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Gravity wave

From Wikipedia, the free encyclopedia
For the phenomenon of general relativity, seeGravitational wave.
Wave where gravity is the main restoring force
Surface gravity wave, breaking on an ocean beach inTučepi, Croatia
Wave clouds overTheresa, Wisconsin, United States
Nonfree image: detailed animation of a water wave
image iconDetailed animation of water wave motion (CC-BY-NC-ND 4.0)
wave clouds observed over the ocean, seen from a satellite
Wind-driven gravity waves in theTimor Sea, Western Australia, as seen from space

Influid dynamics,gravity waves are waves in afluid medium or at theinterface between two media when theforce ofgravity orbuoyancy tries to restore equilibrium. An example of such an interface is that between theatmosphere and theocean, which gives rise towind waves.

A gravity wave results when fluid is displaced from a position ofequilibrium. The restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called awave orbit.[1] Gravity waves on an air–sea interface of the ocean are calledsurface gravity waves (a type ofsurface wave), while gravity waves that arewithin the body of the water (such as between parts of different densities) are calledinternal waves.Wind-generated waves on the water surface are examples of gravity waves, as aretsunamis, oceantides, and thewakes of surface vessels.

The period of wind-generated gravity waves on thefree surface of the Earth's ponds, lakes, seas and oceans are predominantly between 0.3 and 30 seconds (corresponding to frequencies between 3 Hz and .03 Hz). Shorter waves are also affected bysurface tension and are calledgravity–capillary waves and (if hardly influenced by gravity)capillary waves. Alternatively, so-calledinfragravity waves, which are due tosubharmonicnonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.[2]

Atmosphere dynamics on Earth

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Further information:Atmospheric wave
See also:Undular bore

In theEarth's atmosphere, gravity waves are a mechanism that produce the transfer ofmomentum from thetroposphere to thestratosphere andmesosphere. Gravity waves are generated in the troposphere byfrontal systems or by airflow overmountains. At first, waves propagate through the atmosphere without appreciable change inmeanvelocity. But as the waves reach more rarefied (thin) air at higheraltitudes, theiramplitude increases, andnonlinear 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 theQuasi-Biennial Oscillation, and in themesosphere, it is thought to be the major driving force of the Semi-Annual Oscillation. Thus, this process plays a key role in thedynamics of the middleatmosphere.[3]

The effect of gravity waves in clouds can look likealtostratus undulatus clouds, and are sometimes confused with them, but the formation mechanism is different.[citation needed] Atmospheric gravity waves reachingionosphere are responsible for the generation of traveling ionospheric disturbances and could be observed byradars.[4]

Quantitative description

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Further information:Airy wave theory andStokes wave

Deep water

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Thephase velocityc{\displaystyle c} of a linear gravity wave withwavenumberk{\displaystyle k} is given by the formula

c=gk,{\displaystyle c={\sqrt {\frac {g}{k}}},}

whereg is the acceleration due to gravity. When surface tension is important, this is modified to

c=gk+σkρ,{\displaystyle c={\sqrt {{\frac {g}{k}}+{\frac {\sigma k}{\rho }}}},}

whereσ is the surface tension coefficient andρ is the density.

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,(u′(x,z,t),w′(x,z,t)).{\displaystyle (u'(x,z,t),w'(x,z,t)).} Because the fluid is assumed incompressible, this velocity field has thestreamfunction representation

u′=(u′(x,z,t),w′(x,z,t))=(ψz,−ψx),{\displaystyle {\textbf {u}}'=(u'(x,z,t),w'(x,z,t))=(\psi _{z},-\psi _{x}),\,}

where the subscripts indicatepartial derivatives. In this derivation it suffices to work in two dimensions(x,z){\displaystyle \left(x,z\right)}, where gravity points in the negativez-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid staysirrotational, hence∇×u′=0.{\displaystyle \nabla \times {\textbf {u}}'=0.\,} In the streamfunction representation,∇2ψ=0.{\displaystyle \nabla ^{2}\psi =0.\,} Next, because of the translational invariance of the system in thex-direction, it is possible to make theansatz

