The ground is considered as a semi-infinite solid with one-dimensional coordinates, as shown in Fig. 1a. The variation of ground temperature follows the one-dimensional, transient heat conduction equation given by Carslaw and Jaeger (1980):

$$\frac{{\partial^{2} T(y,t)}}{{\partial y^{2} }} = \frac{1}{\alpha }\frac{\partial T(y,t)}{\partial t}.$$

(1)

Solution of the above equation is subjected to the first boundary condition at the ground surface given by Bhardwaj and Bansal (1981):

$$- k\left. {\frac{\partial T}{\partial y}} \right|_{y \,=\, 0} = h(T_{\text{a}} - T_{y \,=\, 0} ) - \varepsilon \Delta R + \alpha_{0} S.$$

(2)

The left side of the above equation shows the conduction through the ground surface. The first term on the right-side equation shows convective heat transfer between ground surface (*T*
_{
y=0}) and air (*T*
_{a}). The second term is thermal radiation (Δ*R*) with emissivity of soil *ε*. The third term denotes solar radiation (*S*) absorbed by the ground surface with an absorptivity of soil *α*
_{0}. The above equation can be written in the form of general convective heat transfer boundary condition as follows:

$$- k\left. {\frac{\partial T}{\partial y}} \right|_{y = 0} = h(T_{\text{e}} - T_{y = 0} ).$$

(3)

The temperature \(T_{\text{e}}\) can be expressed as follows:

$$T_{\text{e}} = T_{\text{a}} + \alpha_{0} S/h - \varepsilon \Delta R/h.$$

(4)

In Eq. (4), *h* and \(\Delta R\) are computed according to Kays and Crawford (1980). The symbol *h* is the total heat transfer coefficient which includes convective and radiative heat transfer coefficients. The convective term depends on air velocity (*v*) and radiative term depends on air temperature. Δ*R* is the thermal radiation which depends on air temperature and sky temperature given in Hillel (1980, 1982, 2004):

$$h = h_{\text{c}} + h_{\text{r}}$$

(5)

$$h_{\text{c}} = 2.8 + 3v$$

(6)

$$h_{\text{r}} = 4\varepsilon \sigma T_{\text{a}}^{3}$$

(7)

$$T_{\text{sky}} = T_{\text{a}} { - 12}$$

(8)

$$\Delta R = \sigma [(T_{\text{a}} + 273.15)^{4} - (T_{\text{sky}} + 273.15)^{4} ].$$

(9)

The second boundary condition is considered as constant temperature which is the annual mean effective temperature (\(\bar{T}_{\text{e}}\)):

$$T_{y \to \infty } = \bar{T}_{\text{e}} .$$

(10)