Internal wave

Internal waves are

), the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

Internal waves, also called internal gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude, and influence of external forces. If propagating horizontally along an interface where the density rapidly decreases with height, they are specifically called interfacial (internal) waves. If the interfacial waves are large amplitude they are called internal solitary waves or internal

with latitude.

Visualization of internal waves

An internal wave can readily be observed in the kitchen by slowly tilting back and forth a bottle of salad dressing - the waves exist at the interface between oil and vinegar.

Atmospheric internal waves can be visualized by

, used by some daredevils to glide along like a surfer riding an ocean wave. Satellites over Australia and elsewhere reveal these waves can span many hundreds of kilometers.

Undulations of the oceanic thermocline can be visualized by satellite because the waves increase the surface roughness where the horizontal flow converges, and this increases the scattering of sunlight (as in the image at the top of this page showing of waves generated by tidal flow through the Strait of Gibraltar).

Buoyancy, reduced gravity and buoyancy frequency

According to

, the weight of an immersed object is reduced by the weight of fluid it displaces. This holds for a fluid parcel of density ${\displaystyle \rho }$ surrounded by an ambient fluid of density ${\displaystyle \rho _{0}}$. Its weight per unit volume is ${\displaystyle g(\rho -\rho _{0})}$, in which ${\displaystyle g}$ is the acceleration of gravity. Dividing by a characteristic density, ${\displaystyle \rho _{00}}$, gives the definition of the reduced gravity:

${\displaystyle g^{\prime }\equiv g{\frac {\rho -\rho _{0}}{\rho _{00}}}}$

If ${\displaystyle \rho >\rho _{0}}$, ${\displaystyle g^{\prime }}$ is positive though generally much smaller than ${\displaystyle g}$. Because water is much more dense than air, the displacement of water by air from a surface gravity wave feels nearly the full force of gravity (${\displaystyle g^{\prime }\sim g}$). The displacement of the thermocline of a lake, which separates warmer surface from cooler deep water, feels the buoyancy force expressed through the reduced gravity. For example, the density difference between ice water and room temperature water is 0.002 the characteristic density of water. So the reduced gravity is 0.2% that of gravity. It is for this reason that internal waves move in slow-motion relative to surface waves.

Whereas the reduced gravity is the key variable describing buoyancy for interfacial internal waves, a different quantity is used to describe buoyancy in continuously stratified fluid whose density varies with height as ${\displaystyle \rho _{0}(z)}$. Suppose a water column is in

and a small parcel of fluid with density ${\displaystyle \rho _{0}(z_{0})}$ is displaced vertically by a small distance ${\displaystyle \Delta z}$. The buoyant restoring force results in a vertical acceleration, given by^[1][2]

${\displaystyle {\frac {d^{2}\Delta z}{dt^{2}}}=-g^{\prime }=-g(\rho _{0}(z_{0})-\rho _{0}(z_{0}+\Delta z))/\rho _{0}(z_{0})\simeq -g\left(-{\frac {d\rho _{0}}{dz}}\Delta z\right)/\rho _{0}(z_{0})}$

This is the spring equation whose solution predicts oscillatory vertical displacement about ${\displaystyle z_{0}}$ in time about with frequency given by the buoyancy frequency:

${\displaystyle N=\left(-{\frac {g}{\rho _{0}}}{\frac {d\rho _{0}}{dz}}\right)^{1/2}.}$

The above argument can be generalized to predict the frequency, ${\displaystyle \omega }$, of a fluid parcel that oscillates along a line at an angle ${\displaystyle \Theta }$ to the vertical:

${\displaystyle \omega =N\cos \Theta }$.

This is one way to write the dispersion relation for internal waves whose lines of constant phase lie at an angle ${\displaystyle \Theta }$ to the vertical. In particular, this shows that the buoyancy frequency is an upper limit of allowed internal wave frequencies.

Mathematical modeling of internal waves

The theory for internal waves differs in the description of interfacial waves and vertically propagating internal waves. These are treated separately below.

