Advanced Ocean Modelling: Using Open-Source Software

80 3 Basics of Nonhydrostatic ModellingFig. 3.47 Tilted coordinate systemdynamical equations can be written as:∂u r∂t∂w r∂t+ u r∂u r∂x r+ w r∂u r∂z r+ u r∂w r∂x r+ w r∂w r∂z r=− 1 ρ o∂ P∂x r+ sin (γ ) ρ′ρ og + Diff(u r ) (3.78)=− 1 ρ o∂ P∂z r− cos (γ ) ρ′ρ og + Diff(w r ) (3.79)∂u r∂x r+ ∂w r∂z r= 0 (3.80)∂ρ∂t + u ∂ρ ∂ρr + w r = ∂ ( )∂ρK h + ∂ ( )∂ρK z∂x r ∂z r ∂x r ∂x r ∂x r ∂z r(3.81)where u r is the bottom-parallel component of velocity, and w r is the velocity componentperpendicular to the sea floor. The diffusion operator can be defined by:Diff(ψ) =∂ ( )∂ψA h + ∂ ( )∂ψA z∂x r ∂x r ∂z r ∂z rwhich is only justifiable for small bottom inclinations. As the reader can see, all thiscoordinate transformation does is to rotate direction-dependent forces such as thereduced-gravity force. The component of the reduced-gravity force normal to thesea floor is sometimes referred to as buoyant-slope effect. In case the buoyant-slopeeffect is fully balanced by a linear bottom-friction force, we yield:sin (γ ) ρ′ρ og − ru r = 0where r is a linear bottom-drag coefficient. This gives an equilibrium flow speed ofu r = sin (γ )g ′ /r, where g ′ is reduced gravity, which depends on the density excesscarried by the flow, bottom inclination and frictional effects. This relation, however,