specific heat of water, cp,w (4217.7 J/kgK) or ice, cp,i ( -13.3 + 7.80Ta J/kgK) depending
is assumed that the precipitation temperature is the same as the air temperature.
4.1.6 Final Top Boundary Equation
Combining Equations (4.2), (4.4), (4.5) (4.8) and (4.11), the top boundary condition is
reconfigured as
F (T ) = (1 - α ) I s ↓ +ε Iι↓ - εσ T 4 + (e0 + ρac p,aCDW )(Ta - T )
h
ρ
∂T
- ρaCDW ′l (qg - qa ) + U pc pTp + κ
e
@ z = 0m.
(4.12)
∂z
ρi ∂θi
+l fus
∆z - vc pT = 0
ρw ∂t
This is solved at each time step for the surface temperature.
4.2 Numerical Solution
Equation (4.1) is solved using a modified second-order Crank-Nicholson approach.
Following the technique presented in Hornbeck (1975), Equation (4.1) is rewritten as
Tj+1,i - Tj,i ⎡ kthj ,i+1 - kthj ,i ⎤ ⎡ Tj,i+1 - Tj,i ⎤
=⎢
⎥+
⎥⎢
∆t
∆z
∆z
⎢
⎥⎣
⎦
⎣
⎦
kthj ,i ⎡ Tj+1,i+1 - 2Tj+1,i + Tj+1,i-1 Tj,i+1 - 2T j,i + Tj,i-1 ⎤
+
⎢
⎥
(4.13)
( ∆z )
( ∆z )
2
2
2 ⎢
⎥
⎣
⎦
⎡ Tj,i+1 - Tj,i ⎤ l fus ρi ( ∆θi ) j,i
c p,w
-
⎥+
.
v j,i ⎢
⎦ c p ρw ∆t
∆z
c p j ,i
⎣
where the subscripts j and i represent time and depth, respectively. Combining like terms
and rearranging so that all terms involving Tj+1 are on the left-hand side of the equation,
Equation (4.13) becomes
T j+1,i-1 + γ j,iT j+1,i + T j+1,i+1 =
(4.14)
-T j,i-1 + T j,i ( β j,i + α j,i - δ j,i ) + T j,i+1(1 - β j,i + δ j,i ) + j,i ≡ φ j,i
where
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