The Alfvén velocity calculator above gives the speed at which a transverse magnetohydrodynamic wave travels along a magnetic field in a plasma. It is the plasma equivalent of the speed of sound: the natural signalling speed of the medium, and the number that decides whether a disturbance can propagate upstream or is swept away.
Arb Digital builds free physics calculators that own a single quantity properly. The magnetic field converter will get your field into tesla and the density converter will get your density into SI, but neither combines the two into a wave speed. This page does, and it handles the unit chaos that makes plasma physics so error-prone in the process.
What This Alfvén Velocity Calculator Does
A magnetic field embedded in a conducting fluid behaves rather like a set of elastic strings threaded through the plasma. Pluck them and a wave runs along the field lines. Hannes Alfvén worked out in 1942 that such waves must exist, which was not obvious at the time and won him a Nobel Prize thirty years later. The speed of those waves is what this tool computes.
The inputs are the magnetic field magnitude and the mass density of the plasma. Because plasma work spans twenty orders of magnitude in both, the tool accepts the field in nanotesla through tesla or in gauss, and accepts density either as a particle count per cubic centimetre with a mean ion mass, or as a mass density in SI directly. The two routes give identical answers when they describe the same plasma.
The grid reports the mass density that was actually used, the speed in metres per second alongside the headline figure in kilometres per second, the speed as a fraction of the speed of light, and the time an Alfvén wave takes to cross a structure whose size you choose. That last figure is often the most useful, because Alfvén crossing time is the natural unit of time in magnetohydrodynamics the way the sound crossing time is in ordinary fluid dynamics.
How to Use It
- Enter the field magnitude and pick its unit. Space physics runs on nanotesla, laboratory and solar work on gauss or tesla. Use the magnitude of the total field, not a single component.
- Choose how you want to give the density. Number density with a mean ion mass is the natural form for space and fusion plasmas; mass density is easier if you already have it in SI.
- Set the mean ion mass honestly. A hydrogen plasma is 1. Adding helium raises it, and a plasma of heavier ions raises it a great deal, which slows Alfvén waves in proportion to the square root.
- Enter a structure size. The transit time it produces tells you how quickly the plasma can communicate across the region you care about.
- Check the fraction of light speed. Once that climbs above about a tenth, the non-relativistic formula starts overstating the speed and the tool applies the relativistic correction for you.
The Formula: How Alfvén Velocity Is Calculated
In SI units the Alfvén speed is vA = B ÷ √(μ0ρ), where B is the magnetic flux density in tesla, ρ is the plasma mass density in kilograms per cubic metre, and μ0 is the vacuum magnetic permeability. The NIST CODATA value for the vacuum magnetic permeability is 1.25663706127 × 10−6 N/A², which is the figure this tool uses.
When the density is supplied as a particle count, the mass density is ρ = n × μ × mp, with n the number density converted to particles per cubic metre, μ the mean ion mass in proton masses and mp = 1.67262192 × 10−27 kg. Electron mass is left out because it contributes less than a twentieth of one per cent.
Space physics usually works in a shorthand. NASA's CDAWeb notes for the OMNI solar wind datasets define the Alfvén speed as 20 × B ÷ √N in kilometres per second, with B in nanotesla and N in particles per cubic centimetre. Substituting the constants into the SI expression for a pure hydrogen plasma gives a coefficient of 21.8 rather than 20; the smaller number in the OMNI convention corresponds to a mean ion mass a little above one proton mass, which is what the helium content of the solar wind supplies. Set the mean ion mass to 1.19 in this tool and it reproduces the OMNI coefficient almost exactly.
Work the defaults. A field of 5 nT is 5 × 10−9 T. A density of 5 particles per cubic centimetre is 5 × 106 m−3, and at one proton mass each that is ρ = 8.363 × 10−21 kg/m³. Then μ0ρ = 1.0509 × 10−26, whose square root is 1.0252 × 10−13, and vA = 5 × 10−9 ÷ 1.0252 × 10−13 = 48,770 m/s, or about 48.8 km/s. The shorthand agrees: 21.8 × 5 ÷ √5 = 48.8.
Why the Formula Looks Like a Guitar String
The wave speed on a stretched string is the square root of tension divided by mass per unit length. The Alfvén speed has exactly that form. A magnetic field in a conducting fluid carries a tension of B² ÷ μ0 along the field lines, and the mass being dragged along is the plasma density. Divide one by the other and take the square root and the formula falls out.
The analogy is not decoration; it is the physics. Field lines in a highly conducting plasma are effectively frozen into the fluid, so moving the fluid sideways bends the field, the field's tension pulls back, and the restoring force sets up a transverse wave. The same magnetic tension is why solar prominences hang in loops instead of falling, and why a plasma jet can stay collimated over enormous distances.
One consequence of the string picture is worth noting. Pure Alfvén waves are transverse and incompressible: they bend the field without squeezing the plasma. That makes them completely different from sound waves, which are longitudinal compressions and travel at the speed given by the speed of sound calculator. A magnetised plasma supports both, plus two hybrid magnetosonic modes that mix them.
