Alfvén Velocity Calculator

Background.

Pluck a magnetic field line in a plasma and the kink travels along it like a wave on a guitar string. The field supplies the tension and the plasma threaded on it supplies the mass, so the wave speed comes out the same way a string's does: the square root of tension over density. That speed is the Alfvén velocity, named after Hannes Alfvén, who predicted these waves in 1942 and later shared the 1970 Nobel Prize in Physics for his work on magnetohydrodynamics.

It is one of the first numbers a space physicist works out for any plasma, because comparing it with the flow speed tells you what kind of medium you are dealing with. Near Earth the solar wind blows at several times its local Alfvén speed, which is why a shock wave stands in front of the planet's magnetic field. Close to the Sun the order reverses; in 2021 Parker Solar Probe became the first spacecraft to fly through the boundary where the two speeds are equal.

The calculation takes three SI inputs: field B in tesla, total mass density ρ in kilograms per cubic metre, and permeability μ, which defaults to the vacuum value. Mass density, not particle count, is asked for deliberately, since converting one to the other means choosing what the plasma is made of. The result belongs to ideal magnetohydrodynamics: no resistivity, no kinetic effects, no relativistic correction.

What is alfvén velocity calculator?

The Alfvén velocity, v_A = B ÷ √(μρ), is the speed at which a transverse disturbance of the magnetic field, an Alfvén wave, travels along the field in a perfectly conducting fluid. It depends only on the field strength and the inertia of the material carried with the field. It is a wave speed, not the speed of the plasma itself, and the ratio of flow speed to Alfvén speed, the Alfvén Mach number, separates sub-Alfvénic from super-Alfvénic flow.

How to use this calculator.

1. 01Enter the magnetic flux density in tesla. Space data usually comes in nanotesla (1 nT = 10⁻⁹ T) and older solar papers in gauss (1 G = 10⁻⁴ T).
2. 02Enter the total mass density in kg/m³. For a hydrogen plasma, multiply the number of protons per cubic metre by 1.6726 × 10⁻²⁷ kg; add the mass of any helium or heavier ions.
3. 03Leave the permeability at the vacuum value for any plasma or liquid metal. Change it only if you are modelling a magnetisable medium.
4. 04Read the speed in m/s or km/s. The calculator refuses inputs whose speed would reach the speed of light; if the result is above a few per cent of that, see the relativistic note in the questions below.

The formula.

In magnetohydrodynamics a bent field line pulls itself straight with a tension of B²/μ per unit area, while the plasma frozen onto the field has density ρ. As with a stretched string, wave speed is √(tension ÷ density), so v_A = √(B²/(μρ)) = B ÷ √(μρ). With B in tesla, μ in henries per metre and ρ in kg/m³ the units reduce to metres per second. The speed rises in direct proportion to the field and falls with the square root of the density, so a field twice as strong doubles it, while four times the density halves it. The default μ is the 2022 CODATA value of the vacuum permeability, 1.25663706127 × 10⁻⁶ H/m; since the 2019 SI revision it is a measured constant rather than exactly 4π × 10⁻⁷, though the two differ by about one part in ten billion. The formula leaves out the displacement current, so it overstates speeds that approach that of light.

A worked example.

Frequently asked questions.

References& sources.

1. [1]U.S. Naval Research Laboratory, 2019 NRL Plasma Formulary, p. 29, Velocities: Alfvén velocity equation.
2. [2]NASA Goddard OMNIWeb, Detailed Derivation of Parameters, Characteristic speeds: independent Alfvén-speed definition.
3. [3]NIST SP 961, 2022 CODATA recommended values, May 2024: vacuum magnetic permeability value and uncertainty.