Alfvén Speed Calculator

What Is the Alfvén Speed?

The Alfvén speed is the characteristic velocity at which low-frequency perturbations of the magnetic field and plasma ions propagate along magnetic field lines. These disturbances, known as Alfvén waves, are fundamental to magnetohydrodynamics (MHD) and appear in environments ranging from the solar corona to laboratory fusion devices.

In a magnetized plasma, magnetic field lines behave in some ways like elastic strings under tension. If you disturb the field in one location, that disturbance travels along the field lines. The Alfvén speed sets how quickly information, energy, and momentum can be transported along those magnetic structures, provided the conditions of ideal MHD are approximately satisfied.

Formula and Units

In SI units, the Alfvén speed v_A for a uniform plasma is given by:

where:

• B is the magnetic field strength in tesla (T).
• ρ is the plasma mass density in kilograms per cubic metre (kg/m³).
• μ₀ is the permeability of free space, equal to 4π × 10⁻⁷ H/m.

The output v_A from this calculator is in metres per second (m/s). For convenience, the tool also reports the same value in kilometres per second (km/s) by dividing by 1000.

Dimensional analysis confirms that the formula is consistent: the numerator has units of T, and the denominator has units of √(μ₀ ρ), which reduces to s/m, giving overall units of m/s.

Physical Interpretation

The Alfvén speed balances the magnetic tension force against the inertia of the plasma. A stronger magnetic field increases the tension along field lines and therefore raises the Alfvén speed. A higher mass density means more inertia that must be accelerated by the same magnetic tension, so the wave travels more slowly.

Conceptually, you can think of three key dependencies:

• Magnetic field dependence: For fixed density, doubling B doubles v_A.
• Density dependence: For fixed B, increasing ρ by a factor of 4 cuts v_A in half, because of the square root in the denominator.
• Plasma regime: Alfvén waves are transverse, low-frequency oscillations where ions and the magnetic field are tightly coupled; electrons, being lighter, largely provide the current that supports the magnetic field perturbations.

The Alfvén speed is one of several characteristic speeds in a plasma. Others include the sound speed and the fast and slow magnetosonic speeds. In many magnetically dominated environments (high magnetic pressure compared with gas pressure), the Alfvén speed is the most important wave speed for energy transport along field lines.

How to Use This Calculator

The form lets you either select a typical plasma environment or enter custom parameters. All inputs are in SI units.

1. Select an environment (optional): Choose a preset such as the solar corona, Earth's magnetosphere, a tokamak edge plasma, or the interstellar medium. The calculator will auto-fill representative values of B and ρ.
2. Enter or adjust magnetic field B (T): Provide the magnetic field strength in tesla. Very weak astrophysical fields may be as small as 10⁻¹⁰ T, while fusion devices can reach several tesla. Only non-negative values are physically meaningful.
3. Enter or adjust mass density ρ (kg/m³): Specify the plasma mass density. Space plasmas can have extremely low densities (e.g., 10⁻²⁰ kg/m³), while laboratory plasmas are usually much denser. Again, only non-negative values are valid.
4. Use scientific notation if needed: You can input numbers like 5e-4 for 5 × 10⁻⁴ T. This is especially convenient for very small densities or fields.
5. Compute the Alfvén speed: Click the button to calculate v_A. The result will be displayed in m/s and km/s. For physical interpretation, the km/s value is often easier to compare across environments.

If you obtain a value that seems unexpectedly large or small, check carefully for unit consistency. For example, sometimes densities are quoted in particles per cubic centimetre (cm⊃−3) rather than kg/m³. You must convert to kg/m³ before using this tool.

Worked Example: Solar Corona Loop

This section walks through a calculation similar to the “Solar Corona Loop” preset, using typical reference values:

• B = 5 × 10⁻⁴ T
• ρ = 1 × 10⁻¹² kg/m³
• μ₀ = 4π × 10⁻⁷ H/m

First, compute the product inside the square root:

μ₀ρ = (4π × 10⁻⁷) × (1 × 10⁻¹²) ≈ 1.26 × 10⁻¹⁸ H·kg/m⁴.

Next, take the square root:

√(μ₀ρ) ≈ √(1.26 × 10⁻¹⁸) ≈ 1.12 × 10⁻⁹ (in the appropriate SI combination giving seconds per metre).

Now divide the magnetic field by this value:

v_A = B / √(μ₀ρ) ≈ (5 × 10⁻⁴ T) / (1.12 × 10⁻⁹) ≈ 4.5 × 10⁵ m/s.

Converting to km/s:

v_A ≈ 450 km/s.

This is consistent with commonly cited Alfvén speeds in active-region coronal loops, illustrating that the calculator reproduces realistic orders of magnitude when given typical parameters.

