New analytical derivation of the mean annual water‐energy balance equation

The coupled water‐energy balance on long‐term time and catchment scales can be expressed as a set of partial differential equations, and these are proven to have a general solution as E / P = F ( E 0 / P , c ), where c is a parameter. The state‐space of ( P , E 0 , E ) is a set of curved faces in P − E 0 − E three‐dimensional space, whose projection into E / P − E 0 / P two‐dimensional space is a Budyko‐type curve. The analytical solution to the partial differential equations has been obtained as E = E 0 P /( P n + E 0 n ) 1/ n (parameter n representing catchment characteristics) using dimensional analysis and mathematic reasoning, which is different from that found in a previous study. This analytical…

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  National Natural Science Foundation of China