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100 5 2D Shallow-Water Modellingv n+1j,k= v n j,k− rΔt vn+1j,k√(un v) 2+(v n j,k) 2/hv(5.10)where the subscripts u and v indicate the location at which a variable is calculated.This is necessary because u, v and h are not evaluated at the same grid point in theArakawa C-grid (see Fig. 5.1). Reorganisation of these equations gives:u n+1j,k= u n j,k / (1 + R x) (5.11)v n+1j,k= v n j,k / ( )1 + R y (5.12)The parameters R x and R y are given by:R x = rΔt√ (u n j,k) 2+(vn u) 2/hu(5.13)R y = rΔt√(un v) 2+(v n j,k) 2/hv(5.14)are always positive quantities, so that bottom friction will gradually decrease speed,as required. Hence, a semi-implicit approach for bottom friction should always beemployed in layer models.5.4.3 Finite-Difference EquationsUsing a semi-implicit approach for bottom friction, the finite-di ference equationsstating momentum conservation are given by:u n+1j,k= ( u n j,k + j,k) Δun / (1 + Rx ) (5.15)= ( v n j,k + ) ( )Δvn j,k / 1 + Ry (5.16)v n+1j,kwhere R x and R y are given by (5.13) and (5.14), andΔu n j,k = Δt { τ windx/ (ρ o h u ) − g ( η n j,k+1 − ) }ηn j,k /Δx) }/ΔyΔv n j,k = Δt { τywind / (ρ o h v ) − g ( η n j+1,k − ηn j,k(5.17)(5.18)