Rossby number

The Rossby number, named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of centrifugal to Coriolis accelerations.cite book |title=Coastal, Estuarial, and Harbour Engineers' Reference Book |author=M. B. Abbott & W. Alan Price |page= p. 16 |url=http://books.google.com/books?id=vmlqje7hr_4C&pg=PA16&dq=centrifugal+Rossby&lr=&as_brr=0&sig=ACfU3U3TWfz-lGQEkqRdGeb-swtTx1gZ5Q
isbn=0419154302 |year=1994 |publisher=Taylor & Francis ] cite book |title=Oceanography for beginners |year=2004 |page= p. 98 |author=Pronab K Banerjee |isbn=8177646532 |publisher=Allied Publishers Pvt. Ltd.|location=Mumbai, India |url=http://books.google.com/books?id=t3pMEnSQlY8C&pg=PA98&dq=centrifugal+Rossby&lr=&as_brr=0&sig=ACfU3U0K4WxAwmDX1Z8u3z-a5MqL8S2ASQ#PPA98,M1 ] 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=p.8 |isbn=0792333713 |year=1995 |publisher=Springer |url=http://books.google.com/books?id=KOmZVfrnlW0C&pg=PA8&dq=Kibel+%22Rossby+number%22&lr=&as_brr=0&sig=ACfU3U3YSSIg6p-J9J_ge7NohVMWW2YfRw ]

The Rossby number is defined as:

:$R\_o=frac\{U\}\{Lf\}$

where "U" and "L" are, respectively, characteristic velocity and length scales of the phenomenon and "f" = 2 Ω sin φ is the Coriolis frequency, where Ω is the angular velocity of planetary rotation and φ the latitude.

A small Rossby number signifies a system which is strongly affected by Coriolis forces, and a large Rossby number signifies a system in which 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 |author=Lakshmi H. Kantha & Carol Anne Clayson |publisher=Academic Press |isbn=0124340687 |year=2000 |page=Table 1.5.1, p. 56 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby&lr=&as_brr=0&sig=ACfU3U0B5edyjk-Vek5TXK1RNsTikBpGJw#PPA56,M1] 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=http://books.google.com/books?id=fhW5oDv3EPsC&pg=PA64&dq=tornado+rossby&lr=&as_brr=0&sig=ACfU3U3_xIjuaBO-7JXZguecMx9a58Xd3A
page= p. 64 |isbn=0123540151 |publisher=Academic Press ] This balance also occurs at the outer eyewall of a tropical cyclone.cite book |title=Mathematics in Nature: Modeling Patterns in the Natural World |author=John A. Adam |isbn=0691114293 |publisher=Princeton University Press |url=http://books.google.com/books?id=2gO2sBp4ipQC&pg=PA134&dq=Coriolis+cyclostrophic+%22low+pressure+%22&lr=&as_brr=0&sig=ACfU3U2wXO6vGwlbVZ-vqjMVyM45tWpf-w#PPA135,M1
page=p. 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=p. 103 |author=Lakshmi H. Kantha & Carol Anne Clayson|isbn=0124340687 |year=2000 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA103&dq=Coriolis+cyclostrophic+%22low+pressure+%22&lr=&as_brr=0&sig=ACfU3U2D17cQw-EqXGqYCoF5R_h44zwYpQ] 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 & Carol Anne Clayson |isbn=0124340687 |year=2000 |title=Figure 1.5.1 p. 55 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby&lr=&as_brr=0&sig=ACfU3U0B5edyjk-Vek5TXK1RNsTikBpGJw#PPA55,M1]

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=http://books.google.com/books?id=MUQOAAAAQAAJ&pg=PA115&dq=Coriolis++%22low+pressure%22&lr=&as_brr=0&sig=ACfU3U2KRehZqWtLiiGi6hvfN3iVBEDxqw#PPA115,M1
page=p. 115 |isbn=0415271711 |year=2003 |publisher=Routledge ]

References and notes

Further reading

For more on numerical analysis and the role of the Rossby number, see:
* cite book |title=Numerical Ocean Circulation Modeling |author=Dale B. Haidvogel & Aike Beckmann |page=p. 27 |url=http://books.google.com/books?id=18MFVdYtJCgC&printsec=frontcover&dq=inauthor:Haidvogel&lr=&as_brr=0#PPA27,M1
year=1998 |publisher=Imperial College Press |isbn=1860941141
* cite book |title=Numerical Modeling of Ocean Dynamics: Ocean Models |url=http://books.google.com/books?id=qiullk0B940C&pg=PA326&vq=Rossby&dq=Murty+inauthor:Kowalik&lr=&as_brr=0&source=gbs_search_s&cad=5&sig=ACfU3U081GcUbNZTrQ020vtLjo47mYoFvA
page=p. 326 |author=Zygmunt Kowalik & T. S. Murty |year=1993 |publisher=World Scientific |isbn=9810213344For an historical account of Rossby's reception in the United States, see
* cite book |title=Eye of the Storm: Inside the World's Deadliest Hurricanes, Tornadoes, and Blizzards|page=p. 108 |author= |url=http://books.google.com/books?id=4H0IeN8OT44C&pg=PA108&dq=tornado+rossby&lr=&as_brr=0&sig=ACfU3U0n0INLhfBuFHbFysNPudapaEMIoA#PPA108,M1
year=2003 |publisher=Basic Books |isbn=0738208914 |author=Jeffery Rosenfeld

See also

* Coriolis effect
* Centrifugal force