Magnetohydrodynamic Waves
The three fundamental linear wave modes of a magnetized conducting fluid: shear Alfven, fast, and slow magnetosonic.
Origin
Linearizing the ideal MHD equations about a uniform state with density rho, pressure p, and magnetic field B yields a dispersion relation supporting three wave families. The restoring forces are magnetic tension along field lines, magnetic pressure, and thermal pressure. Two characteristic speeds appear: the Alfven speed v_A = B / sqrt(mu0 rho) and the sound speed c_s = sqrt(gamma p / rho).
The Three Modes
- Shear Alfven wave: propagates along B with speed v_A cos(theta); driven purely by magnetic tension; incompressible; bends field lines without changing their strength.
- Fast magnetosonic wave: magnetic and thermal pressure add; propagates in all directions with the largest phase speed; compresses both plasma and field together.
- Slow magnetosonic wave: magnetic and thermal pressure oppose; slowest of the three; compresses plasma while field partly rarefies.
Dispersion Relation
For propagation at angle theta to B, the shear Alfven branch has phase speed v_A cos(theta). The fast and slow branches solve v^4 - (v_A^2 + c_s^2) v^2 + v_A^2 c_s^2 cos^2(theta) = 0. When v_A greatly exceeds c_s, as in a hot low-density fusion plasma, the fast wave approaches v_A and the slow wave approaches the sound branch confined near the field direction.
Relevance to Confinement Devices
MHD waves set the fastest timescales in a confined plasma and underlie many diagnostics and heating schemes. Global Alfven eigenmodes can be driven unstable by energetic particles, which matters for any burning plasma where fusion-born ions or beam ions provide a fast population. In the Hyperion spherical-tokamak concept the strong on-axis field and low aspect ratio place the Alfven speed high, so Alfven-eigenmode analysis is part of the design-stage stability assessment; no such behavior has been measured on hardware, which does not yet exist.