What are Alfven waves?

Alfvén waves are a type of magnetohydrodynamic wave that propagates along magnetic field lines in a plasma. The ions carry the inertia while the tension of the magnetic field line acts as the restoring force, causing the magnetic field lines to behave like stretched strings. Alfvén waves were predicted by Hannes Alfvén in 1942. They are studied in the solar atmosphere because they do not exhibit dispersion and cause only small energy losses during propagation, which makes them important for studying energy transport.

How to calculate the Alfven speed?

Divide the magnetic field by the square root of the product of the vacuum permeability and the mass density of the plasma. The Alfven velocity is calculated as: v_A = B / sqrt(mu0 * rho). If the magnetic field is given in teslas and the density in kilograms per cubic meter, then the result will be in meters per second. For example, if the magnetic field is 0.001 tesla and the plasma density is 1.7e-12 kg/m3, then the Alfven velocity would be approximately 690 kilometers per second.

What is the speed of the Alfven wave in the solar corona?

In the solar corona, magnetic field strengths vary between a few gauss and several tens of gauss. As the plasma is extremely thin, Alfvén wave speeds are very high, typically ranging from hundreds to about two thousand kilometres per second. A representative value is about 690 km/s when the magnetic field is ten gauss (0.001 tesla) and the density is close to 1.7e-12 kg/m3. This very high speed is one reason why Alfvén waves are considered a possible mechanism for heating of the corona.

Why can't the Alfven speed exceed the speed of light?

The simple formula v_A = B / sqrt(mu0*x*rho) has no upper speed limit. So it can give values that exceed the speed of light when the magnetic field is very strong or the density is very low. This contradicts physical laws. The relativistic Alfvén velocity c*v_A/sqrt(c^2+v_A^2) is used for correction. At low velocities, it corresponds to normal velocity, and as normal velocity approaches the speed of light, it stabilizes at a value slightly below the speed of light. When the velocity exceeds about one-tenth of the speed of light, the relativistic form should be used.

What is the difference between mass density and particle number?

The mass density is the mass of plasma per unit volume and it is measured in kilograms per cubic meter. It is a quantity that is required for calculating the Alfvén velocity. The particle number is the number of particles per unit volume, usually given in particles per cubic centimeter. Both quantities are related by the mass of the ion, with the mass density being the product of the particle number and the mass of an individual ion. You can convert the particle number to mass density using the "particle number" section of this calculator.

Alfven Velocity Calculator

Magnetic field and density

Magnetic field (B)

Plasma mass density (ρ)

Alfven velocity (v_A)

Enter a magnetic field and a plasma density, and the calculator finds the Alfven velocity from v_A = B / √(μ₀ × ρ). Leave any one of the three boxes blank to solve for it instead. You can also build the density from a particle count, cap the speed relativistically, or time an Alfven wave across a distance, in the sections below.

Build the density from a particle count

Time an Alfven wave across a distance

Please input values into the fields

When a magnetic field is introduced into a plasma it can be made to oscillate like the string of a guitar. Disturbances that propagate along the magnetic field lines are called Alfvén waves and their propagation speed is the Alfvén velocity. This calculator determines this velocity from the magnetic field and plasma density, taking any direction into account. If one of the three values is left blank, it will be calculated.

It can also determine the density from the particle count, show the relativistic speed limit and calculate how long it takes for an Alfvén wave to travel a given distance.

Formula for the Alfvén velocity:

The inertia of an Alfvén wave comes from the ions in the plasma, while the restoring force comes from tension in the magnetic field lines. The balance between these two forces gives the wave speed.

v_A = \dfrac{B}{\sqrt{\mu_0\, \rho}}

Here v_A is the Alfvén velocity, B is the magnetic field strength and rho is the mass density of the plasma. The Greek letter mu-nought represents the vacuum permeability and is a fixed physical constant that has a value of approximately 1.257 times ten to the power minus six Henrys per meter. In SI units, the unit for the magnetic field is Tesla, the unit for the density is kilograms per cubic meter and the unit for the final velocity is meters per second.

Derivation of formula.

Magnetic field lines under tension are like a stretched string. The magnetic tension per unit area is B squared over mu-zero and the mass per unit volume that's oscillating is rho. The speed of a wave on the string is the square root of the ratio of tension to mass. If you take the same analogy for magnetic field lines, then you get the Alfvén velocity.

v_A = \sqrt{\dfrac{B^2/\mu_0}{\rho}} = \dfrac{B}{\sqrt{\mu_0\, \rho}}

So the speed is higher for a stronger magnetic field and lower for a denser plasma. A strong magnetic field in a thin plasma will create very fast waves, which corresponds to conditions in the solar atmosphere.

