Magnetic Helicity

Magnetic helicity, K, is the self linkage of magnetic field lines.
Magnetic Helicity Figure with Equation
Figure 1. Linking magnetic field lines and helicity equation K ≡ ∫A·BdV = 2φψ


Where A is the vector potential, B is the magnetic field and φ and ψ are the magnitudes of the simply linked fluxes.

Like most things in nature, magnetic field lines relax to a state of minimum energy and helicity is the best constant of motion in a magnetized plasma.

The idea of magnetic helicity and relaxation can be seen in a simple experiment with a string. Hold a string at both ends and twist each end in opposite directions. Slowly bring the ends together and the string ‘relaxes’ into loops. The string looped about itself is in a lower energy level than the string twisted about itself.

The same principle holds for the relaxation of magnetic field lines. This can be seen in the figures below. Originally there are longitudinal magnetic field lines from the solenoid coils.


Applying a voltage across the ends of the cylinder causes a current to flow in the plasma.

helicity equation



The plasma current creates a magnetic field in the azimuthal direction.




You can now see the linkage of azimuthal and longitudinal flux.



The magnetic field lines relax to a new configuration, which is in a lower energy level.