84
4 Long Waves in a Channel
where h i,o are undisturbed thicknesses for a flui at rest and η nz+1 = 0 represents a
rigid seafloo . With use of the latter relations, the layer-thickness equations turn into
prognostic equations for interface displacements, given by:
∂η nz
∂t
= −
∂ (u nz h nz )
∂ x
∂η nz−1
∂t
= −
∂ (u nz−1 h nz−1 )
∂ x
+
∂η nz
∂t
. . .
∂η 2
∂t
= −
∂ (u 2 h 2 )
∂ x
+
∂η 3
∂t
∂η 1
∂t
= −
∂ (u 1 h 1 )
∂ x
+
∂η 2
∂t
These equations can be written in the generalised form:
∂η i
∂t
= −
∂ (u i h i )
∂ x
+
∂η i+1
∂t
for i = nz, nz − 1, . . . , 1
(4.26)
with η nz+1 = 0 representing the rigid seafloo . Note that, in contrast to the pressure
iteration, this interaction goes from bottom to top.
4.6 Exercise 7: Long Waves in a Layered Fluid
4.6.1 Aim
The aim of this exercise is to simulate the progression of long gravity waves in a
flui consisting of multiple layers of different densities.
4.6.2 Task Description
We consider a stratifie flui consisting of ten layers of an initial thickness of
10 m each. The density of the top layer is 1025 kg/m
3
and density increases from
1026 kg/m
3
to 1026.5 kg/m
3
from the second layer to the bottom layer. In addition
to this, we add a simple bathymetry including a riff (Fig. 4.17) to test the multi-layer
floodin algorithm. The model is forced by prescribing sinusoidal oscillations of
surface and interface displacements of an amplitude of 1m near the western boundary. Lateral boundaries are closed.
Two different forcing periods are considered. The f rst experiment uses a forcing
period of 10 s. The forcing period in the second experiment is 2 h. Simulations run
over 10 times the respective forcing period and data outputs are produced at intervals
of a tenth of the forcing period. The time step is set to Δt = 0.25 s in both cases.