Advanced Ocean Modelling: Using Open-Source Software

5.2 Numerical Treatment 127Vertical integration of the continuity equation gives a prognostic equation forsurface dynamic pressure, yielding:∂q s∂t( ∂(h 〈u〉)=−ρ o g +∂x)∂(h 〈v〉)∂y(5.5)where q s = ρ o gη with ρ o being surface density and η being sea-surface elevation,h is total local water depth, and 〈.〉 represents a vertical average.5.1.4 Evolution of the Density FieldSeawater density depends on temperature, salinity and pressure. For simplicity, weassume a linear dependence of seawater on temperature and salinity, ignore anypressure effects, and assume that eddy diffusivities for heat and salt are the same. Tothis end, the evolution of the density field can be described by a density-conservationequation, given by:∂ρ∂twhere the diffusion operator is given by:+ Adv(ρ) = Diff(ρ) (5.6)Diff(ρ) = ∂ ( )∂ρK h + ∂ ( )∂ρK h + ∂ ( )∂ρK z∂x ∂x ∂y ∂y ∂z ∂zwhere K h and K z , respectively, are horizontal and vertical eddy diffusivities.5.2 Numerical Treatment5.2.1 The 3d Arakawa C-gridThe index triplet (i, j, k) is used as a pointer to certain grid cells of the ArakawaC-grid (Fig. 5.1), where Δx is the grid spacing in the x-direction, Δy is the gridspacing in the y-direction, and Δz is the grid spacing in the vertical direction. Notethat the i index runs opposite to the z-coordinate. For convenience, we locate theCartesian coordinate system such that the x-axis points to the east, the y-axis tothe north, and the z-axis upward. Grid points of scalars (pressure, density, Eulerianconcentration, eddy viscosity, etc.) are centred between velocity grid points. Withineach grid cell, the u-grid point is located to the east, the v-grid point to the north,and the w-grid point above with respect to the scalar grid point.