The Rossby Number describes the importance of the rotation of the frame on the flow. It is the ratio of the inertial force (e.g. buoyancy) of the fluid to the Coriolis force imposed by rotation. A larger Rossby number will mean the flow’s behavior is less dependent on the rotation.

 

  Ro = \frac{\Upsilon}{2\Omega sin(\varphi)} = \frac{\Upsilon}{fL}

where \Upsilon describes the typical magnitude of the horizontal flow components, \Omega is the rotation rate, \varphi is the latitude of the fluid parcel on a sphere (e.g. Earth), f is the Coriolis parameter, and L is the distance the parcel traverses (characteristic length).

One way to see how the Rossby Number is defined is to consider the approximation of the equation of motion for the rotating fluid¹:

\frac{Du}{dt} + f\hat{z} \times u = \frac{\partial u}{\partial t} + u \cdot \nabla u + f\hat{z} \times u 

  \frac{Du}{dt} + f\hat{z} \times u =   \frac{\Upsilon}{T} + \frac{\Upsilon^2}{L} + f\Upsilon

where T is the characteristic time of the system.

An interesting consequence of increasing the Rossby Number up through unity is that the structure of currents will transition from a quasi-2D (e.g. Taylor Columns) flow to a quasi-3D flow as the ordering influence of the Coriolis force loses its dominance. This can dramatically change the global-scale current patterns of the flow, which can influence the distribution of heat and stresses in the system.

In this example, a Rossby Number much less than unity will have the maximum heat flux at high latitudes (purple) while a Rossby Number larger than unity will have a maximum at low latitudes (red). The concentration of heat closer to the equator correlates well with the preponderance of chaos terrain (which is believed to be formed by either melt through of a thin ice shell or solid state convection in a thick ice shell).

For an deeper look into this example, check out “Ocean-driven Heating of Europa’s Ice Shell at Low Latitudes” by Soderlund et. al in 2014!

¹Marshall, J. & Plumb, R. A., 2012, “Atmosphere, Ocean, and Climate Dynamics: An Introductory Text” 

Image Credit: NASA/JPL/Galileo Project