Plasma (MHD) Field Calculator
Compute Alfven speed, sound speed, plasma beta and fast/slow magnetosonic speeds from density, pressure and magnetic field using ideal MHD. Free, in-browser.
Plasma Magnetohydrodynamic Field Solver
Exact circularly polarized Alfvén benchmark
- B_y
- B_z
- v_y
- v_z
Magnetic pressure, total pressure, and energy
- B²/(2μ0)
- p_total
- total energy density
Fast and slow magnetosonic speeds vs. angle
- v_fast
- v_slow
- V_A
- c_s
Isometric Alfvén helix and vector lattice
Exact benchmark snapshot at t = 0.0040 s
| x (m) | B_y (T) | B_z (T) | v_y (m/s) | v_z (m/s) | |B| (T) | p_B (Pa) |
|---|---|---|---|---|---|---|
| 0.0000 | -1.288e-3 | -3.254e-3 | -3.635e+4 | -9.180e+4 | 1.059e-2 | 4.466e+1 |
| 0.4175 | 2.180e-3 | -2.738e-3 | 6.150e+4 | -7.724e+4 | 1.059e-2 | 4.466e+1 |
| 0.8351 | 3.460e-3 | 5.264e-4 | 9.761e+4 | 1.485e+4 | 1.059e-2 | 4.466e+1 |
| 1.2526 | 1.267e-3 | 3.263e-3 | 3.574e+4 | 9.204e+4 | 1.059e-2 | 4.466e+1 |
| 1.6701 | -2.198e-3 | 2.724e-3 | -6.200e+4 | 7.684e+4 | 1.059e-2 | 4.466e+1 |
| 2.0877 | -3.457e-3 | -5.491e-4 | -9.751e+4 | -1.549e+4 | 1.059e-2 | 4.466e+1 |
| 2.5052 | -1.246e-3 | -3.271e-3 | -3.514e+4 | -9.227e+4 | 1.059e-2 | 4.466e+1 |
| 2.9228 | 2.216e-3 | -2.709e-3 | 6.250e+4 | -7.643e+4 | 1.059e-2 | 4.466e+1 |
| 3.3403 | 3.453e-3 | 5.718e-4 | 9.741e+4 | 1.613e+4 | 1.059e-2 | 4.466e+1 |
| 3.7578 | 1.224e-3 | 3.279e-3 | 3.453e+4 | 9.250e+4 | 1.059e-2 | 4.466e+1 |
| 4.1754 | -2.233e-3 | 2.695e-3 | -6.300e+4 | 7.602e+4 | 1.059e-2 | 4.466e+1 |
| 4.5929 | -3.449e-3 | -5.944e-4 | -9.730e+4 | -1.677e+4 | 1.059e-2 | 4.466e+1 |
A clear path from inputs to results.
Work out how fast waves travel through a magnetized plasma and how the magnetic field competes with gas pressure. This solver uses ideal magnetohydrodynamics to return the Alfvén speed, the sound speed, the plasma beta, and the fast and slow magnetosonic speeds for a given density, pressure, and magnetic field — the core numbers behind solar, space, and fusion plasma analysis.
How to use
- Enter the mass density ρ of the plasma and the thermal pressure p.
- Set the adiabatic index γ (5/3 for a monatomic ideal gas) and the background magnetic field strength B0.
- Enter the perturbation wavelength λ and the propagation angle θ between the wavevector and the magnetic field.
- Read the Alfvén speed, the sound speed, and the plasma beta that compares magnetic to thermal pressure.
- Read the wave frequency and period for the Alfvén mode, plus the fast and slow magnetosonic speeds at your chosen angle.
The formula
Reference: NRL Plasma Formulary (2023), Alfvén and magnetosonic speeds; standard ideal-MHD dispersion relation (e.g., Boyd & Sanderson, The Physics of Plasmas).
Worked example
Low-beta coronal-like plasma, 20° propagation
- Density ρ = 1×10⁻⁹ kg/m³; pressure p = 1×10⁻⁴ Pa; adiabatic index γ = 5/3; magnetic field B0 = 0.01 T; wavelength λ = 2.5 m; propagation angle θ = 20°. μ0 = 4π×10⁻⁷.
