Alfvén Velocity Calculator
Find the Alfvén velocity of a magnetized plasma from field strength and ion density, then compare it with a flow speed or solve for the field.
🧲 What is the Alfvén Velocity?
The Alfvén velocity is the speed at which a small disturbance of the magnetic field travels along the field lines of a magnetized plasma. The field lines behave like stretched elastic strings: they carry a tension, and the plasma ions give them inertia. When you pluck a string it carries a transverse wave at a speed set by tension over mass per length. In a plasma the same idea gives vA = B / √(μ0 ρ), where B is the field strength and ρ is the mass density of the plasma. The result was predicted by Hannes Alfvén in 1942 and is one of the characteristic speeds of magnetohydrodynamics (MHD).
The quantity appears across plasma and space physics. In the solar wind and the corona it sets how fast magnetic disturbances and energy spread, and it controls whether a flow is slow enough for waves to travel upstream. In a tokamak or other fusion device it sets the timescale for fast magnetic rearrangements and the frequencies of Alfvén eigenmodes that can affect fast ion confinement. In planetary magnetospheres and in neutron star environments it marks where magnetic forces start to dominate the dynamics. Liquid metal experiments also reveal Alfvén waves in a conducting fluid threaded by a field.
Several distinctions are worth keeping in mind. The density in the formula is the mass density, which is carried almost entirely by the ions even though the electrons carry much of the current, so the calculator multiplies the ion number density by the ion mass. The formula assumes a fully ionized, quasi-neutral plasma, and neutral atoms in a partially ionized gas add inertia. At very low density or very strong field the classical value can approach or exceed the speed of light, which is not physical, so a relativistic limit is shown next to it. Finally, Alfvén waves are not sound waves: they need a magnetic field and travel along it, with magnetic tension rather than pressure as the restoring force.
This calculator offers three modes. Alfvén speed finds vA from field and ion density, with proton, deuteron, triton, alpha or a custom ion mass, and gives the crossing time over a chosen length. Mach number divides a flow speed by vA and classifies the flow. Required field solves the opposite problem and gives the field needed for a target speed, with a table. The units of field and density are selectable, and the charts show how the speed responds to density and how the Mach number responds to the field.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Tokamak Plasma
B = 5 T, n = 1e20 m⁻³ protons, path length 2 m
Example 2 — Solar Wind at 1 AU
B = 5 nT, n = 5 cm⁻³ protons, path length 1 AU = 1.496e11 m
Example 3 — Strong Field, Low Density (Relativistic Limit)
B = 1 T, n = 1e15 m⁻³ protons, path length 10 m
Example 4 — Alfvén Mach Number of the Solar Wind
B = 5 nT, n = 5 cm⁻³ protons, flow speed 400 km/s
Example 5 — Field Needed for 1000 km/s in Deuterium
Target vA = 1000 km/s, n = 1e19 m⁻³ deuterons
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for Alfvén velocity?
The Alfvén velocity is v_A = B / sqrt(mu0 rho), where B is the magnetic field in tesla, mu0 = 1.25663706212e-6 H/m is the vacuum permeability and rho is the plasma mass density in kg/m3. For B = 5 T and a proton density of 1e20 per cubic metre it is about 1.09e7 m/s.
How do you find the mass density for the Alfvén speed?
Multiply the ion number density by the ion mass, rho = n_i m_i. For protons m is 1.67262e-27 kg, so 1e20 ions per m3 gives 1.673e-7 kg/m3. The electrons add only Z times the electron to ion mass ratio, about 0.054 percent for protons, and are neglected here.
What is the Alfvén Mach number?
The Alfvén Mach number is M_A = V / v_A, the flow speed divided by the Alfvén velocity. Below 1 the flow is sub-Alfvénic and the magnetic field dominates. Above 1 it is super-Alfvénic. M_A squared also equals the ratio of kinetic energy density to magnetic energy density.
What is a typical Alfvén speed in the solar wind?
At 1 AU the field is about 5 nT and the proton density about 5 per cm3, which gives an Alfvén speed near 49 km/s. The solar wind flows at roughly 400 km/s, so its Alfvén Mach number is around 8, well into the super-Alfvénic regime.
How do I convert gauss to tesla for this calculator?
One gauss equals 1e-4 tesla, so 10 G is 1 mT. Pick the unit from the selector beside the field box and type the number as given. The calculator converts internally, and the results panel shows the field used in tesla.
Why does the Alfvén speed exceed the speed of light in some inputs?
The classical formula ignores the displacement current. At low density and strong field, B / sqrt(mu0 rho) can exceed c, which is unphysical. The relativistic form v_A / sqrt(1 + v_A squared / c squared) always stays below c, and the calculator reports it next to the classical value.
How is the Alfvén velocity different from the sound speed?
The sound speed depends on pressure and density, while the Alfvén velocity depends on the magnetic field and density. Alfvén waves are transverse and travel along the field lines with magnetic tension as the restoring force. In a plasma with both effects, fast magnetosonic waves combine the two speeds.
Does this calculator work for partially ionized gas?
It assumes a fully ionized, quasi-neutral plasma where the ions carry the mass. In a partially ionized medium neutral atoms add inertia, so the true Alfvén speed is lower if the neutrals are strongly coupled to the ions. Use the total mass density if the whole gas moves together.
How is this different from the Alfvén Wave Speed Calculator?
Both use v_A = B / sqrt(mu0 rho). This page adds four ion species and a custom mass, field and density unit selectors, the relativistic limit, the crossing time, an Alfvén Mach number mode and a required field mode, plus charts. It sits with the physics tools and the plasma physics page covers the plain speed.
What field do I need for a given Alfvén speed?
Use the required field mode. It solves B = v_A sqrt(mu0 rho). For deuterons at 1e19 per m3, an Alfvén speed of 1000 km/s needs about 0.205 T. Because B is proportional to v_A, doubling the target speed doubles the required field.
What is the Alfvén crossing time?
It is the time an Alfvén wave takes to travel a length L along the field, tau = L / v_A. In a tokamak with B = 5 T and 1e20 protons per m3, crossing 2 m takes about 0.18 microseconds, which sets the natural timescale for fast magnetic rearrangements.