Advanced Ocean Modelling: Using Open-Source Software

3.13 Exercise 7: Lee Waves 55case studies consider variations of the barotropic forcing with the background sealevel varying between 0.1 and 1.5 cm over the length of the model domain.Density diffusion is included in both scenarios with horizontal and vertical densitydiffusivities being set to small uniform values of K h = K z = 1 × 10 −4 m 2 /s.The total simulation time is 60 min with data outputs at one-minute intervals. Thepressure accuracy for the S.O.R. iteration is set to ɛ = 1 × 10 −3 Pa. The time stepis set to Δt = 1 s, using the rigid-lid approximation. Note that neither momentumdiffusion nor boundary friction is yet included in the momentum equations.3.13.2 Results: Continuous Density StratificationFor a weak initial stratification (N 2 = 0.5 × 10 −4 s −2 ), the flow over the sill triggersa standing internal lee wave of an enormous wave height of 40 m on a wavelength ofapproximately 150 m (Fig. 3.28a). Near-bottom water is lifted across the sill whereit becomes subject to vigorous mixing via the breaking of internal waves. This instabilitymechanism is responsible for the localised generation of internal waves in theocean.The situation is different for a stronger initial stratification (N 2 = 5 × 10 −4 s −2 )(Fig. 3.28b). In this case, the sill operates as a barrier for the flow with the consequencethat flow below sill depth is almost absent. Interaction between flow at middepth and bathymetry creates lee waves of a reduced height of 10 m. This demonstratesthat, for strong density stratification, bathymetric obstacles can operate as abarrier for flows.According to Eq. (3.55), the phase speed of internal waves in a fluid of continuousdensity stratification depends on the wavelength of the disturbance. In this situation,a hydraulic jump flow will always generate lee waves of a wavelength correspondingFig. 3.28 Exercise 7.Scenario 1. Densitydistribution (shading) andflow field (arrows) after40 min of simulation.Panel (a) shows resultsfor an initial value ofN 2 = 0.5 × 10 −4 s −2 , panel(b) forN 2 = 5 × 10 −4 s −2 .Flow vectors are averagedover5by5gridcells