View source for Gravity wave

{{for|the phenomenon of general relativity |Gravitational wave}}
{{short description|Wave where gravity is the main restoring force}}
[[Image:Waves.jpg|thumb|right|Surface gravity wave, breaking on an ocean beach in [[Tučepi]], Croatia]]
[[Image:wave clouds.jpg|thumb|right|Wave clouds over [[Theresa, Wisconsin]], United States]]

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[[File:Satellite view of the Timor Sea (cropped).jpg|alt=wave clouds observed over the ocean, seen from a satellite|thumb|Wind-driven gravity waves in the [[Timor Sea]], Western Australia, as seen from space]]
In [[fluid dynamics]], '''gravity waves''' are waves in a [[fluid]] medium or at the [[interface (matter)|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]] and the [[ocean]], which gives rise to [[wind wave]]s.

A gravity wave results when fluid is displaced from a position of [[mechanical equilibrium|equilibrium]]. The restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a '''''wave orbit'''''.<ref>{{Citation | publisher = Cambridge University Press | isbn = 978-0-521-01045-0 | last = Lighthill | first = James |author-link = James Lighthill | title = Waves in fluids | year = 2001 | page = 205 }}</ref> 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 {{em|within}} the body of the water (such as between parts of different densities) are called ''[[internal wave]]s''. [[Wind wave|Wind-generated waves]] on the water surface are examples of gravity waves, as are [[tsunami]]s, ocean [[tide]]s, and the [[Wake (physics)|wakes]] of surface vessels.

The period of wind-generated gravity waves on the [[free surface]] of the Earth's ponds, lakes, seas and oceans are predominantly between 0.3 and 30 seconds (corresponding to frequencies between 3&nbsp;Hz and .03&nbsp;Hz). Shorter waves are also affected by [[surface tension]] and are called ''[[gravity–capillary wave]]s'' and (if hardly influenced by gravity) ''[[capillary wave]]s''. Alternatively, so-called ''[[infragravity wave]]s'', which are due to [[undertone series|subharmonic]] [[nonlinear system|nonlinear]] wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.<ref>{{Citation
  | last1 = Bromirski
  | first1 = Peter D.
  | first2 = Olga V. 
  | last2 = Sergienko 
  | first3 = Douglas R. 
  | last3 = MacAyeal
  | title = Transoceanic infragravity waves impacting Antarctic ice shelves
  | journal = Geophysical Research Letters
  | volume = 37
  | issue = L02502
  | year = 2010
  | pages = n/a
 | doi = 10.1029/2009GL041488
  | postscript = .
| bibcode=2010GeoRL..37.2502B| s2cid = 38071443
 | url = https://escholarship.org/uc/item/3k2933rc
 | doi-access = free
  }}</ref>

==Atmosphere dynamics on Earth==
{{further|Atmospheric wave}}
{{See also|Undular bore}}
In the [[Earth's atmosphere]], gravity waves are a mechanism that produce the transfer of [[momentum]] from the [[troposphere]] to the [[stratosphere]] and [[mesosphere]]. Gravity waves are generated in the troposphere by [[Weather front|frontal systems]] or by airflow over [[mountain]]s. At first, waves propagate through the atmosphere without appreciable change in [[arithmetic mean|mean]] [[velocity]]. But as the waves reach more rarefied (thin) air at higher [[altitude]]s, their [[amplitude]] increases, and [[nonlinearity|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|Quasi-Biennial Oscillation]], 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 (mechanics)|dynamics]] of the middle [[Atmosphere of Earth|atmosphere]].<ref>{{Citation
  | last1 = Fritts
  | first1 = D.C.
  | first2 = M.J.
  | last2 = Alexander
  | title = Gravity wave dynamics and effects in the middle atmosphere
  | journal = Reviews of Geophysics
  | volume = 41
  | issue = 1
  | page = 1003
 | year = 2003
  | doi = 10.1029/2001RG000106
  | postscript = .
| bibcode=2003RvGeo..41.1003F| citeseerx = 10.1.1.470.3839
  | s2cid = 122701606
 }}</ref>

