5.2 Numerical Treatment
129
Fig. 5.2 The three-dimensional control volume
5.2.3 The Nonhydrostatic Solver of the Momentum Equations
As in the vertical ocean-slice model, the S.O.R. scheme is used to derive the pressure
field q. To this end, q is split up into explicit and implicit components; that is,
q ⇒ q
n
+ Δq
n+1
For the fully three-dimensional momentum equations, the Poisson equation for
Δq reads:
a e Δq
n+1
i, j,k+1 + a w Δq
n+1
i, j,k−1 + a n Δq
n+1
i, j+1,k + a s Δq
n+1
i, j−1,k +
+a t Δq
n+1
i−1, j,k + a b Δq
n+1
i+1, j,k − a o Δq
n+1
i, j,k = q
∗
i, j,k
(5.8)
The coefficients in this equation for uniform grid spacings are given by:
a e = a w = Δz/Δx
a n = a s = ΔzΔx/(Δy)
2
a t = a b = Δx/Δz
a o = a e + a w + a n + a s + a t + a b
The right-hand side of the Poisson equation (Eq. 5.8) represents the divergence
of first-guess values of velocity and is given by:
q
∗
i, j,k =
ρ o
Δt
(u
∗
i, j,k − u
∗
i, j,k−1 )Δz+
+ (v
∗
i, j,k − v
∗
i, j−1,k )
ΔxΔz
Δy
+ (w
∗
i, j,k − w
∗
i+1, j,k )Δx
(5.9)
129
Fig. 5.2 The three-dimensional control volume
5.2.3 The Nonhydrostatic Solver of the Momentum Equations
As in the vertical ocean-slice model, the S.O.R. scheme is used to derive the pressure
field q. To this end, q is split up into explicit and implicit components; that is,
q ⇒ q
n
+ Δq
n+1
For the fully three-dimensional momentum equations, the Poisson equation for
Δq reads:
a e Δq
n+1
i, j,k+1 + a w Δq
n+1
i, j,k−1 + a n Δq
n+1
i, j+1,k + a s Δq
n+1
i, j−1,k +
+a t Δq
n+1
i−1, j,k + a b Δq
n+1
i+1, j,k − a o Δq
n+1
i, j,k = q
∗
i, j,k
(5.8)
The coefficients in this equation for uniform grid spacings are given by:
a e = a w = Δz/Δx
a n = a s = ΔzΔx/(Δy)
2
a t = a b = Δx/Δz
a o = a e + a w + a n + a s + a t + a b
The right-hand side of the Poisson equation (Eq. 5.8) represents the divergence
of first-guess values of velocity and is given by:
q
∗
i, j,k =
ρ o
Δt
(u
∗
i, j,k − u
∗
i, j,k−1 )Δz+
+ (v
∗
i, j,k − v
∗
i, j−1,k )
ΔxΔz
Δy
+ (w
∗
i, j,k − w
∗
i+1, j,k )Δx
(5.9)