Advanced Ocean Modelling: Using Open-Source Software

22 3 Basics of Nonhydrostatic ModellingFig. 3.1 Illustration of vertical levels in z-coordinate models and in σ -coordinate models3.2 2D Vertical-Slice Modelling3.2.1 ConfigurationWe consider a vertical ocean slice with the z-axis pointing upward and the x-axisbeing the horizontal coordinate (Fig. 3.2). The free sea surface is located at z =η. The sea floor can be found at z =−h o . The assumption of this chapter isabsence of both motion and gradients of variables perpendicular to this slice. Thisimplies neglect of the Coriolis force. A modification to the latter are so-called2.5-dimensional ocean-slice models which allow for flow normal to the slice andinclusion of the Coriolis force, but this flow is also assumed to have vanishinggradients in the y-direction. Chapter 4 presents applications of such models. Themomentum equations for the two-dimensional ocean slice are given by:∂u∂t + Adv(u) =−1 ∂ Pρ o ∂x + ∂ ( )∂uA h + ∂ (∂x ∂x ∂z∂w+ Adv(w) =− 1 ∂ P∂tρ o ∂z − ρ′g + ∂ (∂wA hρ o ∂x ∂x∂ρ ′+ Adv(ρ ′ ) = ∂ ∂ρ(K ′ )h + ∂ ∂ρ(K ′ )z∂t∂x ∂x ∂z ∂zThe advection operator is given by:Adv(ψ) = u ∂ψ∂x + w ∂ψ∂z)∂uA z∂z)+ ∂ ∂z( )∂wA z∂z(3.1)