ψ(x,z,t)=eik(x−ct)Ψ(z),{\displaystyle \psi \left(x,z,t\right)=e^{ik\left(x-ct\right)}\Psi \left(z\right),\,}

wherek is a spatial wavenumber. Thus, the problem reduces to solving the equation

(D2−k2)Ψ=0, D=ddz.{\displaystyle \left(D^{2}-k^{2}\right)\Psi =0,\,\,\,\ D={\frac {d}{dz}}.}

We work in a sea of infinite depth, so the boundary condition is atz=−∞.{\displaystyle \scriptstyle z=-\infty .} The undisturbed surface is atz=0{\displaystyle \scriptstyle z=0}, and the disturbed or wavy surface is atz=η,{\displaystyle \scriptstyle z=\eta ,} whereη{\displaystyle \scriptstyle \eta } is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

u=DΨ=0,onz=−∞.{\displaystyle u=D\Psi =0,\,\,{\text{on}}\,z=-\infty .}

Hence,Ψ=Aekz{\displaystyle \scriptstyle \Psi =Ae^{kz}} onz∈(−∞,η){\displaystyle \scriptstyle z\in \left(-\infty ,\eta \right)}, whereA and the wave speedc are constants to be determined from conditions at the interface.

The free-surface condition: At the free surfacez=η(x,t){\displaystyle \scriptstyle z=\eta \left(x,t\right)\,}, the kinematic condition holds:

∂η∂t+u′∂η∂x=w′(η).{\displaystyle {\frac {\partial \eta }{\partial t}}+u'{\frac {\partial \eta }{\partial x}}=w'\left(\eta \right).\,}

Linearizing, this is simply

∂η∂t=w′(0),{\displaystyle {\frac {\partial \eta }{\partial t}}=w'\left(0\right),\,}

where the velocityw′(η){\displaystyle \scriptstyle w'\left(\eta \right)\,} is linearized on to the surfacez=0.{\displaystyle \scriptstyle z=0.\,} Using the normal-mode and streamfunction representations, this condition iscη=Ψ{\displaystyle \scriptstyle c\eta =\Psi \,}, the second interfacial condition.

Pressure relation across the interface: For the case withsurface tension, the pressure difference over the interface atz=η{\displaystyle \scriptstyle z=\eta } is given by theYoung–Laplace equation:

p(z=η)=−σκ,{\displaystyle p\left(z=\eta \right)=-\sigma \kappa ,\,}

whereσ is the surface tension andκ is thecurvature of the interface, which in a linear approximation is

κ=∇2η=ηxx.{\displaystyle \kappa =\nabla ^{2}\eta =\eta _{xx}.\,}

Thus,

p(z=η)=−σηxx.{\displaystyle p\left(z=\eta \right)=-\sigma \eta _{xx}.\,}

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

[P(η)+p′(0)]=−σηxx.{\displaystyle \left[P\left(\eta \right)+p'\left(0\right)\right]=-\sigma \eta _{xx}.}

(As usual, The perturbed quantities can be linearized onto the surfacez=0.) Usinghydrostatic balance, in the formP=−ρgz+Const.,{\displaystyle \scriptstyle P=-\rho gz+{\text{Const.}},}

this becomes

p=gηρ−σηxx,on z=0.{\displaystyle p=g\eta \rho -\sigma \eta _{xx},\qquad {\text{on }}z=0.\,}

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearisedEuler equations for the perturbations,

∂u′∂t=−1ρ∂p′∂x{\displaystyle {\frac {\partial u'}{\partial t}}=-{\frac {1}{\rho }}{\frac {\partial p'}{\partial x}}\,}

to yieldp′=ρcDΨ.{\displaystyle \scriptstyle p'=\rho cD\Psi .}

Putting this last equation and the jump condition together,

cρDΨ=gηρ−σηxx.{\displaystyle c\rho D\Psi =g\eta \rho -\sigma \eta _{xx}.\,}

Substituting the second interfacial conditioncη=Ψ{\displaystyle \scriptstyle c\eta =\Psi \,} and using the normal-mode representation, this relation becomesc2ρDΨ=gΨρ+σk2Ψ.{\displaystyle \scriptstyle c^{2}\rho D\Psi =g\Psi \rho +\sigma k^{2}\Psi .}

