Interactive comment on “ An energy-conserving model of freezing variably-saturated soil ”

Introduction Conclusions References

Reply: We have changed the title.
Changes: The new title is: A robust and energy-conserving model of freezing variablysaturated soil Comment: P. 1247 L. 10-11: 'to our knowledge this equation has never been fully derived from a thermodynamical point of view leaving some doubt on its limitations'.The derivation of Zhao et al., 1997 was from a thermodynamical point of view and quite similar.
Reply: Zhao et al., (1997) Eq. ( 12) is referred to as "the maximum liquid water content at sub-zero temperature" and is called the relation as "freezing point depression equation".In our text, however, we call "freezing point depression" the Equation (6) (Equation ( 17) in the new version), which sets the temperature of phase change of an unsaturated soil.Watanabe and Mizogouchi (2002) refer this depressed temperature to the Gibbs-Thomson effect.The liquid water content at sub-zero temperature in our paper is given by Equation (9) (equation ( 23) in the new version) and this takes into account not only the temperature under freezing conditions (as Zhao et al, 1997) but also the depressed melting temperature that depends on the total water content (i.e. on the air entry potential).Furthermore, we consider also the total volumetric water content (ice and water fractions) in deriving the unfrozen water content formulation, differently from Zhao et al (1997) that consider that the unfrozen liquid volume is only a function of temperature, according to Jame (1972).We believe that our approach is more general and less empirical.
Comment: P. 1247, L. 22-23: 'The energy equation with freezing soil in the above considered literature is always written in a non-conservative form'.It seems that all the energy balance equations in this study (Eq.16), or in Zhao et al. (1997, Eq. (1)) or in Hansson et al. (2004, Eq. ( 7)) are similar and energy conservative.The only difference is that this study presented in a more generalized 3D form while the other two presented in a specific 1D form.

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Reply: Actually it is true that we are not the only one to write the equation in a conservative form.We have done a further review (below reported) by which one realizes that, even though most of the authors write it non-conservative, others write it conservative.The difference in our notation is that we always use the conservation of internal energy rather than differentiating between sensible and latent term.This allows to obtain a more general form of the diffusion-advection equation, similar to the mass balance equation.Thus the numerical method used in the energy equation can be further used to the mass balance equation.Finally, we use a more generalized 3D form.
Changes: We removed the text "The energy equation with freezing soil in the above considered literature is always written in a non-conservative form" Comment: Figure 3: Check the legends of line style for 'An' and 'Sim'.They are not differentiable.
Reply: You are right, we have improved the readability of the Figure 3.