Rossby number

The Rossby number (Ro) is a dimensionless number , a similarity criterion used to describe the flow. Named after Carl Gustav Rossby . It is the relationship between the force of inertia and the Coriolis force . In the Navier-Stokes equation , these are the terms $v\cdot \nabla v\sim U^{2}/L$ ${\ displaystyle v \ cdot \ nabla v \ sim U ^ {2} / L}$ ${\ displaystyle v \ cdot \ nabla v \ sim U ^ {2} / L}$ ( inertia force ) and $\Omega \times v\sim U\Omega$ ${\ displaystyle \ Omega \ times v \ sim U \ Omega}$ ${\ displaystyle \ Omega \ times v \ sim U \ Omega}$ ( Coriolis force ) ^[1] ^[2] . Often used to describe geophysical phenomena in the ocean and atmosphere, where it characterizes the importance of the Coriolis acceleration caused by the rotation of the Earth. Also known as the " kibel number " (Ki) ^[3] .

Content

Math expression

The Rossby number is denoted as $\mathrm {Ro}$ ${\ displaystyle \ mathrm {Ro}}$ (not how $R_{o}$ ${\ displaystyle R_ {o}}$ ) and is defined as follows:

\mathrm {Ro} ={\frac {U}{Lf}},

{\ displaystyle \ mathrm {Ro} = {\ frac {U} {Lf}},}

Where $U$ ${\ displaystyle U}$ - the characteristic speed of the geophysical phenomenon ( cyclone , ocean vortex ), $L$ ${\ displaystyle L}$ - the characteristic spatial scale of the geophysical phenomenon, $f=2\Omega \sin \varphi$ ${\ displaystyle f = 2 \ Omega \ sin \ varphi}$ Is the Coriolis parameter , where $\Omega$ ${\ displaystyle \ Omega}$ Is the angular velocity of rotation of the Earth, and $\varphi$ ${\ displaystyle \ varphi}$ - latitude .

Usage

A small Rossby number is a sign of a system that is significantly affected by the Coriolis force . A large number of Rossby is a sign of a system in which inertia and centrifugal force dominate. For example, for a tornado, the Rossby number is large (≈10 ³ , high speed and small spatial scale), and for a low pressure system (such as a cyclone ) it is small (≈0.1-1). For various phenomena in the ocean, the Rossby number can vary on a scale of ≈10 ⁻² –10 ² ^[4] . As a result, the action of the Coriolis force on the tornado is negligible and a balance is achieved between the baric gradient and centrifugal force (cyclostrophic balance) ^[5] ^[6] .

In low pressure systems, centrifugal force is negligible, and a balance is reached between the Coriolis force and the baric gradient ( geostrophic balance ). In the oceans, all three forces are comparable to each other (cyclo-geostrophic balance) ^[6] . In the work of Kanthi ( LH Kantha ) and Clayson ( CA Clayson ) you can see an illustration showing the spatial and temporal scales of phenomena in the atmosphere and ocean ^[7] .

When the Rossby number is large (either because it is small $f$ ${\ displaystyle f}$ as it occurs in the tropics and lower latitudes; or $L$ ${\ displaystyle L}$ small, as is the case with a sink in the sink; or speeds are great), the effect of the Earth's rotation is negligible and can be neglected. When the Rossby number is small, then the effect of the Earth's rotation is significant and the overall acceleration is relatively small, allowing the use of the geostrophic approximation ^[8] .

Notes

↑ MB Abbott & W. Alan Price. Coastal, Estuarial, and Harbor Engineers' Reference Book . - Taylor & Francis, 1994. - P. 16. - ISBN 0419154302 .
↑ Pronab K Banerjee. Oceanography for beginners . - Mumbai, India: Allied Publishers Pvt. Ltd., 2004. - P. 98. - ISBN 8177646532 .
↑ Boubnov BM, Golitsyn GS Convection in Rotating Fluids . - Springer, 1995. - P. 8. - ISBN 0792333713 .
↑ Lakshmi H. Kantha & Carol Anne Clayson. Numerical Models of Oceans and Oceanic Processes . - Academic Press, 2000. - P. Table 1.5.1, p. 56. - ISBN 0124340687 .
↑ James R. Holton. An Introduction to Dynamic Meteorology . - Academic Press, 2004. - P. 64. - ISBN 0123540151 .
↑ ¹ ² Lakshmi H. Kantha & Carol Anne Clayson. p. 103 . - 2000. - ISBN 0124340687 .
↑ Lakshmi H. Kantha & Carol Anne Clayson. Figure 1.5.1 p. 55 . - 2000. - ISBN 0124340687 .
↑ Roger Graham Barry & Richard J. Chorley. Atmosphere, Weather and Climate . - Routledge, 2003. - P. 115. - ISBN 0415271711 .

Literature

Robert Stewart Introduction to physical oceanography. Chapter 10, Geostrophic Currents . - 2005. Archived on January 3, 2011. Archived January 3, 2011 on Wayback Machine

[Abbot-1] MB Abbott & W. Alan Price. Coastal, Estuarial, and Harbor Engineers' Reference Book . - Taylor & Francis, 1994. - P. 16. - ISBN 0419154302 .

[Banerjee-2] Pronab K Banerjee. Oceanography for beginners . - Mumbai, India: Allied Publishers Pvt. Ltd., 2004. - P. 98. - ISBN 8177646532 .

[Boubnov-3] Boubnov BM, Golitsyn GS Convection in Rotating Fluids . - Springer, 1995. - P. 8. - ISBN 0792333713 .

[Kantha1-4] Lakshmi H. Kantha & Carol Anne Clayson. Numerical Models of Oceans and Oceanic Processes . - Academic Press, 2000. - P. Table 1.5.1, p. 56. - ISBN 0124340687 .

[Holton-5] James R. Holton. An Introduction to Dynamic Meteorology . - Academic Press, 2004. - P. 64. - ISBN 0123540151 .

[Kantha2-6] ¹ ² Lakshmi H. Kantha & Carol Anne Clayson. p. 103 . - 2000. - ISBN 0124340687 .

[Kantha3-7] Lakshmi H. Kantha & Carol Anne Clayson. Figure 1.5.1 p. 55 . - 2000. - ISBN 0124340687 .

[Barry-8] Roger Graham Barry & Richard J. Chorley. Atmosphere, Weather and Climate . - Routledge, 2003. - P. 115. - ISBN 0415271711 .