The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

This paper presents both rigorous results and physical theory on the breakdown of magnetic flux conservation for ideal plasmas by nonlinear effects. Our analysis is based upon an effective equation for magnetohydrodynamic (MHD) modes at length scales >ℓ>ℓ, with smaller scales eliminated, as in renormalization-group methodology. We prove that flux conservation can be violated at an instant of time for an arbitrarily small length scale ℓℓ, and in the absence of any nonideality, but only if at least one of three necessary conditions is satisfied. These conditions are (i) nonrectifiability of advected loops, (ii) unbounded velocity or magnetic fields, (iii) singular current sheets and vortex sheets that both exist and intersect in sets of large enough dimension. This result gives analytical support to and rigorous constraints on theories of fast turbulent reconnection. Mathematically, our theorem is analogous to Onsager’s result on energy dissipation anomaly in hydrodynamic turbulence. As a physical phenomenon, the breakdown of magnetic flux conservation in ideal MHD is similar to the decay of magnetic flux through a narrow superconducting ring, by phase-slip of quantized flux lines. The effect should be observable both in numerical MHD simulations and in laboratory plasma experiments at moderately high magnetic Reynolds numbers.