Ferraro Isorotation Law

Let us consider a rotating blob of plasma. The rotation does not have to be uniform, but we do assume no change in the azimuthal direction, so the velocity field can be written in cylindrical coordinates as

${\vec {u}}=r\Omega \left(r,z\right){\hat {\varphi }}$

where $r$ is the distance from the rotation axis, $z$ is the direction along the rotation axis, $\Omega$ is the angular velocity and ${\hat {\varphi }}$ is a unit vector in the azimuthal direction. Substituting this velocity field in the ideal induction equation yields

${\frac {\partial {\vec {B}}}{\partial t}}=\nabla \times \left[{\vec {u}}\times {\vec {B}}\right]=r\nabla \Omega \cdot {\vec {B}}{\hat {\varphi }}$

Hence, in steady state

$\nabla \Omega \cdot {\vec {B}}=0$