Using the solutionΨ=ekz{\displaystyle \scriptstyle \Psi =e^{kz}}, this gives

c=gk+σkρ.{\displaystyle c={\sqrt {{\frac {g}{k}}+{\frac {\sigma k}{\rho }}}}.}

Sincec=ω/k{\displaystyle \scriptstyle c=\omega /k} is the phase speed in terms of the angular frequencyω{\displaystyle \omega } and the wavenumber, the gravity wave angular frequency can be expressed as

ω=gk.{\displaystyle \omega ={\sqrt {gk}}.}

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

cg=dωdk,{\displaystyle c_{g}={\frac {d\omega }{dk}},}

and thus for a gravity wave,

cg=12gk=12c.{\displaystyle 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.

Shallow water

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Gravity waves traveling in shallow water (where the depth is much less than the wavelength), arenondispersive: the phase and group velocities are identical and independent of wavelength and frequency. When the water depth ish,

cp=cg=gh.{\displaystyle c_{p}=c_{g}={\sqrt {gh}}.}

Generation of ocean waves by wind

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Main article:Wind wave

Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, andcapillary-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,[5] 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 fluctuatingstresses (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(ω,k){\displaystyle \scriptstyle \left(\omega ,k\right)} 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.[6] This is the so-called critical-layer mechanism. Acritical layer forms at a height where the wave speedc equals the mean turbulent flowU. 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.

ThisMiles–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 asfetch length.

Analog gravity models and surface gravity waves

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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.[7]

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.

See also

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Notes

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  1. ^Lighthill, James (2001),Waves in fluids, Cambridge University Press, p. 205,ISBN 978-0-521-01045-0
  2. ^Bromirski, Peter D.; Sergienko, Olga V.; MacAyeal, Douglas R. (2010),"Transoceanic infragravity waves impacting Antarctic ice shelves",Geophysical Research Letters,37 (L02502): n/a,Bibcode:2010GeoRL..37.2502B,doi:10.1029/2009GL041488,S2CID 38071443.
  3. ^Fritts, D.C.; Alexander, M.J. (2003), "Gravity wave dynamics and effects in the middle atmosphere",Reviews of Geophysics,41 (1): 1003,Bibcode:2003RvGeo..41.1003F,CiteSeerX 10.1.1.470.3839,doi:10.1029/2001RG000106,S2CID 122701606.
  4. ^Günzkofer, F.; Pokhotelov, D.; Stober, G.; Mann, I.; Vadas, S.L.; Becker, E.; et al. (2023-10-18)."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".Annales Geophysicae.41 (2):409–428.doi:10.5194/angeo-41-409-2023.hdl:10037/32314.
  5. ^Phillips, O. M. (1957), "On the generation of waves by turbulent wind",J. Fluid Mech.,2 (5):417–445,Bibcode:1957JFM.....2..417P,doi:10.1017/S0022112057000233,S2CID 116675962
  6. ^Miles, J. W. (1957), "On the generation of surface waves by shear flows",J. Fluid Mech.,3 (2):185–204,Bibcode:1957JFM.....3..185M,doi:10.1017/S0022112057000567,S2CID 119795395
  7. ^Rozenman, Georgi Gary; Ullinger, Freyja; Zimmermann, Matthias; Efremov, Maxim A.; Shemer, Lev; Schleich, Wolfgang P.; Arie, Ady (2024-07-16)."Observation of a phase space horizon with surface gravity water waves".Communications Physics.7 (1): 165.doi:10.1038/s42005-024-01616-7.ISSN 2399-3650.

References

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  • 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)ISBN 978-0-07-004860-7Free 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

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External links

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