Internal waves (marked with arrows) caused by the Strait of Gibraltar

Internal waves are gravity waves that oscillate within, rather than on the surface of, a fluid medium. They are one of many types of wave motion in stratified fluids (another example being Lee waves). A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water. Internal wave motions are ubiquitous in both the ocean and atmosphere, where they create wave clouds. Nonlinear solitary internal waves are called solitons.

## Gravity versus buoyancy

Suppose a water column is in hydrostatic equilibrium and displace a small packet of fluid with density $\rho_0$ vertically by a distance $\Delta z$. The balance between the force of gravity and the buoyant restoring force is disturbed, and the motion of the packet obeys the equation[1]

$\rho_0 \frac{d^2\Delta z}{dt^2} = g \frac{d\rho_0}{dz}\Delta z,$

where $g$ is the gravitational acceleration. The packet oscillates at the Brunt–Väisälä frequency:

$N = \left(-\frac{g}{\rho_0} \frac{d\rho_0}{dz}\right)^{1/2}.$

The Brunt–Väisälä frequency is an upper limit for the frequency in actual internal waves.[1]

## Internal waves in the ocean

Internal Wave trains around Trinidad, as seen from space

Most people think of waves as a surface phenomenon, which acts between water (as in lakes or oceans) and the air. Where low density water overlies high density water in the ocean, internal waves propagate along the boundary. They are especially common over the continental shelf regions of the world oceans and where brackish water overlies salt water at the outlet of large rivers. There is typically little surface expression of the waves, aside from slick bands that can form over the trough of the waves.

Internal waves are the source of a curious phenomenon called dead water, first reported by the Norwegian oceanographer Fridtjof Nansen, in which a boat may experience strong resistance to forward motion in apparently calm conditions. This occurs when the ship is sailing on a layer of relatively fresh water whose depth is comparable to the ship's draft. This causes a wake of internal waves that dissipates a lot of energy.[2]

## Properties of internal waves

Internal waves typically have much lower frequencies and higher amplitudes than surface gravity waves because the density differences (and therefore the restoring forces) within a fluid are usually much smaller than the density of the fluid itself. Wavelengths vary from centimetres to kilometres with periods of seconds to hours respectively.

The atmosphere and ocean are continuously stratified: potential density generally increases steadily downward. Internal waves in a continuously stratified medium may propagate vertically as well as horizontally. The dispersion relation for such waves is curious: For a freely-propagating internal wave packet, the direction of propagation of energy (group velocity) is perpendicular to the direction of propagation of wave crests and troughs (phase velocity). An internal wave may also become confined to a finite region of altitude or depth, as a result of varying stratification or wind. Here, the wave is said to be ducted or trapped, and a vertically standing wave may form, where the vertical component of group velocity approaches zero. A ducted internal wave mode may propagate horizontally, with parallel group and phase velocity vectors, analogous to propagation within a waveguide.

At large scales, internal waves are influenced both by the rotation of the Earth as well as by the stratification of the medium. The frequencies of these geophysical wave motions vary from a lower limit of the Coriolis frequency (inertial motions) up to the Brunt–Väisälä, or buoyancy frequency (buoyancy oscillations). Above the Brunt–Väisälä frequency may exist evanescent internal wave motions, for example those resulting from partial reflection. Internal waves at tidal frequencies are produced by tidal flow over topography/bathymetry, and are known as internal tides. Similarly, Atmospheric tides arise from, for example, non-uniform solar heating associated with diurnal motion.

## Onshore transport of planktonic larvae

Cross-shelf transport, the exchange of water between coastal and offshore environments, is of particular interest for its role in delivering meroplanktonic larvae to often disparate adult populations from shared offshore larval pools.[3] Several mechanisms have been proposed for the cross-shelf of planktonic larvae by internal waves. The prevalence of each type of event depends on a variety of factors including bottom topography, stratification of the water body and tidal influences.

### Internal tidal bores

Similarly to surface waves, internal waves change as they approach the shore. As the ratio of wave amplitude to water depth becomes such that the wave “feels the bottom,” water at the base of the wave slows down due to friction with the sea floor. This causes the wave to become asymmetrical, the face of the wave to steepen and finally the wave will break, propagating forward as an internal bore.[4][5] Internal waves are often formed as tides pass over a shelf break.[6] The largest of these waves are generated during springtides and those of sufficient magnitude break and progress across the shelf as bores.[7][8] These bores are evidenced by rapid, step-like changes in temperature and salinity with depth, the abrupt onset of upslope flows near the bottom and packets of high frequency internal waves following the fronts of the bores.[9]

The arrival of cool, formerly deep water associated with internal bores into warm, shallower waters corresponds with drastic increases in phytoplankton and zooplankton concentrations and changes in plankter species abundances.[10] Additionally, while both surface waters and those at depth tend to have relatively low primary productivity, thermoclines are often associated with a chlorophyll maximum layer. These layers in turn attract large aggregations of mobile zooplankton[11] that internal bores subsequently push inshore. Many taxa can be almost absent in warm surface waters, yet plentiful in these internal bores.[10]

