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In
fluid dynamics, gravity waves are waves generated in a
fluid medium or at the
interface
Interface or interfacing may refer to:
Academic journals
* ''Interface'' (journal), by the Electrochemical Society
* '' Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Linguistics''
* '' Int ...
between two media when the
force of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
or
buoyancy
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 ...
tries to restore equilibrium. An example of such an interface is that between the
atmosphere and the
ocean
The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the wo ...
, which gives rise to
wind waves.
A gravity wave results when fluid is displaced from a position of
equilibrium. The restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a ''wave orbit''. Gravity waves on an air–sea interface of the ocean are called surface gravity waves (a type of
surface wave), while gravity waves that are the body of the water (such as between parts of different densities) are called ''
internal wave
Internal waves are gravity waves that oscillate 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 ...
s''.
Wind-generated waves on the water surface are examples of gravity waves, as are
tsunami
A tsunami ( ; from ja, 津波, lit=harbour wave, ) is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater exp ...
s and ocean
tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another.
Tide tables ...
s.
The period of wind-generated gravity waves on the
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
of the Earth's ponds, lakes, seas and oceans are predominantly between 0.3 and 30 seconds (corresponding to frequencies predominantly between 3 Hz and 30 mHz). Shorter waves are also affected by
surface tension and are called ''
gravity–capillary waves'' and (if hardly influenced by gravity) ''
capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The wav ...
s''. Alternatively, so-called ''
infragravity wave
Infragravity waves are surface gravity waves with frequencies lower than the wind waves – consisting of both wind sea and swell – thus corresponding with the part of the wave spectrum lower than the frequencies directly generated by forc ...
s'', which are due to
subharmonic nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.
Atmosphere dynamics on Earth
In the
Earth's atmosphere
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing fo ...
, gravity waves are a mechanism that produce the transfer of
momentum from the
troposphere
The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphere, 99% of the total mass of water vapour and aerosols, and is where most weather phenomena occur. From ...
to the
stratosphere and
mesosphere. Gravity waves are generated in the troposphere by
frontal systems or by airflow over
mountain
A mountain is an elevated portion of the Earth's crust, generally with steep sides that show significant exposed bedrock. Although definitions vary, a mountain may differ from a plateau in having a limited summit area, and is usually highe ...
s. At first, waves propagate through the atmosphere without appreciable change in
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
. But as the waves reach more rarefied (thin) air at higher
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
s, their
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 am ...
increases, and
nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere. For example, this momentum transfer is partly responsible for the driving of the
Quasi-Biennial Oscillation
The quasi-biennial oscillation (QBO) is a quasiperiodic oscillation of the equatorial zonal wind between easterlies and westerlies in the tropical stratosphere with a mean period of 28 to 29 months. The alternating wind regimes develop at the to ...
, and in the
mesosphere, it is thought to be the major driving force of the Semi-Annual Oscillation. Thus, this process plays a key role in the
dynamics of the middle
atmosphere.
The effect of gravity waves in clouds can look like
altostratus undulatus cloud
The altostratus undulatus is a type of altostratus cloud with signature undulations within it. These undulations may be visible (usually as "wavy bases"), but frequently they are indiscernible to the naked eye. These formations will generally a ...
s, and are sometimes confused with them, but the formation mechanism is different.
Quantitative description
Deep water
The
phase velocity of a linear gravity wave 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 te ...
is given by the formula
where ''g'' is the acceleration due to gravity. When surface tension is important, this is modified to
where ''σ'' is the surface tension coefficient and ''ρ'' is the density.
The gravity wave represents a perturbation around a stationary state, in which there is no velocity. Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,
Because the fluid is assumed incompressible, this velocity field has the
streamfunction representation
:
where the subscripts indicate
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
. In this derivation it suffices to work in two dimensions
, where gravity points in the negative ''z''-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays
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 ...
, hence
In the streamfunction representation,
Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
:
where ''k'' is a spatial wavenumber. Thus, the problem reduces to solving the equation
:
We work in a sea of infinite depth, so the boundary condition is at
The undisturbed surface is at
, and the disturbed or wavy surface is at
where
is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition
:
Hence,
on
, where ''A'' and the wave speed ''c'' are constants to be determined from conditions at the interface.
''The free-surface condition:'' At the free surface
, the kinematic condition holds:
:
Linearizing, this is simply
:
where the velocity
is linearized on to the surface
Using the normal-mode and streamfunction representations, this condition is
, the second interfacial condition.
