The energy balance presented in Chapter 1 (Equation 1.1) is the key equation in land surface parameterization. It represents the conversion of radiative energy to other forms of energy and it is solved numerically to derive the surface temperature. LSMs solve the energy balance at each tile representing each land surface type and then compute the fluxes and state variables as a weighted average of the tiles of a gridbox.
In JULES the equation takes the form shown in Eq. 2.1. All fluxes are positive downward execept for sensible and latent heat fluxes, which are positive upward. The energy balance is solved at the surface and the term on the left represents the excess of energy. This energy varies the surface temperature T∗ accordingly with the surface heat capacity Cs.
Cs
δT∗
δt = (1 − α)Rs↓ +Rl↓ −εσ(T∗)
4− H − LE − G (2.1)
where Rs ↓ and Rl ↓ are the incoming shortwave and longwave radiations, α the surface
albedo, ε the canopy emissivity, σ the Stefan Boltzmann constant. The sensible heat flux (H) is a function of the gradient of temperature between the surface (T∗) and the lowest
layer of the atmosphere (T1), regulated by the aerodynamic resistance ra, specific heat
capacity cp, and density ρ of air.
H = ρcp ra
The latent heat flux (LE ) is the energy consumed in evaporation (E ). L is the latent heat of vaporisation (J kg−1) and evaporation E is a function of the gradient of moisture between the saturated specific humidity at the surface temperature and the specific humidity at the lowest layer of the atmosphere (q1). The flux is regulated by the air density, the
aerodynamic resistance and the surface resistance, which can be that of soil or vegetation.
E = ρ
ra+ rs
q1− qsat(T∗) (2.3)
The aerodynamic resistance used in the computation of the turbulent fluxes depends on atmospheric stability via exchange coefficients, which take into account roughness lengths and windspeed.
The ground heat flux (G) in JULES is expressed as follows: G = ν σεεs (T∗)4− (Ts1)4 + ρcp racan (T∗− Ts1) + (1 − ν)λsoil(T∗− Ts1) (2.4)
It is composed of three channels for energy transmission towards the ground: radiative, turbulent, and conductive. The transmission in the vegetated (ν represents the vegetated fraction) part has a radiative component for the radiative loss from the first layer of soil, at temperature Ts1, and its reflection in the canopy and the radiative emission from the
canopy, at temperature T∗, and its reflection in the first layer of soil. ε is the canopy
emissivity and εs the soil emissivity. The second component is the turbulent transmission
of energy from the canopy to the ground, with racanrepresenting an aerodynamic resistance
between the canopy and the soil. The last term accounts for the thermal conduction within the soil, only in the non vegetated area (1-ν) with λsoil being the thermal conductivity.
CTESSEL uses an energy balance equation similar to Eq. 2.1, but the equation is solved at the skin layer, defined as the interface between land and atmosphere. The temperature at this layer is the skin temperature (Tsk).
0 = (1 − fRs)(1 − α)Rs↓ +ε(Rl ↓) − εσ(Tsk)4+ H + LE − G (2.5)
Contrary to JULES there is no term for surface thermal inertia (Cs = 0 in Eq. 2.1).
However, CTESSEL accounts for a small fraction of net shortwave energy transmitted directly to the soil or snow, (fRs). This fraction is tile dependent and is only transmitted
The latent heat and sensible fluxes are calculated similarly to JULES but making use of the skin temperature.
E = ρ ra+ rs q1− qsat(Tsk) (2.6) H = ρcp ra (Ts1+ gZL/Cp− Tsk) (2.7)
where the extra term contains g for the acceleration of gravity and is ZL the lowest
atmospheric model level.
Finally, the ground heat flux represents the flux of energy from the skin layer to the top soil layer.
G = Λsk(Tsk− Ts1) (2.8)
The ground heat flux is only described with a conduction term determined by the temper- ature difference between the skin layer and the top soil layer. The skin conductivity Λsk
establishes the thermal connection between the skin level and the soil or snow deck. It varies for stable or unstable stratification of the temperature gradient in the case of high vegetation. This difference is considered to represent the asymmetric coupling between the ground surface and the tree canopy layer: an effective convective transport within the tree trunk space for unstable conditions, and a limited turbulent exchange for stable strat- ification (ECMWF, 2015). Although there are no explicit terms for radiative or turbulent transfer as in JULES (Eq. 2.4), this transmission is implicitly accounted for.
Subsoil heat transfer
The transfer of heat vertically through the soil is described by the Fourier law of diffusion with an additional term to account for the thermal effects related to phase changes of water. Csoil ∂T ∂t = ∂ ∂z λT ∂T ∂z + Lf usρw ∂θI ∂t (2.9)
Csoil is the volumetric soil heat capacity (J m−3 K−1), T is the soil temperature, z is the
vertical coordinate and λT is the thermal conductivity, which depends on the soil water
content. The last term represents thermal effects of latent heat of fusion or freezing. Lf us
is the latent heat of fusion, ρw is the density of water and θI is the volumetric ice water
content which depends on temperature and soil moisture content. The energy used to melt the frozen soil in spring delays the surface warming and the freezing of the soil in autumn
or winter delays soil cooling (Viterbo et al., 1999). JULES solves a similar equation for the soil heat transfer. The effects of the water phase changes are accounted for using an apparent capacity, and additional term accounts for the transfer of heat via the water flow (Cox et al., 1999). This transfer is neglected in CTESSEL.
To solve this differential equation the soil is discretised into horizontal layers. Both JULES and CTESSEL divide the soil into four layers with exponentially increasing depths. JULES layers are 0.1, 0.25, 0.65 and 2 m thick and CHTESSEL layers are 0.07, 0.21, 0.72 and 1.89 m thick; adding up to a total depth of 3 and 2.89 m respectively. This depth allows to capture the seasonal signal in temperature variation (Deardorff, 1978).
Although at the surface the gridbox is divided into several tiles, the subsoil is described as a single type. At the top, the boundary condition is the soil heat flux at the surface computed as a weighted average over the tiles, plus, in the case of CTESSEL, the fraction of solar radiation which was transmitted directly (fRs in Equation 2.5) and snow basal
flux when present. At the bottom, the boundary condition is of zero heat flux, to ensure conservation of energy.