Interfacial waves

In the simplest case, one considers a two-layer fluid in which a slab of fluid with uniform density ${\displaystyle \rho _{1}}$ overlies a slab of fluid with uniform density ${\displaystyle \rho _{2}}$. Arbitrarily the interface between the two layers is taken to be situated at ${\displaystyle z=0.}$ The fluid in the upper and lower layers are assumed to be irrotational. So the velocity in each layer is given by the gradient of a velocity potential, ${\displaystyle {{\vec {u}}=\nabla \phi ,}}$ and the potential itself satisfies Laplace's equation:

${\displaystyle \nabla ^{2}\phi =0.}$

Assuming the domain is unbounded and two-dimensional (in the ${\displaystyle x-z}$ plane), and assuming the wave is periodic in ${\displaystyle x}$ with wavenumber ${\displaystyle k>0,}$ the equations in each layer reduces to a second-order ordinary differential equation in ${\displaystyle z}$. Insisting on bounded solutions the velocity potential in each layer is

${\displaystyle \phi _{1}(x,z,t)=Ae^{-kz}\cos(kx-\omega t)}$

and

${\displaystyle \phi _{2}(x,z,t)=Ae^{kz}\cos(kx-\omega t),}$

with ${\displaystyle A}$ the amplitude of the wave and ${\displaystyle \omega }$ its angular frequency. In deriving this structure, matching conditions have been used at the interface requiring continuity of mass and pressure. These conditions also give the dispersion relation:^[3]

${\displaystyle \omega ^{2}=g^{\prime }k}$

in which the reduced gravity ${\displaystyle g^{\prime }}$ is based on the density difference between the upper and lower layers:

${\displaystyle g^{\prime }={\frac {\rho _{2}-\rho _{1}}{\rho _{2}+\rho _{1}}}\,g,}$

with ${\displaystyle g}$ the

by setting ${\displaystyle g^{\prime }=g.}$

Internal waves in uniformly stratified fluid

The structure and dispersion relation of internal waves in a uniformly stratified fluid is found through the solution of the linearized conservation of mass, momentum, and internal energy equations assuming the fluid is incompressible and the background density varies by a small amount (the Boussinesq approximation). Assuming the waves are two dimensional in the x-z plane, the respective equations are

${\displaystyle \partial _{x}u+\partial _{z}w=0}$

${\displaystyle \rho _{00}\partial _{t}u=-\partial _{x}p}$

${\displaystyle \rho _{00}\partial _{t}w=-\partial _{z}p-\rho g}$

${\displaystyle \partial _{t}\rho =-wd\rho _{0}/dz}$

in which ${\displaystyle \rho }$ is the perturbation density, ${\displaystyle p}$ is the pressure, and ${\displaystyle (u,w)}$ is the velocity. The ambient density changes linearly with height as given by ${\displaystyle \rho _{0}(z)}$ and ${\displaystyle \rho _{00}}$, a constant, is the characteristic ambient density.

Solving the four equations in four unknowns for a wave of the form ${\displaystyle \exp[i(kx+mz-\omega t)]}$ gives the dispersion relation

${\displaystyle \omega ^{2}=N^{2}{\frac {k^{2}}{k^{2}+m^{2}}}=N^{2}\cos ^{2}\Theta }$

in which ${\displaystyle N}$ is the buoyancy frequency and ${\displaystyle \Theta =\tan ^{-1}(m/k)}$ is the angle of the wavenumber vector to the horizontal, which is also the angle formed by lines of constant phase to the vertical.

The phase velocity and group velocity found from the dispersion relation predict the unusual property that they are perpendicular and that the vertical components of the phase and group velocities have opposite sign: if a wavepacket moves upward to the right, the crests move downward to the right.

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

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 in 1893 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 huge amount of energy.^[4]

Properties of internal waves

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Internal waves typically have much lower frequencies and higher amplitudes than

because the density differences (and therefore the restoring forces) within a fluid are usually much smaller. Wavelengths vary from centimetres to kilometres with periods of seconds to hours respectively.

The atmosphere and ocean are continuously stratified:

.

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

.

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.^[5] 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 and the face of the wave to steepen, and finally the wave will break, propagating forward as an internal bore.

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.^[12] 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^[13] that internal bores subsequently push inshore. Many taxa can be almost absent in warm surface waters, yet plentiful in these internal bores.^[12]

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.

Predictable downwellings

Thermoclines are often associated with chlorophyll maximum layers.

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.

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

Other

• Sutherland, Bruce (October 2010). Internal Gravity Waves.
  ISBN 978-0-52-183915-0
  . Retrieved 7 June 2013.
• Cushman-Roisin, Benoit; Beckers, Jean-Marie (October 2011). Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects (Second ed.).
  ISBN 978-0-12-088759-0
  .
• Pedlosky, Joseph (1987). Geophysical Fluid Dynamics (Second ed.).
  ISBN 978-0-387-96387-7
  .
• ISBN 978-0-19-854489-0
  .
• Thomson, R.E. (1981). Oceanography of the British Columbia Coast (Canadian Special Publication of Fisheries & Aquatic Sciences). Gordon Soules Book Pub.
  ISBN 978-0-660-10978-7
  .

External links

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