the limiting value ρ3 (δ+) of ρ3 as ξ approaches δ, while ξ is in R2 is given (Nakano 1994a) as
ρ (δ+) = ρ Vo [r(δ+) + Vo ]-1
(26)
3
30
where r (δ+) is the rate of heave at δ+ given as
-1
-1
r(δ+) = d2 fο + d2 s2 [ρ10 - ρ30 ν{T(δ)}]Vο .
(27)
The heat flux is discontinuous at n1 and the jump condition is given as
q(δ+) = q(δ) + f1(δ)[L + (c1 c2)T1]
(28)
where q(δ ) and f1(δ ) are limiting values of q and f1, respectively, as ξ approaches δ, while
ξ is in R1 and T1 is T (δ).
We will reduce eq 17 and eq 18 to a simpler form. Using eq 13, 14, and 15, we obtain
cV = coVo (c1 c2)Λ + c1(fo f1)
(29)
where
co = c1ρ10 + c3ρ30
(30)
ξ
Λ(ξ) = ∫ λ 2dξ ,
ξ ≥ 0.
(31)
0
Using eq 29, neglecting a sensible heat term, and integrating eq 17, we obtain
-k T ′ = koα o + LΛ,
0<ξ<δ
(32)
where ko and k are the thermal conductivities of Ro and R1, respectively, and αo ≥ 0 is the
absolute value of the temperature gradient at ξ = 0. Using eq 32 and neglecting sensible
heat terms, we will reduce eq 28 to
[
]
k1α1 - koα o = f1(δ-) + Λ(δ-) L
(33)
Using the principle of mass and heat conservation, we have derived equations that must
be satisfied by a traveling wave solution of soil freezing. Clearly these equations are not
sufficient to solve the problem. We need a model of a frozen fringe that specifies f1 (ξ) and Λ
(ξ).
MODEL STUDY
A model of a frozen fringe called M1 was introduced by Nakano (1990) to explain empir-
ical findings on the growth condition of an ice layer in freezing soils. The model has been
modified as its empirical evaluation has progressed (Takeda and Nakano 1990, Nakano
and Takeda 1991, Takeda and Nakano 1993, Nakano and Takeda 1994). The latest version
assumes the validity of equations in R1 given as
k / ko = η ≥ 1
k = constant,
(34)
6
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