(m) and hs,melt (m) are the heads due to melting ice and snow, respectively, and ∆t (sec) is
the time step. If the ground is sloped, no water accumulates and any water that falls on
the surface, but which does not infiltrate, becomes runoff.
The evaporation and condensation rates depend on the moisture difference between the
air and the surface. They are quantified as the latent heat flux. Following the procedure
outlined in Chapter 4, Section 4.1.6, the modified surface energy balance is rearranged
such that
(1 - α ) I s ↓ +ε Ii↓ - εσ T 4 + ρac p,aCDW (Ta - T )
r
latent heat =
.
(5.4)
∂T
+ U pc pTp + κ
∂z
Reference should be made to Chapter 6 for explanation of the individual terms. The
moisture flux to/from the surface is then
latent heat
moisture flux =
(5.5)
ρwl
and l (J/kg) is the latent heat of evaporation if the air temperature is above freezing and
the latent heat of sublimation otherwise. If the latent heat flux is positive, evaporation is
occurring, otherwise condensation is happening.
5.1.2 Numerical Solution
Equation (5.2) is solved numerically using an explicit scheme such that
θwj+1,i - θwj ,i
⎡ v j +1,i+1 - v j +1,i ⎤
= -⎢
⎥ + sources(i) - losses(i)
(5.6)
∆t
∆zi
⎣
⎦
where
⎡h - h ⎤
vi = Ki-1/ 2 ⎢ i i-1 ⎥
⎣ zi - zi-1 ⎦
.
(5.7)
⎡h -h ⎤
vi+1 = Ki+1/ 2 ⎢ i+1 i ⎥
⎣ zi+1 - zi ⎦
hi = zi cos ϕ - Ψi , ∆zi = ( zi+1 - zi-1 ) / 2 , ni is the porosity at i and the subscripts j and i
represent time and depth, respectively. The change in soil moisture content due to
changes in the ice content, i.e., freezing/thawing, is incorporated into the source and sink
terms. In Equation (5.2) it is the second term on the right-hand side. Equation (5.6) is
solved for ψi using a Newton-Raphson technique so that the final matrix equation
becomes
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