Just as mechanical waves propagate along a tensioned string, transverse magnetic tensions propagate along magnetic field lines embedded in an elastic conducting fluid. Discovered by Hannes Alfvén in 1942, these Alfvén Waves are the fundamental carrier of electromagnetic energy in astrophysical plasmas. In this article, we linearize the ideal MHD equations to derive the three fundamental MHD wave modes: the incompressible Shear Alfvén Wave, and the compressive Fast and Slow Magnetosonic Waves.
1. Linearization of the Ideal MHD Equations
Consider an unperturbed, static, uniform plasma with density (
ho_0), pressure (P_0), and magnetic field (mathbf{B}_0 = B_0 hat{mathbf{z}}). Let perturbations have plane-wave dependence (xp(i(mathbf{k} cdot mathbf{r} – omega t))), where the wavevector is aligned without loss of generality in the (x ext{-}z) plane: (mathbf{k} = k_perp hat{mathbf{x}} + k_parallel hat{mathbf{z}} = k sin heta hat{mathbf{x}} + k cos heta hat{mathbf{z}}).
The linearized continuity, momentum, induction, and adiabatic energy equations are:
[
-iomega
ho_1 + i
ho_0 (mathbf{k} cdot mathbf{v}_1) = 0
]
[
-iomega
ho_0 mathbf{v}_1 = -imathbf{k} P_1 + rac{1}{mu_0} (imathbf{k} imes mathbf{B}_1) imes mathbf{B}_0
]
[
-iomega mathbf{B}_1 = imathbf{k} imes (mathbf{v}_1 imes mathbf{B}_0)
]
[
-iomega P_1 = -igamma P_0 (mathbf{k} cdot mathbf{v}_1)
]
Defining the acoustic sound speed (c_s quiv sqrt{gamma P_0 /
ho_0}) and the Alfvén Speed (v_A quiv rac{B_0}{sqrt{mu_0
ho_0}}), we eliminate (
ho_1, P_1), and (mathbf{B}_1) to construct the vector wave equation for velocity perturbation (mathbf{v}_1):
[
-omega^2 mathbf{v}_1 = -c_s^2 mathbf{k}(mathbf{k} cdot mathbf{v}_1) + mathbf{v}_A imes left[ mathbf{k} imes (mathbf{k} imes (mathbf{v}_1 imes mathbf{v}_A))
ight]
]
2. Decoupling the Shear Alfvén Wave
Expanding the vector components, the (y)-component (perpendicular to both (mathbf{B}_0) and (mathbf{k})) decouples completely from density and pressure fluctuations ((
abla cdot mathbf{v}_1 = 0)):
[
omega^2 v_{1y} = k_parallel^2 v_A^2 v_{1y} = k^2 v_A^2 cos^2 heta , v_{1y}
]
This gives the exact dispersion relation for the Shear (Torsional) Alfvén Wave:
[
omega = k_parallel v_A = k v_A cos heta
]
The phase and group velocities are:
[
mathbf{v}_phi = rac{omega}{k} hat{mathbf{k}} = v_A cos heta hat{mathbf{k}}, quad mathbf{v}_g =
abla_mathbf{k} omega = v_A hat{mathbf{z}} = mathbf{v}_A
]
Crucially, the group velocity points strictly along the background magnetic field (mathbf{B}_0), regardless of the propagation angle ( heta). Energy in shear Alfvén waves is channeled purely along magnetic field lines, acting as the primary mechanism heating the solar corona and transferring angular momentum in accretion disks.
3. Fast and Slow Magnetosonic Waves
The remaining coupled system for (v_{1x}) and (v_{1z}) yields the compressive dispersion relation:
[
omega^4 – k^2 (v_A^2 + c_s^2)omega^2 + k^4 v_A^2 c_s^2 cos^2 heta = 0
]
Solving this biquadratic equation yields the phase speeds (v_phi = omega/k) of the Fast (+) and Slow (-) Magnetosonic Modes:
[
v_{phi,pm}^2 = rac{1}{2}left( v_A^2 + c_s^2 pm sqrt{(v_A^2 + c_s^2)^2 – 4 v_A^2 c_s^2 cos^2 heta}
ight)
]
In the Fast Magnetosonic Wave, plasma thermal pressure and magnetic pressure oscillate in phase, reinforcing each other. In the Slow Magnetosonic Wave, thermal and magnetic pressures oscillate out of phase, partially canceling. The spatial radiation pattern is visualized by Friedrichs group-velocity polar diagrams, demonstrating how magnetic fields introduce profound mechanical anisotropy into cosmic fluid dynamics.
