Elsasser number

Remarks

With the magnetic diffusivity $\eta = 1/(\mu_0 \sigma)$, where $\mu_0$ is the vacuum permeability, the same number is $\Lambda = B^2/(\mu_0 \rho \, \eta \, \Omega)$. A flow of speed $U$ induces currents of order $\sigma U B$ and so a Lorentz force of order $\sigma U B^2$ per unit volume. The Coriolis force per unit volume is of order $\rho \Omega U$. So $\Lambda$ estimates the ratio of the two forces, and the speed $U$ cancels. Some authors use $2\Omega$ in place of $\Omega$. Example: for the Earth's core, $\sigma \approx 10^6$ S/m, $\rho \approx 10^4$ kg/m³, $\Omega \approx 7.3 \times 10^{-5}$ s⁻¹ and $B \approx 2 \times 10^{-3}$ T give $\Lambda \approx 5$, of order one. It is one of the dimensionless numbers of fluid mechanics used in rotating magnetohydrodynamics.