# Force-free, axisymmetric magnetic field in MHD

• ergospherical
ergospherical
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?

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 $$\mathbf{B} \cdot \nabla \alpha = 0.$$ Mestel, Stellar Magnetism (2nd ed) at 58ff gives the equations satisfied by the poloidal and toroidal components of $\mathbf{B}$ in the axisymmetric case.

Last edited:
vanhees71 and ergospherical

## What is a force-free, axisymmetric magnetic field in MHD?

A force-free, axisymmetric magnetic field in magnetohydrodynamics (MHD) refers to a magnetic field configuration where the Lorentz force is zero, meaning the magnetic forces are balanced internally. This occurs when the current density is parallel to the magnetic field. Axisymmetry implies that the system is symmetric around a central axis, simplifying the mathematical description of the field.

## Why are force-free magnetic fields important in astrophysics?

Force-free magnetic fields are crucial in astrophysics because they often model the magnetic structures in various cosmic environments, such as the solar corona, neutron stars, and accretion disks around black holes. These fields help in understanding the stability, dynamics, and energy distribution in such high-energy astrophysical systems.

## How is the force-free condition mathematically expressed?

The force-free condition is mathematically expressed by the equation ∇ × B = αB, where B is the magnetic field and α is a scalar function that can vary spatially but is constant along magnetic field lines. This equation indicates that the current density J (given by ∇ × B) is parallel to the magnetic field B.

## What are the challenges in solving the force-free, axisymmetric magnetic field equations?

Solving these equations is challenging due to their non-linear nature and the complexity added by the axisymmetric condition. Analytical solutions are rare and often require simplifying assumptions. Numerical methods are typically employed to find solutions, but these can be computationally intensive and sensitive to boundary conditions and initial configurations.

## What are some common applications of force-free, axisymmetric magnetic field models?

Common applications include modeling the magnetic fields in the solar corona to understand solar flares and coronal mass ejections, simulating the magnetospheres of neutron stars, and studying the dynamics of magnetized accretion disks around black holes. These models are also used in laboratory plasma experiments to replicate and study astrophysical phenomena.

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