📂Fluid Mechanics

Definition of the Ekman Number

Definition¹

For a fluid on a celestial body, when the kinematic viscosity is $\nu$, the Coriolis frequency is $f$, and the characteristic length is $L$, the dimensionless quantity defined as follows is called the Ekman number.

$$ \mathrm{Ek} := {\frac{ \nu }{ f L^{2} }} $$

Explanation

The Ekman number is derived as the ratio of viscous forces to Coriolis forces; put simply, a large Ekman number means that the influence of the fluid itself is large. $$ \mathrm{Ek} = {\frac{ \text{viscous forces} }{ \text{Coriolis forces} }} = {\frac{ \rho \nu U / L^{2} }{ \rho f U }} = {\frac{ \nu }{ f L^{2} }} $$

$$ \mathrm{Ro} := {\frac{ U }{ f L }} $$ $$ \mathrm{Re} := \frac{\rho v d}{\mu} $$

Meanwhile, the Ekman number can also be derived as the ratio of the Rossby number $\mathrm{Ro}$ to the Reynolds number $\mathrm{Re}$, as follows. $$ \mathrm{Ek} = {\frac{ \mathrm{Ro} }{ \mathrm{Re} }} = {\frac{ U / f L }{ U L / \nu }} = {\frac{ \nu }{ f L^{2} }} $$