Advanced Ocean Modelling: Using Open-Source Software

3.9 Exercise 5: Internal Waves 47Fig. 3.20 Initial density distribution (shading and contours) and bathymetry for Exercise 5near the left boundary (Fig. 3.20). The ambient ocean is initially at rest and linearlystratified characterised by a stability frequency of N = 0.1 s −1 . Surface densityis set to ρ s = 1,028 kg/m 3 . The minimum period of internal waves according toEq. (3.54) is about 63 s. Such a strong density stratification does rarely exist inthe ocean. The sole purpose here is to minimise the total simulation time whilstcapturing about 10 wave periods.An initial density disturbance is added in the centre of the model domain overa width of 25 m (five adjacent grid cells). Density is increased by 20 kg/m 3 in thewater column in this region, but the maximum density is limited by the bottom densityfound in ambient water. The purpose of this treatment is to prevent creation of adensity-driven bottom-arrested flow which already has been studied in the previousexercise.Horizontal and vertical density diffusivities are set to small uniform values ofK h = K z = 1 × 10 −4 m 2 /s. The rigid-lid version of the nonhydrostatic verticalocean-slice model is applied with a time step of Δt = 1 s and a pressure accuracyof ɛ = 0.01 Pa. The simulation time is 10 mins with data outputs every 10 secs.3.9.3 ResultsOwing to reflection at closed boundaries, wave disturbances in closed domains triggerthe formation of standing waves. Standing waves are waves of zero horizontalpropagation that, at certain locations called nodes, exhibit no vertical displacements.Exclusively vertical displacements are found between nodes and near lateral boundaries.The initial density disturbance creates an internal wave pattern that becomesreflected at the closed lateral boundaries (Fig. 3.21). At times, internal waves breakand induce vertical mixing. Reflection at closed boundaries gives rise to a complexwave pattern. Vertical profiles of vertical velocity (Fig. 3.22) reveal wave shapes ofone or two maxima in the water column corresponding to mode numbers n = 1and n = 2 (see Fig. 3.19), demonstrating that the vertical boundaries of fluidsoperate as a waveguide. Notice that the vertical speed of wave motions exceeds40 cm/s.