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 inertial to Coriolis force, terms 
and 
in the Navier–Stokes equations, respectively.[1][2] 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.[3]
The Rossby number (Ro and not 
) is defined as:
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 frequency 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 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).[4] As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces (called cyclostrophic balance).[5][6] This balance also occurs at the outer eyewall of a tropical cyclone.[7] 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).[6] For a figure showing spatial and temporal scales of motions in the atmosphere and oceans, see Kantha and Clayson [8]
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.[9]
[edit] See also
[edit] References and notes
- ^ M. B. Abbott & W. Alan Price (1994). Coastal, Estuarial, and Harbour Engineers' Reference Book. Taylor & Francis. p. 16. ISBN 0419154302. http://books.google.com/books?id=vmlqje7hr_4C&pg=PA16&dq=centrifugal+Rossby.
- ^ Pronab K Banerjee (2004). Oceanography for beginners. Mumbai, India: Allied Publishers Pvt. Ltd.. p. 98. ISBN 8177646532. http://books.google.com/books?id=t3pMEnSQlY8C&pg=PA98&dq=centrifugal+Rossby#PPA98,M1.
- ^ B. M. Boubnov, G. S. Golitsyn (1995). Convection in Rotating Fluids. Springer. p. 8. ISBN 0792333713. http://books.google.com/books?id=KOmZVfrnlW0C&pg=PA8&dq=Kibel+%22Rossby+number%22.
- ^ Lakshmi H. Kantha & Carol Anne Clayson (2000). Numerical Models of Oceans and Oceanic Processes. Academic Press. Table 1.5.1, p. 56. ISBN 0124340687. http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby#PPA56,M1.
- ^ James R. Holton (2004). An Introduction to Dynamic Meteorology. Academic Press. p. 64. ISBN 0123540151. http://books.google.com/books?id=fhW5oDv3EPsC&pg=PA64&dq=tornado+rossby.
- ^ a b Lakshmi H. Kantha & Carol Anne Clayson (2000). p. 103. ISBN 0124340687. http://books.google.com/books?id=Gps9JXtd3owC&pg=PA103&dq=Coriolis+cyclostrophic+%22low+pressure+%22.
- ^ John A. Adam (2003). Mathematics in Nature: Modeling Patterns in the Natural World. Princeton University Press. p. 135. ISBN 0691114293. http://books.google.com/books?id=2gO2sBp4ipQC&pg=PA134&dq=Coriolis+cyclostrophic+%22low+pressure+%22#PPA135,M1.
- ^ Lakshmi H. Kantha & Carol Anne Clayson (2000). Figure 1.5.1 p. 55. ISBN 0124340687. http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby#PPA55,M1.
- ^ Roger Graham Barry & Richard J. Chorley (2003). Atmosphere, Weather and Climate. Routledge. p. 115. ISBN 0415271711. http://books.google.com/books?id=MUQOAAAAQAAJ&pg=PA115&dq=Coriolis++%22low+pressure%22#PPA115,M1.
[edit] Further reading
For more on numerical analysis and the role of the Rossby number, see:
- Dale B. Haidvogel & Aike Beckmann (1998). Numerical Ocean Circulation Modeling. Imperial College Press. p. 27. ISBN 1860941141. http://books.google.com/books?id=18MFVdYtJCgC&printsec=frontcover&dq=inauthor:Haidvogel#PPA27,M1.
- Zygmunt Kowalik & T. S. Murty (1993). Numerical Modeling of Ocean Dynamics: Ocean Models. World Scientific. p. 326. ISBN 9810213344. http://books.google.com/books?id=qiullk0B940C&pg=PA326&vq=Rossby&dq=Murty+inauthor:Kowalik&cad=5.
For an historical account of Rossby's reception in the United States, see
- Jeffery Rosenfeld (2003). Eye of the Storm: Inside the World's Deadliest Hurricanes, Tornadoes, and Blizzards. Basic Books. p. 108. ISBN 0738208914. http://books.google.com/books?id=4H0IeN8OT44C&pg=PA108&dq=tornado+rossby#PPA108,M1.
