Advanced Ocean Modelling: Using Open-Source Software

4.1 The Basis 101The speed of frontal flows decreases exponentially (on a spatial scale of R) withincreasing distance from the surface outcrop of density surfaces. In the general case,a counterflow establishes in the water column underneath a surface density front.Density fronts in the coastal ocean attain typical widths of 1–20 km.4.1.6 The 2.5d Shallow-Water ModelOn the basis of vanishing gradients of variables in the y-direction and inclusion ofthe Coriolis force, the horizontal momentum equations for a vertical ocean slice canbe written as:∂u∂t + u ∂u∂x + w ∂u∂z − f v =−1 ∂(p + q)+ Diff(u) (4.14)ρ o ∂x∂v∂t + u ∂v∂x + w ∂v + fu = Diff(v) (4.15)∂zwhere f is the Coriolis parameter and Diff(ψ) is the diffusion operator, being specifiedin Sect. 3.15. The other model equations are the same as in Exercise 12.4.1.7 Implementation of the Coriolis ForceIn the 2.5d version of the Arakawa C-grid, the v-components of velocity are calculatedat pressure grid points (see Fig. 3.3). Hence, interpolation of velocity valuesis required for calculation of the Coriolis force. It can be shown that explicit formulationof the Coriolis force is numerically unstable. The nonhydrostatic solver ofthe Navier-Stokes equations is already formulated in an implicit manner in termsof dynamic pressure (see Sect. 3.4). Using another semi-implicit approach for theCoriolis force would lead to an even more complex solver. To avoid this, the Coriolisforce is treated here by the local-rotation approach (described in Sect. 3.14 of Kämpf(2009)). With the sole presence of the Coriolis force, this approach leads to thefinite-difference equations:u n+1 = cos(α)u n + sin(α)v n ,v n+1 = cos(α)v n − sin(α)u nwhere the rotation angle is α = 2arcsin(0.5Δtf). For sufficiently small numericaltime steps of Δt | f |