One-dimensional diffusion equations for the vertical transport in an oscillating stratified lake of varying cross-section

Lateral uniformity of stratified lakes permits the description of vertical diffusion in them by a one-dimensional equation, with full regard to their three-dimensional nature. Such an equation should not be obtained simply by neglecting the horizontal divergence, because then significant terms are discarded. The complete equation is obtained by integration of the full three-dimensional equation over the volume under isopycnals. Two functions appear in it that do not appear in the simple equation (without horizontal divergence): the horizontal cross-section of the lake, and the surface sources. These correspond to two effects: the variation of the horizontal cross-section of the lake with depth and the existence of sources or sinks on the lake-bottom at different depths. The same equation can be applied to oscillating lakes, by describing temperatures and concentrations as functions of a new independent variable, the volume under isopycnals. The diffusivities and sources in the equation must then be interpreted as weighted averages, influenced by the correlation between the quantities appearing in them.