Rossby number

File:Rossbynumber.png

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 $|\mathbf{v} \cdot \nabla \mathbf{v}| \sim U^2 / L$ and $\Omega \times \mathbf{v} \sim U\Omega$ in the Navier–Stokes equations respectively.{{cite book |title=Coastal, Estuarial, and Harbour Engineers' Reference Book |author=M. B. Abbott & W. Alan Price |page= 16 |url=https://books.google.com/books?id=vmlqje7hr_4C&dq=centrifugal+Rossby&pg=PA16

|isbn=0-419-15430-2 |year=1994 |publisher=Taylor & Francis}}{{cite book |title=Oceanography for beginners |year=2004 |page= 98 |author=Pronab K Banerjee |isbn=81-7764-653-2 |publisher=Allied Publishers Pvt. Ltd. |location=Mumbai, India |url=https://books.google.com/books?id=t3pMEnSQlY8C&dq=centrifugal+Rossby&pg=PA98}} 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.{{cite book |title=Convection in Rotating Fluids |author=B. M. Boubnov, G. S. Golitsyn |page=8 |isbn=0-7923-3371-3 |year=1995 |publisher=Springer |url=https://books.google.com/books?id=KOmZVfrnlW0C&dq=Kibel+%22Rossby+number%22&pg=PA8}}

Definition and theory

The Rossby number (Ro, not R_o) is defined as

: $\text{Ro} = \frac{U}{Lf},$

where U and L are respectively characteristic velocity and length scales of the phenomenon, and $f = 2\Omega \sin \phi$ is the Coriolis frequency, with $\Omega$ being the angular frequency of planetary rotation, and $\phi$ the latitude.

A small Rossby number signifies a system strongly affected by Coriolis forces, and a large Rossby number signifies a system in which inertial and centrifugal forces dominate. For example, in tornadoes, the Rossby number is large (≈ 10³), in low-pressure systems it is low (≈ 0.1–1), and in oceanic systems it is of the order of unity, but depending on the phenomena can range over several orders of magnitude (≈ 10⁻²–10²).{{cite book |title=Numerical Models of Oceans and Oceanic Processes |author1=Lakshmi H. Kantha |author2= Carol Anne Clayson|author2-link=Carol Anne Clayson |publisher=Academic Press |isbn=0-12-434068-7 |year=2000 |page=56 (Table 1.5.1) |url=https://books.google.com/books?id=Gps9JXtd3owC&dq=tornado+rossby&pg=PA56}} As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces (called cyclostrophic balance).{{cite book |title=An Introduction to Dynamic Meteorology |year=2004 |author=James R. Holton |url=https://books.google.com/books?id=fhW5oDv3EPsC&dq=tornado+rossby&pg=PA64

|page= 64 |isbn=0-12-354015-1 |publisher=Academic Press}} Cyclostrophic balance also commonly occurs in the inner core of a tropical cyclone.{{cite book |title=Mathematics in Nature: Modeling Patterns in the Natural World |author=John A. Adam |isbn=0-691-11429-3 |publisher=Princeton University Press |url=https://books.google.com/books?id=2gO2sBp4ipQC&dq=Coriolis+cyclostrophic+%22low+pressure+%22&pg=PA134 |page=135 |year=2003}} In low-pressure systems, centrifugal force is negligible, and balance is between Coriolis and pressure forces (called geostrophic balance). In the oceans all three forces are comparable (called cyclogeostrophic balance).{{cite book |title=Numerical Models of Oceans and Oceanic Processes |page=103 |author1=Lakshmi H. Kantha |author2= Carol Anne Clayson|author2-link=Carol Anne Clayson |isbn=0-12-434068-7 |year=2000 |publisher=Elsevier |url=https://books.google.com/books?id=Gps9JXtd3owC&dq=Coriolis+cyclostrophic+%22low+pressure+%22&pg=PA103}} For a figure showing spatial and temporal scales of motions in the atmosphere and oceans, see Kantha and Clayson.{{cite book |author=Lakshmi H. Kantha |author2= Carol Anne Clayson |author2-link=Carol Anne Clayson|isbn=0-12-434068-7 |year=2000 |title=Numerical Models of Oceans and Oceanic Processes |page=55 (Figure 1.5.1) |publisher=Elsevier |url=https://books.google.com/books?id=Gps9JXtd3owC&dq=tornado+rossby&pg=PA56}}

When the Rossby number is large (either because f is small, such as in the tropics and at lower latitudes; or because L is small, that is, for small-scale motions such as flow in a bathtub; or for large speeds), the effects of planetary rotation are unimportant and can be neglected. When the Rossby number is small, then the effects of planetary rotation are large, and the net acceleration is comparably small, allowing the use of the geostrophic approximation.{{cite book |title=Atmosphere, Weather and Climate |author=Roger Graham Barry & Richard J. Chorley |url=https://books.google.com/books?id=MUQOAAAAQAAJ&dq=Coriolis++%22low+pressure%22&pg=PA115 |page=115 |isbn=0-415-27171-1 |year=2003 |publisher=Routledge}}

Rossby number

Definition and theory

See also

References and notes

Further reading