Example calculation for the solar corona:

We take a magnetic field of about ten Gauss, so thousandths of Tesla, and a density of about 1.7 times 10 to the minus twelve kilograms per cubic meter. We divide the magnetic field by the square root of the product of mu-nought and the density.

The square root of the product of mu-nought and the density is about 1.45 times 10 to the minus nine, so the Alfvén speed is about 690 kilometers per second. One reason for studying whether Alfvén waves can heat the corona and drive the solar wind is that Alfvén waves in the corona are actually traveling at speeds of several hundred kilometers per second.

Instructions for using this calculator.

If you fill in two of the three input fields (magnetic field, density, Alfven speed), the calculator will calculate the third value. If you leave the field for the speed empty, it will be calculated from the magnetic field and the density. Conversely, you can also leave a field for the magnetic field or the density blank and back-calculate the values based on a known Alfven speed.

Each input field has separate options for unit conversion. Density can be entered in kilograms per cubic meter or grams per cubic centimeter, while speed can be specified in kilometers per second, meters per second, or as a ratio to the speed of light. All conversions are done by the calculator.

Magnetic field or density determines:

Since these three quantities are related by a single equation, the third value can be determined if two of them are known. By rearranging the formula vA=B/√(μ0ρ), we obtain two reciprocal forms.

B = v_A\,\sqrt{\mu_0\, \rho} \qquad \rho = \dfrac{B^2}{\mu_0\, v_A^2}

The first form allows one to calculate the required magnetic field to create a given Alfvén velocity while the second form can be used to determine the density from the magnetic field and velocity. If you leave any of the fields blank, the calculator will automatically select the correct conversion.

Replacing bulk density with particle density.

Plasma physicists usually measure the particle density, or number density n, rather than mass density. The two quantities are related by the mass of a single ion. Multiplying the number of ions per cubic meter by the mass of an ion gives the mass density.

\rho = n\, m_i \qquad v_A = \dfrac{B}{\sqrt{\mu_0\, n\, m_i}}

If you open the "Number of Particles" section and enter the number density and ion mass, the calculator will calculate the mass density and plug it into the Alfvén velocity formula. The ion mass is set to the mass of a proton by default, which is suitable for hydrogen plasmas. For heavier ions, you can switch to atomic mass units. Note that the number density is usually given in cubic centimeters, so you need to convert it to cubic meters by multiplying by one million.

Relativistic Alfvén velocity

The simple formula itself contains no limit on the speed; if the magnetic field is strong enough or the density low enough, one can easily obtain speeds that exceed the speed of light, which is obviously wrong. For highly magnetized thin plasmas a correction must be made to ensure that the Alfvén velocity remains below the speed of light.

v_{A,\text{rel}} = \dfrac{c\, v_A}{\sqrt{c^2 + v_A^2}}

When the normal Alfvén speed is small compared to light speed, then the correction does not matter and both values are equal. When the normal speed approaches or exceeds the speed of light, the relativistic speed gradually stabilizes at a value only slightly below c. The calculator shows this corrected speed next to its ratio to the speed of light, and issues a warning when a relativistic form should be used.

Alfvén waves do not have dispersion.

Alfvén waves transport energy along a magnetic field at the Alfvén velocity. Their group, or the speed with which a wave packet propagates, is also equal to the phase velocity. Since all wavelengths travel at the same speed, an impulse retains its shape as it propagates. This means there is no dispersion and is why Alfvén waves are such an efficient way of transporting energy in magnetised plasmas.

Since the speed is a well-defined single number, one can calculate how long it takes for a wave to travel across a structure. If you enter in the length of a coronal loop, for example, under "distance," then a tool calculates the time required by dividing that length by the Alfvén speed.

Units of individual physical quantities:

The calculator tracks the units so you can mix and match different units together. The unit for entering the magnetic field is Tesla. Astronomers often use Gauss where one Gauss is ten thousandths of a Tesla.

