# MAGNETOHYDRODYNAMICS USING PATH OR STREAM FUNCTIONS Academic Article

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### abstract

• Non-diffusive flows can be defined by three path functions $\Lambda_\alpha$ or, for a steady flow, by two stream functions $\lambda_\kappa$ and an auxiliary field such as the mass density $\rho$ or the velocity $v$. While typical computations of a frozen magnetic field $\boldsymbol{B}$ involve non-local gradients of the fluid element position $\boldsymbol{x}(t)$, we derive a local solution $\boldsymbol{B}=(\partial\boldsymbol{x} / \partial\Lambda_\alpha)_t \tilde{B}_\alpha \rho/\tilde{\rho}$, where Lagrangian constants denoted by a tilde are fixed at a reference hypersurface $\tilde{H}$ on which $\boldsymbol{B}$ is known. For a steady flow, this becomes $\tilde{\rho}\boldsymbol{B} / \rho = (\partial\boldsymbol{x} / \partial\lambda_\kappa)_{\Delta t}\tilde{B}_\kappa + \boldsymbol{v}\tilde{B}_3/\tilde{v}$, where $\Delta t$ is the travel time from $\tilde{H}$; 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 $\lambda$ and $\tilde{H}$ parameters. Illustrative solutions are derived for compressible axisymmetric flows and incompressible flows around a sphere, showing that viscosity and compressibility enhance the magnetization, and lead to thicker boundary layers.

### publication date

• September 9, 2015