''Pressure relation across the interface'': For the case with
surface tension, the pressure difference over the interface at
is given by the
Young–Laplace equation:
:
where ''σ'' is the surface tension and ''κ'' is the
curvature of the interface, which in a linear approximation is
:
Thus,
:
However, this condition refers to the total pressure (base+perturbed), thus
:
(As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using
hydrostatic balance
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 ...
, in the form
this becomes
:
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
for the perturbations,
:
to yield
Putting this last equation and the jump condition together,
:
Substituting the second interfacial condition
and using the normal-mode representation, this relation becomes
Using the solution
, this gives
Since
is the phase speed in terms of the angular frequency
and the wavenumber, the gravity wave angular frequency can be expressed as
The
group velocity of a wave (that is, the speed at which a wave packet travels) is given by
and thus for a gravity wave,
The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.
Shallow water
Gravity waves traveling in shallow water (where the depth is much less than the wavelength), are
nondispersive: the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is ''h'',
:
Generation of ocean waves by wind
Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, and
capillary-gravity waves play an essential role in this effect. There are two distinct mechanisms involved, called after their proponents, Phillips and Miles.
In the work of Phillips, the ocean surface is imagined to be initially flat (''glassy''), and a
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
wind blows over the surface. When a flow is turbulent, one observes a randomly fluctuating velocity field superimposed on a mean flow (contrast with a laminar flow, in which the fluid motion is ordered and smooth). The fluctuating velocity field gives rise to fluctuating
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
es (both tangential and normal) that act on the air-water interface. The normal stress, or fluctuating pressure acts as a forcing term (much like pushing a swing introduces a forcing term). If the frequency and wavenumber
of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied Periodic function, periodic force (or a Fourier analysis, Fourier component of it) is equal or close to a natural frequency of the system ...
, and the wave grows in amplitude. As with other resonance effects, the amplitude of this wave grows linearly with time.
The air-water interface is now endowed with a surface roughness due to the capillary-gravity waves, and a second phase of wave growth takes place. A wave established on the surface either spontaneously as described above, or in laboratory conditions, interacts with the turbulent mean flow in a manner described by Miles.
This is the so-called critical-layer mechanism. A
critical layer forms at a height where the wave speed ''c'' equals the mean turbulent flow ''U''. As the flow is turbulent, its mean profile is logarithmic, and its second derivative is thus negative. This is precisely the condition for the mean flow to impart its energy to the interface through the critical layer. This supply of energy to the interface is destabilizing and causes the amplitude of the wave on the interface to grow in time. As in other examples of linear instability, the growth rate of the disturbance in this phase is exponential in time.
This Miles–Phillips Mechanism process can continue until an equilibrium is reached, or until the wind stops transferring energy to the waves (i.e., blowing them along) or when they run out of ocean distance, also known as
fetch
Fetch may refer to:
Books
* ''Fetch'', a 2012 book by Alan MacDonald and David Roberts
* ''The Fetch'', a 2006 book by Chris Humphreys
* ''The Fetch'', a 2009 book by Laura Whitcomb
* ''The Fetch'', a 1991 book by Robert Holdstock
* ''Fazbear ...
length.
See also
*
Acoustic wave
*
Asteroseismology
*
Green's law
*
Horizontal convective rolls
Horizontal convective rolls, also known as horizontal roll vortices or cloud streets, are long rolls of counter-rotating air that are oriented approximately parallel to the ground in the planetary boundary layer. Although horizontal convective ...
*
Lee wave
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 ...
*
Lunitidal interval
The lunitidal interval measures the time lag from lunar culmination to the next high tide at a given location. It is also called the high water interval (HWI). Sometimes a term is not used for the time lag, but instead the terms ''age'' or ''est ...
*
Mesosphere#Dynamic features
*
Morning Glory cloud
*
Orr–Sommerfeld equation The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become ...
*
Rayleigh–Taylor instability
The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
Drazin ( ...
*
Rogue wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
*
Skyquake
A skyquake is a phenomenon where a loud booming sound is reported to originate from the sky. The sound may cause noticeable vibration in a building or across a particular area. Those who experience skyquakes typically do not have a clear explanati ...
Notes
References
* Gill, A. E.,
Gravity wave. ''Glossary of Meteorology''. American Meteorological Society (15 December 2014).
* Crawford, Frank S., Jr. (1968). ''Waves'' (Berkeley Physics Course, Vol. 3), (McGraw-Hill, 1968)
Free online version
Further reading
*
*
External links
*
*
*
*
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