k s
∂ T
∂ z
= EH, z = h b ,
ð4Þ
where EH = H + LE + R is the thermal flux through the ice cover; H and LE are the
vertical turbulent fluxes of sensible and latent heat, respectively; and R is the
longwave radiation balance (the shortwave radiation is neglected). H and LE are
calculated using bulk aerodynamic equations widely used to estimate energy
exchange between the sea and the atmosphere in the presence of ice [14]. The
calculation of R was performed according to the Ångström’s equation, linearized
with respect to T b − T a
ð
Þ, where the subscript “a” indicates air.
At the interface between snow and ice (index “0”), we assume continuity for
temperatures and heat fluxes:
T
− = T
+
≡ T 0 , k i
∂ T
+
∂ z
− k s
∂ T
−
∂ z
= 0, z = 0,
ð5a; bÞ
where the negative and positive signs denote upper and lower sides of the interface.
At the moving interface between the water and ice phases (subscript “1”), and at
z = h, we use temperature continuity and thermodynamic equilibrium (liquidus)
conditions, alongside the classical Stefan condition for prescribing sea ice growth:
T
− = T
+ = T eq − αS ≡ T 1 , z ∈ h 1 , h 2
ð
Þ,
ð6aÞ
ρ i L
d h 1
d t
= k i
∂ T
−
∂ z
, t > 0
ð6bÞ
Similarly, the Stefan and the continuity for temperatures and water fraction
conditions, together with liquidus condition, were implemented at the interface
between the two-phase and thawed zones (z = h 2 ðtÞ, subscript “2”):
m 1 − ν 3
ð
Þρ i L
dh 2
dt
= k m
∂T
∂z
−
− Q th ,
ð7aÞ
ν
− = ν 2 , ν
+ = 1, T
− = T
+
≡ θ = T eq − αS 0 , t > 0
ð7bÞ
where θ is the sea water freezing temperature, and Q th is the geothermal heat flux.
Diffusion of salt is neglected here.
We use the following initial conditions for the model:
t = 0: Tðz, 0Þ = T 0 , Sðz, 0Þ = S 0 , z ∈ ½0, hŠ,
ν = ν ∞ = 1, h 1, 0 = 0, h 2, 0 = h, T = T 0 , S = S 0 , z ∈ ½h, ∞Þ,
The equivalent water content ν eq , which serves as a measure of the thermodynamic state of the two-phase zone [12], can be estimated by integrating (2b) within
the mushy layer over time.
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P. V. Bogorodskii et al.