6 Ground surface

The grand temperature is calculated by the 1D thermal conduction equation.

$$\rho _{g}c_{p,g}\DP{T_{g}}{t} = k_{g}\DP[2]{T_{g}}{z}.$$ (55)

where $T_{g}$ is the grand temperature (K), $\rho _{g}$ is the soil density (kgm${}^{-3}$), $c_{p,g}$ is the specific heat of soil (Jkg${}^{-1}$K${}^{-1}$), and $k_{g}$ is the thermal conductivity (Wm${}^{-1}$K${}^{-1}$). The surface temperature $T_{sfc}$ is given by $T_{sfc} = T_{g}\vert _{z=0}$.

The boundary condition at the surface is given as follows.

$$- k\left.\DP{T}{z}\right\vert _{z=0} = - F_{SR}(1-A) + F_{IR,net} + H ,$$ (56)

where $F_{SR}$ is the solar radiative flux at the surface (the sign of downward flux is positive), $A$ is the surface albedo, $F_{IR,net}$ is the net infrared radiative flux emitted from the surface and $H$ is the sensible heat flux (the sign of upward flux is positive). The lower boundary of the grand surface is given as a insulation boundary.

Parameters

The values of soil density, thermal conductivity and specific heat are same as those of standard model of Kieffer et al. (1977).

表 11: Parameters of ground surface model

Parameters	Standard values	Note
$A$	0.25	Kieffer et al. (1977)
$\rho _{g}$	1650 kgm${}^{-3}$	〃
$c_{p,g}$	588 JK${}^{-1}$kg${}^{-1}$	〃
$k_{g}$	7.63 $\times 10^{-2}$JK${}^{-1}$m${}^{-1}$sec${}^{-1}$	〃

By using these values, the thermal inertia $I\equiv \sqrt{\rho _{g}c_{p,g}k_{g}}$ is 272 Wm${}^{-2}$sec${}^{1/2}$K${}^{-1}$ and the diurnal skin depth of $\delta _{d}\equiv\sqrt{k_{g}t_{d}/(\rho_{g}c_{p,g})}$ is about 8.2 cm.