# Axisymmetry (DD-054)

> Modelers often assume the gas around a black hole looks the same at every angle around the spin axis. Quantities then depend only on distance and polar angle, and a 3D simulation becomes a much cheaper 2D one. The shortcut has known limits. Cowling's anti-dynamo theorem (1933) says an axisymmetric magnetized flow can't regenerate its magnetic field, so 2D simulations lose the turbulence that drives accretion unless an artificial 'sub-grid' dynamo is added. And because infalling gas generally doesn't share the black hole's spin direction, many disks are expected to be tilted. Tilted disks break the symmetry and must be simulated in full 3D. The classic prediction that a thin disk's inner part aligns with the spin (Bardeen–Petterson, 1975) didn't appear in some GRMHD simulations of moderately thick disks.

- Department: Space (also Science)
- First published: 2025-05-21
- Last updated: 2025-05-21
- Permalink: https://detailenjoyer.com/case/axisymmetry/
- Author: Marcus Easterling (DetailEnjoyer), The Detail Department

## On the timeline

- **1933**: Cowling publishes the anti-dynamo theorem
- **1975**: Bardeen and Petterson predict that a thin disk's inner part aligns with the black hole's spin

Full timeline: https://detailenjoyer.com/timeline/?case=axisymmetry

## Quick facts

- **DD-054-F01**: 2D axisymmetric simulations can't sustain a black hole disk's magnetic field on their own, because of Cowling's anti-dynamo theorem. _(Write-up · Exhibit B (The dynamo problem))_
- **DD-054-F02**: Tilted accretion disks break axisymmetry and have to be simulated in full 3D. _(Write-up · Exhibit D (Tilted disks))_

## Write-up

Why black hole models assume symmetry. Treating the disk as identical all the way around its axis turns a 3D problem into 2D, at a cost.

1. **Why assume it**: Axisymmetric, two-dimensional simulations are much cheaper to run than full 3D ones, so researchers can explore a wider range of black hole accretion systems.
2. **The dynamo problem**: Cowling's anti-dynamo theorem (1933) says axisymmetric ideal-MHD systems can't regenerate their magnetic fields. In 2D simulations the field decays and accretion dies out, unless a sub-grid dynamo is added to stand in for the missing 3D turbulence.
3. **Near a spinning black hole**: Numerical tests in the Kerr geometry of a rotating black hole also found that an axisymmetric field couldn't be sustained, suggesting Cowling's theorem still holds there.
4. **Tilted disks**: Infalling gas doesn't 'know' which way the black hole spins, so many disks are expected to be tilted. Tilted disks break the symmetries simulations rely on and must be modeled in full 3D.
5. **Alignment isn't guaranteed**: Theory predicts that the inner part of a thin tilted disk aligns with the black hole's spin (the Bardeen–Petterson effect, 1975). But GRMHD simulations of moderately thick disks didn't find this alignment.

## Related case files

- [Neutron Spin (DD-058)](https://detailenjoyer.com/case/neutron-spin/): Another case about spinning collapsed objects: neutron stars, the collapsed cores of massive stars.
- [Alcubierre Drive (DD-060)](https://detailenjoyer.com/case/alcubierre-drive/): More general relativity: Alcubierre showed Einstein's equations allow a warp bubble.

## Reel vs. record

- **Confirmed**: an axisymmetric model treats quantities as independent of azimuth, so they vary only with radius and polar angle, which reduces a 3D problem to 2D. [Sądowski et al.; tilted-disk review]
- **Missing context**: the shortcut has a known flaw. By Cowling's anti-dynamo theorem, axisymmetric MHD can't regenerate magnetic fields, so 2D accretion runs die out unless an artificial dynamo is added. Self-consistent turbulence needs 3D. [Sądowski et al.; arXiv 2024; Brandenburg]
- **Note**: Nuance on 'typically aligned with the spin axis': that's an assumption. Infalling gas is generally misaligned with the spin, so tilted disks are expected, and they can't be modeled axisymmetrically. [Liska et al.; tilted-disk review]
- **Nuance**: the predicted Bardeen–Petterson alignment of a thin disk's inner part with the spin wasn't found in GRMHD simulations of moderately thick disks, so alignment can't be taken for granted. [Liska et al.]
- **Context**: the assumption is chosen mainly for cost. Axisymmetric 2D simulations are far cheaper than 3D, which lets researchers survey many more disk models. [Sądowski et al.]

## Video: Axisymmetry: Why Black Hole Models Assume Symmetry

- Duration: 22.3 s · Published: 2025-05-21
- File: https://detailenjoyer.com/assets/videos/DJ606LpxRzG.mp4

### Transcript

DID YOU KNOW? An axisymmetric distribution of matter around black holes refers to a configuration where physical properties such as density, pressure, and velocity are symmetric about a central axis, typically aligned with the black hole's spin axis. This assumption simplifies the complex geometry of accretion disks and relativistic jets, allowing for models that only vary with radial distance and polar angle, not azimuth. NOW YOU KNOW!

## Questions & answers

**Q: In an axisymmetric model, quantities don't vary with which coordinate?**  
A: They vary only with radius and polar angle. _(Source: Write-up · Exhibit A (Why assume it) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421))_

**Q: Which theorem says axisymmetric MHD can't regenerate magnetic fields?**  
A: Cowling stated it in 1933. _(Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236))_

**Q: What do 2D simulations add to work around it?**  
A: It stands in for the missing 3D turbulence. _(Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236))_

**Q: Why must tilted disks be simulated in 3D?**  
A: No axial or reflection symmetry is left to exploit. _(Source: Write-up · Exhibit D (Tilted disks) · Tilted accretion disks (review, arXiv:2404.10052); Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019))_

**Q: The predicted alignment of a thin disk's inner part with the spin is named after whom?**  
A: It was predicted in 1975. _(Source: Write-up · Exhibit E (Alignment isn't guaranteed) · Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019))_

## Sources

- [Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421)](https://arxiv.org/abs/1407.4421)
- [GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236)](https://arxiv.org/abs/2403.16236)
- [Testing Cowling's antidynamo theorem near a rotating black hole (Brandenburg, ApJ Letters)](https://doi.org/10.1086/310142)
- [Tilted accretion disks (review, arXiv:2404.10052)](https://arxiv.org/abs/2404.10052)
- [Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019)](https://doi.org/10.1093/mnras/stz834)

## Cite this case file

Easterling, Marcus. "Axisymmetry." The Detail Department (DetailEnjoyer), case file DD-054, last updated May 21, 2025. https://detailenjoyer.com/case/axisymmetry/