# Beta plane approximation

The **beta plane approximation**, in oceanography and general fluid dynamics, is a simplified coordinate system for the equations of motion where the variation of the Coriolis parameter *f *with latitude is approximated by

*f *= f_{0} + **β**y

where f_{0} is the value of f at the mid-latitude of the region and **β** the latitudinal gradient of f at that same latitude. This formalism is used to investigate both equatorial and mid-latitude phenomena (for which there are slightly different beta plane approximations) where f varies significantly over a few tens of degrees latitude. The beta plane approximation allows considerable simplification of the governing equations and therefore the use of analytical investigation methods. See Gill [1982].

The beta plane equations are obtained by introducing a background stratification into the shallow water equations, expanding them around a reference latitude **θ**_{0} with respect to **ε**~ **θ** − **θ**_{0}, and keeping terms up to first order in ". This approximation introduces the horizonal coordinates

x = r_{0} cos **θ**_{0}(**φ** − **φ**_{0})

y = r_{0}(**θ** − **θ**_{0})

and expands the Coriolis parameter as

*f* = *f*_{0} + **β**_{0}y + • • •

where **β**_{0} is the beta paramter at the reference latitude. The resulting equations (after Muller [1995])

are:

Image:Img62.png + Image:Img64.png - Image:Img66.png = Image:Img67.png

Image:Img68.png + Image:Img69.png + Image:Img70.png

0 = Image:Img72.png

Image:Img73.png = Image:Img74.png

Image:Img75.png + Image:Img76.png

where (u, v,w) are the velocity components in the (x, y, z) directions, r_{0} is the mean radius of the Earth, **θ**_{0} is the reference latitude, *f*_{0} = 2**Ωsin ****θ _{0}** is the Coriolis parameter at the reference latitude,
Image:Img81.png
is the beta parameter at the reference latitude,

**ρ**is a constant reference density,

_{*}**δ**p and

**δ**ρ are motionally induced deviations from prescribed background fields, and N is the buoyancy frequency.

This article is written at a definitional level only. Authors wishing to expand this entry are inivited to expand the present treatment, which additions will be peer reviewed prior to publication of any expansion. |

## Further Reading

- J.R.Holton. 2004. An introduction to dynamical meteorology, Academic Press. ISBN 978-0123540157.
- Physical Oceanography Index

## 0 Comments