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 the mass density $\rho$, velocity $v$, or travel time $\Delta t$. While typical analyses of a frozen magnetic field $\boldsymbol{B}$ are problem-specific and involve nonlocal gradients of the fluid element position $\boldsymbol{x}(t)$, we derive the general, local (in $\Lambda$ or $\lambda$ space) solution $\boldsymbol{B}=(\partial\boldsymbol{x}/\partial\Lambda_\alpha)_t \tilde{B}_\alpha \rho/\tilde{\rho}$, where Lagrangian constants denoted by a tilde are directly fixed at a boundary hypersurface $\tilde{H}$ on which $\boldsymbol{B}$ is known. For a steady flow, $\tilde{\rho}\boldsymbol{B}/\rho=(\partial\boldsymbol{x}/\partial\lambda_\kappa)_{\Delta t}\tilde{B}_\kappa+\boldsymbol{v}\tilde{B}_3/\tilde{v}$; here the electric field $\boldsymbol{E}\sim(\tilde{B}_2\boldsymbol{\nabla}\lambda_1-\tilde{B}_1\boldsymbol{\nabla}\lambda_2)/\tilde{\rho}$ 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 incorporated into existing hydrodynamic codes that are based on stream or path functions. (Abridged)