Alfvén Speed

Alfvén Speed

Alfvén waves are low-frequency, transverse electromagnetic waves that propagate along magnetic field lines in a magnetized plasma. They are one of the fundamental wave modes in magnetohydrodynamics (MHD).

Alfvén Speed

The speed at which Alfvén waves propagate is:

$v_A = \frac{B}{\sqrt{\mu_0 \rho}}$

where:

• (B) is the magnetic field strength (T)
• (\mu_0 = 4\pi \times 10^{-7}) H/m is the permeability of free space
• (\rho = n m_i) is the mass density of the plasma (kg/m³), with (n) the number density and (m_i) the ion mass

The Alfvén speed is the plasma analog of the speed of sound in a gas, with magnetic pressure playing the role of thermal pressure.

Alfvén Mach Number

The Alfvén Mach number measures how fast a flow is relative to the Alfvén speed:

$M_A = \frac{v}{v_A}$

When (M_A < 1), the flow is sub-Alfvénic. When (M_A > 1), it is super-Alfvénic.

Sound Speed

For comparison, the sound speed in a plasma is:

$v_s = \sqrt{\frac{\gamma k_B T}{m_i}}$

where (\gamma = 5/3) is the adiabatic index and (k_B = 1.38 \times 10^{-23}) J/K is the Boltzmann constant.

Physical Significance

• Solar wind: The Alfvén point marks where the solar wind becomes super-Alfvénic
• Fusion plasmas: Alfvén waves can heat plasma via wave-particle resonance
• Space weather: Alfvénic fluctuations carry energy from the Sun to Earth's magnetosphere