Advanced Ocean Modelling: Using Open-Source Software

4 1 Introduction0 =− ∂ P∂z− ρg (1.5)The hydrostatic approximation forms the basis of the so-called shallow-watermodel and it is employed in so-called hydrostatic level models, being frequentlyapplied by fluid modellers. In contrast, models developed in this volume are basedon the full equations (Eq. 1.1) to enable the predictions of both hydrostatic and nonhydrostaticprocesses, a method called nonhydrostatic modelling. Nonhydrostaticprocesses include those in which horizontal and vertical scales are of similar orderof magnitude.1.1.5 The Stability FrequencyThe degree of density stratification in the ocean can be characterised by the so-calledBrunt-Väisälä frequency N, defined by:N 2 =− g ρ o∂ρ∂z(1.6)where ρ o is mean density. This frequency, which appears as a characteristic scalefor many stratified processes, is referred to as stability frequency in the following.1.2 Numerical Methods1.2.1 Finite DifferencesModels developed in this book are based on finite-difference versions of the Navier-Stokes equations. The basis is that dynamic variables (velocity components, sealevel, density, and dynamic pressure) are calculated at certain discrete grid points inspace. This requires an accurate representation of both gradients (the first derivative)and curvature (the second derivative) of the spatial distribution of a variable. Thesederivatives can be approximated from Taylor series. The first spatial derivative of avariable f , for instance, with respect to a space coordinate x can be approximatedin three different ways:• ∂ f/∂x ≈ ( f k+1 − f k )/Δx (forward difference)• ∂ f/∂x ≈ ( f k − f k−1 )/Δx (backward difference)• ∂ f/∂x ≈ ( f k+1 − f k−1 )/(2Δx) (centred difference)where Δx is the spacing between adjacent grid points, called grid spacing, and theindex k points to a certain grid cell along the x-axis. A finite-difference representationof the second spatial derivative of a function f is: