Magnetic helicity

Magnetic helicity is a topological invariant of the magnetic field, quantifying the degree of knottedness and linkage of magnetic field lines within a volume. Defined as the volume integral of the dot product of the magnetic vector potential and the magnetic field, helicity is conserved in ideal MHD — it cannot be created or destroyed by smooth fluid motions, only redistributed between scales or dissipated by resistive effects at very small scales. This conservation makes helicity a fundamental constraint on magnetic field evolution, governing the inverse cascade of magnetic energy in turbulent plasmas and the stability of magnetized structures from laboratory spheromaks to astrophysical jets. The interplay between helicity conservation and magnetohydrodynamic turbulence determines the terminal state of magnetic relaxation: the system evolves toward a minimum-energy subject to constant helicity, a configuration known as a Taylor state, which is the magnetic analog of a thermal equilibrium constrained by a conserved topological charge.

Magnetic helicity is not a derived quantity; it is the conserved charge of the magnetic field, and any theory of plasma evolution that ignores it is conserving the wrong invariant.