi)) so every spreadsheet engine evaluates it natively): AET_Budyko = P * SQRT(phi * ((1 - EXP(-2 / phi)) / (1 + EXP(-2 / phi))) * (1 - EXP(-phi)))! When radiation PET is unavailable, Lucien Turc’s (1954) empirical watershed formula estimates L_T = 300 + 25 * T_mean + 0.05 * (T_mean^3) and AET_Turc = P / SQRT(0.9 + (P / L_T)^2)!
| Catchment Climate Regime | Annual P & PET (mm/yr) | Aridity Index phi = PET / P | Budyko AET & Annual Runoff Q = P - AET |
|---|---|---|---|
| Humid Temperate Basin | P = 1,200, PET = 720 | phi = 0.60 (Energy-Limited) | AET = 602.5 mm | Runoff Q = 597.5 mm |
| Balanced Semi-Humid Basin | P = 800, PET = 800 | phi = 1.00 (Equilibrium) | AET = 555.1 mm | Runoff Q = 244.9 mm |
| Semi-Arid Steppe Basin | P = 400, PET = 1,200 | phi = 3.00 (Water-Limited) | AET = 382.9 mm | Runoff Q = 17.1 mm |
Budyko (1974) Mean Annual Actual Evapotranspiration AET (mm/yr)
Evaluates P * SQRT((PET/P) * TANH(P/PET) * (1 - EXP(-PET/P))) using exact exponential identity TANH(u) = (1 - EXP(-2*u)) / (1 + EXP(-2*u)) for Precipitation B2 > 0 and PET B3 > 0 (mm/yr).
=IF(OR(B2 <= 0, B3 <= 0), "Requires Annual Precipitation P > 0 and PET > 0 mm/yr", ROUND(B2 * SQRT((B3 / B2) * ((1 - EXP(-2 * (B2 / B3))) / (1 + EXP(-2 * (B2 / B3)))) * (1 - EXP(-(B3 / B2)))), 1))Turc (1954) Temperature-Based Annual AET (mm/yr) & Budyko Runoff Q = P - AET
Computes Turc annual AET from Precipitation B2 (mm/yr) and Mean Annual Air Temperature B4 (°C, > -5°C) using L_T = 300 + 25*T + 0.05*T^3.
=IF(OR(B2 <= 0, B4 <= -5, B4 > 40), "Requires P > 0 and Mean Annual Temp -5 < T <= 40 °C", ROUND(B2 / SQRT(0.9 + ((B2 / (300 + 25 * B4 + 0.05 * (B4 ^ 3))) ^ 2)), 1))Try the example
Use the original values or change them to test your understanding.