- 1
The Alfvén speed is V_A = B0 / √(μ0ρ) = 0.01 / √(4π×10⁻⁷ × 10⁻⁹) = 2.82095×10⁵ m/s. The sound speed is c_s = √(γp/ρ) = √((5/3)(10⁻⁴)/10⁻⁹) = 408.25 m/s. The plasma beta is β = 2μ0p/B0² = 2 × 4π×10⁻⁷ × 10⁻⁴ / (0.01)² = 2.513×10⁻⁶, so the field dominates the dynamics. For an Alfvén wave the wavenumber is k = 2π/λ = 2.5133 rad/m and the frequency is ω = kV_A = 7.0898×10⁵ rad/s, giving a period of 2π/ω = 8.862×10⁻⁶ s. Feeding c_s, V_A, and cos²20° into the magnetosonic relation gives a fast speed of 2.82095×10⁵ m/s and a slow speed of 383.63 m/s.
Alfvén speed 282.09 km/s; sound speed 408.25 m/s; plasma beta 2.51×10⁻⁶ (strongly magnetically dominated); Alfvén-wave frequency 7.09×10⁵ rad/s with an 8.86 µs period; fast magnetosonic speed 282.09 km/s, slow magnetosonic speed 383.63 m/s.
Interpreting the result
The Alfvén speed tells you how fast magnetic tension carries transverse disturbances; it dominates when the plasma beta is well below one, as here. Plasma beta is the single most useful summary number: β ≪ 1 means the magnetic field controls the plasma (typical of the solar corona and tokamak edges), while β ≫ 1 means gas pressure dominates. The fast magnetosonic speed approaches the larger of the Alfvén and sound speeds and the slow speed approaches the smaller; at the chosen angle the fast mode here tracks the Alfvén speed and the slow mode tracks a value near the sound speed, which is the expected low-beta behaviour.
Assumptions & limitations
- Ideal MHD is assumed: the plasma is a perfectly conducting, single, quasi-neutral fluid with no resistivity, viscosity, or heat conduction.
- The wave is a small-amplitude linear perturbation on a uniform background; nonlinear steepening, shocks, and large-amplitude effects are not modeled.
- The adiabatic index γ is taken as constant (5/3 by default); the pressure follows an adiabatic equation of state.
- The magnetosonic speeds depend on the angle θ between the wavevector and the background field; at θ = 0 and θ = 90° the fast and slow modes reduce to the familiar parallel and perpendicular limits.
- SI units are used throughout (kg/m³, Pa, tesla, metres); the result is a dispersion-relation calculation, not a time-domain simulation of the plasma.
Frequently asked questions
What is the Alfvén speed?
It is the speed at which magnetic tension propagates transverse waves along the field lines, V_A = B0/√(μ0ρ). A stronger field or a thinner plasma gives a faster Alfvén speed. In the worked example the 0.01 T field in a very tenuous plasma gives V_A ≈ 282 km/s, far above the 408 m/s sound speed, which is characteristic of a magnetically dominated plasma.
What is the difference between the fast and slow magnetosonic waves?
Both are compressive waves that mix magnetic and gas pressure, but they differ in how those pressures combine. The fast mode has magnetic and thermal pressure reinforcing each other and travels at or above the larger of the Alfvén and sound speeds; the slow mode has them partly opposing and travels at or below the smaller. Their speeds depend on the propagation angle θ relative to the field.
Does this model include resistivity or collisions?
No. This is ideal MHD, which treats the plasma as a perfect conductor with no resistive dissipation, viscosity, or thermal conduction. That is an excellent approximation for large-scale, low-collision plasmas such as the corona and the solar wind, but it does not capture reconnection, resistive damping, or kinetic effects, which require extended or kinetic models.
What does plasma beta tell me?
Plasma beta β = 2μ0p/B0² is the ratio of thermal (gas) pressure to magnetic pressure. When β is much less than one the magnetic field governs the plasma's motion; when β is much greater than one the gas pressure does. The 2.5×10⁻⁶ value here means the field overwhelmingly controls the dynamics, as it does in the solar corona.
Why does the propagation angle matter?
The magnetosonic speeds are anisotropic: they depend on the angle θ between the wavevector and the magnetic field through the cos²θ term in the dispersion relation. Along the field (θ = 0) the modes decouple into a pure Alfvén wave and a pure sound wave; across the field (θ = 90°) only the fast mode propagates. Intermediate angles like the 20° used here give a blend of the two.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.