4.5 The Multi-Layer Shallow-Water Model
83
Fig. 4.16 Configuratio of a multi-layer shallow-water model. The total number of layers is nz
P 3 = P 2 + (ρ 3 − ρ 2 ) g η 3
. . .
P nz−1 = P nz−2 + (ρ nz−1 − ρ nz−2 ) g η nz−1
P nz = P nz−1 + (ρ nz − ρ nz−1 ) g η nz
which can be written in generalised form as:
P i = P i−1 − (ρ i − ρ i−1 ) g η i for i = 1, 2, 3, . . . , nz
(4.23)
where i is the layer index and η i are interface displacements with reference to certain
equilibrium levels. The latter equations require iteration from top to bottom with the
boundary setting P 0 = 0 and ρ 0 = 0, which disables atmospheric pressure and sets
air density to zero.
Conservation of volume in each layer corresponds to the prognostic equations
for layer-thickness:
∂h i
∂t
= −
∂ (u i h i )
∂ x
(4.24)
Layer thicknesses are given by:
h i = h i,o + η i − η i+1
(4.25)
83
Fig. 4.16 Configuratio of a multi-layer shallow-water model. The total number of layers is nz
P 3 = P 2 + (ρ 3 − ρ 2 ) g η 3
. . .
P nz−1 = P nz−2 + (ρ nz−1 − ρ nz−2 ) g η nz−1
P nz = P nz−1 + (ρ nz − ρ nz−1 ) g η nz
which can be written in generalised form as:
P i = P i−1 − (ρ i − ρ i−1 ) g η i for i = 1, 2, 3, . . . , nz
(4.23)
where i is the layer index and η i are interface displacements with reference to certain
equilibrium levels. The latter equations require iteration from top to bottom with the
boundary setting P 0 = 0 and ρ 0 = 0, which disables atmospheric pressure and sets
air density to zero.
Conservation of volume in each layer corresponds to the prognostic equations
for layer-thickness:
∂h i
∂t
= −
∂ (u i h i )
∂ x
(4.24)
Layer thicknesses are given by:
h i = h i,o + η i − η i+1
(4.25)