**Chapter 6**

**Ground State Model**

**6.0 Introduction**

The goal of the Ground State Module is to provide a very simple formulation and very

rapid calculation of freezing and thawing that still accounts for fundamental mechanisms

and conditions in individual cases. The overall strategy is to use simple flux balance

relations to evaluate the interface temperatures in terms of energy input and medium

properties; to then use those temperatures to express the fluxes; and then use fluxes at

depth to express rate of freeze or thaw. This rate, multiplied by the time step, then gives

the depth of freeze or thaw. This depth of freeze or thaw can be continuous or discon-

tinuous, depending on the temperature and moisture profiles of the soil.

**6.1 Physical Setup**

The phase change process is assumed to be isothermal. The main principle behind the

freezethaw model is to compare the energy needed to freeze or thaw the soil to that

available at a given depth as was done by Guyman et al. (1993). Partial freezing or

thawing can occur. If the ice content at a given node is greater than zero, the node is

assumed frozen for the purposes of calculating the thickness of the frozen soil layer.

The energy extracted (freezing) or available (thawing) at a node (*J/m*3) during a time step

can be expressed as

*node energy *= ∆*Q*1 = *T*i - 273.15 ∑θk ρk c pk

(6.1)

where *T*i is the node temperature (*K*), and the subscript *k *is for the soil components dirt,

water, ice and air, θk is the volume fraction of component *k*, ρk (*kg/m*3) is the density of

component *k*, and *c * pk (*J/kg*) is the specific heat of component *k*. Calculation of *T*i is

discussed in Chapter 4. Determination of the various volume fractions is discussed in

Chapter 5.

The energy remaining before the node can completely freeze or thaw (*J/m*3) is defined as

*freeze energy *= ∆*Q*2 = *l * fus (θw - θr ) ∑θk ρk ck

(6.2)

*thaw energy *= ∆*Q*2 = -*l * fusθi ∑θk ρk ck

where *l*fus is the latent heat of fusion of water (3.335 105 J/kg) and θr is the node residu-

al volumetric water content as discussed in Chapter 7. Freezing can occur only if the node

*Ti *≤ 273.15 *K *and thawing if *T*i > 273.15 *K*.

The change in volumetric ice content per time step is therefore

ρw min ( ∆*Q*1, ∆*Q*2 )

∆θi =

(6.3)

ρi

*l fus*

and the nodal volumetric soil moisture and ice contents are updated accordingly.

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