Gravity wave

Encyclopedia

In fluid dynamics

,

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

s while internal gravity waves are called internal wave

s. Wind-generated waves on the water surface are examples of gravity waves, and tsunami

s and ocean tide

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 0.03 Hz). Shorter waves are also affected by surface tension

and are called gravity–capillary waves and (if hardly influenced by gravity) capillary wave

s. 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.

medium, a traveling disturbance will result. In the Earth's atmosphere

, gravity waves are important for transferring momentum

from the troposphere

to the stratosphere

. Gravity waves are generated in the troposphere by frontal systems

or by airflow over mountain

s. At first waves propagate through the atmosphere without affecting its mean

velocity

. But as the waves reach more rarefied air at higher altitude

s, 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.

The clouds in gravity waves can look like Altostratus undulatus cloud

s, and are sometimes confused with them, but the formation mechanism is different.

where

where

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, . Because the fluid is assumed incompressible, this velocity field has the streamfunction representation

where the subscripts indicate partial derivatives. In this derivation it suffices to work in two dimensions , where gravity points in the negative

where

We work in a sea of infinite depth, so the boundary condition is at . The undisturbed surface is at , and the disturbed or wavy surface is at , where is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

.

Hence, on , where

Linearizing, this is simply

where the velocity is linearized on to the surface . Using the normal-mode and streamfunction representations, this condition is , the second interfacial condition.

, the pressure difference over the interface at is given by the Young–Laplace equation:

where

of the interface, which in a linear approximation is

Thus,

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

(As usual, The perturbed quantities can be linearized onto the surface

this becomes

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

for the perturbations,

to yield .

Putting this last equation and the jump condition together,

Substituting the second interfacial condition and using the normal-mode representation, this relation becomes .

Using the solution , this gives

Since is the phase speed in terms of the angular frequency and the wavenumber, the gravity wave angular frequency can be expressed as

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

and thus for a gravity wave,

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

In the work of Phillips, the ocean surface is imagined to be initially flat (

, 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

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.

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

,

**gravity waves**are waves generated in a fluidFluid

In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

medium or at the interface

Interface (chemistry)

An interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...

between two media (e.g., the atmosphere

Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

and the ocean

Ocean

An ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...

) which has the restoring force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

of gravity or buoyancy

Buoyancy

In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

.

When a fluid element is displaced on an interface or internally to a region with a different density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, gravity tries to restore the parcel toward equilibrium

Mechanical equilibrium

A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....

resulting in an oscillation

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

about the equilibrium state or wave orbit. Gravity waves on an air–sea interface are called

**surface gravity waves**or surface waveSurface wave

In physics, a surface wave is a mechanical wave that propagates along the interface between differing media, usually two fluids with different densities. A surface wave can also be an electromagnetic wave guided by a refractive index gradient...

s while internal gravity waves are called internal wave

Internal wave

Internal waves are gravity waves that oscillate within, rather than on the surface of, a fluid medium. They are one of many types of wave motion in stratified fluids . A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water...

s. Wind-generated waves on the water surface are examples of gravity waves, and tsunami

Tsunami

A tsunami is a series of water waves caused by the displacement of a large volume of a body of water, typically an ocean or a large lake...

s and ocean tide

Tide

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....

s are others.

Wind-generated gravity waves on the free surface

Free surface

In physics, a free surface is the surface of a fluid that is subject to constant perpendicular normal stress and zero parallel shear stress,such as the boundary between two homogenous fluids,for example liquid water and the air in the Earth's atmosphere...

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

Surface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

and are called gravity–capillary waves and (if hardly influenced by gravity) capillary wave

Capillary wave

A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics are dominated by the effects of surface tension.Capillary waves are common in nature and the home, and are often referred to as ripples...

s. Alternatively, so-called infragravity waves, which are due to subharmonic

Subharmonic

Subharmonic frequencies are frequencies below the fundamental frequency of an oscillator in a ratio of 1/n, with n a positive integer number. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz and 110 Hz...

nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.

