The Frozen-In Flux Theorem

Alfven's theorem: in a perfectly conducting fluid, magnetic field lines move with the plasma.

Statement

The frozen-in flux theorem, due to Alfven, states that in ideal MHD the magnetic flux through any surface that moves with the fluid is constant in time. Equivalently, two fluid elements initially on the same field line remain on the same field line forever. The plasma and field are locked together: the field is dragged by flows and, reciprocally, the field's tension resists fluid motion across it.

Derivation Sketch

Starting from the ideal induction equation dB/dt = curl(u x B), one computes the rate of change of flux through a comoving surface. The advective and stretching terms exactly cancel the boundary contribution, leaving d(flux)/dt = 0. The key requirement is zero resistivity, so that E + u x B = 0 holds; any finite resistivity introduces a diffusion term that breaks the freezing.

Consequences

Flux freezing underlies flux amplification by compression, the winding-up of field by shear flows, and the very concept of a flux surface in a confined plasma. It is why field topology is conserved in ideal MHD and why reconnection, which changes topology, requires non-ideal physics. It also explains why plasma cannot cross field lines freely, the basis of magnetic confinement.

Relevance

Confinement itself relies on flux freezing keeping plasma tied to closed nested surfaces. Where the theorem locally breaks, at rational surfaces under finite resistivity, islands form and confinement degrades. For the Hyperion breeder concept, the high Lundquist number means flux is well frozen over dynamical timescales, with slow resistive slippage tracked separately in design-stage modeling of a simulated machine.