Advanced Ocean Modelling: Using Open-Source Software

28 3 Basics of Nonhydrostatic Modellingcorrection Δq is calculated implicitly from the requirement that the new velocityfield has to be free of divergence according to Eq. (3.17). The first guess of velocityis hereby calculated from the time-forward scheme:ui,k ∗ = un i,k − Δtρ o Δx (qn i,k+1 − qn i,k ) (3.20)wi,k ∗ = wn i,k − Δtρ o Δz (qn i−1,k − qn i,k ) (3.21)where i and k are the cell references for the Arakawa C-grid (see Fig. 3.3). Consequently,the finite-difference form of the momentum equations can be written as:u n+1i,k= ui,k ∗ − Δt (Δqn+1i,k+1ρ o Δx− )Δqn+1 i,kw n+1i,k= w ∗ i,k − Δtρ o Δz(Δqn+1i−1,k − )Δqn+1 i,k(3.22)(3.23)Insertion of the latter equations in the continuity equation (Eq. 3.17) and multiplicationwith the product ΔzΔx gives:a e Δq n+1i,k+1 + a wΔq n+1i,k−1 + a tΔq n+1i−1,k + a bΔq n+1i+1,k − a oΔq n+1i,k= q ∗ i,k (3.24)This equation is mathematically called a Poisson equation (Poisson, 1813). Thecoefficients for uniform grid spacings are given by:anda e = Δz/Δx , a w = Δz/Δx , a t = Δx/Δz , a b = Δx/Δza o = a e + a w + a t + a bFigure 3.5 shows the locations of these coefficients in the Arakawa C-grid. Thesource term on the right-hand side of Eq. (3.24) contains the divergence of the firstguess of the velocity field (u ∗ , w ∗ ) and is given by:qi,k ∗ = ρ o [(u∗Δt i,k − ui,k−1∗ ) (Δz + w∗i,k − wi+1,k) ∗ ]Δx(3.25)Once a solution of Eq. (3.24) is found, which implies that the new velocity fieldis free of divergence, the new pressure field is given by:q n+1 = q n + Δq n+1 (3.26)