Atmospheric gravity waves reaching [[ionosphere]] are responsible for the generation of traveling ionospheric disturbances and could be observed by [[radars]].<ref>{{Cite journal|title=Inferring neutral winds in the ionospheric transition region from atmospheric-gravity-wave traveling-ionospheric-disturbance (AGW-TID) observations with the EISCAT VHF radar and the Nordic Meteor Radar Cluster|journal=Annales Geophysicae|date=2023-10-18 |doi= 10.5194/angeo-41-409-2023|language=en|last1= Günzkofer|first1= F.|last2= Pokhotelov|first2= D.|last3= Stober|first3=G.|last4=Mann|first4=I.|last5=Vadas|first5=S.L.|last6=Becker|first6=E.|display-authors=etal|volume= 41|issue= 2|pages= 409–428|doi-access= free|hdl=10037/32314|hdl-access=free}}</ref>

==Quantitative description==
{{Further|Airy wave theory|Stokes wave}}

===Deep water===
The [[phase velocity]] <math>c</math> of a linear gravity wave with [[wavenumber]] <math>k</math> is given by the formula

<math>c=\sqrt{\frac{g}{k}},</math>

where ''g'' is the acceleration due to gravity. When surface tension is important, this is modified to

<math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}},</math>

where ''σ'' is the surface tension coefficient and ''ρ'' is the density.

{{hidden begin
|toggle = left
|title = Details of the phase-speed derivation
}}

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, <math>(u'(x,z,t),w'(x,z,t)).</math> Because the fluid is assumed incompressible, this velocity field has the [[streamfunction]] representation

:<math>\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,</math>

where the subscripts indicate [[partial derivatives]]. In this derivation it suffices to work in two dimensions <math>\left(x,z\right)</math>, where gravity points in the negative ''z''-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays [[irrotational]], hence <math>\nabla\times\textbf{u}'=0.\,</math> In the streamfunction representation, <math>\nabla^2\psi=0.\,</math> Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the [[ansatz]]

:<math>\psi\left(x,z,t\right)=e^{ik\left(x-ct\right)}\Psi\left(z\right),\,</math>

where ''k'' is a spatial wavenumber. Thus, the problem reduces to solving the equation

:<math>\left(D^2-k^2\right)\Psi=0,\,\,\,\ D=\frac{d}{dz}.</math>

We work in a sea of infinite depth, so the boundary condition is at <math>\scriptstyle z=-\infty.</math> The undisturbed surface is at <math>\scriptstyle z=0</math>, and the disturbed or wavy surface is at <math>\scriptstyle z=\eta,</math> where <math>\scriptstyle\eta</math> is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

:<math>u=D\Psi=0,\,\,\text{on}\,z=-\infty.</math>

Hence, <math>\scriptstyle\Psi=Ae^{k z}</math> on <math>\scriptstyle z\in\left(-\infty,\eta\right)</math>, 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 <math>\scriptstyle z=\eta\left(x,t\right)\,</math>, the kinematic condition holds:

:<math>\frac{\partial\eta}{\partial t}+u'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,</math>

Linearizing, this is simply

:<math>\frac{\partial\eta}{\partial t}=w'\left(0\right),\,</math>

where the velocity <math>\scriptstyle w'\left(\eta\right)\,</math> is linearized on to the surface <math>\scriptstyle z=0.\,</math> Using the normal-mode and streamfunction representations, this condition is <math>\scriptstyle c \eta=\Psi\,</math>, the second interfacial condition.

''Pressure relation across the interface'': For the case with [[surface tension]], the pressure difference over the interface at <math>\scriptstyle z=\eta</math> is given by the [[Young–Laplace]] equation:

:<math>p\left(z=\eta\right)=-\sigma\kappa,\,</math>

where ''σ'' is the surface tension and ''κ'' is the [[curvature]] of the interface, which in a linear approximation is

:<math>\kappa=\nabla^2\eta=\eta_{xx}.\,</math>

Thus,

:<math>p\left(z=\eta\right)=-\sigma\eta_{xx}.\,</math>

However, this condition refers to the total pressure (base+perturbed), thus

:<math>\left[P\left(\eta\right)+p'\left(0\right)\right]=-\sigma\eta_{xx}.</math>

(As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using [[hydrostatic balance]], in the form <math>\scriptstyle P=-\rho g z+\text{Const.},</math>

this becomes

:<math>p=g\eta\rho-\sigma\eta_{xx},\qquad\text{on }z=0.\,</math>

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised [[Euler equations]] for the perturbations,

