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 = E0P/(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 solution is a useful theoretical tool to evaluate the effect of climate and land use changes on the hydrologic cycle. Mathematical comparisons between the two analytical equations showed that they were approximately equivalent, and their parameters had a perfectly significant linear correlation relationship, while the small difference may be a result of the assumption about derivatives in the previous study.