Advanced Ocean Modelling: Using Open-Source Software

102 4 2.5D Vertical Slice Modellingu ∗ i,k = cos (α)un i,k + sin (α)vn u + Δt Fn uv n+1i,k= cos (α)v n i,k − sin (α)un v + Δt Fn v (4.16)w ∗ i,k = wn i,k + Δt Fn wwhere α = Δtf, u v are u-values interpolated to v grid points, v u are v-valuesinterpolated to u grid points, and the remaining terms are given by:Fu n =− 1ρ o Δx (qn i,k+1 − qn i,k ) − Adv(u) + Diff(u)F n v=−Adv(v) + Diff(v) (4.17)Fw n =− 1ρ o Δz (qn i−1,k − qn i,k ) − Adv(w) + Diff(w)where Adv(ψ) denotes the nonlinear terms. As before, the first-guess arrays u ∗ andw ∗ are input to the right-hand side of the Poisson equation (Eq. 3.25) to be solvedby means of the S.O.R. iteration scheme.4.1.8 Potential ProblemsThe step structure of the sea floor in z-coordinate models gives rise to a number ofpotential problems, summarised in the following.• Each bottom step acts like an impermeable boundary and flows can only climb ordescent these steps via creation of vertical velocity. For gravity currents cascadingdownward on the slope, this implies artificial situations of unstable densitystratification.• The condition of no flow through a vertical face of a bottom step requires useof zero-gradient conditions for dynamic pressure across this face. This, however,implies absence of geostrophic flow running parallel to a bottom step, which cansubstantially disturb the near-bottom geostrophic flow field.• In the Arakawa C-grid, calculations of both the Coriolis force and bottom-stressterms require interpolation of surrounding velocity components (see Fig. 3.3).Discontinuities in vicinity of bottom steps lead to a bias in the dynamics.• Vertical friction operates on horizontal section of the sea floor, whereas lateralfriction occurs at vertical faces of bottom steps. The use of different valuesof horizontal and vertical turbulent viscosities creates dynamical irregularities.This can be avoided when disabling lateral momentum diffusion near bottomsteps.