Alfven Velocity Calculator
Enter the magnetic field strength and plasma mass density to find the Alfven velocity - the speed at which magnetohydrodynamic waves travel through a magnetized plasma. The calculator also computes plasma beta, the Alfven Mach number (when a flow velocity is provided), the ion inertial length, and what fraction of the speed of light the wave reaches. All steps are shown so you can follow the working.
What is the Alfven velocity?
The Alfven velocity (often written v_A) is the characteristic speed at which low-frequency electromagnetic waves travel through a magnetized plasma along the direction of the magnetic field. These waves, called Alfven waves, are a type of magnetohydrodynamic (MHD) wave in which the ions oscillate back and forth across the field lines while the magnetic field itself is carried along like a wave on a taut string. The restoring force is the magnetic tension, and the inertia resisting acceleration is the plasma mass density, so the wave speed scales in exactly the same way as a transverse wave on a string: higher tension (stronger field) means faster propagation, and higher mass density means slower propagation. The formula is v_A = B / sqrt(mu_0 * rho), where B is the magnetic flux density, mu_0 is the permeability of free space, and rho is the plasma mass density. Hannes Alfven derived this relationship in 1942 and was awarded the Nobel Prize in Physics in 1970 partly for this work.
How to use this calculator
Choose what you want to solve for with the 'Solve for' selector at the top. In the default mode (Alfven velocity) you supply the magnetic field strength and plasma mass density; the calculator returns the Alfven velocity in both metres per second and kilometres per second, plus its fraction of the speed of light. You can switch to magnetic field or density mode to back-calculate the field or density needed for a target Alfven speed. The optional flow velocity input enables the Alfven Mach number, and the optional pressure input enables plasma beta. The ion species selector determines the ion inertial length, which is the smallest scale at which Hall MHD effects matter. Magnetic field entries accept Tesla, millitesla, microtesla, or Gauss; density entries accept kg/m3, g/cm3, or g/m3; and velocity entries accept m/s or km/s. All conversions happen automatically before the formulas are applied.
Key derived quantities: plasma beta, Mach number, and ion inertial length
Plasma beta (denoted beta) is the ratio of thermal pressure to magnetic pressure: beta = 2 * mu_0 * p / B^2. When beta is much less than 1 the magnetic field dominates and Alfven waves propagate cleanly; when beta exceeds 1 thermal effects dominate and wave behaviour becomes more complex. The Alfven Mach number (M_A = v_flow / v_A) compares the bulk plasma flow to the Alfven speed: a sub-Alfvenic flow (M_A < 1) cannot push upstream against an Alfven wave, while a super-Alfvenic flow can. This distinction controls whether the bow shocks of planets and stellar objects are mediated by MHD waves or behave more like ordinary fluid shocks. The ion inertial length (d_i = c / omega_pi) is the length below which ions decouple from the magnetic field and Hall currents arise. Below this scale the approximation of a single magnetohydrodynamic fluid breaks down and two-fluid or kinetic treatments are needed.
Applications in astrophysics, space weather, and fusion research
In the solar corona, Alfven velocities reach thousands of kilometres per second, and Alfven wave heating is one of the leading proposed mechanisms for why the corona is millions of degrees hotter than the photosphere below it. In the solar wind, the Alfven speed is comparable to the wind outflow speed; the point where the two become equal is the Alfven critical surface, which the Parker Solar Probe crossed for the first time in 2021. In Earth's magnetosphere and ionosphere, Alfven waves transport energy from the magnetopause to auroral acceleration regions, powering the auroral displays visible at high latitudes. In magnetic confinement fusion devices such as tokamaks, understanding Alfven eigenmode instabilities is critical to confining energetic particles, since they can resonate with these modes and be ejected before giving up their energy to heat the plasma. In laboratory plasma jet experiments, the Alfven velocity sets the timescale for magnetic field diffusion and reconnection, determining how quickly plasma can detach from a gun source.