:<math>\frac{\partial u'}{\partial t} = - \frac{1}{\rho}\frac{\partial p'}{\partial x}\,</math>

to yield <math>\scriptstyle p'=\rho c D\Psi.</math>

Putting this last equation and the jump condition together,

:<math>c\rho D\Psi=g\eta\rho-\sigma\eta_{xx}.\,</math>

Substituting the second interfacial condition <math>\scriptstyle c\eta=\Psi\,</math> and using the normal-mode representation, this relation becomes <math>\scriptstyle c^2\rho D\Psi=g\Psi\rho+\sigma k^2\Psi.</math>

Using the solution <math>\scriptstyle \Psi=e^{k z}</math>, this gives

<math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}}.</math>

{{hidden end}}

Since <math>\scriptstyle c=\omega/k</math> is the phase speed in terms of the angular frequency <math>\omega</math> and the wavenumber, the gravity wave angular frequency can be expressed as

<math>\omega=\sqrt{gk}.</math>

The [[group velocity]] of a wave (that is, the speed at which a wave packet travels) is given by

<math>c_g=\frac{d\omega}{dk},</math>

and thus for a gravity wave,

<math>c_g=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}c.</math>

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 [[dispersion (water waves)|nondispersive]]: the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is ''h'',

:<math>c_p = c_g = \sqrt{gh}.</math>

==Generation of ocean waves by wind==
{{main|Wind wave}}
Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, and [[Capillary wave|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,<ref>{{citation | first=O. M. | last=Phillips | year=1957 | title=On the generation of waves by turbulent wind | journal=J. Fluid Mech. | volume=2 | issue=5 | pages=417–445 | doi=10.1017/S0022112057000233 |bibcode = 1957JFM.....2..417P | s2cid=116675962 }}</ref> the ocean surface is imagined to be initially flat (''glassy''), and a [[turbulent]] 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 (mechanics)|stress]]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 <math>\scriptstyle\left(\omega,k\right)</math> of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a [[resonance]], 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.<ref>{{citation | first=J. W. | last=Miles | author-link=John W. Miles | year=1957 | title=On the generation of surface waves by shear flows | journal=J. Fluid Mech. | volume=3 | issue=2 | pages=185–204 | doi=10.1017/S0022112057000567 |bibcode = 1957JFM.....3..185M | s2cid=119795395 }}</ref> 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|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 (geography)|fetch]] length.

==Analog gravity models and surface gravity waves==

Surface gravity waves have been recognized as a powerful tool for studying analog gravity models, providing experimental platforms for phenomena typically found in black hole physics. In an experiment, surface gravity waves were utilized to simulate phase space horizons, akin to event horizons of black holes. This experiment observed logarithmic phase singularities, which are central to phenomena like Hawking radiation, and the emergence of Fermi-Dirac distributions, which parallel quantum mechanical systems.<ref>{{Cite journal |last1=Rozenman |first1=Georgi Gary |last2=Ullinger |first2=Freyja |last3=Zimmermann |first3=Matthias |last4=Efremov |first4=Maxim A. |last5=Shemer |first5=Lev |last6=Schleich |first6=Wolfgang P. |last7=Arie |first7=Ady |date=2024-07-16 |title=Observation of a phase space horizon with surface gravity water waves |journal=Communications Physics |volume=7 |issue=1 |pages=165 |doi=10.1038/s42005-024-01616-7 |issn=2399-3650 |doi-access=free }}</ref>

By propagating surface gravity water waves, researchers were able to recreate the energy wave functions of an inverted harmonic oscillator, a system that serves as an analog for black hole physics. The experiment demonstrated how the free evolution of these classical waves in a controlled laboratory environment can reveal the formation of horizons and singularities, shedding light on fundamental aspects of gravitational theories and quantum mechanics.

