Hong Life CalculatorsUnderstanding the Alfvén velocity calculator — concept and principle
Alfvén waves are oscillations of magnetic field lines together with the plasma frozen onto them, like waves on a string whose tension is the magnetic field. This calculator finds the speed at which they travel, the Alfvén speed, from the field strength, ion density and ion mass. It is used in space physics, solar physics and fusion research.
How it is calculated
- Mass density ρ = ni × mi (ion number density × ion mass; electrons add only about 0.05% for hydrogen).
- Alfvén speed vA = B ÷ √(μ₀ ρ).
- Field conversions: 1 G = 10⁻⁴ T, 1 nT = 10⁻⁹ T; density in per cm³ is converted to per m³ (× 10⁶).
- Magnetic pressure = B² ÷ (2μ₀); ratio to light speed = vA ÷ c.
- Constants (CODATA 2022): u = 1.66053906892 × 10⁻²⁷ kg, μ₀ = 1.25663706127 × 10⁻⁶ N/A².
The principle
A wave on a string travels at √(tension ÷ mass per length). In a magnetised plasma, field lines behave like elastic strings with tension B² ÷ μ₀ per unit area, and the plasma's mass loads them. Putting tension B²/μ₀ and density ρ into the string formula gives vA = B ÷ √(μ₀ρ). Stronger fields raise the speed; denser plasma lowers it. The simple formula assumes vA is much less than the speed of light.
Worked example
Default inputs (typical solar wind near Earth): B = 5 nT, n = 5 per cm³, ion mass 1.007276 u (proton).
- B = 5 × 10⁻⁹ T; n = 5 × 10⁶ per m³.
- ρ = 5 × 10⁶ × 1.007276 × 1.66054 × 10⁻²⁷ ≈ 8.363 × 10⁻²¹ kg/m³.
- vA = 5 × 10⁻⁹ ÷ √(1.25664 × 10⁻⁶ × 8.363 × 10⁻²¹) ≈ 48,770 m/s = 48.77 km/s.
- vA ÷ c ≈ 0.0001627, so the non-relativistic formula is fine.
- Magnetic pressure = (5 × 10⁻⁹)² ÷ (2μ₀) ≈ 0.009947 nPa.
| Magnetic field strength | Alfvén speed (km/s) |
|---|---|
| 1.25 | 12.19 |
| 2.5 | 24.39 |
| 3.75 | 36.58 |
| 5 | 48.77 |
| 6.25 | 60.97 |
| 7.5 | 73.16 |
| 10 | 97.55 |
How to read the result
Comparing the Alfvén speed with the flow speed tells you whether a plasma flow is super-Alfvénic. The solar wind usually flows at several hundred km/s, well above vA, which is why a bow shock forms in front of Earth's magnetosphere. In the solar corona, where fields are stronger and density lower, Alfvén speeds are much higher.
Common mistakes
- Entering density per m³ when the field expects per cm³ (a factor of 10⁶).
- Mixing gauss and tesla.
- Using the electron mass instead of the ion mass.
- Applying the formula when vA approaches c without the relativistic correction.
Frequently asked questions
What is the Alfvén speed formula?
vA = B ÷ √(μ₀ρ), with ρ the ion mass density.
What is the Alfvén speed in the solar wind?
With typical near-Earth values (5 nT, 5 per cm³), around 50 km/s.
Who discovered Alfvén waves?
Hannes Alfvén predicted them in 1942; he received the 1970 Nobel Prize in Physics for work on magnetohydrodynamics.
Sources: Alfvén, H. (1942). Existence of Electromagnetic-Hydrodynamic Waves. Nature, 150, 405–406. · NRL Plasma Formulary, US Naval Research Laboratory