Where the Alfvén Speed Actually Decides Something
The ratio of flow speed to Alfvén speed is called the Alfvén Mach number, and like the ordinary Mach number from the Mach number calculator it separates two regimes. Below one, information can travel upstream against the flow. Above one, it cannot, and the flow must form a shock when it meets an obstacle.
This is why Earth has a bow shock. The solar wind arrives at several hundred kilometres per second against an Alfvén speed of a few tens, so it is strongly super-Alfvénic and cannot be told about the magnetosphere in advance. A shock forms and slows it. The same arithmetic governs whether a spacecraft near a comet, a moon or a planet sits inside or outside a shocked region.
In fusion devices the Alfvén speed sets the timescale of the most dangerous instabilities. A magnetohydrodynamic mode that grows on the Alfvén crossing time is measured in microseconds, far faster than any mechanical response, so machines are designed so that such modes are stable rather than merely controllable. Enter a metre-scale structure with a tesla-scale field into this tool and you will see why.
Relativistic Limits and Where the Formula Breaks
Taken literally, the non-relativistic expression allows the Alfvén speed to exceed the speed of light whenever the field is strong enough or the plasma thin enough. It cannot, of course, and the resolution is that the magnetic field itself carries energy, and therefore inertia. The relativistic form divides by a factor √(1 + vA² ÷ c²), which leaves ordinary plasmas untouched and correctly saturates the speed at c in the extreme.
This calculator applies that correction automatically and tells you when it has made a visible difference. In the solar wind it changes nothing at all. Near a magnetar or in the magnetosphere of a pulsar, where the field is enormous and the density tiny, it is the whole story.
Two other limits are worth keeping in mind. Ideal magnetohydrodynamics assumes the plasma is well magnetised and highly conducting; in a weakly ionised gas the neutrals do not follow the field and an effective density that includes them is more appropriate. And below the ion cyclotron frequency the simple Alfvén wave picture holds, but above it the wave becomes dispersive and splits into modes with different speeds.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Forgetting the ion mass — number density alone is not mass density. A helium or deuterium plasma at the same particle count is heavier and its Alfvén speed is lower by the square root of the mass ratio.
- Mixing gauss and tesla — one gauss is 10−4 tesla, and getting that factor wrong scales the answer by ten thousand. Use the unit selector rather than converting in your head.
- Using a single field component — the formula wants the magnitude of the total field. Using only the component along one axis understates the speed, sometimes badly.
- Assuming it is the only wave speed — a magnetised plasma also carries sound waves and two magnetosonic modes. The Alfvén speed is one of several characteristic speeds, not the whole picture.
- Ignoring the relativistic ceiling — in low-density high-field environments the naive formula can return a speed above light. That is a signal the correction is needed, not a physical result.
Related Free Tools From Arb Digital
Get the field into the right units with the magnetic field converter, or compute one from a source with the solenoid magnetic field calculator or the magnetic field of a wire calculator. For the other characteristic speeds and scales of a plasma, use the cyclotron frequency calculator and the Debye length calculator. Compare against the neutral-gas case with the speed of sound calculator and the Mach number calculator, and handle density units with the density converter. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the speed at which a transverse magnetohydrodynamic wave travels along a magnetic field in a plasma, given by the field strength divided by the square root of the vacuum permeability times the mass density. It plays the same role in a magnetised plasma that the speed of sound plays in an ordinary gas.
With a field of about 5 nanotesla and roughly 5 protons per cubic centimetre, the Alfvén speed near Earth works out at around 40 to 50 kilometres per second. The solar wind flows at several hundred kilometres per second, so it is strongly super-Alfvénic.
Because the formula uses mass density, not particle count. Doubling the mean ion mass at the same number density doubles the density and lowers the Alfvén speed by a factor of the square root of two. A deuterium plasma is slower than a hydrogen one at the same particle count.
No. The simple formula can return such a value in very strong fields or very thin plasmas, but that signals the non-relativistic approximation has broken down. The relativistic form divides by the square root of one plus the ratio of the squared speed to the squared speed of light, which correctly saturates at light speed.
The ratio of the flow speed to the Alfvén speed. Below one, disturbances can travel upstream against the flow; above one they cannot, and an obstacle in the flow produces a shock. It is the magnetic counterpart of the ordinary Mach number.
It is the same relation with the constants folded in, for field in nanotesla and density in particles per cubic centimetre, giving kilometres per second. The coefficient for pure hydrogen is 21.8; the rounder 20 used in NASA's OMNI datasets corresponds to a mean ion mass slightly above one proton mass, reflecting solar wind helium.
Pure Alfvén waves are not. They bend the field lines transversely without changing the plasma density, which makes them quite unlike sound waves. Magnetised plasmas also support fast and slow magnetosonic modes, and those do compress the plasma.
This tool is provided for educational and study use. It computes the ideal magnetohydrodynamic Alfvén speed for a uniform, fully ionised, well-magnetised plasma, and does not model dispersion above the ion cyclotron frequency, partial ionisation, anisotropy or kinetic effects, so treat its output as a physics estimate rather than a modelled result.
