Gravity wave

For the different concept in general relativity, see gravitational wave.

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Surface gravity wave, breaking on an ocean beach.

File:Wave clouds.jpg

Wave clouds over Theresa, Wisconsin, United States.

File:GravityWaves ArabianSea.MODIS.2005may23.jpg

Atmospheric gravity waves as seen from space.

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NASA satellite image of a gravity wave cloud pattern formed in the wake of the Île Amsterdam, a volcanic island in the southern Indian Ocean.

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media (e.g., the atmosphere and the ocean) which has the restoring force of gravity or buoyancy.

When a fluid element is displaced on an interface or internally to a region with a different density, gravity tries to restore the parcel toward equilibrium resulting in an oscillation about the equilibrium state or wave orbit. Gravity waves on an air–sea interface are called surface gravity waves or surface waves while internal gravity waves are called internal waves. Wind-generated waves on the water surface are examples of gravity waves, and tsunamis and ocean tides 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 0.03 Hz). Shorter waves are also affected by surface tension 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

See also: Undular bore

In the Earth's atmosphere, gravity waves are a mechanism for the transfer of momentum from the troposphere to the stratosphere. Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first, waves propagate through the atmosphere without appreciable change in mean velocity. But as the waves reach more rarefied air at higher altitudes, their amplitude 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.^{[citation needed]}

The clouds in gravity waves can look like altostratus undulatus clouds, and are sometimes confused with them, but the formation mechanism is different.^{[citation needed]}

Quantitative description

Further information: Airy wave theory

The phase speed \(\scriptstyle c\) of a linear gravity wave with wavenumber \(\scriptstyle k\) 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

\(c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}},\)

where σ is the surface tension coefficient, ρ is the density, and k is the wavenumber (spatial frequency) of the disturbance.

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

\[\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,\]

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

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

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

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

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

\[u=D\Psi=0,\,\,\text{on}\,z=-\infty\].

Hence, \(\scriptstyle\Psi=Ae^{k z}\) 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:

\[\frac{\partial\eta}{\partial t}+u'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,\]

Linearizing, this is simply

\[\frac{\partial\eta}{\partial t}=w'\left(0\right),\,\]

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, 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 of the interface, which in a linear approximation is

\[\kappa=\nabla^2\eta=\eta_{xx}.\,\]

Thus,

\[p\left(z=\eta\right)=-\sigma\eta_{xx}.\,\]

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

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

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

this becomes

\[p=g\eta\rho-\sigma\eta_{xx},\qquad\text{on }z=0.\,\]

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

\[\frac{\partial u'}{\partial t} = - \frac{1}{\rho}\frac{\partial p'}{\partial x}\,\]

to yield \(\scriptstyle p'=\rho c D\Psi\).

Putting this last equation and the jump condition together,

\[c\rho D\Psi=g\eta\rho-\sigma\eta_{xx}.\,\]

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 \(\scriptstyle \Psi=e^{k z}\), this gives

\(c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}}.\)

Since \(\scriptstyle c=\omega/k\) is the phase speed in terms of the angular frequency \(\scriptstyle\omega\) and the wavenumber, the gravity wave angular 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

\(c_g=\frac{d\omega}{dk},\)

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.

Lunitidal Waves

Further information: Lunitidal interval

High tide does not occur when the moon is at its zenith, but about 6 hours later, which means the moon is always maximally out of phase with the tides that it causes. This delay is called the lunitidal interval. The main cause is that the extremely long wavelength gravity waves that transport the water don't travel fast enough to keep up with the earth's rotation. The oceans have a depth \(h < L / 20\), where \(L\) is the length of these waves; hence the case of propagation in shallow water applies and their speed is proportional to \(\sqrt{h}\). These lunitidal waves travel at over 500 mph, but they would need to travel at slightly more than 1000 mph to keep up with the earth's rotation, since the equator moves 25,000 miles in 24 hours. If the oceans were four times as deep the lunitidal interval would vanish and the tides would be in phase with the moon.

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, 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, 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.

Rogue waves

Further information: Rogue wave

When the wavelength of a gravity wave \(L\) is less than twice the depth of the ocean or lake its behavior is governed by the nonlinear Schrodinger equation and its speed is proportional to \(\sqrt{L}\). This means that different waves can travel at different speeds. Just by chance it can happen that several peaks, traveling at different speeds arrive simultaneously at the location of an unfortunate ship. If the combined height of the different peaks is a little more than twice that of normal storm waves it is considered a rogue wave and will usually sink the ship. This is one way in which rogue waves can arise. By contrast, the more familiar sound and light waves satisfy linear differential equations, so their speed is independent of wavelength. Experiments in large tanks with wave generating machines are used with models of ships to study rogue waves. The wave machines can generate waves of several different wavelengths simultaneously, and when these wavelengths are less than twice the depth of the tank the above phenomenon can be observed.

See also

• Cloud street
• Lee waves
• Rayleigh–Taylor instability
• Orr–Sommerfeld equation

References

• Gill, A. E., "Gravity wave". Atmosphere Ocean Dynamics, Academic Press, 1982.

External links

• Koch, Steven; Cobb, III, Hugh D.; Stuart, Neil A., Notes on Gravity Waves – Operational Forecasting and Detection of Gravity Waves Weather and Forecasting, NOAA, http://www.erh.noaa.gov/er/akq/GWave.htm, retrieved 2010-11-11
• Gallery of cloud gravity waves over Iowa, http://www.kcrg.com/news/local/3195031.html, retrieved 2010-11-11
• Time-lapse video of gravity waves over Iowa, http://www.youtube.com/watch?v=yXnkzeCU3bE, retrieved 2010-11-11
• Water Waves Wiki, http://www.wikiwaves.org/index.php/Main_Page, retrieved 2010-11-11

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