Advanced Ocean Modelling: Using Open-Source Software

2 1 IntroductionFig. 1.1 The Cartesian coordinate systemwhere (x, y, z) is location in the Cartesian coordinate system, (u,v,w) is the velocityvector, t is time, f is the Coriolis parameter, P is dynamic pressure, ρ is density,mean density is ρ o , and g is acceleration due to gravity. Density is weight of seawaterper unit volume. The operator Adv() denotes the advection terms and is givenby:Adv(ψ) = u ∂ψ∂x + v ∂ψ∂y + w ∂ψ∂zwhere ψ is the property subject to advection. Momentum advection is also referredto as the nonlinear terms. Diffusion of any of the three velocity components is givenby:Diff(ψ) = ∂ ( )∂ψA h + ∂ ( )∂ψA h + ∂ ( )∂ψA z∂x ∂x ∂y ∂y ∂z ∂zwhere A h and A z are horizontal and vertical eddy viscosities parameterising theeffects of turbulence. Dynamic pressure includes only pressure parts that have adynamical consequence. The pressure field associated with uniform density and aplane sea surface does not contribute to the horizontal pressure-gradient force and itcan therefore be subtracted from the true pressure field.The Boussinesq approximation, used in the above equation, is based on theassumption that density fluctuations are small compared with mean density, which isthe case for oceanic applications. To this end, density can be expressed by a constantvalue except when multiplied with gravity.The essence of the momentum equations is that an imbalance of forces actingon a fluid parcel causes an acceleration or deceleration of the parcel. On the otherhand, motions remain steady if the residual force vanishes, a situation referred to assteady state.For an incompressible fluid, mass conservation turns in a conservation principlefor volume, which can be expressed by the continuity equation, given by:∂u∂x + ∂v∂y + ∂w∂z = 0 (1.2)