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 (ΔR) 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. ΔR 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)