## Atmosphere dynamics on Earth

Since the fluid is, to close approximation, a continuousContinuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

medium, a traveling disturbance will result. In the Earth's atmosphere

Earth's atmosphere

The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

, gravity waves are important for transferring momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

from the troposphere

Troposphere

The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....

to the stratosphere

Stratosphere

The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

. Gravity waves are generated in the troposphere by frontal systems

Weather front

A weather front is a boundary separating two masses of air of different densities, and is the principal cause of meteorological phenomena. In surface weather analyses, fronts are depicted using various colored lines and symbols, depending on the type of front...

or by airflow over mountain

Mountain

Image:Himalaya_annotated.jpg|thumb|right|The Himalayan mountain range with Mount Everestrect 58 14 160 49 Chomo Lonzorect 200 28 335 52 Makalurect 378 24 566 45 Mount Everestrect 188 581 920 656 Tibetan Plateaurect 250 406 340 427 Rong River...

s. At first waves propagate through the atmosphere without affecting its mean

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. But as the waves reach more rarefied air at higher altitude

Altitude

Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

s, their amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

increases, and nonlinear effects

Nonlinearity

In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...

cause the waves to break, transferring their momentum to the mean flow.

This process plays a key role in controlling the dynamics

Dynamics (mechanics)

In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...

of the middle atmosphere.

The clouds in gravity waves can look like Altostratus undulatus cloud

Altostratus undulatus cloud

The altostratus undulatus is a type of low altocumulus cloud with signature undulations within it. These undulations may be visible , but frequently they are indiscernible to the naked eye...

s, and are sometimes confused with them, but the formation mechanism is different.

## Quantitative description

The phase speed of a linear gravity wave with wavenumber is given by the formulawhere

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

*σ*is the surface tension coefficient,*ρ*is the density, and*k*is the wavenumber (spatial frequency) of the disturbance.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, . Because the fluid is assumed incompressible, this velocity field has the streamfunction representation

where the subscripts indicate partial derivatives. In this derivation it suffices to work in two dimensions , 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 . In the streamfunction representation, Next, because of the translational invariance of the system in the*x*-direction, it is possible to make the ansatzAnsatz

Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...

where

*k*is a spatial wavenumber. Thus, the problem reduces to solving the equationWe work in a sea of infinite depth, so the boundary condition is at . The undisturbed surface is at , and the disturbed or wavy surface is at , where is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

.

Hence, on , 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 , the kinematic condition holds:Linearizing, this is simply

where the velocity is linearized on to the surface . Using the normal-mode and streamfunction representations, this condition is , the second interfacial condition.

*Pressure relation across the interface*: For the case with surface tensionSurface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

, the pressure difference over the interface at is given by the Young–Laplace equation:

where

*σ*is the surface tension and*κ*is the curvatureCurvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

of the interface, which in a linear approximation is

Thus,

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

(As usual, The perturbed quantities can be linearized onto the surface

*z=0*.) Using hydrostatic balance, in the formthis becomes

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

Euler equations

In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...

for the perturbations,

to yield .

Putting this last equation and the jump condition together,

Substituting the second interfacial condition and using the normal-mode representation, this relation becomes .

Using the solution , this gives

Since is the phase speed in terms of the angular frequency and the wavenumber, the gravity wave angular frequency can be expressed as

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

and thus for a gravity wave,

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, 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 of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a resonanceResonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

, 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

Fetch (geography)

The fetch, often called the fetch length, is the length of water over which a given wind has blown. It is used in geography and meteorology and is usually associated with coastal erosion. It plays a large part in longshore drift as well....

length.

## See also

- Cloud street
- Lee wavesLee wavesIn meteorology, lee waves are atmospheric standing waves. The most common form is mountain waves, which are atmospheric internal gravity waves...
- Rayleigh–Taylor instability
- Orr–Sommerfeld equation