(18)
resistance in water transport was reported by Van
The mass conservation equation can be written as
/ z[K(θ) ( hm / z + 1)] + ϑ v / ρl z + Sm
den Honert (1948). Under steady-state condi-
tions, the pathway of water movement through
the soilrootstemleaves could be expressed as
= θ / t + ρi θi / ρ1 t
(19)
resistances in series (Fig. 5) using the equation
T = Ψs Ψr/Rs = Ψr Ψl/Rp
where ρ = density of liquid water
(17)
c = specific heat capacity of water
where T = transpiration rate, Ψs, Ψr, and Ψl are the
v = downward liquid water flux
water potentials in the soil matrix, at the root sur-
Cs = volumetric heat capacity of soil
ρi = density of ice
face, and in the leaves, respectively; and Rs and
Lf = latent heat of fusion
θi = volumetric ice content
Rp is considered to be dominant over Rs and de-
ρv = vapor density in soil pore space
creases with increased transpiration rate (Feddes
ϑv = downward water vapor flux through
1981).
Steady-state conditions rarely exist in the field.
soil
Furthermore, the plant root system is dynamic
Sh = heat sink/source
(root senescence occurs and new roots emerge),
Sm = water sink/source
θ = volumetric soil water content
root geometry is time-dependent, and water per-
meability varies with position along the root and
t = time
with time. Instead of considering water flow to
T = temperature
single roots, a more suitable approach might be
z = soil depth
h = thermal conductivity of soil.
the macroscopic one, in which a sink term repre-
senting water extraction by a homogeneous and
isotropic element of the root system (volume of
Several other workers used eq 18 and 19 and
water per volume of soil per unit of time) is add-
developed the following types of water transport
ed to the conservation mass equation (eq 4).
models (Kung and Steenhuis 1986).
Variation in root diameter has long been ob-
1. Models based on an analogy between the
served, which further complicates the mathemat-
mechanism of water transport in unsaturated
ical analysis of root function. Models of the trans-
soil and in frozen soil (Harlan 1971).
port of ions, water, oxygen, and other materials
2. Models based on the theory that the pressure
from the soil to the root interior assume the root
jump is due to the curvature in the liquid
has constant diameter, and transfer coefficients
water/ice/air interfaces (Bresler and Miller
remain constant with time.
1975; Miller 1973, 1977, 1978; Miller et al.
The simulation of water transport through a
1975).
root system is complicated in cold regions be-
3. Models based on the theory of irreversible
cause of the presence of permafrost and seasonal-
thermodynamics (Kay and Groenevelt 1974,
ly frozen soils. Furthermore, the soil freezing
Groenevelt and Kay 1974, O'Neill and Miller
front and ice formed in the frozen ground en-
1982).
hance runoff and reduce groundwater recharge.
4. Models simulating temperature distribution
in a partially frozen soil (Miller et al. 1984).
A few workers (Taylor and Luthin 1978, Guymon
These models were developed without consid-
et al. 1980, Hromadka et al. 1981) studied the
ering the plant root system. During thawing sea-
complex processes of characterizing simulta-
son, soils without vegetation in cold regions are
neous heat and soil water transport in a freezing
susceptible to degradation processes of soil ero-
soil without considering vegetation. The physics
sion, nutrient losses, and organic matter deple-
of a frozen heterogeneous soil profile include the
tion.
terms of the soil energy balance and soil water
balance. The one-dimensional energy conserva-
tion equation for potentially freezing soils can be
Soil air
Roots and soil organisms capture energy from
written as
the oxidation of organic substances in a series of
[
]
/ z h(θ) T / z + ρlcl ( vlT / z) + Sh = Cs T / t
enzyme-catalyzed reactions. Plants need molecu-
lar oxygen to respire and convert carbohydrates
- ρiLf ( θi / t) + Lv ( ρv / t + ϑv / z).
to carbon dioxide and water. This is an exother-
11
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