# Alfvén Velocity Calculator

Created by Miłosz Panfil, PhD
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Jan 24, 2023

Using the Alfvén velocity calculator, you can compute the velocity of magnetohydrodynamic waves, which are a type of plasma wave. For example, this kind of wave appears in the Sun's corona or the ionosphere.

## Magnetohydrodynamic wave (plasma wave)

The Alfvén wave is a type of plasma wave. You might know that plasma is an ionized gas, which means it consists of charged particles. A few types of waves can propagate in plasma, one of which is the magnetohydrodynamic wave known as an Alfvén wave.

If you want to learn more about the waves in general, make sure to check our harmonic wave equation calculator and wave velocity calculator.

If we place a plasma in an external magnetic field, we will observe the Alfvén waves. They come from balancing two factors. One is the inertia of plasma particles, and the second is the restoring force due to the magnetic field. Plasma particles are charged and in constant motion. This means that the external magnetic field acts on them with the Lorentz force. This Lorentz force is the restoring force that opposes the particle's inertia. You can check out the Lorentz force calculator to learn more about the Lorentz force.

## Alfvén velocity calculator formula

The formula for the Alfvén velocity is:

$v = \frac{B}{\sqrt{\mu_0\rho}},$

where:

• $B\ \rm[T]$ – Magnetic field;
• $\mu_0 = 4 \pi \times 10^{-7}\ \rm[T \cdot m / A]$ – Magnetic permeability of free space;
• $\rho\ \rm [kg/m^3]$ – Density of ions; and
• $v\ \rm [m/s]$ – Alfvén velocity.

For example, the Earth's ionosphere consists of plasma of the density around $\rho = 1 \times 10^{-15}\ \rm kg/m^3$. The magnetic field in the ionosphere is $B = 10\ \text{μT}$. The resulting Alfvén velocity is $v = 282.1\ \rm km/s$.

Try out this example and others in the Alfvén velocity calculator.

## FAQ

### What are Alfvén waves?

Alfvén waves are disturbances in a plasma, a type of magnetohydrodynamic wave where massive ions (with inertia) feel a restoring force due to tension in the magnetic field lines. This type of phenomenon was theorized and later accepted as a way to explain the behavior of the plasma surrounding the Sun: Alfvén waves are excellent means of transporting energy, being dispersionless.

### How do I calculate the group velocity of Alfvén waves?

To calculate the group velocity of Alfvén waves:

1. Measure the strength of the magnetic field B.
2. Measure the density of the plasma ρ.
3. Divide the magnetic field by the square root of the density multiplied by the magnetic permeability of the vacuum, µ0:
v = B/sqrt(µ0 · ρ)

That's it! Remember that Alfvén waves propagate in the direction of the magnetic field.

### What is the velocity of Alfvén waves in the Solar corona?

Assuming a magnetic field strength of 500 mT at the surface of the Sun and a density 10E-12 kg/m³ in the corona, we find that the velocity of Alfvén waves is:
v = B/sqrt(µ0 · ρ) = 500E-3/(4πE-7 T · m/A · 10E-12 kg/m³ = 4,460 km/s
This is, of course, a pretty high value: the Sun is a place of violent phenomena and immense forces!

### Are Alfvén waves responsible for the aruoras?

Recent discoveries showed a correlation between auroras (the Northern and Southern lights) and Alfvén waves: it is possible that electrons "surfing" on the plasma waves created by disturbances in Earth's magnetic field accelerate using the energy transported by the wave itself till they collide with oxygen molecules in the higher atmosphere.

Miłosz Panfil, PhD
Magnetic field
µT
Ion density
×10⁻¹⁵
kg/m³
Velocity
mi/s
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