Rossby number#

Named after: Carl-Gustaf Rossby (1898-1957).

$$\text{Ro} \stackrel{\text{def}}{=} \frac{U}{\Omega L} \sim \frac{\text{inertia}}{\text{rotation}}$$

Measures inertial motion relative to rotational or Coriolis effects. It indicates whether rotation controls the flow dynamics.

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
angular velocity	$\Omega$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Rotating flow

Range	Regime	Description
0 – 0.1	rotation dominated	Coriolis effects dominate inertia. Flow tends toward geostrophic or columnar balance.
0.1 – 1	mixed	Rotation and inertia both influence trajectories, waves, and instabilities.
1 – ∞	inertia dominated	Inertial motion dominates over rotation. Coriolis deflection is a secondary correction.

Name	Symbol	SI units	Dimension
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
angular velocity	\(\Omega\)	\(\mathrm{Hz}\)	\(\text T^{-1}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)