The Ekman number (Ek) is a

dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

used in fluid dynamics to describe the ratio of

viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...

forces to

Coriolis force In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

s. It is frequently used in describing

geophysical Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...

phenomena in the

oceans The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the worl ...

and

atmosphere An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...

in order to characterise the ratio of viscous forces to the Coriolis forces arising from

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

ary

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

. It is named after the

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

oceanographer Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamics ...

Vagn Walfrid Ekman Vagn Walfrid Ekman (3 May 1874 – 9 March 1954) was a Swedish oceanographer. Born in Stockholm to Fredrik Laurentz Ekman, himself an oceanographer, he became committed to oceanography while studying physics at the University of Uppsala and ...

. When the Ekman number is small, disturbances are able to propagate before decaying owing to low frictional effects. The Ekman number also describes the order of magnitude for the thickness of an

Ekman layer The Ekman layer is the layer in a fluid where there is a force balance between pressure gradient force, Coriolis force and turbulent drag. It was first described by Vagn Walfrid Ekman. Ekman layers occur both in the atmosphere and in the ocean. ...

, a

boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...

in which viscous diffusion is balanced by Coriolis effects, rather than the usual convective

inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...

Definitions

It is defined as: :

\mathrm=\frac

- where ''D'' is a characteristic (usually vertical) length scale of a phenomenon; ''ν'', the kinematic

eddy viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

; Ω, the

angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

ary

; and φ, the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

. The term 2 Ω sin φ is the

Coriolis frequency The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi. :f = 2 \Omega \sin \varphi.\, The rotation rate o ...

. It is given in terms of the kinematic viscosity, ''ν''; the angular velocity, Ω; and a characteristic length scale, ''L''. There do appear to be some differing conventions in the literature. Tritton gives: :

\mathrm = \frac.

In contrast, the NRL Plasma Formulary gives: :

\mathrm = \sqrt = \sqrt.

where Ro is the

Rossby number The Rossby number (Ro), named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms , \mathbf \cdot \nabla \mathbf, \sim U^2 / L and \Omega ...

and Re is the

Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...

. These equations can generally not be used in oceanography. An estimation of the viscous terms of Navier-Stokes equation (with eventually the

Eddy Viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

) and of the Coriolis terms needs to be done.

References

Dimensionless numbers of fluid mechanics Fluid dynamics {{fluiddynamics-stub