Equilibrium Evaporation and the Convective Boundary Layer

A theory is developed for surface energy exchanges in well-mixed, partlyopen systems, embracing fully open and fully closed systems as limits.Conservation equations for entropy and water vapour are converted intoan exact rate equation for the potential saturation deficit D in a well-mixed, partly open region. The main contributions to changes in D arise from (1) the flux of D at the surface, dependent on a conductance g_q that is a weighted sum of the bulk aerodynamic and surface conductances; and (2) the exchange flux of D with the external environment by entrainment or advection, dependent on a conductance g_e that is identifiable with the entrainment velocity when the partly open region is a growing convective boundary layer (CBL). The system is fully open when g_e/g_q , and fully closed when g_e/g_q 0. The equations determine the steady state surface energy balance (SEB) in a partly open system, the associated steady-state deficit, and the settling time scale needed to reach the steady state. The general result for the steady-state SEB corresponds to the equations of conventional combination theory for the SEB of a vegetated surface, with the surface-layer deficit replaced by the external deficit and with g_q replaced by the series sum (g_q^-1 + g_e^-1)^-1. In the fully open limit D is entirely externally prescribed, while in the fully closed limit, D is internally determined and the SEB approaches thermodynamic equilibrium energy partition. In the case of the CBL, the conductances g_q and g_e are themselves functions of D through short-term feedbacks, induced by entrainment in the case of g_e and by both physiological and aerodynamic (thermal stability) processes in the case of g_q. The effects of these feedbacks are evaluated. It is found that a steady-state CBL is physically achievable only over surfaces with at least moderate moisture availability; that entrainment has a significant accelerating effect on equilibration; that the settling time scale is well approximated by h/(g_q + g_e), where h is the CBL depth; and that this scale is short enough to allow a steady state to evolve within a semi-diurnal time scale only when h is around 500 m or less.