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In
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, gravity waves are waves in a
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
medium or at the interface between two media when the
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
or
buoyancy Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...
tries to restore equilibrium. An example of such an interface is that between the
atmosphere An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
and the
ocean The ocean is the body of salt water that covers approximately 70.8% of Earth. The ocean is conventionally divided into large bodies of water, which are also referred to as ''oceans'' (the Pacific, Atlantic, Indian Ocean, Indian, Southern Ocean ...
, which gives rise to
wind wave In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is ...
s. 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 In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...
), while gravity waves that are the body of the water (such as between parts of different densities) are called '' internal waves''. Wind-generated waves on the water surface are examples of gravity waves, as are
tsunami A tsunami ( ; from , ) 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 underwater explosions (including detonations, ...
s, 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, and the wakes of surface vessels. 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 between 3 Hz and .03 Hz). Shorter waves are also affected by
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
and are called '' gravity–capillary waves'' and (if hardly influenced by gravity) '' capillary waves''. Alternatively, so-called '' infragravity waves'', which are due to subharmonic nonlinear 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 composed of a layer of gas mixture that surrounds the Earth's planetary surface (both lands and oceans), known collectively as air, with variable quantities of suspended aerosols and particulates (which create weathe ...
, gravity waves are a mechanism that produce the transfer of
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
from the
troposphere The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the Atmosphere, planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the ...
to the
stratosphere The stratosphere () is the second-lowest layer of the atmosphere of Earth, located above the troposphere and below the mesosphere. The stratosphere is composed of stratified temperature zones, with the warmer layers of air located higher ...
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 higher t ...
s. At first, waves propagate through the atmosphere without appreciable change in
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
. But as the waves reach more rarefied (thin) air at higher
altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...
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, 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 An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
. The effect of gravity waves in clouds can look like altostratus undulatus clouds, and are sometimes confused with them, but the formation mechanism is different. Atmospheric gravity waves reaching
ionosphere The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays ...
are responsible for the generation of traveling ionospheric disturbances and could be observed by
radars Radar is a system that uses radio waves to determine the distance ('' ranging''), direction (azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track aircr ...
.


Quantitative description


Deep water

The
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, t ...
c of a linear gravity wave with
wavenumber In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...
k is given by the formula c=\sqrt, where ''g'' is the acceleration due to gravity. When surface tension is important, this is modified to c=\sqrt, 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, (u'(x,z,t),w'(x,z,t)). Because the fluid is assumed incompressible, this velocity field has the streamfunction representation :\textbf'=(u'(x,z,t),w'(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\right), 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 path between two points does not chan ...
, hence \nabla\times\textbf'=0.\, In the streamfunction representation, \nabla^2\psi=0.\, 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 ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be ...
:\psi\left(x,z,t\right)=e^\Psi\left(z\right),\, where ''k'' is a spatial wavenumber. Thus, the problem reduces to solving the equation :\left(D^2-k^2\right)\Psi=0,\,\,\,\ D=\frac. We work in a sea of infinite depth, so the boundary condition is at \scriptstyle z=-\infty. The undisturbed surface is at \scriptstyle z=0, and the disturbed or wavy surface is at \scriptstyle z=\eta, where \scriptstyle\eta is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition :u=D\Psi=0,\,\,\text\,z=-\infty. Hence, \scriptstyle\Psi=Ae^ on \scriptstyle z\in\left(-\infty,\eta\right), 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 \scriptstyle z=\eta\left(x,t\right)\,, the kinematic condition holds: :\frac+u'\frac=w'\left(\eta\right).\, Linearizing, this is simply :\frac=w'\left(0\right),\, where the velocity \scriptstyle w'\left(\eta\right)\, is linearized on to the surface \scriptstyle z=0.\, Using the normal-mode and streamfunction representations, this condition is \scriptstyle c \eta=\Psi\,, the second interfacial condition. ''Pressure relation across the interface'': For the case with
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
, the pressure difference over the interface at \scriptstyle z=\eta is given by the Young–Laplace equation: :p\left(z=\eta\right)=-\sigma\kappa,\, where ''σ'' is the surface tension and ''κ'' is the
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of the interface, which in a linear approximation is :\kappa=\nabla^2\eta=\eta_.\, Thus, :p\left(z=\eta\right)=-\sigma\eta_.\, However, this condition refers to the total pressure (base+perturbed), thus :\left \left(\eta\right)+p'\left(0\right)\right-\sigma\eta_. (As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using hydrostatic balance, in the form \scriptstyle P=-\rho g z+\text, this becomes :p=g\eta\rho-\sigma\eta_,\qquad\textz=0.\, The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised
Euler equations In mathematics and physics, many topics are eponym, 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 their own unique function, e ...
for the perturbations, :\frac = - \frac\frac\, to yield \scriptstyle p'=\rho c D\Psi. Putting this last equation and the jump condition together, :c\rho D\Psi=g\eta\rho-\sigma\eta_.\, Substituting the second interfacial condition \scriptstyle c\eta=\Psi\, and using the normal-mode representation, this relation becomes \scriptstyle c^2\rho D\Psi=g\Psi\rho+\sigma k^2\Psi. Using the solution \scriptstyle \Psi=e^, this gives c=\sqrt. Since \scriptstyle c=\omega/k is the phase speed in terms of the angular frequency \omega and the wavenumber, the gravity wave angular frequency can be expressed as \omega=\sqrt. The
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
of a wave (that is, the speed at which a wave packet travels) is given by c_g=\frac, and thus for a gravity wave, c_g=\frac\sqrt=\fracc. 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'', :c_p = c_g = \sqrt.


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 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 stresses (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 \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 a
resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...
, 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 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. 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

*
Acoustic wave Acoustic waves are types of waves that propagate through matter—such as gas, liquid, and/or solids—by causing the particles of the medium to compress and expand. These waves carry energy and are characterized by properties like acoustic pres ...
*
Asteroseismology Asteroseismology is the study of oscillations in stars. Stars have many Resonance, resonant modes and frequencies, and the path of sound waves passing through a star depends on the local speed of sound, which in turn depends on local temperature a ...
* Green's law * Horizontal convective rolls * Lee wave *
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 ''esta ...
* Mesosphere#Dynamic features * Morning Glory cloud * Orr–Sommerfeld equation * Rayleigh–Taylor instability *
Rogue wave A rogue wave is an abnormally large ocean wave. Rogue wave may also refer to: * Optical rogue waves, are rare pulses of light analogous to rogue or freak ocean waves. * Rogue Wave Software, a software company * Rogue Wave (band), an American in ...
* Skyquake


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
* 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

* *


External links

* * * * * {{DEFAULTSORT:Gravity Wave