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/m3) during a time step
can be expressed as
node energy = ∆Q1 = Ti - 273.15 ∑θk ρk c pk
(6.1)
where Ti 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/m3) is the density of
component k, and c pk (J/kg) is the specific heat of component k. Calculation of Ti 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/m3) is defined as
freeze energy = ∆Q2 = l fus (θw - θr ) ∑θk ρk ck
(6.2)
thaw energy = ∆Q2 = -l fusθi ∑θk ρk ck
where lfus 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 Ti > 273.15 K.
The change in volumetric ice content per time step is therefore
ρw min ( ∆Q1, ∆Q2 )
∆θi =
(6.3)
ρi
l fus
and the nodal volumetric soil moisture and ice contents are updated accordingly.
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