Advanced Ocean Modelling: Using Open-Source Software

10 2 1D Models of Ekman LayersFig. 2.1 Examples of flow paths induced by inertial oscillations (a) in the absence of ambient flow,and (b) for an ambient flow of u amb = v amb = 0.02 m/s. The open circle shows the starting positionof a float, the closed circle the end position2.1.2 Semi-implicit Treatment of the Coriolis ForceAdequate formulation of the Coriolis force in a finite-difference model can beachieved by means of a semi-implicit approach. For the momentum equations governinginertial oscillations (Eqs. 2.1 and 2.2), this approach gives:u n+1 = u n + 0.5 α(v n + v n+1 )v n+1 = v n − 0.5 α(u n + u n+1 )where n is the current time level, n + 1 refers to the future value (one time stepΔt ahead), and α = Δtf. Cross-combination of the latter equations yields the finalform:u n+1 = [ (1 − β)u n + αv n] /(1 + β)v n+1 = [ (1 − β)v n − αu n] /(1 + β)where β = 0.25 α 2 . This scheme requires numerical time steps small compared withthe rotation period; that is, |α|