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

The Rossby parameter β is given by the equation:

Where is the latitude, Ω is the angular speed of the Earth's rotation, and is the mean radius of the Earth.

Although both involve the Coriolis effect, the Rossby parameter describes the variation of the effect with latitude (hence the latitudinal derivative), and should not be confused with the Rossby number.

The Rossby parameter has units of inverse length inverse time. Below is a table of Rossby parameters as a function of latitude [1].

Variation of β with Latitude
Latitude (°)	β (1013 cm-1sec-1)
90	0
75	0.593
60	1.145
45	1.619
30	1.983
15	2.12
0	2.29

The Rossby parameter increases near the poles, and is zero at the equator. The Rossby parameter is used to determine the speed of a Rossby wave. The wave speed is given by

where c is the wave speed, u is the mean westerly flow, β is the Rossby parameter, and k is the total wave number.

The Rossby parameter is named after C. G. Rossby, who used it to describe zonal circulation in the atmosphere [1].



Derivation of Rossby parameter and Rossby wave [1]
Rossby waves exist for the simplified case of ideal (i.e. frictionless) homogenous, incompressible atmosphere in purely horizontal motion. Therefore, the velocity of the atmosphere is represented by:

Where x points eastward and y points northward. We assume there is no vertical velocity.

The motion is then defined by the following equation, which expresses the conservation of absolute vorticity:


Where f is the Coriolis parameter, which is the force compelling the horizontal motion of the atmosphere due to the rotation of the Earth. The vertical component of the vorticity is also due to the rotation of the Earth, and ζ is the vertical component of the vorticity of motion of the air relative to the Earth’s surface.

So we have the Coriolis force f

And the vertical component of the vorticity ζ

It follows that, by taking the time derivative of the equation of conservation of absolute vorticity, that the constant vanishes and we have

Since the Coriolis’ parameter does not depend on longitude or time, it follows that

Then the new variable β is defined as

And hence, the vorticity equation becomes

We see that β may be computed from the equation

Where is the mean radius of the Earth. The parameter β represents the rate at which the Coriolis’ parameter increases northward.



Non-terrestial Rossby waves and parameters
Rossby waves are not limited to the Earth’s fluid motion, but can be extended to other planets as well. Below is a table of planetary constants as well as Rossby parameters for various planets in our solar system [3].

Jupiter’s Great Red Spot has been suggested to be a Rossby wave at the latitude of 22°S [4]. Using the table above, the Rossby parameter for the Great Red Spot should be:

However, Rossby waves are only a shallow model, based on terrestrial atmospheric waves which arise due to turbulence between the atmosphere and the surface. However, the atmosphere of Jupiter is drastically different, most notably because the interior of Jupiter is fluid and lacks any solid surface. So models have also been derived for deep models, see Great Red Spot Dynamics.



Bibliography:
[1] Rossby, C.-G., and Collaborators, Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action, Journal of Marine Research, 1939. P38-55.
[2] Rossby, C-G., Planetary flow patterns in the atmosphere, Quarterly journal of the Royal Meteorological Society.
[3] Milivoj B. Gavrilov, Aleksandar D. Prodanov, The characteristics of Rossby waves frequencies on planets of the Solar system, Planetary and Space Science, Volume 56, Issue 11, October 2008, Pages 1480-1484, ISSN 0032-0633, DOI: 10.1016/j.pss.2008.05.020. (http://www.sciencedirect.com/science/article/B6V6T-4SR716N-2/2/d17fae813749ea99013fd1f4418e42cf)[4] Nezlin, M. V., Snezhkin, E. N. Rossby Vortices, Spiral Structures, Solitons.Springer-Verlag, Berlin, 1993.