Gravity Waves – A Closer Look

13 04 2009

Gravity Waves
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 0.033 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.

Atmospheric Dynamics on Earth
Since the fluid is a continuous medium, a traveling disturbance will result. In the earth’s atmosphere, gravity waves are important for transferring momentum from the troposphere to the mesosphere. Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first waves propagate through the atmosphere without affecting its 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.

The clouds in gravity waves can look like Altostratus undulatus clouds, and are sometimes confused with them, but the formation mechanism is different.

Quantitive Description
The phase speed c of a linear gravity wave with wavenumber k is given by the formula

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

where g is the acceleration due to gravity, σ is the surface tension coefficient, ρ is the density, and k is the wavenumber of the disturbance.
Since c = ω / k is the phase speed in terms of the frequency ω and the wavenumber, the gravity wave 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[1], 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’ is a forcing term for a swing). 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 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[2]. 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 is 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.

This article was not my own. I do not take credit for this article, in whole or part. Article was derived from a collaborative effort from Wikipedia.

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