4.2 Long Surface Gravity Waves
71
Fig. 4.4 Snapshot of a shallow-water wave of 1 m in amplitude and 500 m in wavelength in a 20 m
deep ocean. Shown are sea-level elevation and vertical displacements of flui parcels at selected
depths
4.2.7 Numerical Grid
We use a spatial grid of constant grid spacing in which velocity grid points are
located halfway between adjacent sea-level grid points (Fig. 4.5).
Fig. 4.5 The staggered grid. The cell index k refers to a certain grid cell. Water depth is calculated
at sea-level grid points
4.2.8 Finite-Difference Scheme
On the basis of the staggered grid (see Fig. 4.5), the momentum equation (4.12) can
be written in finite-di ference form as:
u
n+1
k
= u
n
k − Δt g
η
n
k+1 − η
n
k
/Δx
(4.17)
where n is time level, k is grid index, Δt is time step, and Δx is grid spacing. A
control volume (Fig. 4.6) is used to discretise equation (4.3). Accordingly, we can
write this equation as:
η
∗
k = η
n
k − Δt
u
n+1
k
h e − u
n+1
k−1 h w
/Δx
(4.18)
71
Fig. 4.4 Snapshot of a shallow-water wave of 1 m in amplitude and 500 m in wavelength in a 20 m
deep ocean. Shown are sea-level elevation and vertical displacements of flui parcels at selected
depths
4.2.7 Numerical Grid
We use a spatial grid of constant grid spacing in which velocity grid points are
located halfway between adjacent sea-level grid points (Fig. 4.5).
Fig. 4.5 The staggered grid. The cell index k refers to a certain grid cell. Water depth is calculated
at sea-level grid points
4.2.8 Finite-Difference Scheme
On the basis of the staggered grid (see Fig. 4.5), the momentum equation (4.12) can
be written in finite-di ference form as:
u
n+1
k
= u
n
k − Δt g
η
n
k+1 − η
n
k
/Δx
(4.17)
where n is time level, k is grid index, Δt is time step, and Δx is grid spacing. A
control volume (Fig. 4.6) is used to discretise equation (4.3). Accordingly, we can
write this equation as:
η
∗
k = η
n
k − Δt
u
n+1
k
h e − u
n+1
k−1 h w
/Δx
(4.18)