Magnetohydrodynamics using path or stream functions

Magnetization in highly conductive plasmas is ubiquitous to astronomical systems. Flows in such media can be described by three path functions Lambda(alpha), or, for a steady flow, by two stream functions lambda(kappa), and an additional field such as mass density rho, velocity nu, or travel time Delta t. While typical analyses of a frozen magnetic field B are problem-specific and involve nonlocal gradients of the fluid element position x (t), we derive the general, local (in Lambda or lambda space) solution B = (partial derivative x/partial derivative Lambda(alpha))t (B) over tilde (alpha)rho/(rho) over tilde, where Lagrangian constants denoted by a tilde are directly fixed at a boundary hypersurface (H) over tilde on which B is known. For a steady flow, (rho) over tildeB/rho = (partial derivative x/partial derivative lambda(kappa))(Delta t)(B) over tilde (kappa) + nu(B) over tilde (3)/(nu) over tilde; here the electric field E similar to((B) over tilde (2) del lambda(1) - (B) over tilde (1) del lambda(2))/(rho) over tilde depends only on lambda(kappa) and the boundary conditions. Illustrative special cases include compressible axisymmetric flows and incompressible flows around a sphere, showing that viscosity and compressibility enhance the magnetization and lead to thicker boundary layers. Our method is especially useful for directly computing electric fields, and for addressing upstream magnetic fields that vary in spacetime. We thus estimate the electric fields above heliospheres and magnetospheres, compute the draping of magnetic substructure around a planetary body, and demonstrate the resulting inverse polarity reversal layer. Our analysis can be immediately incorporated into existing hydrodynamic codes that are based on stream or path functions, to passively evolve the electromagnetic fields in a simulated flow. Furthermore, in such a prescription the electromagnetic fields are frozen onto the grid, so it may be developed into a fully magnetohydrodynamic, efficient simulation.