Alfven velocities in natural and laboratory plasmas
| Plasma environment | B (typical) | Density (kg/m3) | v_A (km/s) | Plasma beta |
|---|---|---|---|---|
| Solar corona | 1-100 mT | 1e-12 to 1e-10 | 1000-10000 | < 0.1 |
| Solar wind (1 AU) | 5-10 nT | 1e-20 to 1e-19 | 30-80 | ~1 |
| Earth ionosphere | 25-65 uT | 1e-13 to 1e-11 | 100-3000 | < 1 |
| Tokamak plasma | 1-10 T | 1e-7 to 1e-5 | 1000-30000 | ~0.1 |
| Magnetosphere | 10-100 nT | 1e-21 to 1e-17 | 100-2000 | < 1 |
| Lab plasma jet | 0.01-1 T | 1e-4 to 1e-2 | 1-100 | ~0.1 to 1 |
Representative values of the Alfven velocity for common plasma environments. Real conditions vary widely within each environment.
Frequently asked questions
What is the Alfven velocity formula?
The Alfven velocity is v_A = B / sqrt(mu_0 * rho), where B is the magnetic flux density in Tesla, mu_0 is the permeability of free space (4*pi * 10^-7 H/m), and rho is the plasma mass density in kg/m3. The formula can also be written using ion number density: v_A = B / sqrt(mu_0 * n_i * m_i), where n_i is the number density of ions and m_i is their mass. Both forms give the same result when rho = n_i * m_i.
Why does higher magnetic field increase the Alfven velocity?
Alfven waves propagate like transverse waves on a magnetic field line, with the tension in the field acting as the restoring force. A stronger magnetic field means greater tension, so wave energy is transmitted faster, increasing the wave speed. The relationship is linear: doubling the field strength doubles the Alfven velocity.
Why does higher plasma density decrease the Alfven velocity?
Greater plasma density means more mass must be accelerated for each unit of wave displacement. The restoring force (magnetic tension) stays the same, so the wave moves more slowly, just as a heavier string vibrates more slowly at the same tension. The dependence goes as the inverse square root of density: quadrupling the density halves the Alfven velocity.
What is plasma beta and why does it matter?
Plasma beta (beta) is the ratio of thermal gas pressure to magnetic pressure: beta = 2 * mu_0 * p / B^2. It indicates which force dominates the plasma behavior. Low-beta plasmas (beta much less than 1) are magnetically dominated: the field strongly guides the plasma and Alfven waves propagate with little damping. High-beta plasmas (beta greater than 1) are thermally dominated: kinetic pressure becomes comparable to or larger than the field tension, wave propagation becomes more complex, and MHD instabilities are more likely.
What is the Alfven Mach number?
The Alfven Mach number M_A is the ratio of the bulk plasma flow speed to the Alfven velocity: M_A = v_flow / v_A. It is analogous to the ordinary Mach number in gas dynamics. A sub-Alfvenic flow (M_A < 1) cannot outrun Alfven waves propagating upstream; perturbations generated by an obstacle can propagate ahead of the flow and 'warn' the incoming plasma. A super-Alfvenic flow (M_A > 1) outruns those waves, so a standing bow shock forms upstream of the obstacle, similar to a sonic boom. Planetary magnetospheres and stellar winds transition from sub- to super-Alfvenic flow at the Alfven critical surface.
What is the ion inertial length?
The ion inertial length d_i = c / omega_pi (where omega_pi = sqrt(n_i e^2 / epsilon_0 m_i) is the ion plasma frequency) sets the scale below which ions decouple from the electron fluid and from the magnetic field. At scales larger than d_i, the standard MHD approximation is valid and the Alfven velocity formula applies directly. At scales smaller than d_i, Hall MHD, two-fluid, or fully kinetic (particle-in-cell) treatments are required to capture the physics correctly.
Can the Alfven velocity exceed the speed of light?
The non-relativistic Alfven velocity formula v_A = B / sqrt(mu_0 * rho) can formally give results exceeding c for extremely high fields or very low densities. In reality this simply means the non-relativistic approximation has broken down; the correct relativistic Alfven velocity is v_A,rel = v_A / sqrt(1 + (v_A / c)^2), which always stays below c. In most natural and laboratory settings the Alfven velocity is a small fraction of c, so the non-relativistic formula is accurate.