Advanced Ocean Modelling: Using Open-Source Software

40 3 Basics of Nonhydrostatic ModellingWith inclusion of the nonlinear terms (required for subsequent exercises), thefirst-guess value of velocity is now calculated from:ui,k ∗ = un i,k − ΔtAdv(u) − Δtρ o Δx (pn i,k+1 − pn i,k + qn i,k+1 − qn i,k ) (3.45)w ∗ i,k = wn i,k − ΔtAdv(w) −where Adv(u) and Adv(w) represent the nonlinear terms.Δtρ o Δz (pn i−1,k − pn i,k + qn i−1,k − qn i,k ) (3.46)3.7 Exercise 4: Density-Driven Flows3.7.1 AimThe aim of this exercise is to apply the vertical ocean-slice model in a study ofbottom-arrested density-driven flows over variable bottom topography.3.7.2 Task DescriptionConsider a closed channel, 500 m long and 100 m deep, resolved by grid spacingsof Δx = 5 m and Δz = 2 m. This configuration is the same as in Exercise 3. Themodel is forced via prescription of a layer of dense water that initially leans againstthe left boundary, as shown in Fig. 3.14. This layer is initially 100 m thick and 50 mwide. Its density is 1 kg/m 3 greater compared with ambient water having a densityof ρ o = 1,028 kg/m 3 . Owing to initially unbalanced lateral pressure gradients, thislayer will spread along the sea floor with the aim to achieve a final state at rest voidof any horizontal density gradients.Horizontal and vertical density diffusivities are set to small uniform values ofK h = K z = 1 × 10 −4 m 2 /s. The total simulation time is 50 min with data outputsevery 30 secs. The author used a time step of Δt = 0.1 s, which satisfies the CFLFig. 3.14 Initial density field for Exercise 4