Internal wave

File:InternalWaves Gibraltar ISS009-E-09952 54.jpg

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

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

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.

Internal waves in the ocean

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.

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.

External links

• Discussion and videos of internal waves made by an oscillating cylinder.
• Atlas of Oceanic Internal Waves - Global Ocean Associates

de:Interne Wellen