Magnetorotational Instability and Angular Momentum Transport

How can a weak magnetic field help gas lose angular momentum? Magnetic tension couples gas that rotates at different rates. Small disturbances can draw energy from this shear and grow. Correlated velocities and the bent field then carry angular momentum outward, helping gas farther in move inward.

Change a vertical magnetic field and the disk shear. Watch a local channel grow or oscillate, then compare its magnetic and velocity stresses.

Analytic exponential growth rate $\gamma/\Omega=0.750$

Signed outward stress $T_{xy}=4.52\times10^{-6}$

Perturbation energy density $E=4.52\times10^{-6}$

Local time $\Omega t=1.80$

Let magnetic tension act

Vertical magnetic coupling

0.968

K/\Omega=kv_A/\Omega

Differential rotation

1.50

q=-\mathrm d\ln\Omega/\mathrm d\ln R

Drag the gold field tip to change its strength, or drag the background to turn the view.

A growing representative channel is ready.

Units, small amplitudes and display

\begin{gathered}L=\rho_0=\Omega=1,\\k=2\pi/L,\\v_A=B_0/\sqrt{\mu_0\rho_0}\end{gathered}

Energy and stress use density times box length squared times angular frequency squared. Height is periodic; the horizontal directions are homogeneous. The vertical field is fixed during each run.

Field-line transverse displacement is enlarged thirty times. Teal arrows show velocity perturbations; the three-dimensional view also uses blue arrows for background shear. Arrow lengths do not measure speed. Changing a control prepares a new small disturbance.

Growth stops at the small-amplitude limit or at time eight. The final frame is the last accepted linear state.

\Delta E-W=0

Model assumptions

This is a linear, unstratified, incompressible ideal-MHD channel in a co-rotating local Cartesian box. Radial and orbital directions are homogeneous, height is periodic, density is uniform and the vertical background field is constant. Rotation and box height set the units. The browser integrates four real sine/cosine amplitudes with RK4 and independently integrates shear work from stage stresses. It stops before either the horizontal magnetic amplitude exceeds eight percent of the background field or the velocity amplitude exceeds five percent of box length times angular frequency, and ends at dimensionless time eight. Cases with growth above one hundred-thousandth of the rotation frequency start in the analytic growing eigenmode; the other cases start with a radial velocity kick. Changing a parameter prepares a new small initial perturbation. Field-line transverse displacement is enlarged thirty times. Teal arrows show horizontal velocity perturbations; the three-dimensional view also uses blue arrows for the prescribed background shear. Arrow lengths do not measure speed. The analytic growth rate is separate from the evolving state. There is no stratification, compressibility, nonideal magnetic effect, nonlinear turbulence, saturation, global inflow or emitted spectrum. The separate offline replay evolves actual vertical mesh fields and retains its own sampled convergence limits.

Who carries angular momentum outward?

Velocity correlationMagnetic tension

\begin{gathered}T_{\rm Re}=\rho\langle u_xu_y\rangle,\\T_{\rm Max}=-\rho\langle b_xb_y\rangle\end{gathered}

Where the shear energy goes

Kinetic energyMagnetic energy

\begin{gathered}E_K=\rho\langle u_x^2+u_y^2\rangle/2,\\E_B=\rho\langle b_x^2+b_y^2\rangle/2\end{gathered}

Replay the spatial fields and read the equations

The downloadable Python standard-library example evolves four periodic vertical mesh fields. Eleven fixed cases test spatial and temporal refinement, analytic growth and stress, stable limits, shear work and the divergence of this channel. It includes 277 saved field frames.

Download the source, fields and independent checks

Balbus and Hawley 1991, linear analysis, equations 2.2–2.14b; Stone and Gardiner 2010, local rotating-box equations

The replay is a linear spatial calculation. The original papers and third-party simulation packages are linked separately.

Focus question: Which stress points outward as the channel grows?
One-sentence intuition: The growing disturbance draws energy from differential rotation. Outward transport appears in the signs of both velocity and magnetic correlations.

Core mathematical model

The local rotation law

\begin{gathered}v_{0y}=-q\Omega x,\\\kappa^2=2(2-q)\Omega^2\end{gathered}

Radial x points outward, orbital y follows rotation, and z is vertical. The epicyclic frequency describes a small radial orbital oscillation. Keplerian rotation has shear index one and a half.

The exponential-mode test

\begin{gathered}K=kv_A,\\s^4+(\kappa^2+2K^2)s^2\\{}+K^2(K^2-2q\Omega^2)=0\end{gathered}

The magnetic coupling frequency combines wave number and Alfven speed. A positive real root gives exponential growth. At a fixed wavelength, weak coupling grows slowly and strong tension restores oscillation. The displayed rate comes from this polynomial.

Signed angular-momentum transport

\begin{gathered}T_{xy}=\rho\langle u_xu_y-b_xb_y\rangle,\\b=\delta B/\sqrt{\mu_0\rho}\end{gathered}

Angle brackets average over height. Positive stress carries angular momentum outward. The sine velocity and cosine magnetic fields each contribute their own height average.

Energy supplied by shear

\begin{gathered}\frac{\mathrm dE}{\mathrm dt}=q\Omega T_{xy},\\E=\frac{\rho}{4}(u_x^2+u_y^2+b_x^2+b_y^2)\end{gathered}

The amplitudes in the energy expression are the four evolving real coefficients. A separate stage-stress integral checks the energy change. Solid-body rotation supplies no shear work.

Common difficulties

Field strength and wavelength belong together

Typical misconceptionA stronger magnetic field always makes MRI grow faster.

Better mental modelAt the fixed vertical wavelength here, growth exists only within a finite coupling band. Maximizing over all wavelengths asks a different question.

The stopped channel is still linear

Typical misconceptionThe last frame gives the turbulent saturation level.

Better mental modelThe calculation stops at its declared small-amplitude boundary. Following saturation requires nonlinear equations and separately validated calculations.

Run the experiment

01
Watch the growing channel
Start with Fastest Keplerian growth and compare the two stress curves below the scene.
What to observe: The magnetic contribution is positive and larger than the velocity contribution. In this analytic mode their ratio is five-thirds.
02
Change the background field
Drag the gold field tip vertically. Compare slow growth with the tension-restored preset.
What to observe: Each change starts a new small perturbation at the same representative time. The fixed wavelength makes the two edges of the unstable band visible.
03
Remove the shear
Choose Solid-body rotation and follow the sum of the two energy curves.
What to observe: Magnetic and kinetic energy can exchange while their total stays fixed to integration accuracy.
04
Save the physical channel
Pause and save. Compare the four real amplitudes, their sampled fields and signed stresses.
What to observe: The saved binary64 words retain the exact accepted amplitudes. The separate offline package evolves mesh fields and tests spatial convergence.

Magnetorotational Instability and Angular Momentum Transport

How a bent field carries angular momentum

Core mathematical model

The local rotation law

The exponential-mode test

Signed angular-momentum transport

Energy supplied by shear

Common difficulties

Field strength and wavelength belong together

The stopped channel is still linear

Run the experiment

Watch the growing channel

Change the background field

Remove the shear

Save the physical channel

Related experiments and articles

Articles

Magnetic tension and the growth of a disk disturbance