A Topological Problem in Magnetohydrodynamics

(Abstract)

In magnetohydrodynamics, a magnetic field B evolves in time with no change in the topology T(B) of its lines of force defined by

.

In this physical sense, a field in static equilibrium governed by the force-free equations:

B = B

. B = 0,

is subject to a prescribed T (B) the field happens to possess from its evolutionary past. I will describe the challenge of expressing T (B) as integral equations, the mixed (hyperbolic & elliptic) nature of the force-free pdes, and, the general need for this nonlinear, nonlocal problem to admit weak solutions in which B is tangentially discontinuous at suitably placed surfaces. It is instructive to contrast this kind of weak solutions with those containing shocks associated with intersecting characteristics in compressive hydrodynamics as a purely hyperbolic pde system. The construction of such equilibrium states by numerical simulation, via 3D time-dependent topology-preserving relaxation, will be briefly discussed, pointing out an interesting break through. The formation of magnetic discontinuities as proposed by E. N. Parker is a promising explanation of the million-degree temperature of the outer atmosphere of the Sun.

(Dr. Boon Chye Low is an accomplished mathematical physicist. He is currently the section head for solar atmosphere and heliosphere, High Altitude Observatory, National Center for Atmospheric Research (NCAR)

at Colorado, USA. Dr. Low received his Ph.D. in Physics in 1972 from University of Chicago. His main research interests are magnetohydro-dynamics, gas dynamics and plasma physics, the physics of magnetic fields in the solar atmosphere, interplanetary space, and related astrophysical systems. Currently Dr. Low is a member of the editorial board of Asia Pacific Journal of Climate Change.)