==Clear-air turbulence==

Gravity waves are generated as air is forced to rise and sink over smaller mountain ranges, creating oscillations in buoyancy and vertical wind. These waves typically remain stable in gently stratified conditions like those found over ranges such as the Appalachians. However, as terrain height increases, the resulting vertical displacement amplifies these perturbations and enhances vertical wind shear. This destabilization can be quantified using the [[Richardson number]], the ratio of [[buoyancy]] and [[flow shear]]. If the shear becomes strong enough, the stretching forces from wind gradients are more positive than the buoyant resistance which causes air layers to roll over each other. This can be further analyzed with the [[Kelvin-Helmholtz instability]] equation. This breakdown of wave structure results in [[clear-air turbulence]], which is smaller scale atmospheric movement typically felt on aircraft.<ref>{{cite journal |last1=Canuto |first1=V.M |title=Critical Richardson Numbers and Gravity Waves |journal=Astronomy and Astrophysics |date=2002 |page=1119–1123 |doi=10.1051/0004-6361:20011773 |url=https://www.aanda.org/articles/aa/full/2002/12/aa1711/aa1711.right.html}}</ref>

==See also==
{{Div col|small=no}}
* {{annotated link|Acoustic wave}}
* {{annotated link|Asteroseismology}}
* {{annotated link|Green's law}}
* {{annotated link|Horizontal convective rolls}}
* {{annotated link|Lee wave}}
* {{annotated link|Lunitidal interval}}
* {{annotated link|Mesosphere#Dynamic features}}
* {{annotated link|Morning Glory cloud}}
* {{annotated link|Orr–Sommerfeld equation}}
* {{annotated link|Rayleigh–Taylor instability}}
* {{annotated link|Rogue wave}}
* {{annotated link|Skyquake}}
{{Div col end}}

==Notes==
<references/>

==References==
* Gill, A. E., "[http://glossary.ametsoc.org/wiki/Gravity_wave 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)   {{ISBN|978-0-07-004860-7}} [https://archive.org/details/Waves_371 Free online version]
* Alexander, P., A. de la Torre, and P. Llamedo (2008), Interpretation of gravity wave signatures in GPS radio occultations, J. Geophys. Res., 113, D16117, doi:10.1029/2007JD009390.

==Further reading==
* {{cite web | first1=Steven | last1=Koch | first2=Hugh D. III | last2=Cobb | first3=Neil A. | last3=Stuart | url=http://www.erh.noaa.gov/er/akq/GWave.htm | title=Notes on Gravity Waves – Operational Forecasting and Detection of Gravity Waves Weather and Forecasting | publisher=[[National Oceanic and Atmospheric Administration|NOAA]] | access-date=2010-11-11 }}
* {{cite book|last=Nappo|first=Carmen J.|title=An Introduction to Atmospheric Gravity Waves, Second Ed|year=2012|publisher=Elsevier Academic Press (International Geophysics Volume 102)|location=Waltham, Massachusetts|isbn=978-0-12-385223-6}}

==External links==
{{Commons category|Gravity waves}}
* {{cite web | url=https://www.youtube.com/watch?v=vxzavcDBsvQ  |archive-url=https://ghostarchive.org/varchive/youtube/20211211/vxzavcDBsvQ| archive-date=2021-12-11 |url-status=live| title=Time Lapse of Gravity Wave Courtesy The Weather Nutz |website=[[YouTube]]|date=13 December 2018 | access-date=2018-12-13 }}{{cbignore}}
* {{cite web | url=http://www.kcrg.com/news/local/3195031.html | title=Gallery of cloud gravity waves over Iowa | access-date=2010-11-11  | archive-url=https://web.archive.org/web/20110524074238/http://www.kcrg.com/news/local/3195031.html | archive-date=2011-05-24 }}
* {{APOD|date=20 November 2022|title=Airglow Ripples over Tibet}}
* {{cite web | url=https://www.youtube.com/watch?v=yXnkzeCU3bE  |archive-url=https://ghostarchive.org/varchive/youtube/20211211/yXnkzeCU3bE| archive-date=2021-12-11 |url-status=live| title=Time-lapse video of gravity waves over Iowa |website=[[YouTube]]|date=6 May 2007 | access-date=2010-11-11 }}{{cbignore}}
* {{cite web | url=http://www.wikiwaves.org/index.php/Main_Page | title=Water Waves Wiki | access-date=2010-11-11  | archive-url=https://web.archive.org/web/20101113225116/http://www.wikiwaves.org/index.php/Main_Page | archive-date=2010-11-13 }}

{{physical oceanography}}

{{DEFAULTSORT:Gravity Wave}}
[[Category:Gravity waves| ]]