Internal wave

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 Earth's gravity. Note that the dispersion relation is the same as that for deep water surface waves by setting ${\displaystyle g^{\prime }=g.}$

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.

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.[6][7] Internal waves are often formed as tides pass over a shelf break.[8] The largest of these waves are generated during springtides and those of sufficient magnitude break and progress across the shelf as bores.[9][10] 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.[11]

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]

While internal waves of higher magnitudes will often break after crossing over the shelf break, smaller trains will proceed across the shelf unbroken.[10][14] 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.[15][16] Waters above an internal wave converge and sink in its trough and upwell and diverge over its crest.[15] The convergence zones associated with internal wave troughs often accumulate oils and flotsam that occasionally progress shoreward with the slicks.[17][18] These rafts of flotsam can also harbor high concentrations of larvae of invertebrates and fish an order of magnitude higher than the surrounding waters.[18]

Thermoclines are often associated with chlorophyll maximum layers.[13] Internal waves represent oscillations of these thermoclines and therefore have the potential to transfer these phytoplankton rich waters downward, coupling benthic and pelagic systems.[19][20] 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.[21] Periodic depression of the thermocline and associated downwelling may also play an important role in the vertical transport of planktonic larvae.

Large steep internal waves containing trapped, reverse-oscillating cores can also transport parcels of water shoreward.[22] These non-linear waves with trapped cores had previously been observed in the laboratory[23] and predicted theoretically.[24] 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.[22] 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.