Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2013 / Paper 1 / 2 / a

An Alfvén wave is a transverse disturbance restored by magnetic tension. In a uniform conducting fluid with background ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a small transverse displacement ${\mathbf{\xi }}_{\perp }(z,t)$ bends the magnetic field lines. Magnetic flux freezing gives ${\mathbf{b}}_{\perp }=B_0{\partial }_z{\mathbf{\xi }}_{\perp }$, and the linear transverse Lorentz force density is $(B_0/{\mu }_0){\partial }_z{\mathbf{b}}_{\perp }$. Thus the Linear Alfvén wave equations reduce toFor this shear polarization, $\delta \rho =\delta p=0$ at linear order. The disturbance transports energy along the background magnetic field; a single traveling wave has equal perturbation magnetic energy and kinetic energy, with ${\mathbf{b}}_{\perp }=\mp \sqrt{{\mu }_0\rho }{\mathbf{u}}_{\perp }$ for propagation along $\pm {\mathbf{B}}_0$. These properties distinguish it from compressive magnetosonic waves.

The ideal magnetohydrodynamics approximation requires strong electrical coupling and wavelengths large compared with microscopic scales. Finite electrical resistivity and viscosity dissipate the wave's energy. In stellar winds and other astrophysical plasmas, Alfvén waves carry disturbances and can contribute to heating when dissipated. The Lundquist liquid-metal Alfvén wave experiment provides a laboratory counterpart: a conducting liquid in a strong applied magnetic field transmits a transverse magnetic disturbance, making the field's mechanical restoring action measurable.