What is the Rossby number?

The Rossby number is used to describe whether a phenomenon is large-scale, i.e. if it is affected by earth’s rotation. But do we actually quantify if a fluid flow is affected by earth’s rotation?

Consider two quantities $L$ and $U$ , with $L$ being a characteristic scale-length of the phenomenon (ex. distance between two peaks, distance between two isobars, length of simulation domain) and $U$ the horizontal velocity scale of the motion. The ratio $\frac{L}{U}$ is the time it takes to the motion to cover a distance $L$ with velocity $U$ . If this time is bigger than the period of earth’s rotation, then the phenomenon IS affected by the rotation.

So if $\frac{L}{U} \geq \frac{1}{\Omega}$ , then the phenomenon IS a large-scale one. Thus we can define $\epsilon = \frac{U}{2L \Omega}$ and say that for $\epsilon \leq 1$ a phenomenon is large scale. Phenomena with small Rossby number are dominated by Coriolis force behavior, while those with large Rossby number are dominated by inertial forces (ex: a tornado). However, rotational effects are more evident for low latitudes (i.e. near the equator), so the Rossby number can be different depending on where on earth we are.

(Notice that $\Omega$ is in theory equal to $2 \Omega \sin(\phi)$ , with $\Omega$ being the earth rotational velocity and $\phi$ the angle between the axis of rotation and the direction of fluid movement. In the geophysical context, flows are mostly horizontal (also due to density stratification in both atmosphere and ocean), so $\sin(\phi)$ can be approximated with 1. There is a bunch of different notation, but this $\Omega$ is also referred to as $f$ , called the Coriolis frequency.)

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