Advanced Ocean Modelling: Using Open-Source Software

3.8 Internal Waves 45which implies purely horizontal propagation. This corresponds to the period ofoscillations experienced by a buoyant object in a stratified fluid (see Sect. 3.10 ofKämpf (2009)). Consequently, such internal waves propagate at a phase speed of:c = λ T=±λNcos (θ)2πwhere λ is the wavelength. The fact that two signs are allowed indicates that thewave can travel into one of two directions (see Fig. 3.18). If the frequency of aninternal wave is imposed via external forcing, regardless of wavelength, all wavespropagate at a certain fixed angle from the horizontal. The longer the period thesteeper the direction.Fig. 3.18 Vertical structure of an internal wave. Adapted from Cushman-Roisin (1994)3.8.2 Normal Wave ModesIn the ocean interior, internal wave motion can induce large vertical excursions ofdensity interfaces of several tens of metres. Vertical boundaries (sea surface andsea floor) do not permit such large-amplitude vertical oscillations. Consequently,vertical velocity inherent with internal waves has to vanish at the sea floor and ithas to become very small at the sea surface. Since the resultant waves can onlypropagate horizontally, the vertical boundaries operate as a waveguide.Only a discrete set of wave solutions, so-called normal modes, satisfies theconditions of vanishing vertical velocity at vertical boundaries. It can be shownthat, for a constant stability frequency N, possible wave frequencies are (Pond andPickard, 1983):Nσ =± √1 + (0.5nλ/h)2where n = 1, 2, 3, ··· is the mode number, λ is horizontal wavelength, and h istotal water depth.