Ekman number

The Ekman number is a dimensionless number used in describing geophysical phenomena in the oceans and atmosphere. It characterises the ratio of viscous forces in a fluid to the fictitious forces arising from planetary rotation. It is named after the Swedish oceanographer Vagn Walfrid Ekman.

More generally, in any rotating flow, the Ekman number \( \mathrm{Ek} \) is the ratio of viscous forces to Coriolis forces. When the Ekman number is small, disturbances are able to propagate before decaying owing to frictional effects. The Ekman number describes the order of magnitude for the thickness of an Ekman layer, a boundary layer in which viscous diffusion is balanced by Coriolis effects, rather than the usual convective inertia.

Definitions

It is defined as:

\[Ek=\frac{\nu}{2D^2\Omega\sin\varphi}\]

- where D is a characteristic (usually vertical) length scale of a phenomenon; ν, the kinematic eddy viscosity; Ω, the angular velocity of planetary rotation; and φ, the latitude. The term 2 Ω sin φ is the Coriolis frequency. It is given in terms of the kinematic viscosity \( \nu \), the angular velocity \( \Omega \), and a characteristic lengthscale \( L \).

There do appear to be some differing conventions in the literature.

Tritton gives:

\[ \mathrm{Ek} = \frac{\nu}{\Omega L^2}. \]

In contrast, the NRL Plasma Formulary gives:

\[ \mathrm{Ek} = \sqrt{\frac{\nu}{2\Omega L^2}}. \]

NRL states that this latter definition is equivalent to the root of the ratio of Rossby number to Reynolds number. There are various definitions for the Rossby number as well.

References

ar:عدد إيكمان