Inertial waves, also known as inertial oscillations, are a type of mechanical wave

Wave

In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

 possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves travel through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength

Wavelength

In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

 and frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents

Geostrophic

A geostrophic current is an oceanic flow in which the pressure gradient force is balanced by the Coriolis force. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the...

, and geostrophic wind

Geostrophic wind

The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars . This balance seldom holds exactly in nature...

s are examples of inertial waves. Inertial waves are also likely to exist in the core of the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

.

Restoring force

To understand the idea of a restoring force, imagine a guitar string. In equilibrium, the string is taut and straight, held stationary between its ends. Plucking the string moves it away from this equilibrium position. The tension

Tension (mechanics)

In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...

 in the string immediately pulls it back toward equilibrium, but soon overshoots, so that the string bows in the opposite direction. Next, tension again pulls the string back toward equilibrium, but again overshoots, and the cycle continues until the string finally comes to rest. Since tension restores the string to equilibrium (overshooting many times along the way), it is called the restoring force. Without it, the string would not vibrate, and no wave could exist.

Likewise, the open ocean is in equilibrium when it is level and at rest. If something (like wind

Wind

Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

) causes part of the ocean to rise and form a crest

Crest (physics)

A crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle.-Interference:...

, the crest is immediately pulled back toward equilibrium by gravity. Soon gravity overshoots, and the crest becomes a trough, displacing water and forming other crests nearby. They, in turn, are pulled back toward equilibrium by gravity, and the cycle continues. So gravity is the restoring force for wind waves on the open ocean, often called gravity wave

Gravity wave

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....

s.

Inertial waves are restored to equilibrium by the Coriolis force, a result of rotation. To be precise, the Coriolis force arises (along with the centrifugal force

Centrifugal force

Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

) in a rotating frame to account for the fact that such a frame is always accelerating. Inertial waves, therefore, cannot exist without rotation. More complicated than tension on a string, the Coriolis force acts at a 90° angle to the direction of motion, and its strength depends on the rotation rate of the fluid. These two properties lead to the peculiar characteristics of inertial waves.

Characteristics

Inertial waves are possible only when a fluid is rotating, and exist in the bulk of the fluid, not at its surface. Like light waves, inertial waves are transverse, which means that their vibrations occur perpendicular to the direction of wave travel. (The opposite of a transverse wave is a longitudinal wave

Longitudinal wave

Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

, where the vibrations are in the same direction as the wave travel. Sound waves, for example, are longitudinal.) One peculiar geometrical characteristic of inertial waves is that their phase velocity

Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

, which tells about the movement of the crests and troughs of the wave, is perpendicular to their group velocity

Group velocity

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

, which tells about the propagation of energy.

Whereas a sound wave or an electromagnetic wave of any frequency is possible, inertial waves can exist only over the range of frequencies from zero to twice the rotation rate of the fluid. Moreover, the frequency of the wave is determined by its direction of travel. Waves traveling perpendicular to the axis of rotation have zero frequency and are sometimes called the geostrophic

Geostrophic

A geostrophic current is an oceanic flow in which the pressure gradient force is balanced by the Coriolis force. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the...

 modes. Waves traveling parallel to the axis have maximum frequency (twice the rotation rate), and waves at intermediate angles have intermediate frequencies. In free space, an inertial wave can exist at any frequency between 0 and twice the rotation rate. A closed container, however, can impose restrictions on the possible frequencies of inertial waves, as it can for any kind of wave. Inertial waves in a closed container are often called inertial modes. In a sphere, for example, the inertial modes are forced to take on discrete frequencies, leaving gaps where no modes can exist.

Examples of inertial waves

Any kind of fluid can support inertial waves: water, oil, liquid metals, air, and other gases. Inertial waves are observed most commonly in planetary atmospheres (Rossby waves, geostrophic wind

Geostrophic wind

The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars . This balance seldom holds exactly in nature...

s) and in oceans and lakes (geostrophic currents

Geostrophic

A geostrophic current is an oceanic flow in which the pressure gradient force is balanced by the Coriolis force. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the...

), where they are responsible for much of the mixing that takes place. Inertial waves affected by the slope of the ocean floor are often called Rossby waves. Inertial waves can be observed in laboratory experiments or in industrial flows where a fluid is rotating. Inertial waves are also likely to exist in the liquid outer core of the Earth, and at least one group http://www.nature.com/nature/journal/v325/n6103/abs/325421a0.html has claimed evidence of them. Similarly, inertial waves are likely in rotating astronomical flows like accretion disks, planetary rings, and galaxies.

Mathematical description

Fluid flow is governed by the momentum equation (often called the Navier-Stokes equation) which is essentially a statement of Newton's second law for the fluid. The velocity in a fluid with viscosity under pressure and rotating at rate changes over time according to

To be precise, is the velocity of the fluid as observed in the rotating frame of reference. Since a rotating frame of reference is accelerating (i.e. non-inertial frame), two additional (pseudo)forces (as mentioned above) emerges as the result of this coordinate transformation: the centripetal force and the Coriolis force. In the equation above, the centripetal force is included as a part of the generalized pressure , that is, is related to the usual pressure , depending on the distance from the rotation axis , by

The last term on the right side of the momentum equation is the Coriolis term. The first term on the right accounts for pressure, and the second accounts for viscous diffusion.

In the case where the rotation rate is large, the Coriolis force and the centripetal force become large compared to the other terms. Being small in comparison, diffusion and the "convective derivative" (second term on the left) can be left out. Taking a curl of both sides and applying a few vector identities, the result is

Solutions to this equation are waves that satisfy two conditions. First, if is the wave vector

Wave vector

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

,

that is, the waves must be transverse, as mentioned above. Second, solutions are required to have a frequency that satisfies the dispersion relation


where is the angle between the axis of roation and the direction of the wave. These particular solutions are known as inertial waves.

The dispersion relation looks much like the Coriolis term in the momentum equation—notice the rotation rate and the factor of two. It immediately implies the range of possible frequencies for inertial waves, as well as the dependence of their frequency on their direction.

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