Lagrange Multiplier Formulation of Ideal Magnetohydrodynamics (IMHD)*

This talk will be about the continuation of work concieved of and done with R. L. Dewar.

In the standard formulation of IMHD the electric field is eliminated by taking the curl of the Ideal (i.e. zero resistivity) Ohm’s Law (IOL) and using the Faraday-Maxwell induction equation. This builds in the “frozen-in flux” topological constraint [1] that forbids reconnection and hence magnetic island creation and destruction in magnetic toroidal confinement, which leads to singular behavior at rational magnetic surfaces in 3-D equilibria when regarded as the long-time limit of a damped evolution away from an initially axisymmetric

state.

A recent paper [2] derives a new version of IMHD by enforcing (weakly or strongly) the IOL constraint using a Lagrange multiplier field that can be identified physically as a time-space-varying electrostatic polarization vector (cf. [3]). This affords the possibility of regularizing IMHD by relaxing the frozen-in flux constraint using a sequence of approximating Lagrange multipliers.

References

[1] W.A. Newcomb, Ann. Phys. 3, 347–385 (1958) “Motion of Magnetic Field Lines of Force”

[2] R.L. Dewar & Z.S. Qu, J. Plasma Phys. 88, 835880101-1–37 (2022) “Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint”

[3] M.G. Calkin, Can. J. Phys. 41, 2241–2251 (1963) “An Action Principle for Magnetohydrodynamics”