- Gravity wave
In

fluid dynamics ,**gravity waves**are waves generated in afluid medium or at the interface between two media (e.g. theatmosphere orocean ) which has the restoringforce ofgravity orbuoyancy .When a

fluid parcel is displaced on an interface or internally to a region with a differentdensity , gravity tries to restore the parcel toward equilibrium resulting in anoscillation about theequilibrium state or wave orbit. Gravity waves on an air-sea interface are called**surface gravity waves**orsurface wave s while internal gravity waves are calledinternal wave s. Ocean waves generated by wind are examples of gravity waves, andtsunami s and oceantide s are others.Wind-generated gravity waves on the

free surface of the Earth's ponds, lakes, seas and oceans have a period of between 0.3 and 30 seconds (3 Hz to 0.033 Hz). Shorter waves are also affected bysurface tension and are calledgravity-capillary wave s and (if hardly influenced by gravity)capillary wave s. Alternatively, so-calledinfragravity waves — which are due tosubharmonic nonlinear wave interaction with the wind waves — have periods longer than the accompanying wind-generated waves.fact|date=October 2008**Atmosphere dynamics on Earth**Since the fluid is a continuous medium, a traveling disturbance will result. In the earth's atmosphere, gravity waves are important for transferring

momentum from thetroposphere to themesosphere . Gravity waves are generated in the troposphere by frontal systems or by airflow overmountain s. At first waves propagate through the atmosphere without affecting its meanvelocity . But as the waves reach more rarefied air at higheraltitude s, theiramplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This process plays a key role in controlling the dynamics of the middle atmosphere.The clouds in gravity waves can look like

Altostratus undulatus cloud s, and are sometimes confused with them, but the formation mechanism is different.**Quantitative description**The phase speed $c$ of a linear gravity wave with wavenumber $k$ is given by the formula

$c=sqrt\{frac\{g\}\{k,$

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

$c=sqrt\{frac\{g\}\{k\}+frac\{sigma\; k\}\{\; ho,$

where "g" is the acceleration due to gravity, "σ" is the surface tension coefficient, "ρ" is the density, and "k" is the wavenumber of the disturbance.

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title = Details of the the phase-speed derivationThe 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, $(u\text{'}(x,z,t),w\text{'}(x,z,t)).,$ Because the fluid is assumed incompressible, this velocity field has the

streamfunction representation:$extbf\{u\}\text{'}=(u\text{'}(x,z,t),w\text{'}(x,z,t))=(psi\_z,-psi\_x),,$

where the subscripts indicate

partial derivatives . In this derivation it suffices to work in two dimensions $left(x,z\; ight)$, where gravity points in the negative "z"-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid staysirrotational , hence $abla\; imes\; extbf\{u\}\text{'}=0,$. In the streamfunction representation, $abla^2psi=0.,$ Next, because of the translational invariance of the system in the "x"-direction, it is possible to make theansatz :$psileft(x,z,t\; ight)=e^\{ikleft(x-ct\; ight)\}Psileft(z\; ight),,$

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

:$left(D^2-k^2\; ight)Psi=0,,,,\; D=frac\{d\}\{dz\}.$

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

:$u=DPsi=0,,,\; ext\{on\},z=-infty$.

Hence, $Psi=Ae^\{k\; z\}$ on $zinleft(-infty,eta\; ight)$, 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 $z=etaleft(x,t\; ight),$, the kinematic condition holds:

:$frac\{partialeta\}\{partial\; t\}+u\text{'}frac\{partialeta\}\{partial\; x\}=w\text{'}left(eta\; ight).,$

Linearizing, this is simply

:$frac\{partialeta\}\{partial\; t\}=w\text{'}left(0\; ight),,$

where the velocity $w\text{'}left(eta\; ight),$ is linearized on to the surface$z=0,$. Using the normal-mode and streamfunction representations, this condition is $c\; eta=Psi,$, the second interfacial condition.

