Strong operator topology: Difference between revisions

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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 [[inertia|inertial]] to [[Coriolis force]], terms <math>v\cdot\nabla v\sim U^2 / L</math> and <math>\Omega\times v\sim U\Omega</math> in the [[Navier–Stokes equations]], respectively.<ref name=Abbot>{{cite book |title=Coastal, Estuarial, and Harbour Engineers' Reference Book |author=M. B. Abbott & W. Alan Price |page= 16 |url=http://books.google.com/books?id=vmlqje7hr_4C&pg=PA16&dq=centrifugal+Rossby
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|isbn=0-419-15430-2 |year=1994 |publisher=Taylor & Francis  }}</ref><ref name=Banerjee>{{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=http://books.google.com/books?id=t3pMEnSQlY8C&pg=PA98&dq=centrifugal+Rossby#PPA98,M1  }}</ref> It is commonly used in [[geophysics|geophysical]] phenomena in the [[oceans]] and [[earth's atmosphere|atmosphere]], where it characterizes the importance of [[Coriolis effect|Coriolis accelerations]] arising from [[planet]]ary [[rotation]]. It is also known as the '''Kibel number'''.<ref name=Boubnov>{{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=http://books.google.com/books?id=KOmZVfrnlW0C&pg=PA8&dq=Kibel+%22Rossby+number%22
  }}</ref>
 
The Rossby number (Ro and not R<sub>o</sub>) is defined as:
 
:<math>\mathrm{Ro}=\frac{U}{Lf}</math>
 
where ''U'' and ''L'' are, respectively, characteristic velocity and length scales of the phenomenon and ''f'' = 2 Ω sin φ is the [[Coriolis effect|Coriolis frequency]], where Ω is the [[angular frequency]] of [[planet]]ary [[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 [[tornado]]es, the Rossby number is large (≈ 10<sup>3</sup>), in [[low-pressure system]]s 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<sup>−2</sup> – 10<sup>2</sup>).<ref name=Kantha1>{{cite book |title=Numerical Models of Oceans and Oceanic Processes |author=Lakshmi H. Kantha & Carol Anne Clayson |publisher=Academic Press |isbn=0-12-434068-7 |year=2000 |page=Table 1.5.1, p. 56 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby#PPA56,M1 |nopp=true
}}</ref> As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces (called ''cyclostrophic balance'').<ref name=Holton>{{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
|page= 64 |isbn=0-12-354015-1 |publisher=Academic Press  }}</ref><ref name=Kantha2/> Cyclostrophic balance also commonly occurs in the inner core of a [[tropical cyclone]].<ref name=Adam>{{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=http://books.google.com/books?id=2gO2sBp4ipQC&pg=PA134&dq=Coriolis+cyclostrophic+%22low+pressure+%22#PPA135,M1
|page=135 |year=2003 }}</ref> 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]]'').<ref name=Kantha2>{{cite book |title=p. 103 |author=Lakshmi H. Kantha & Carol Anne Clayson|isbn=0-12-434068-7  |year=2000 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA103&dq=Coriolis+cyclostrophic+%22low+pressure+%22
}}</ref> For a figure showing spatial and temporal scales of motions in the atmosphere and oceans, see Kantha and Clayson <ref Name=Kantha3>{{cite book |author=Lakshmi H. Kantha & Carol Anne Clayson |isbn=0-12-434068-7 |year=2000 |title=Figure 1.5.1 p. 55 |url=http://books.google.com/books?id=Gps9JXtd3owC&pg=PA56&dq=tornado+rossby#PPA55,M1
}}</ref>  
 
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 [[Coriolis_force#Draining_in_bathtubs_and_toilets|flow in a bathtub]]; or for large speeds), the effects of [[planet]]ary [[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 wind|geostrophic approximation]].<ref name=Barry>{{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#PPA115,M1
|page=115 |isbn=0-415-27171-1 |year=2003 |publisher=Routledge  }}</ref>
 
== See also ==
* [[Coriolis effect]]
* [[Centrifugal force]]
 
== References and notes ==
{{reflist|2}}
 
== 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=27 |url=http://books.google.com/books?id=18MFVdYtJCgC&printsec=frontcover&dq=inauthor:Haidvogel#PPA27,M1
|year=1998 |publisher=Imperial College Press |isbn=1-86094-114-1}}
* {{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&cad=5
|page=326 |author=Zygmunt Kowalik & T. S. Murty |year=1993 |publisher=World Scientific |isbn=981-02-1334-4}}
For 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=108 |author=Jeffery Rosenfeld|url=http://books.google.com/books?id=4H0IeN8OT44C&pg=PA108&dq=tornado+rossby#PPA108,M1
|year=2003 |publisher=Basic Books |isbn=0-7382-0891-4}}
 
{{NonDimFluMech}}
[[Category:Atmospheric dynamics]]
[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Fluid dynamics]]
[[Category:Geophysics]]

Latest revision as of 13:00, 16 September 2014

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