Quantity	Symbol	SI unit
Magnetic field	B	tesla (T); 1 gauss = 1e-4 T
Mass density	ρ	kilogram per cubic metre (kg/m3)
Number density	n	per cubic metre (1/m3); 1 per cm3 = 1e6 per m3
Ion mass	m_i	kilogram (kg); proton = 1.67e-27 kg
Alfven velocity	v_A	metre per second (m/s); often km/s
Permeability	μ₀	henry per metre (H/m); 1.257e-6

Applications of the Alfvén speed:

Alfvén waves were predicted by Hannes Alfvén in 1942 and he was later awarded the Nobel Prize for this work. They are found wherever plasmas interact with magnetic fields, and on the Sun they are a leading candidate mechanism to heat the corona to millions of degrees and drive the solar wind into the Solar System.

In our vicinity, during geomagnetic storms, Alfvén waves travel through the Earth's magnetosphere. They are also important in laboratories for hot plasmas in Tokamak type fusion reactors since certain instabilities propagate with the Alfvén velocity. In all these cases, it is the Alfvén velocity that sets the rhythm.

Common mistakes:

A common mistake is to confuse mass density with particle density. In this formula, you need to use the mass density in kilograms per cubic meter. If you only have the number of particles, first convert it to a mass density using the section on particle density.

Also note the units of magnetic field. Since one Gauss is ten thousandths of a Tesla, a solar magnetic field of ten Gauss is only one thousandth of a Tesla. Also note the value relative to the speed of light. If the normal Alfvén velocity exceeds a significant fraction (say about a tenth) of the speed of light then relativistic velocities are physically correct.

This tool is for general education and everyday problem solving. For research work, engineering tasks or anything where safety is important, verify the values against the standards and data required by your project.

Frequently asked questions

What are Alfven waves?: Alfvén waves are a type of magnetohydrodynamic wave that propagates along magnetic field lines in a plasma. The ions carry the inertia while the tension of the magnetic field line acts as the restoring force, causing the magnetic field lines to behave like stretched strings. Alfvén waves were predicted by Hannes Alfvén in 1942. They are studied in the solar atmosphere because they do not exhibit dispersion and cause only small energy losses during propagation, which makes them important for studying energy transport.
How to calculate the Alfven speed?: Divide the magnetic field by the square root of the product of the vacuum permeability and the mass density of the plasma. The Alfven velocity is calculated as: v_A = B / sqrt(mu0 * rho). If the magnetic field is given in teslas and the density in kilograms per cubic meter, then the result will be in meters per second. For example, if the magnetic field is 0.001 tesla and the plasma density is 1.7e-12 kg/m3, then the Alfven velocity would be approximately 690 kilometers per second.
What is the speed of the Alfven wave in the solar corona?: In the solar corona, magnetic field strengths vary between a few gauss and several tens of gauss. As the plasma is extremely thin, Alfvén wave speeds are very high, typically ranging from hundreds to about two thousand kilometres per second. A representative value is about 690 km/s when the magnetic field is ten gauss (0.001 tesla) and the density is close to 1.7e-12 kg/m3. This very high speed is one reason why Alfvén waves are considered a possible mechanism for heating of the corona.
Why can't the Alfven speed exceed the speed of light?: The simple formula v_A = B / sqrt(mu0*x*rho) has no upper speed limit. So it can give values that exceed the speed of light when the magnetic field is very strong or the density is very low. This contradicts physical laws. The relativistic Alfvén velocity c*v_A/sqrt(c^2+v_A^2) is used for correction. At low velocities, it corresponds to normal velocity, and as normal velocity approaches the speed of light, it stabilizes at a value slightly below the speed of light. When the velocity exceeds about one-tenth of the speed of light, the relativistic form should be used.
What is the difference between mass density and particle number?: The mass density is the mass of plasma per unit volume and it is measured in kilograms per cubic meter. It is a quantity that is required for calculating the Alfvén velocity. The particle number is the number of particles per unit volume, usually given in particles per cubic centimeter. Both quantities are related by the mass of the ion, with the mass density being the product of the particle number and the mass of an individual ion. You can convert the particle number to mass density using the "particle number" section of this calculator.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Alfven wave
Definition of Alfven waves, the Alfven velocity v_A = B/sqrt(mu0 rho), the relativistic correction, and their role in astrophysical plasmas.
NRL Plasma Formulary
The standard reference for plasma parameters, including the Alfven speed in terms of magnetic field and ion mass density.
Wikipedia: Vacuum permeability
The constant mu-nought used in the Alfven velocity formula, its value and units.