Alfvén Velocity Calculator — Magnetohydrodynamic Wave Speed
The Alfvén velocity is the speed at which Alfvén waves — low-frequency transverse magnetohydrodynamic (MHD) waves — travel along magnetic field lines in a conducting plasma. Enter the magnetic field strength B (in Tesla) and the plasma mass density ρ (in kg/m³) to get the Alfvén speed.
T
kg/m³
Speed at which Alfvén waves propagate along the magnetic field
- 1
μ₀ × ρ
1.2566×10⁻⁶ × 1.0000e-5 = 1.2566e-11Product of the permeability of free space and the plasma mass density. - 2
√(μ₀ρ)
√(1.2566e-11) = 3.5449e-6 - 3
Alfvén velocity (m/s)
1 ÷ 3.5449e-6 = 2.8209e+5 - 4
Alfvén velocity (km/s)
2.8209e+5 ÷ 1000 = 282.095
How does this calculator work?
The Alfvén velocity is v_A = B / √(μ₀ ρ), where B is the magnetic field in Tesla, μ₀ = 4π × 10⁻⁷ H/m is the permeability of free space, and ρ is the plasma mass density in kg/m³. It gives the speed of transverse magnetohydrodynamic (Alfvén) waves propagating along B-field lines in a conducting plasma.
Formula
How this is calculated
Alfvén waves are transverse oscillations of ions in a magnetized plasma, analogous to waves on a stretched string, where the magnetic tension B²/μ₀ plays the role of string tension and the plasma density ρ plays the role of linear mass density. The Alfvén velocity v_A = B / √(μ₀ ρ) sets the characteristic speed for these waves. It was predicted by Hannes Alfvén in 1942 — work that earned him the 1970 Nobel Prize in Physics — and has since been confirmed in laboratory plasmas, Earth's magnetosphere, and the solar wind.
In the solar wind near Earth (~1 AU), typical values are B ≈ 5 nT and ρ ≈ 8 × 10⁻²¹ kg/m³, giving v_A ≈ 50 km/s, comparable to the local solar wind speed (~400 km/s) and explaining why Alfvénic fluctuations are routinely detected by spacecraft. In tokamak fusion plasmas (B ≈ 3–5 T, ρ ≈ 10⁻⁴ kg/m³), v_A reaches tens of thousands of km/s, making Alfvén waves relevant to instabilities and energy transport. In the solar corona, the high B and very low density combine to give v_A > 1000 km/s.
The formula assumes an ideal single-fluid MHD plasma: quasi-neutral, fully ionized, with ρ equal to the total ion mass density. It neglects relativistic effects (valid when v_A ≪ c), finite-Larmor-radius effects, and multi-species corrections. When B or ρ vary with position (as in real space plasmas or fusion devices), v_A is evaluated locally.
Frequently asked questions
The restoring force is magnetic tension: when a flux tube is bent by plasma motion perpendicular to B, the tension (B²/μ₀ per unit area) acts like a stretched string to pull the plasma back, creating oscillations that propagate at v_A along the field.
Both are characteristic MHD speeds. The sound speed in plasma is c_s = √(γkT/m). When v_A ≫ c_s (magnetically dominated), Alfvén waves dominate; when c_s ≫ v_A (thermally dominated), acoustic modes dominate. The ratio β = c_s²/v_A² ∝ nkT/(B²/2μ₀) is called plasma beta.
Convert carefully: number density n (m⁻³) × mean ion mass m_i (kg) gives ρ (kg/m³). For solar wind with n = 5 × 10⁶ protons/m³ and m_p = 1.67 × 10⁻²⁷ kg: ρ = 8.35 × 10⁻²¹ kg/m³. Magnetic field in nT must be converted to Tesla (1 nT = 10⁻⁹ T).
Also known as
TG we-Calculate Editorial Team. (2026). Alfvén Velocity Calculator — Magnetohydrodynamic Wave Speed [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/alfven-velocity-calculator
TG we-Calculate Editorial Team. "Alfvén Velocity Calculator — Magnetohydrodynamic Wave Speed." TG we-Calculate. 2026. https://we-calculate.com/calculator/alfven-velocity-calculator.
TG we-Calculate Editorial Team, "Alfvén Velocity Calculator — Magnetohydrodynamic Wave Speed," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/alfven-velocity-calculator
@misc{wecalculate_alfven_velocity_calculator, title = {Alfvén Velocity Calculator — Magnetohydrodynamic Wave Speed}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/alfven-velocity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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