28
3 Basics of Nonhydrostatic Modelling
correction Δq is calculated implicitly from the requirement that the new velocity
field has to be free of divergence according to Eq. (3.17). The first guess of velocity
is hereby calculated from the time-forward scheme:
u
∗
i,k = u
n
i,k −
Δt
ρ o Δx
(q
n
i,k+1 − q
n
i,k )
(3.20)
w
∗
i,k = w
n
i,k −
Δt
ρ o Δz
(q
n
i−1,k − q
n
i,k )
(3.21)
where i and k are the cell references for the Arakawa C-grid (see Fig. 3.3). Consequently, the finite-difference form of the momentum equations can be written as:
u
n+1
i,k = u
∗
i,k −
Δt
ρ o Δx
Δq
n+1
i,k+1 − Δq
n+1
i,k
(3.22)
w
n+1
i,k = w
∗
i,k −
Δt
ρ o Δz
Δq
n+1
i−1,k − Δq
n+1
i,k
(3.23)
Insertion of the latter equations in the continuity equation (Eq. 3.17) and multiplication with the product ΔzΔx gives:
a e Δq
n+1
i,k+1 + a w Δq
n+1
i,k−1 + a t Δq
n+1
i−1,k + a b Δq
n+1
i+1,k − a o Δq
n+1
i,k = q
∗
i,k
(3.24)
This equation is mathematically called a Poisson equation (Poisson, 1813). The
coefficients for uniform grid spacings are given by:
a e = Δz/Δx , a w = Δz/Δx , a t = Δx/Δz , a b = Δx/Δz
and
a o = a e + a w + a t + a b
Figure 3.5 shows the locations of these coefficients in the Arakawa C-grid. The
source term on the right-hand side of Eq. (3.24) contains the divergence of the first
guess of the velocity field (u
∗ , w
∗ ) and is given by:
q
∗
i,k =
ρ o
Δt
u
∗
i,k − u
∗
i,k−1
Δz +
w
∗
i,k − w
∗
i+1,k
Δx
(3.25)
Once a solution of Eq. (3.24) is found, which implies that the new velocity field
is free of divergence, the new pressure field is given by:
q
n+1
= q
n
+ Δq
n+1
(3.26)