### Surface slicks

While internal waves of higher magnitudes will often break after crossing over the shelf break, smaller trains will proceed across the shelf unbroken.[8][12] At low wind speeds these internal waves are evidenced by the formation of wide surface slicks, oriented parallel to the bottom topography, which progress shoreward with the internal waves.[13][14] Waters above an internal wave converge and sink in its trough and upwell and diverge over its crest.[13] The convergence zones associated with internal wave troughs often accumulate oils and flotsam that occasionally progress shoreward with the slicks.[15][16] These rafts of flotsam can also harbor high concentrations of larvae of invertebrates and fish an order of magnitude higher than the surrounding waters.[16]

### Predictable downwellings

Thermoclines are often associated with chlorophyll maximum layers.[11] Internal waves represent oscillations of these thermoclines and therefore have the potential to transfer these phytoplankton rich waters downward, coupling benthic and pelagic systems.[17][18] Areas affected by these events show higher growth rates of suspension feeding ascidians and bryozoans, likely due to the periodic influx of high phytoplankton concentrations.[19] Periodic depression of the thermocline and associated downwelling may also play an important role in the vertical transport of planktonic larvae.

### Trapped cores

Large steep internal waves containing trapped, reverse-oscillating cores can also transport parcels of water shoreward.[20] These non-linear waves with trapped cores had previously been observed in the laboratory[21] and predicted theoretically.[22] These waves propagate in environments characterized by high shear and turbulence and likely derive their energy from waves of depression interacting with a shoaling bottom further upstream.[20] The conditions favorable to the generation of these waves are also likely to suspend sediment along the bottom as well as plankton and nutrients found along the benthos in deeper water.

## References

### Footnotes

1. ^ a b (Tritton 1990, pp. 208–214)
2. ^ (Cushman-Roisin and Beckers 2011, pp. 7)
3. ^ Botsford LW, Moloney CL, Hastings A, Largier JL, Powell TM, Higgins K, Quinn JF (1994) The influence of spatially and temporally varying oceanographic conditions on meroplanktonic metapopulations. Deep Sea Research II 41:107–145
4. ^ Defant A (1961) Physical Oceanography, 2nd edn. Pergamon Press, New York
5. ^ Cairns JL (1967) Asymmetry of internal tidal waves in shallow coastal waters. Journal of Geophysical Research 72:3563–3565
6. ^ Rattray MJ (1960) On coastal generation of internal tides. Tellus 12:54–62
7. ^ Winant CD, Olson JR (1976) The vertical structure of coastal currents. Deep Sea Research 23:925–936
8. ^ a b Winant CD (1980) Downwelling over the Southern California shelf. Journal of Physical Oceanography 10:791–799
9. ^ Shanks AL (1995) Mechanisms of cross-shelf dispersal of larval invertebrates and fish. In: McEdward L (ed) Ecology of marine invertebrate larvae. CRC Press, Boca Raton, FL, p 323–336
10. ^ a b Leichter JJ, Shellenbarger G, Genovese SJ, Wing SR (1998) Breaking internal waves on a Florida (USA) coral reef: a plankton pump at work? Marine Ecology Progress Series 166:83–97
11. ^ a b Mann KH, Lazier JRN (1991) Dynamics of marine ecosystems. Blackwell, Boston
12. ^ Cairns JL (1968) Thermocline strength fluctuations in coastal waters. Journal of Geophysical Research 73:2591–2595
13. ^ a b Ewing G (1950) Slicks, surface films and internal waves. Journal of Marine Research 9:161–187
14. ^ LaFond EC (1959) Sea surface features and internal waves in the sea. Indian Journal of Meteorology and Geophysics 10:415–419
15. ^ Arthur RS (1954) Oscillations in sea temperature at Scripps and Oceanside piers. Deep Sea Research 2:129–143
16. ^ a b Shanks AL (1983) Surface slicks associated with tidally forces internal waves may transport pelagic larvae of benthic invertebrates and fishes shoreward. Marine Ecology Progress Series 13:311–315
17. ^ Haury LR, Brisco MG, Orr MH (1979) Tidally generated internal wave packets in Massachusetts Bay. Nature 278:312–317
18. ^ Haury LR, Wiebe PH, Orr MH, Brisco MG (1983) Tidally generated high-frequency internal wave-packets and their effects on plankton in Massachusetts Bay. Journal of Marine Research 41:65–112
19. ^ Witman JD, Leichter JJ, Genovese SJ, Brooks DA (1993) Pulsed Phytoplankton Supply to the Rocky Subtidal Zone: Influence of Internal Waves. Proceedings of the National Academy of Sciences 90:1686–1690
20. ^ a b Scotti A, Pineda J (2004) Observation of very large and steep internal waves of elevation near the Massachusetts coast. Geophysical Research Letters 31:1–5
21. ^ Manasseh R, Chin CY, Fernando HJ (1998) The transition from density-driven to wave-dominated isolated flows. Journal of Fluid Mechanics 361:253–274
22. ^ Derzho OG, Grimshaw R (1997) Solitary waves with a vortex core in a shallow layer of stratified fluid. Physics of Fluids 9:3378–3385