Analytical solution of the coupled 2-D turbulent heat and vapor transport equations and the complementary relationship of evaporation

The existence of a symmetrical complementary relationship (CR) in evaporation has been hypothesized in the hydrologic literature but the conditions required have not been investigated in much detail. In this study it is shown that under near-neutral atmospheric conditions and a constant energy term at the evaporating surface, the analytical solution of the coupled turbulent diffusion equations of heat and vapor transport across a moisture discontinuity of the surface yields a symmetrical CR between the evaporation rate of the uniform drying land upwind of the discontinuity and the mean evaporation rate of the wet area provided, the latter has a proper fetch (i.e., along-the-wind extent). This fetch is a function of the air stability parameter and the assumed uniform surface roughness value, and it is in the order of a 100 m for a smooth surface under near-neutral atmospheric conditions. The analytically derived mean evaporation rates of such a smooth wet surface compare well, i.e., within 10%, to the Penman equation estimates, most frequently employed within the CR framework.