Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow 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 and 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, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.
Restoring force 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) 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. One peculiar geometrical characteristic of inertial waves is that their phase velocity, which describes the movement of the crests and troughs of the wave, is perpendicular to their group velocity, which is a measure of 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 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 winds) and in oceans and lakes (geostrophic currents), 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 [1] has claimed evidence of them. Similarly, inertial waves are likely in rotating astronomical flows like stars, accretion disks, planetary rings, and galaxies.
Mathematical description Fluid flow is governed by the Navier-Stokes equation for momentum. The flow velocity u → {\displaystyle {\vec {u}}} of a fluid with viscosity ν {\displaystyle \nu } under pressure P {\displaystyle P} and rotating at rate Ω {\displaystyle \Omega } changes over time t {\displaystyle t} according to
∂ u → ∂ t + ( u → ⋅ ∇ → ) u → = − 1 ρ ∇ → P + ν ∇ 2 u → − 2 Ω → × u → . {\displaystyle {\frac {\partial {\vec {u}}}{\partial t}}+({\vec {u}}\cdot {\vec {\nabla }}){\vec {u}}=-{\frac {1}{\rho }}{\vec {\nabla }}P+\nu \nabla ^{2}{\vec {u}}-2{\vec {\Omega }}\times {\vec {u}}.}
… excerpt ends here. Continue reading the full article.