Typical Alfvén Speeds in Different Plasmas

The presets in the form correspond roughly to the following representative conditions and Alfvén speeds:

These values are order-of-magnitude estimates. Real plasmas can be highly structured, with local variations in both magnetic field and density that cause the Alfvén speed to vary in space and time.

Relation to Other Characteristic Speeds

The Alfvén speed is closely related to, but distinct from, several other important plasma speeds:

• Sound speed c_s: The speed of ordinary pressure (acoustic) waves in the plasma. It depends on temperature and composition rather than on the magnetic field.
• Fast magnetosonic speed: A combined magnetic and pressure wave that generally travels faster than both v_A and c_s, except in certain parameter regimes.
• Slow magnetosonic speed: A wave mode that typically travels slower than both the sound speed and the Alfvén speed, often guided along magnetic field lines in high-beta plasmas.

Comparing v_A with the sound speed helps determine whether magnetic or gas pressure effects dominate dynamics. This comparison is captured by the plasma beta parameter, which is proportional to the ratio of gas pressure to magnetic pressure.

Interpreting the Results

Once you compute an Alfvén speed for your chosen parameters, consider the following points:

• Order of magnitude: For space plasmas, values from tens to thousands of km/s are common. In very weakly magnetized or extremely dense plasmas, the Alfvén speed may drop below 1 km/s.
• Propagation direction: The formula applies to waves propagating along the magnetic field. Oblique or perpendicular propagation involves more complex magnetosonic modes.
• Comparison with flows: Comparing bulk flow speeds with v_A indicates whether the flow is sub-Alfvénic or super-Alfvénic, which has strong implications for shock formation and energy dissipation.
• Time scales: Over a magnetic structure of length L, the Alfvén transit time is roughly τ ≈ L / v_A. This sets a characteristic time scale for rearranging magnetic structures or transmitting disturbances.

Applications in Astrophysics and Fusion

Alfvén waves and the associated speed are central to many plasma-physics problems:

• Solar coronal heating: Alfvén waves generated in the lower solar atmosphere can transport energy upward along magnetic loops. Dissipation of this wave energy is one of the proposed mechanisms to explain why the solar corona is much hotter than the underlying photosphere.
• Space weather and aurorae: In Earth's magnetosphere, field-aligned Alfvén waves can accelerate charged particles into the upper atmosphere, contributing to auroral emissions and influencing satellite environments.
• Magnetic confinement fusion: In tokamaks and stellarators, energetic particles can drive Alfvén eigenmodes. If these modes grow too strong, they can enhance particle transport, degrade confinement, or damage internal components.
• Interstellar and intergalactic media: On galactic scales, Alfvén waves affect the transport of cosmic rays, the support of gas against gravity, and the cascade of turbulent energy across scales.

Accurate estimates of the Alfvén speed help set stability criteria, design diagnostics, and interpret observations across these very different environments.

Assumptions and Limitations

The calculator is based on the simplest, ideal-MHD expression for the Alfvén speed. It is important to understand the assumptions behind this formula to avoid misinterpretation:

• Ideal MHD: The plasma is treated as a single conducting fluid with negligible resistivity. Effects such as finite resistivity, Hall physics, and electron inertia are ignored.
• Low-frequency limit: The wave frequency is assumed low enough that ions and the magnetic field remain tightly coupled. Very high-frequency waves may require kinetic or multi-fluid models.
• Uniform, isotropic plasma: The formula assumes a scalar mass density and a uniform background magnetic field. Strong spatial gradients, anisotropic pressure, or multi-component plasmas can modify the effective Alfvén speed.
• Non-relativistic regime: Relativistic corrections are not included. If the computed Alfvén speed is a substantial fraction of the speed of light, or if the plasma is strongly relativistic, more advanced models are required.
• Single-fluid mass density: The density input should correspond to the total mass density of the coupled ion species. Situations with neutral components that are only weakly coupled to the plasma may need a more careful treatment.
• SI units only: All inputs must be in tesla and kg/m³. If your data are in other units (e.g., gauss, cm⊃−3), convert them before using the calculator.

Because of these assumptions, the calculator is intended as an educational and approximate engineering tool, not a replacement for full MHD simulations or kinetic plasma models. For precision-critical work, consult domain-specific literature and numerical codes.

Related Calculators and Further Exploration

The Alfvén speed connects naturally to several other plasma parameters. For a broader picture of your system, you may also want to compute:

• The plasma frequency, which describes the natural oscillation rate of charge carriers.
• The magnetic Reynolds number, which quantifies the relative importance of advection versus diffusion of magnetic fields.

Together, these quantities help you classify regimes such as magnetically dominated versus gas-pressure dominated plasmas, strongly versus weakly conducting flows, and collisional versus collisionless behaviour. Use them alongside the Alfvén speed to build an integrated view of your plasma system.