Advanced Ocean Modelling: Using Open-Source Software

56 3 Basics of Nonhydrostatic Modellingto a Froude number of unity. We anticipate that stronger density stratification leadsto smaller amplitudes of lee waves.3.13.3 Results: Two-Layer StratificationExistence of a pycnocline supports the creation of long interfacial waves of a phasespeed of (e.g. Pond and Pickard, 1983):c = √ g ′ h ∗ (3.63)where reduced gravity is g ′ = Δρ/ρ o g with Δρ being the difference in layerdensities, and the reduced depth scale is given by: h ∗ = H 1 H 2 /(H 1 + H 2 ), whereH 1 and H 2 , are the initial thicknesses of the upper and lower layers.In contrast to long internal waves in a fluid of continuous density stratification,the phase speed of such waves is independent of wavelength. In this situation, thecreation of a hydraulic jump exclusively depends on whether the incident flow isswift enough to become supercritical over the sill. For the density configurationof Scenario 2, the phase speed of interfacial waves is about 50 cm/s upstream thetopographic obstacle, but decreases to 28 cm/s over the hill. The occurrence of ahydraulic jump and lee wave formation requires an ambient flow of speed exceeding30 cm/s.The choice of a sea-level gradient of 0.1 cm over the length of the model domaincreates an ambient flow of maximum speed of 16 cm/s over the sill, which is notsufficient to create a hydraulic jump (not shown). The reader can easily verifythis statement with own simulations. For a increased sea-level gradient of 0.5 cm,the flow speed increases to values of 35 cm/s above the sill. This is sufficient toproduce interfacial waves of a height of 10 m in the lee of the sill (Fig. 3.29a).Fig. 3.29 Exercise 7.Scenario 2. Densitydistribution (shading andlines) andflowfield(arrows)after 35 min of simulation.Panel (a) shows result for abackground sea-level gradientof 0.5 cm over the length ofthe model domain, panel (b)for 1.5 cm. Flow vectors areaveraged over 5 by 5 gridcells