# Beta plane

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In geophysical fluid dynamics, an approximation whereby the Coriolis parameter, f, is set to vary linearly in space is called a beta plane approximation.

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,

On a rotating sphere such as the Earth, f varies with the sine of latitude; in the so-called f-plane approximation, this variation is ignored, and a value of f appropriate for a particular latitude is used throughout the domain. This approximation can be visualized as a tangent plane touching the surface of the sphere at this latitude.

In geophysical fluid dynamics, the f-plane approximation is an approximation where the Coriolis parameter, denoted f, is set to a constant value.

A more accurate model is a linear Taylor series approximation to this variability about a given latitude ${\displaystyle \phi _{0}}$:

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

${\displaystyle f=f_{0}+\beta y}$, where ${\displaystyle f_{0}}$ is the Coriolis parameter at ${\displaystyle \phi _{0}}$, ${\displaystyle \beta =(\mathrm {d} f/\mathrm {d} y)|_{\phi _{0}}=2\Omega \cos(\phi _{0})/a}$ is the Rossby parameter, ${\displaystyle y}$ is the meridional distance from ${\displaystyle \phi _{0}}$, ${\displaystyle \Omega }$ is the angular rotation rate of the Earth, and ${\displaystyle a}$ is the Earth's radius. [1]

The Rossby parameter is a number used in geophysics and meteorology which arises due to the meridional variation of the Coriolis force caused by the spherical shape of the Earth. It is important in the generation of Rossby waves. The Rossby parameter is given by

In analogy with the f-plane, this approximation is termed the beta plane, even though it no longer describes dynamics on a hypothetical tangent plane. The advantage of the beta plane approximation over more accurate formulations is that it does not contribute nonlinear terms to the dynamical equations; such terms make the equations harder to solve. The name 'beta plane' derives from the convention to denote the linear coefficient of variation with the Greek letter β.

The beta plane approximation is useful for the theoretical analysis of many phenomena in geophysical fluid dynamics since it makes the equations much more tractable, yet retains the important information that the Coriolis parameter varies in space. In particular, Rossby waves, the most important type of waves if one considers large-scale atmospheric and oceanic dynamics, depend on the variation of f as a restoring force; they do not occur if the Coriolis parameter is approximated only as a constant.

The Coriolis frequencyƒ, also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate Ω of the Earth multiplied by the sine of the latitude φ.

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces, quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.

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Rossby waves, also known as planetary waves, are a natural phenomenon in the atmospheres and oceans of planets that largely owe their properties to rotation of the planet. Rossby waves are a subset of inertial waves. They were first identified by Carl-Gustaf Arvid Rossby.

The Rossby number (Ro) named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms and in the Navier–Stokes equations, respectively. It is commonly used in geophysical phenomena in the oceans and atmosphere, where it characterizes the importance of Coriolis accelerations arising from planetary rotation. It is also known as the Kibel number.

The Ekman number (Ek) is a dimensionless number used in fluid dynamics to describe the ratio of viscous forces to Coriolis forces. It is frequently used in describing geophysical phenomena in the oceans and atmosphere in order to characterise the ratio of viscous forces to the Coriolis forces arising from planetary rotation. It is named after the Swedish oceanographer Vagn Walfrid Ekman.

The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis force 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. The true wind almost always differs from the geostrophic wind due to other forces such as friction from the ground. Thus, the actual wind would equal the geostrophic wind only if there were no friction and the isobars were perfectly straight. Despite this, much of the atmosphere outside the tropics is close to geostrophic flow much of the time and it is a valuable first approximation. Geostrophic flow in air or water is a zero-frequency inertial wave.

A Kelvin wave is a wave in the ocean or atmosphere that balances the Earth's Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase speed of the wave crests is equal to the group speed of the wave energy for all frequencies. This means that it retains its shape as it moves in the alongshore direction over time.

In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are rarely used: capital A, B, E, Z, H, I, K, M, N, O, P, T, Y, X. Small ι, ο and υ are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for ε/ϵ and π/ϖ. The archaic letter digamma (Ϝ/ϝ/ϛ) is sometimes used.

The Voigt effect is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium. Unlike many other magneto-optical effects such as the Kerr or Faraday effect which are linearly proportional to the magnetization, the Voigt effect is proportional to the square of the magnetization and can be seen experimentally at normal incidence. There are several denominations for this effect in the literature: the Cotton–Mouton effect, the Voigt effect, and magnetic-linear birefringence. This last denomination is closer in the physical sense, where the Voigt effect is a magnetic birefringence of the material with an index of refraction parallel and perpendicular ) to the magnetization vector or to the applied magnetic field.

Potential vorticity (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers.

Equatorial waves are oceanic and atmospheric waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds.

Rossby-gravity waves are equatorially trapped waves, meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius. They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency waves found near the equator and are derived using the equatorial beta plane approximation.

Q-vectors are used in atmospheric dynamics to understand physical processes such as vertical motion and frontogenesis. Q-vectors are not physical quantities that can be measured in the atmosphere but are derived from the quasi-geostrophic equations and can be used in the previous diagnostic situations. On meteorological charts, Q-vectors point toward upward motion and away from downward motion. Q-vectors are an alternative to the omega equation for diagnosing vertical motion in the quasi-geostrophic equations.

Rossby Wave Instability (RWI) is a concept related to astrophysical discs. In non-self-gravitating discs, for example around newly forming stars, the instability can be triggered by an axisymmetric bump, at some radius , in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation [ , ] in the vicinity of consisting of anticyclonic vortices. These vortices are regions of high pressure and consequently act to trap dust particles which in turn can facilitate planetesimal growth in proto-planetary discs. The Rossby vortices in the discs around stars and black holes may cause the observed quasi-periodic modulations of the disc's thermal emission.

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

## References

1. Holton, James R.; Hakim, Gregory J. (2013). An Introduction to Dynamic Meteorology (fifth ed.). Academic Press. p. 160.