Gravity wave

In fluid dynamics, gravity waves are waves in a fluid medium or at the 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 .

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 . Wind wave on the water surface are examples of gravity waves, as are , ocean , and the 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 Hz and .03 Hz). Shorter waves are also affected by surface tension and are called gravity–capillary waves and (if hardly influenced by gravity) . Alternatively, so-called infragravity waves, which are due to undertone series nonlinear system wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.

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Atmosphere dynamics on Earth 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 or by airflow over . At first, waves propagate through the atmosphere without appreciable change in arithmetic mean velocity. But as the waves reach more rarefied (thin) air at higher , their amplitude increases, and nonlinearity 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.

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 are responsible for the generation of traveling ionospheric disturbances and could be observed by radars.

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Quantitative description

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Deep water The phase velocity $c$ of a linear gravity wave with wavenumber $k$ is given by the formula

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

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

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

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

$\backslash textbf\{u\}\text{'}=(u\text{'}(x,z,t),w\text{'}(x,z,t))=(\backslash psi\_z,-\backslash psi\_x),\backslash ,$

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

$\backslash psi\backslash left(x,z,t\backslash right)=e^\{ik\backslash left(x-ct\backslash right)\}\backslash Psi\backslash left(z\backslash right),\backslash ,$

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

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

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

$u=D\backslash Psi=0,\backslash ,\backslash ,\backslash text\{on\}\backslash ,z=-\backslash infty.$

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

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

Linearizing, this is simply

$\backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; t\}=w\text{'}\backslash left(0\backslash right),\backslash ,$

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

Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at $\backslash scriptstyle\; z=\backslash eta$ is given by the Young–Laplace equation:

$p\backslash left(z=\backslash eta\backslash right)=-\backslash sigma\backslash kappa,\backslash ,$

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

$\backslash kappa=\backslash nabla^2\backslash eta=\backslash eta\_\{xx\}.\backslash ,$

Thus,

$p\backslash left(z=\backslash eta\backslash right)=-\backslash sigma\backslash eta\_\{xx\}.\backslash ,$

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

$\backslash leftP\backslash left(\backslash eta\backslash right)+p\text{'}\backslash left(0\backslash right)\backslash right=-\backslash sigma\backslash eta\_\{xx\}.$

(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form $\backslash scriptstyle\; P=-\backslash rho\; g\; z+\backslash text\{Const.\},$

this becomes

$p=g\backslash eta\backslash rho-\backslash sigma\backslash eta\_\{xx\},\backslash qquad\backslash text\{on\; \}z=0.\backslash ,$

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

$\backslash frac\{\backslash partial\; u\text{'}\}\{\backslash partial\; t\}\; =\; -\; \backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\; p\text{'}\}\{\backslash partial\; x\}\backslash ,$

to yield $\backslash scriptstyle\; p\text{'}=\backslash rho\; c\; D\backslash Psi.$

Putting this last equation and the jump condition together,

$c\backslash rho\; D\backslash Psi=g\backslash eta\backslash rho-\backslash sigma\backslash eta\_\{xx\}.\backslash ,$

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

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

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

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

$\backslash omega=\backslash sqrt\{gk\}.$

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

$c\_g=\backslash frac\{d\backslash omega\}\{dk\},$

and thus for a gravity wave,

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

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

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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,

$c\_p\; =\; c\_g\; =\; \backslash sqrt\{gh\}.$

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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 wave 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 $\backslash scriptstyle\backslash left(\backslash omega,k\backslash right)$ 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. 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.

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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.

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See also

• Acoustic wave
• Asteroseismology
• Green's law
• Horizontal convective rolls
• Lee wave
• Lunitidal interval
• Mesosphere#Dynamic features
• Morning Glory cloud
• Orr–Sommerfeld equation
• Rayleigh–Taylor instability
• Rogue wave
• Skyquake

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Notes

• 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.

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Further reading

• Carmen J. Nappo (2025). 9780123852236, Elsevier Academic Press (International Geophysics Volume 102). ISBN 9780123852236

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External links

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