Force-free, axisymmetric magnetic field in MHD

AI Thread Summary
The discussion focuses on the characteristics of force-free, axisymmetric magnetic fields in magnetohydrodynamics (MHD). It establishes that a force-free condition implies the relationship \(\nabla \times \mathbf{B} = \alpha \mathbf{B}\), where \(\alpha\) is a scalar field. The axisymmetric magnetic field can be expressed as \(\mathbf{B} = \nabla \psi \times \nabla \phi + B_{\phi} \mathbf{e}_{\phi}\). The complexity of inserting this expression into the force-free condition raises the question of whether a simpler approach exists. The reference to Mestel's work highlights the equations governing the poloidal and toroidal components of the magnetic field in this context.
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Homework Statement
Show that a force-free, axisymmetric field has ##B_{\phi} = f(\psi)/R## in cylindrical polars, where ##f## is an arbitrary function and ##\psi(R,z) = rA_{\phi}(R,z)## is the poloidal flux function, and find the equation satisfied by ##\psi##.
Relevant Equations
##-\nabla^2 \mathbf{B} = \lambda^2 \mathbf{B}##
Force free: ##\mathbf{J} \times \mathbf{B} \sim (\nabla \times \mathbf{B}) \times \mathbf{B} = 0##
(N.B. MHD applies so ##\epsilon_0|\partial \mathbf{E}/\partial t|/|\mathbf{J}| \ll 1##).

Axisymmetric: can write ##\mathbf{B} = \nabla \psi \times \nabla \phi + B_{\phi} \mathbf{e}_{\phi}##
(##\phi## is the azimuthal coordinate i.e. ##\nabla \phi = \mathbf{e}_{\phi}/r##)

Inserting into ##(\nabla \times \mathbf{B}) \times \mathbf{B} = 0## gives a bit of a mess. Is there an easier route?
 
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If the cross-product of two vectors is zero, then those vectors are scalar multiples of each other. Hence a force-free field satisfies \nabla \times \mathbf{B} = \alpha\mathbf{B} for some scalar field \alpha, which (taking the divergence of the above with \nabla \cdot \mathbf{B} = 0) must satisfy <br /> \mathbf{B} \cdot \nabla \alpha = 0. Mestel, Stellar Magnetism (2nd ed) at 58ff gives the equations satisfied by the poloidal and toroidal components of \mathbf{B} in the axisymmetric case.
 
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