Theory and equations

Surface energy balance

With the option Evaporation and heat flux from the surface energy balance (page Meteorological data) the evaporation is not prescribed but follows, together with the soil heat flux, from the energy balance at the soil surface Saito et al. (2006); van Bavel and Hillel (1976):

R_n − H − L E − G = 0

Symbol	Meaning
R_n	net radiation [W/m²]
H	sensible heat flux [W/m²]
L E	latent heat flux [W/m²]: the evaporation E times the latent heat of vaporization L
G	soil heat flux [W/m²], the upper boundary condition of the heat transport

It requires heat transport and is usually combined with vapor flow.

Net radiation

R_n = (1 − a) R_s + R_L↓ − R_L↑

R_L↑ = ε_s σ T_s⁴, R_L↓ = ε_a σ T_a⁴

the surface albedo a depends on the water content θ of the surface van Bavel and Hillel (1976): 0.25 for θ ≤ 0.1, 0.35 − θ for 0.1 < θ ≤ 0.25, 0.1 for θ > 0.25;
the soil emissivity ε_s = min(0.9 + 0.18 θ, 1) van Bavel and Hillel (1976);
the clear-sky emissivity of the atmosphere ε_ac = 1.24 (e_a/T_a)^1/7 Brutsaert (1975), corrected for the cloud cover c: ε_a = (1 − 0.84 c) ε_ac + 0.84 c Campbell and Norman (1998);
daily radiation is distributed over the day with the sine of the solar elevation.

Sensible and latent heat

H = C_a (T_s − T_a) / r_h

E = (ρ_vs − ρ_va) / (r_v + r_s)

C_a is the volumetric heat capacity of the air, T_s and T_a the surface and air temperatures, ρ_vs the water vapor density at the soil surface (the saturated vapor density times the relative humidity of the soil air, from the pressure head and temperature of the top node), ρ_va that of the air, r_h = r_v the aerodynamic resistances to heat and vapor transfer, and r_s the soil surface resistance to evaporation.

The aerodynamic resistance follows from the wind speed with stability corrections for unstable and stable atmospheric conditions (Monin–Obukhov similarity, iterated) Campbell and Norman (1998); Saito et al. (2006); with no wind, heat is transferred by diffusion only. The soil surface resistance increases as the surface dries van de Griend and Owe (1994):

r_s = 10 exp[35.63 (0.15 − θ)] for θ < 0.15; r_s = 10 s/m otherwise

Coupling

The energy balance is solved together with the heat transport for the surface temperature; the evaporation becomes the flux condition of the water flow, limited by hCritA (atmospheric boundary condition). With vapor flow, the latent heat carried through the surface by the soil vapor flux is returned to the top node. The time step is limited so that the surface temperature changes smoothly.

Literature cited in this topic

Brutsaert, W. (1975). On a derivable formula for long-wave radiation from clear skies. Water Resources Research 11(5): 742–744. doi:10.1029/WR011i005p00742
Campbell, G. S., and J. M. Norman (1998). An Introduction to Environmental Biophysics. 2nd ed., Springer, New York, 286 pp.
Saito, H., J. Šimůnek, and B. P. Mohanty (2006). Numerical analysis of coupled water, vapor, and heat transport in the vadose zone. Vadose Zone Journal 5(2): 784–800. doi:10.2136/vzj2006.0007
van Bavel, C. H. M., and D. Hillel (1976). Calculating potential and actual evaporation from a bare soil surface by simulation of concurrent flow of water and heat. Agricultural Meteorology 17(6): 453–476. doi:10.1016/0002-1571(76)90022-4
van de Griend, A. A., and M. Owe (1994). Bare soil surface resistance to evaporation by vapor diffusion under semiarid conditions. Water Resources Research 30(2): 181–188. doi:10.1029/93WR02747

All sources: References.