crystallization in the temperature spectrum: the range determined by the local
condition of liquidus (see [2, 25, 27, 28] for details).
The model assumes a two-layer structure for freezing soils, consisting of a
two-phase (partially frozen) zone and an underlying, completely thawed layer
(Fig. 2b). The layers of fast ice and water are considered a monophase, meaning
there is no liquid water in the ice layer, and vice versa. In addition, ice does not
exist in the thawed ground layer. It is assumed that the freezing soil is a porous
medium, saturated only with ice and sea water; the ice skeleton (subscript “g”) is
incompressible and immobile, and all dissolved salt formed during freezing is
rejected in the volume of unfrozen liquid. The properties of sea and pore ice are
considered the same, and the heat fluxes in all layers of the system are stationary. In
the mushy zone, phase composition is determined by the thermodynamic equilibrium (liquidus condition) of ice and an unhardened solution. This solution is
characterized by water fraction ν, whose value at the interface between the mushy
and thawed zones changes discontinuously.
The propagation of heat in the snow − h s ≤ z ≤ 0
ð
Þ , ice − h s ≤ z ≤ 0
ð
Þ , and
thawed soil h < z < h 2
ð
Þlayers are described by the equations of heat transfer:
ρC
ð Þ s, i, th
∂T
∂t
=
∂
∂z
k s, i, th
∂T
∂z
.
ð1Þ
In the two-phase zone h < z < h 2
ð
Þ , the heat and mass transfer equations can be
written as:
∂
∂t
ρC
ð Þ m T
Â
Ã
+ mρ i L
∂ν
∂t
=
∂
∂z
k m
∂ T
∂z
,
ð2aÞ
∂
∂t
ðνSÞ = D
∂
∂ z
ν
∂ S
∂z
, z ∈ h, h 2
ð
Þ, T = T eq − αS.
ð2b; cÞ
where
ρ C
ð Þ m = ð1 − mÞ ρ C
ð Þ g + mν ρ C
ð Þ w + m ð1 − νÞ ρ C
ð Þ i ,
ρ C
ð Þ th = ð1 − mÞ
ρ C
ð Þ g + m ρC
ð Þ w , k m = ð1 − mÞk g + mν k w + mð1 − νÞk i ; k th = ð1 − mÞk g + m k w , T is
temperature, S is salinity, C is specific heat, ρ is density, k is the heat conduction
coefficient, L is the latent heat of fusion of ice, D is salt diffusivity, and m is
porosity.
The sub-ice water layer h 1 < z < h
ð
Þis supposed to be homogeneous due to
convective mixing with temperatures equal to the freezing point, which decreases
with ice growth due to salt rejection.
T ≡ Θ = T eq − αS, S = S 0 h − h 1, 0
ð
Þ ̸ h − h 1 ðtÞ
ð
Þ , z ∈ h 1 , h
ð
Þ,
ð3a; bÞ
where Θ is the freezing point of seawater, α and T eq are constants, and subscript “0”
indicates the initial time t = 0.
At the upper interface of the system (subscript “b”), the boundary conditions
describe total heat fluxes to the atmosphere:
Seasonal Freezing of a Subwater Ground Layer …
615
condition of liquidus (see [2, 25, 27, 28] for details).
The model assumes a two-layer structure for freezing soils, consisting of a
two-phase (partially frozen) zone and an underlying, completely thawed layer
(Fig. 2b). The layers of fast ice and water are considered a monophase, meaning
there is no liquid water in the ice layer, and vice versa. In addition, ice does not
exist in the thawed ground layer. It is assumed that the freezing soil is a porous
medium, saturated only with ice and sea water; the ice skeleton (subscript “g”) is
incompressible and immobile, and all dissolved salt formed during freezing is
rejected in the volume of unfrozen liquid. The properties of sea and pore ice are
considered the same, and the heat fluxes in all layers of the system are stationary. In
the mushy zone, phase composition is determined by the thermodynamic equilibrium (liquidus condition) of ice and an unhardened solution. This solution is
characterized by water fraction ν, whose value at the interface between the mushy
and thawed zones changes discontinuously.
The propagation of heat in the snow − h s ≤ z ≤ 0
ð
Þ , ice − h s ≤ z ≤ 0
ð
Þ , and
thawed soil h < z < h 2
ð
Þlayers are described by the equations of heat transfer:
ρC
ð Þ s, i, th
∂T
∂t
=
∂
∂z
k s, i, th
∂T
∂z
.
ð1Þ
In the two-phase zone h < z < h 2
ð
Þ , the heat and mass transfer equations can be
written as:
∂
∂t
ρC
ð Þ m T
Â
Ã
+ mρ i L
∂ν
∂t
=
∂
∂z
k m
∂ T
∂z
,
ð2aÞ
∂
∂t
ðνSÞ = D
∂
∂ z
ν
∂ S
∂z
, z ∈ h, h 2
ð
Þ, T = T eq − αS.
ð2b; cÞ
where
ρ C
ð Þ m = ð1 − mÞ ρ C
ð Þ g + mν ρ C
ð Þ w + m ð1 − νÞ ρ C
ð Þ i ,
ρ C
ð Þ th = ð1 − mÞ
ρ C
ð Þ g + m ρC
ð Þ w , k m = ð1 − mÞk g + mν k w + mð1 − νÞk i ; k th = ð1 − mÞk g + m k w , T is
temperature, S is salinity, C is specific heat, ρ is density, k is the heat conduction
coefficient, L is the latent heat of fusion of ice, D is salt diffusivity, and m is
porosity.
The sub-ice water layer h 1 < z < h
ð
Þis supposed to be homogeneous due to
convective mixing with temperatures equal to the freezing point, which decreases
with ice growth due to salt rejection.
T ≡ Θ = T eq − αS, S = S 0 h − h 1, 0
ð
Þ ̸ h − h 1 ðtÞ
ð
Þ , z ∈ h 1 , h
ð
Þ,
ð3a; bÞ
where Θ is the freezing point of seawater, α and T eq are constants, and subscript “0”
indicates the initial time t = 0.
At the upper interface of the system (subscript “b”), the boundary conditions
describe total heat fluxes to the atmosphere:
Seasonal Freezing of a Subwater Ground Layer …
615