"Pressure relation across the interface:" For the case with

surface tension , the pressure difference over the interface at $z=eta$ is given by theYoung–Laplace equation::$pleft(z=eta\; ight)=-sigmakappa,,$

where "σ" is the surface tension and "κ" is the

curvature of the interface, which in a linear approximation is:$kappa=\; abla^2eta=eta\_\{xx\}.,$

Thus,

:$pleft(z=eta\; ight)=-sigmaeta\_\{xx\}.,$

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

:$left\; [Pleft(eta\; ight)+p\text{'}left(0\; ight)\; ight]\; =-sigmaeta\_\{xx\}.$

(As usual, The perturbed quantities can be linearized onto the surface "z=0".) Using

hydrostatic balance , in the form $P=-\; ho\; g\; z+\; ext\{Const.\},$this becomes

:$p=geta\; ho-sigmaeta\_\{xx\},qquad\; ext\{on\; \}z=0.,$

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised

Euler equations for the perturbations,:$frac\{partial\; u\text{'}\}\{partial\; t\}\; =\; -\; frac\{1\}\{\; ho\}frac\{partial\; p\text{'}\}\{partial\; x\},$

to yield $p\text{'}=\; ho\; c\; DPsi$.

Putting this last equation and the jump condition together,

:$c\; ho\; DPsi=geta\; ho-sigmaeta\_\{xx\}.,$

Substituting the second interfacial condition $ceta=Psi,$ and using the normal-mode representation, this relationbecomes $c^2\; ho\; DPsi=gPsi\; ho+sigma\; k^2Psi$.

Using the solution $Psi=e^\{k\; z\}$, this gives

$c=sqrt\{frac\{g\}\{k\}+frac\{sigma\; k\}\{\; ho.$

Since $c=omega/k$ is the phase speed in terms of the frequency $omega$ and the wavenumber, the gravity wave frequency can be expressed as

$omega=sqrt\{gk\}.$

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

$c\_g=frac\{domega\}\{dk\},$

and thus for a gravity wave,

$c\_g=frac\{1\}\{2\}sqrt\{frac\{g\}\{k=frac\{1\}\{2\}c.$

The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.

**The generation of 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 [

*O. M. Phillips (1957) On the generation of waves by turbulent wind, J. Fluid Mech. 2, p. 417.*] , the ocean surface is imagined to be initially flat ('glassy'), and aturbulent 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 fluctuatingstress 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' is a forcing term for a swing). If the frequency and wavenumber $left(omega,k\; ight)$ of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is aresonance , 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 [

*J. W. Miles (1957) On the generation of surface waves by shear flows, J. Fluid Mech. 3, p. 185.*] . 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 is 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 length.

**ee also***

Cloud street

*Lee waves

*Rayleigh–Taylor instability

*Orr–Sommerfeld equation **Notes****References*** Dr. Steven Koch, Hugh D. Cobb, III and Neil A. Stuart, " [

*http://www.erh.noaa.gov/er/akq/GWave.htm Notes on Gravity Waves - Operational Forecasting and Detection of Gravity Waves Weather and Forecasting*] ".NOAA , Eastern Region Site Server.

* Gill, A. E., " [*http://amsglossary.allenpress.com/glossary/search?id=gravity-wave1 Gravity wave*] ". Atmosphere Ocean Dynamics, Academic Press, 1982.**External links*** [

*http://www.kcrg.com/news/local/3195031.html Gallery of cloud gravity waves over Iowa*]

* [*http://www.youtube.com/watch?v=yXnkzeCU3bE Time-lapse video of gravity waves over Iowa*]

* [*http://www.wikiwaves.org/index.php/Main_Page Water Waves Wiki*]

* [*http://digilander.libero.it/gravitazioneonde/WAVES%20GRAVITAZIONALI.htm Gravitational tidal waves*]

*Wikimedia Foundation.
2010.*

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