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4.5 The Multi-Layer Shallow-Water Model 83Fig. 4.16 Configuratio of a multi-layer shallow-water model. The total number of layers is nzP 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−1P nz = P nz−1 + (ρ nz − ρ nz−1 ) g η nzwhich 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 certainequilibrium levels. The latter equations require iteration from top to bottom with theboundary setting P 0 = 0 and ρ 0 = 0, which disables atmospheric pressure and setsair density to zero.Conservation of volume in each layer corresponds to the prognostic equationsfor layer-thickness:∂h i∂tLayer thicknesses are given by:=− ∂ (u i h i )∂x(4.24)h i = h i,o + η i − η i+1 (4.25)