Alfvén Waves

As discussed in Section 2.2.3 the Lorentz force can be interpreted as the sum of a magnetic tension (B²/μ₀) and a magnetic pressure (B²/2μ₀). By analogy with a string under tension, this will permit transverse waves with an Alfvén velocity, v_A, given by,

When the magnetic field is the only restoring force, Eqn. 49 reduces to,

which can be reduced by vector identity to,

which has two distinct solutions. For k.v₁ = 0, Eqn. 51 (after multiplying out the vector products) becomes

where θ_B is the angle between k and B₀. This is the shear Alfvén wave, with a phase velocity which is just the Alfvén velocity along the magnetic field and zero perpendicular to it, and a group velocity which is always directed along the magnetic field. The k.v₁ = 0 property means that the velocity is perpendicular to the direction of propagation, hence the waves are transverse. The second solution, ω =kv_A, is the compressional Alfvén wave, which propagates isotropically.