Magnetorotational Instability Growth Rate Calculator

Introduction

The magnetorotational instability, usually shortened to MRI, is one of the most important ideas in modern accretion-disk physics. It explains how a rotating disk of gas can become turbulent even when the magnetic field is weak. That turbulence matters because it helps move angular momentum outward, which in turn allows matter to drift inward toward a young star, white dwarf, neutron star, or black hole. In plain language, the MRI is a mechanism that turns smooth differential rotation into growing disturbances and then into sustained turbulent transport.

This calculator focuses on the classic textbook version of the problem: an axisymmetric perturbation in an incompressible disk threaded by a uniform vertical magnetic field. In that idealized setting, the fastest-growing MRI mode has a simple and famous result. Once you enter the local angular velocity Ω, the vertical magnetic field strength B, and the mass density ρ, the calculator estimates four quantities: the Alfvén speed, the most unstable wavelength, the maximum linear growth rate, and the corresponding e-folding time. These outputs give a quick sense of whether the MRI can grow and how quickly it does so.

The tool is intentionally narrow in scope. It does not attempt to model the full nonlinear turbulence of a real disk, nor does it include non-ideal magnetohydrodynamic effects such as resistivity, Hall drift, or ambipolar diffusion. Instead, it gives a clean first estimate based on the standard vertical-field MRI result that is widely used for order-of-magnitude reasoning.

How to Use

Use the form below by entering three positive quantities in SI units. The first input is the angular velocity Ω in radians per second. This describes how quickly the disk rotates at the location you care about. In a Keplerian disk, Ω decreases with radius, and that outward decline is exactly the kind of rotational shear that makes MRI possible.

The second input is the vertical magnetic field B in tesla. The field does not need to be strong; in fact, the MRI is often discussed in the weak-field limit. What matters is that the field can magnetically couple neighboring fluid elements. The third input is the mass density ρ in kilograms per cubic meter. Density enters because it sets the inertia of the gas and therefore affects the Alfvén speed, which is the characteristic speed of magnetic communication through the fluid.

After you press the compute button, the calculator reports the Alfvén speed v_A, the most unstable wavelength λ_max, the maximum growth rate γ_max, and the e-folding time t_growth. The e-folding time is especially useful for interpretation. If the MRI is growing exponentially as e^γt, then one e-folding time is the time required for the perturbation amplitude to increase by a factor of about 2.718. A shorter e-folding time means the instability develops more rapidly.

For reliable results, keep the units consistent and enter strictly positive values. If any input is zero or negative, the calculator will reject it because the ideal MRI formulas used here require positive angular velocity, magnetic field strength, and density.

Formula

The full linear MRI problem leads to a quartic dispersion relation. In the standard axisymmetric, incompressible, vertical-field case, perturbations of the form exp(γt + ikz) satisfy the relation below. This page preserves that full expression because it shows where the simplified calculator formulas come from.

Here, κ is the epicyclic frequency, k is the vertical wavenumber, and v_A is the Alfvén speed. For a Keplerian disk, the epicyclic frequency equals the orbital angular velocity, so κ = Ω. In that common case, the fastest-growing MRI mode has a maximum growth rate of

γ_max = (3/4)Ω

and occurs at a wavenumber

k_max = Ω / (√2 v_A).

The corresponding wavelength is λ_max = 2π / k_max, and the e-folding time is t_growth = 1 / γ_max. The Alfvén speed itself is given by the standard magnetohydrodynamic expression

where μ₀ is the permeability of free space. These are the exact relations used by the calculator script. In other words, the page is not fitting data or using a hidden approximation beyond the assumptions of the ideal MRI model itself.

Physical Interpretation

The outputs are easiest to understand if you connect each one to a physical question. The Alfvén speed tells you how strongly the magnetic field can communicate forces through the gas. A larger field or a lower density gives a larger Alfvén speed. The most unstable wavelength tells you the characteristic vertical scale of the fastest-growing MRI disturbance. If that wavelength is much larger than the local disk thickness, then the ideal fastest-growing mode may not fit comfortably inside the real system. The growth rate tells you how quickly the instability amplifies, while the e-folding time translates that rate into a more intuitive timescale.

