Internal waves are
gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s that
oscillate
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
within a fluid medium, rather than on its surface. To exist, the fluid must be
stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance (as in the case of the
thermocline
A thermocline (also known as the thermal layer or the metalimnion in lakes) is a thin but distinct layer in a large body of fluid (e.g. water, as in an ocean or lake; or air, e.g. an atmosphere) in which temperature changes more drastically with ...
in lakes and oceans or an
atmospheric inversion
In meteorology, an inversion is a deviation from the normal change of an atmospheric property with altitude. It almost always refers to an inversion of the air temperature lapse rate, in which case it is called a temperature inversion. N ...
), 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
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s. If moving vertically through the atmosphere where substantial changes in air density influences their dynamics, they are called anelastic (internal) waves. If generated by flow over topography, they are called
Lee waves
In meteorology, lee waves are atmospheric stationary waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above ...
or
mountain waves
In meteorology, lee waves are atmospheric stationary waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above ...
. If the mountain waves break aloft, they can result in strong warm winds at the ground known as
Chinook winds
Chinook winds, or simply Chinooks, are two types of prevailing warm, generally westerly winds in western North America: Coastal Chinooks and interior Chinooks. The coastal Chinooks are persistent seasonal, wet, southwesterly winds blowing in from ...
(in North America) or
Foehn
A Foehn or Föhn (, , ), is a type of dry, relatively warm, downslope wind that occurs in the lee (downwind side) of a mountain range.
It is a rain shadow wind that results from the subsequent adiabatic warming of air that has dropped most of ...
winds (in Europe). If generated in the ocean by tidal flow over submarine ridges or the continental shelf, they are called internal tides. If they evolve slowly compared to the Earth's rotational frequency so that their dynamics are influenced by the
Coriolis effect
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
, they are called
inertia gravity waves or, simply,
inertial waves
Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not ...
. Internal waves are usually distinguished from
Rossby wave
Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby. They are observed in the atmospheres and ...
s, which are influenced by the change of
Coriolis frequency
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate o ...
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
wave cloud
A wave cloud is a cloud form created by atmospheric internal waves.
Formation
The atmospheric internal waves that form wave clouds are created as stable air flows over a raised land feature such as a mountain range, and can form either direc ...
s: at the wave crests air rises and cools in the relatively lower pressure, which can result in water vapor condensation if the
relative humidity
Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present.
Humidity depe ...
is close to 100%. Clouds that reveal internal waves launched by flow over hills are called
lenticular clouds
Lenticular clouds (Latin: ''Lenticularis'' lentil-shaped, from ''lenticula'' lentil) are stationary clouds that form mostly in the troposphere, typically in parallel alignment to the wind direction. They are often comparable in appearance to ...
because of their lens-like appearance. Less dramatically, a train of internal waves can be visualized by rippled cloud patterns described as
herringbone sky or
mackerel sky
A mackerel sky is a common term for clouds made up of rows of cirrocumulus or altocumulus clouds displaying an undulating, rippling pattern similar in appearance to fish scales; this is caused by high altitude atmospheric waves.
Cirrocumulus ap ...
. The outflow of cold air from a thunderstorm can launch large amplitude internal solitary waves at an
atmospheric inversion
In meteorology, an inversion is a deviation from the normal change of an atmospheric property with altitude. It almost always refers to an inversion of the air temperature lapse rate, in which case it is called a temperature inversion. N ...
. In northern Australia, these result in
Morning Glory clouds, 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
The Strait of Gibraltar ( ar, مضيق جبل طارق, Maḍīq Jabal Ṭāriq; es, Estrecho de Gibraltar, Archaic: Pillars of Hercules), also known as the Straits of Gibraltar, is a narrow strait that connects the Atlantic Ocean to the Medi ...
).
Buoyancy, reduced gravity and buoyancy frequency
According to
Archimedes principle
Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimedes' ...
, the weight of an immersed object is reduced by the weight of fluid it displaces. This holds for a fluid parcel of density
surrounded by an ambient fluid of density
. Its weight per unit volume is
, in which
is the acceleration of gravity. Dividing by a characteristic density,
, gives the definition of the reduced gravity:
:
If
,
is positive though generally much smaller than
. Because water is much more dense than air, the displacement of water by air from a surface
gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
feels nearly the full force of gravity (
). The displacement of the
thermocline
A thermocline (also known as the thermal layer or the metalimnion in lakes) is a thin but distinct layer in a large body of fluid (e.g. water, as in an ocean or lake; or air, e.g. an atmosphere) in which temperature changes more drastically with ...
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
. Suppose a water column is in
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
and a small parcel of fluid with density
is displaced vertically by a small distance
. The
buoyant
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
restoring force results in a vertical acceleration, given by
[(Sutherland 2010, pp 141-151)]
:
This is the spring equation whose solution predicts oscillatory vertical displacement about
in time about with frequency given by the
buoyancy frequency
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
:
:
The above argument can be generalized to predict the frequency,
, of a fluid parcel that oscillates along a line at an angle
to the vertical:
:
.
This is one way to write the dispersion relation for internal waves whose lines of constant phase lie at an angle
to the vertical. In particular, this shows that the
buoyancy frequency
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
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
overlies a slab of fluid with uniform density
. Arbitrarily the interface between the two layers is taken to be situated at
The fluid in the upper and lower layers are assumed to be
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
. So the velocity in each layer is given by the gradient of a
velocity potential,
and the potential itself satisfies
Laplace's equation:
:
Assuming the domain is unbounded and two-dimensional (in the
plane), and assuming the wave is
periodic in
with
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
the equations in each layer reduces to a second-order ordinary differential equation in
. Insisting on bounded solutions the velocity potential in each layer is
:
and
:
with
the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of the wave and
its
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. 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
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
:
:
in which the reduced gravity
is based on the density difference between the upper and lower layers:
:
with
the
Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector quantity ...
. Note that the dispersion relation is the same as that for deep water
surface waves
In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at th ...
by setting
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
:
:
:
:
in which
is the perturbation density,
is the pressure, and
is the velocity. The ambient density changes linearly with height as given by
and
, a constant, is the characteristic ambient density.
Solving the four equations in four unknowns for a wave of the form