Magnetic flux freezing

Magnetic flux freezing states that every material loop in an ideal conducting fluid encloses constant magnetic flux. A deformation perpendicular to a uniform field that preserves mass per unit length therefore preserves $B/\rho$.

Combine the material surface element evolution law with the product rule for $\mathbf{B}\cdot d\mathbf{S}$. The result isExpanding the curl gives the equivalent title formula. The ideal magnetohydrodynamic induction equation and Gauss's law for magnetism make the integrand zero and prove magnetic flux freezing, without requiring incompressible flow.

Suppose the Cartesian magnetic flux function is advected, the initial axial velocity is zero, $B_y=f(\psi )$ initially, and $\nabla \cdot \mathbf{u}=g(\psi ,t)$. Then $B_y=f(\psi )\exp [-{\int }_0^{t}g(\psi ,s)ds]$ remains a function of the flux label and time. The axial Lorentz force vanishes because its transverse derivative along a magnetic field line vanishes. Smooth ideal evolution therefore preserves zero axial motion.

The mass-to-flux ratio is the matter carried per unit magnetic flux by a bundle of field lines. Ideal magnetohydrodynamics preserves it for a material flux tube.

For fixed cloud shape, gravitational energy scales as $-C_{g}G{M}^2/L$ and magnetic energy as $C_B{\Phi }^2/({\mu }_{0}L)$. The magnetic-criticality threshold for negligible pressure is $M/\Phi >\sqrt{C_B/C_g}/\sqrt{{\mu }_{0}G}$. Ideal flux freezing keeps this ratio fixed during homologous collapse; geometry determines the numerical coefficient.