One useful rule of thumb is that the maximum MRI growth rate in a Keplerian disk is always tied to the orbital timescale. Since γ_max = 0.75Ω, the instability grows on a dynamical timescale rather than a slow viscous one. That is why the MRI is so effective as a trigger for turbulence: once the conditions are right, it does not take many orbits for small perturbations to become dynamically important.

Example

Consider a simple protoplanetary-disk example. Suppose the local angular velocity is Ω = 2 × 10⁻⁷ rad/s, the vertical magnetic field is B = 1 × 10⁻⁷ T, and the density is ρ = 1 × 10⁻⁹ kg/m³. These are rough order-of-magnitude values often used for illustrative work near 1 AU in a young stellar disk.

First compute the Alfvén speed. With μ₀ = 4π × 10⁻⁷ H/m, the result is about 2.82 m/s. Next compute the most unstable wavenumber using k_max = Ω/(√2 v_A). That gives a wavelength λ_max of roughly 1.25 × 10⁸ m. The maximum growth rate is simply γ_max = 0.75Ω = 1.5 × 10⁻⁷ s⁻¹, so the e-folding time is about 6.67 × 10⁶ s, or roughly 77 days.

What does that mean physically? It means that if a small MRI-unstable perturbation is present, its amplitude grows by a factor of e in about 77 days under these ideal assumptions. After several e-foldings, the disturbance can become large enough that linear theory no longer applies and nonlinear turbulence takes over. In a disk that lives for thousands to millions of years, that is extremely fast.

Reference Scenarios

The table below gives a few illustrative environments. These values are not meant to be exact models of any one object. Instead, they show how the same MRI framework can be applied across very different astrophysical settings, from cool protoplanetary disks to compact-object accretion flows.

Limitations and Assumptions

This calculator assumes the ideal, axisymmetric, vertical-field MRI in a local Keplerian setting. That is a useful and standard approximation, but it is still an approximation. Real disks can be vertically stratified, compressible, partially ionized, and threaded by magnetic fields that are not purely vertical. In some environments, especially protoplanetary disks, non-ideal effects such as Ohmic resistivity, Hall drift, and ambipolar diffusion can strongly alter or even suppress MRI growth. If the gas is poorly ionized, the magnetic field may not couple well enough to the fluid for the ideal formulas to remain accurate.

The calculator also reports the fastest-growing wavelength from local linear theory. In a real disk, that wavelength must fit within the available geometry. If λ_max is larger than the local scale height or larger than the region over which the background conditions are roughly uniform, then the ideal fastest-growing mode may not be physically realized. Likewise, the growth rate shown here applies only during the linear phase. Once the perturbation becomes large, nonlinear saturation determines the eventual turbulence level, and that requires simulations or more advanced modeling.

Another practical limitation is that the result assumes a Keplerian-like rotation law with outwardly decreasing angular velocity. If the rotation profile differs substantially, the exact growth rate and unstable range of wavelengths can change. The page therefore works best as a quick estimate for standard accretion-disk conditions rather than as a complete stability solver for arbitrary rotating plasmas.

Why This Calculator Is Useful

Despite those limitations, a compact MRI calculator is valuable because it turns abstract theory into immediate scale estimates. Researchers, students, and science communicators often want to know whether a chosen field strength is dynamically relevant, whether the unstable wavelength is microscopic or macroscopic, and whether the instability grows in seconds, days, or years. Those questions can be answered quickly with just three inputs. The result is especially helpful when comparing MRI to other timescales such as the orbital period, sound-crossing time, or viscous evolution time.

In short, this page gives a practical entry point into one of the central instabilities of astrophysical fluid dynamics. If the output shows a short e-folding time and a wavelength that fits within the system, then the MRI is likely to be dynamically important. If not, you may need to consider whether the field is too weak, the density too high, the geometry too constrained, or whether non-ideal effects dominate the local plasma behavior.