Progressive Internal Wave Rays in a Continuous Stratification

We have conveniently described standing waves in a continuous stratification in terms of wave modes. A continuous stratification will also support progressive sub-basin scale internal waves; however, it is more insightful to describe these waves in terms of rays (both methods of analysis can be shown to be equivalent). Unlike NLIWs, which require a thermo-cline waveguide, progressive waves in a continuous stratification are described by linear equations and occur in regions of the water column where N > 0 and slowly varying (e.g., Figure 1(e,f)). A disturbance in the flow (e.g., flow over rough topography) with a particular excitation frequency will generate a range of wavelengths that will radiate from the source at the same frequency. The wave rays will propagate through the fluid at a fixed angle to the horizontal b given by the dispersion relation

where o is the wave frequency and the wavenumber vector K = Vk2 + m2 has horizontal k and vertical m components (Figure 13(b)). The angle at which the rays propagate is chosen such that the vertical component of their frequency matches N, leading to a four-ray St. Andrew's cross pattern (Figure 13(a)). From the dispersion relation, the wave frequency is independent of the magnitude of the wavelength and only depends upon b. This property is quite different than for interfacial waves, where the wave frequency and period depend only on the magnitude of the wavelength.

Andrews Cross Internal Waves

z

/f

(-k,-m) y"

^ (k,-m)

JK<

CA\

Figure 13 Propagation of internal wave rays in a stratified fluid with constant N. (a) Laboratory images showing internal wave rays propagation from an oscillating cylinder. The light and dark bands are lines of constant phase (wave crests and troughs). In (i) and (ii) o = 0.4N and 0.9N giving b = 25° and 64°, respectively. From Mowbray DE and Rarity BSH (1967) A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. Journal of Fluid Mechanics 28: 1-16. (b) Schematic showing the direction of the group velocity cg, wavenumber vectors (showing the direction of phase propagation), and angle of inclination of the rays relative to the horizontal b, for the experiments in (a). Adapted from Thorpe SA (2005) The Turbulent Ocean. Cambridge, UK: Cambridge University Press.

Equations for the velocity and density perturbations induced by the wave passage are beyond the scope of this review and can be found in physical oceanography texts.

Another property of the dispersion relation is that the wave frequency must lie in the range 0 < o < N, mathematically showing that N is indeed the cut-off frequency for internal waves. Excitation at frequencies o > N generates motions that are exponentially damped. Wave energy will propagate from an excitation region at the group velocity cg of the wave envelope, which is perpendicular to the phase velocity c; the wave rays carry the energy at right angles to the motion of the crests and troughs! These waves are difficult to visualize, internal waves generated by a local source do not have the concentric circle pattern of crests and troughs familiar to those who observe a stone thrown into a pond, but are composed of crests that stretch outward as spokes (rays) carrying energy radially from the source. The wave crests and troughs slide perpendicularly across the rays, seeming to appear from and disappear to nowhere.

From the discussion above, it is not surprising that internal wave rays do interesting things as they reflect from sloping topography and propagate into regions of variable N(z). The intrinsic frequency o is always conserved causing rays propagating into depths of diminishing N(z) to refract towards the vertical and be totally reflected at the turning depth where o = N(z). If generated in the seasonal thermocline these waves can be trapped between the upper and lower surfaces of the metalimnion where o <N(z). Waves propagating into depths of increasing N(z), such as toward the seasonal thermocline, will refract toward the horizontal.

Upon reflection from the lake surface or a sloping bottom with angle a, the intrinsic wave frequency is conserved and, by the dispersion relation, the reflected ray must propagate at the same angle b as the incident ray (Figure 14). The wavelength and group velocity will change as is evident in the change in concentration of wave rays upon refection. If a = b, the rays are reflected parallel to the slope and have zero wavelength and group velocity. A turbulent bore will from and propagate along the slope and wave energy is rapidly converted into local dissipation and mixing. In this case, both the slope angle and wave frequency are considered critical.

Subcritical waves b < a will be reflected back in the direction from which they came and the may escape to deeper water (Figure 14(c,d)). However, supercritical waves b > a will continue in the same direction (Figure 14(a)) and if propagating towards shallower water in the littoral zone may thus be trapped, repeatedly reflecting off the surface, lake-bed and turning depths (Figure 15). Eventually the rays will break when on a critical slope a = b, where they have a critical frequency o = Nsin a.

Progressive internal waves are produced by small localized disturbances such as flow over rough topography, patches of shear and turbulence and wave-wave/wave-flow interactions, when the excitation frequency o < N. Progressive waves are found ubiquitously in lakes and those in the 10—5 to 10—3Hz bandwidth generally have critical frequencies relative to the sloping boundaries found where the metalim-nion intersects the lake bed.

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