Ekman number, abbreviated as \(Ek\), a dimensionless number, used in fluid dynamics to describe the ratio of viscous forces to Coriolis forces in a rotating fluid system.
Ekman Number formula |
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\( Ek \;=\; \dfrac{ \nu }{ 2 \cdot \omega \cdot L_c^2 }\) (Ekman Number) \( \nu \;=\; 2 \cdot Ek \cdot \omega \cdot L_c^2 \) \( \omega \;=\; \dfrac{ \nu }{ 2 \cdot Ek \cdot L_c^2 }\) \( L_c \;=\; \sqrt{ \dfrac{ \nu }{ 2 \cdot \omega \cdot Ek } }\) |
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| Symbol | English | Metric |
| \( Ek \) = Ekman Number | \( dimensionless \) | \( dimensionless \) |
| \( \nu \) (Greek symbol nu) = Fluid Kinematic Viscosity | \(ft^2 \;/\; sec\) | \(m^2 \;/\; s\) |
| \( \omega \) (Greek symbol omega) = Angular Velocity of the Rotation System | \(deg \;/\; sec\) | \(rad \;/\; s\) |
| \( L_c \) = Characteristic Length (Scale of the System) | \(ft\) | \(m\) |
Ekman Number Interpretation
Ek << 1 (very small, e.g., ~10^{-4} or smaller in many geophysical cases) - Viscous (frictional) forces are much weaker than Coriolis forces. This implies that rotation dominates over friction in the interior flow, allowing for nearly geostrophic balance away from boundaries. Disturbances can propagate with relatively low frictional decay. The Ekman layer (a thin boundary layer where viscous diffusion balances Coriolis effects) is much thinner than the characteristic vertical scale of the flow.
Ek ~ O(1) - Viscous and Coriolis forces are comparable; significant frictional influence throughout the flow domain.
Ek >> 1 - Viscous forces dominate Coriolis forces; rotation effects are negligible, and the flow behaves more like a non-rotating viscous flow.
Ek ~ O(1) - Viscous and Coriolis forces are comparable; significant frictional influence throughout the flow domain.
Ek >> 1 - Viscous forces dominate Coriolis forces; rotation effects are negligible, and the flow behaves more like a non-rotating viscous flow.
Typical geophysical values (oceans, atmosphere) are very small due to large length scales and rotation, e.g., ~10^{-4} or lower in the ocean for H ~1000 m.
Engineering Significance
Boundary layers and pumping - Critical for modeling Ekman layers in oceans (surface/bottom), atmosphere, and rotating machinery. Explains wind-driven ocean currents, Ekman transport, and upwelling/downwelling (Ekman pumping).Geophysical modeling - Allows simplification of equations (neglect interior friction for small Ek) while capturing essential boundary effects. Used in ocean circulation, atmospheric boundary layers, and lab simulations of rotating flows.
Engineering - Relevant in rotating fluid systems (turbines, centrifuges, precessing cylinders), where small Ek indicates thin boundary layers and potential for secondary flows.
Common Misconceptions or Limitations
Definition Variations - Different conventions for the factor of 2 or sqrt can lead to order-of-magnitude confusion; always check the exact form used in a paper.Assumes Constant Eddy Viscosity - Real flows have variable turbulence/stratification; analytical solutions are approximations.
Not Always Negligible - Even for small bulk Ek, boundary layers matter; neglecting friction entirely loses higher-order derivatives and some boundary conditions.
Turbulence and Stratification - Geophysical applications often require eddy viscosity and may break down in convective or strongly stratified conditions.
