with Tp the precipitation temperature defined as the wet-bulb temperature, cp either the
specific heat of water, cp,w (4217.7 J/kgK) or ice, cp,i ( -13.3 + 7.80Ta J/kgK) depending
on Tp, Up the mass precipitation flux (kg/m2s), and γ p the precipitation density (kg/m3). It
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
F (T ) = (1 - α ) I s ↓ +ε Iι↓ - εσ T 4 + (e0 + ρac p,aCDW )(Ta - T )
- ρaCDW ′l (qg - qa ) + U pc pTp + κ
@ z = 0m.
∆z - vc pT = 0
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 ⎤
kthj ,i ⎡ Tj+1,i+1 - 2Tj+1,i + Tj+1,i-1 Tj,i+1 - 2T j,i + Tj,i-1 ⎤
( ∆z )
( ∆z )
⎡ Tj,i+1 - Tj,i ⎤ l fus ρi ( ∆θi ) j,i
v j,i ⎢
⎦ c p ρw ∆t
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 =
-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