Sensible and latent heat content of a partially frozen soil

A change in sensible heat content in the soil, H, results in a new soil temperature, which in turn gives rise to an energy flux that affects the energy storage and so forth. Thus the soil temperature is a function of the sensible heat:

                                                                    (1.23)

where H is the sensible heat content and C_f is the heat capacity of the frozen soil, eq.(1.29). The phase change takes place in a temperature interval from 0 °C to T_f, which is the threshold temperature below which the soil is assumed to be completely frozen. In this temperature range, the sensible heat content is not equal to the total energy content in the soil, E, and therefore has to be calculated specifically as:

                                                    (1.24)

where r is the freezing-point depression, eq.(1.30), and E is the total heat content of the soil (i.e. left hand side of eq.(1.2)). f_lat is the ratio of latent heat of ice to the total heat content of the soil, E_f, at the temperature T_f:

                                                              (1.25)

where L_f is the latent heat of freezing, E_f is the total heat content of the soil at the temperature T_f (see below) and w_ice is the mass of water available for freezing calculated as:

                                                    (1.26)

where w is the total mass of water, θ_lf is the residual amount of water and ρ_water is the density of water.

The simplified assumption is made that all water at the temperature, T_f, is frozen except of a residual unfrozen amount, θ_lf calculated as:

                                                                (1.27)

where d₁ is a constant and θ_wilt is volumetric water content at a soil water potential corresponding to pF 4.2.

The heat content of soil, E_f, at the temperature T_f  is a function of latent and sensitive heat:

                                                      (1.28)

For temperatures between 0 ^oC and T_f  the soil heat capacity, C_f , is calculated as:

                                               (1.29)

where C_s is the heat capacity of solid material, C_i is the heat capacity of ice and C_w is the heat capacity of water. θ _i  is the water content in the ice and f_s is the volumetric content of the solid material (i.e. 1 - θ_s ).

 

Figure 1.3          Soil temperature (T) as a function of heat content (E) for different degrees of freezing-point depression, i.e. different values of d_2λ+d₃ (see eq. (1.30)   ). Both axes are distorted for the sake of clarity. With a completely frozen soil temperature (T_f ) of -5° C the ratio between sensible and latent heat is approximately 1:24.

Freezing-point depression (Beskow, 1935), which depends on soil texture (see Figure 1.3), is expressed by the ratio between latent heat contents of E at temperature T (when the temperature is between 0 °C and T_f) and E_f  at temperature T_f:

                             (1.30)

where d₂ and d₃ are empirical constants and λ is the pore size distribution index. The second factor in eq. (1.30) is inserted to ensure that temperatures close to T_f  never exceed free water temperatures at equivalent heat contents. See viewing function Freezing Temperature Function.