SANE and Magnetically Arrested Disks

SANE and MAD describe different magnetic-flux states of a black-hole accretion flow. A standard-and-normal-evolution disk carries comparatively little coherent net poloidal flux to the horizon; a magnetically arrested disk accumulates enough flux for magnetic pressure and tension to obstruct inflow, producing interchange and episodic accretion. The distinction is dynamical. One dimensionless flux number is indispensable, but a threshold without topology, time averaging, and a closed energy budget is not a classification.

Required background. Fluid and Kinetic Theory: continuum kinematics and balance laws supplies ideal-MHD flux freezing, stress, and conservation.

Helpful background. Black-Hole Geometry supplies the Kerr horizon and regular fluxes; Relativistic Matter and Compact Objects supplies relativistic plasma closure.

Horizon flux and arrested inflow

For a horizon-penetrating coordinate system, define the unsigned magnetic flux through one effective horizon hemisphere by

$$\Phi_{\rm BH} =\frac12\int_H |B^r|\sqrt{-g}\,d\theta\,d\phi .$$

Here $B^r={}^*F^{rt}$ is the coordinate magnetic-flux density in Gaussian units, so $\sqrt{-g}B^r\,d\theta\,d\phi$ is the radial flux element. It is not an orthonormal field component multiplied by an unrelated coordinate area. The factor $1/2$ and the use of unsigned flux are part of the definition. A common dimensionless diagnostic is

$$\phi_{\rm BH} =\frac{\Phi_{\rm BH}} {\sqrt{\langle\dot M\rangle r_g^2c}}, \qquad r_g=\frac{GM}{c^2},$$

where the mass rate and magnetic flux must be averaged over compatible windows. In this convention, three-dimensional MAD simulations often saturate around $\phi_{\rm BH}\sim40$–$60$, whereas SANE controls remain substantially lower. That interval is empirical and setup dependent, not a universal constant. Tchekhovskoy, Narayan, and McKinney 2011, §§ 2–3 and equations (1)–(2) states the flux and efficiency conventions for a canonical high-flux calculation.

Flux saturation is only the first diagnostic. A MAD also shows magnetic support against gravity, nonaxisymmetric interchange, episodic penetration of dense gas, and persistent competition between supplied poloidal flux and inward mass transport. The classification should therefore report:

• $\phi_{\rm BH}(t)$ and its distribution over an inflow-equilibrated window;
• radial magnetic-flux transport and the initial net-flux topology;
• shell-integrated mass, angular-momentum, and energy fluxes;
• magnetic-to-gas pressure and stress near the horizon;
• interchange structure in a full $2\pi$ three-dimensional domain.

A transient flux spike in a short run does not demonstrate saturation. Conversely, a fixed numerical threshold cannot erase differences caused by one-versus-two-hemisphere flux, Heaviside–Lorentz versus Gaussian field units, or a different normalization of $\dot M$.

Energy extraction and observables

Organize the horizon energy flux so that

$$\eta_{\rm out} =\frac{\langle\dot M\rangle c^2-\langle\dot E_H\rangle} {\langle\dot M\rangle c^2}.$$

Take $\dot E_H$ positive into the hole and include matter, rest mass, and electromagnetic energy. Then $\eta_{\rm out}\gt1$ means $\langle\dot E_H\rangle\lt0$: the black hole loses mass-energy despite accreting rest mass. Interpreting this horizon diagnostic as power delivered to large radius additionally requires a statistically steady exterior budget with storage, floors, boundaries, and radiation accounted for. It is not proof that all extracted power becomes a narrow observable jet: the budget must separate electromagnetic, matter, radiation, wind, and bound turbulent flux.

MADs can make strong Blandford–Znajek jets and can alter variability, disk thickness, polarization, and image morphology. SANE flows can also launch outflows, and either state can be made dim or bright by electron thermodynamics. Thus “MAD” is a simulation-state statement, not a label read directly from one spectrum or image. Radiative inference must forward-model the same fluid solution and marginalize electron heating, inclination, mass loading, optical depth, and foreground propagation.

A reproducible GRMHD experiment

A production comparison can be built with the open iharm3D v3.7 release, archived as Prather et al. 2021, software version 3.7. Pin its source archive and compiler/MPI/HDF5 environment. Use the same Kerr spin, Fishbone–Moncrief torus, adiabatic index, radial and polar mappings, reconstruction, Riemann solver, floors, and boundary conditions in both runs. Change only the seed field topology and net poloidal flux: weak alternating loops for the SANE control, and a coherent loop with enough flux supply for the MAD case.

At minimum, archive parameters.h, param.dat, the build archive, initial torus and vector potential, executable hash, dump cadence, and analysis commit. Run the three-dimensional domain long enough that the outer edge of the reported radial interval has reached inflow equilibrium. Density and internal-energy floors must enter the reported energy budget as explicit source terms. Narayan et al. 2012, §§ 2–4 shows how topology, flux accumulation, and arrested dynamics are distinguished in global calculations.

The independent benchmark is twofold. First run relativistic linear waves, magnetized shock tubes, and a stationary magnetized Bondi problem to verify the solver. Then reproduce the common weak-field torus comparison of Porth et al. 2019, §§ 4–6, which found improving agreement among nine GRMHD codes as resolution increased. Agreement in that SANE-like benchmark does not certify the MAD result; it verifies the shared algorithm before the high-flux experiment.

Refine radial, polar, and azimuthal resolution together, extend the duration, and repeat at least one field topology. Require the time-averaged $\phi_{\rm BH}$, stress, accretion rate, and total-energy residual to stabilize over the same inflow-equilibrated region. The table below is the output of the diagnostic teaching script in the next section, not a production GRMHD result. Its manufactured signal has baseline $\phi_{\rm BH}=48$ and an analytically known finite-window mean $48.009544\ldots$ after the resolution bias vanishes. The finite averaging interval does not contain an integer number of both sinusoidal periods, so its exact mean is not $48$.

Polar resolution $N_\theta$	MAD $\phi_{\rm BH}$ result	Change from previous resolution	Energy residual
96	47.7382	—	0.1200
144	47.8889	0.1508	0.0533
192	47.9417	0.0528	0.0300
288	47.9794	0.0377	0.0133

The decreasing change and residual verify this harness only. A real acceptance table must replace these manufactured inputs with shell-integrated outputs from every production resolution and report sampling uncertainty from correlated turbulence.

Diagnostic harness

Use Python 3.12.5 and NumPy==2.1.3. Save the following program as sane_mad_diagnostic.py. It constructs a deterministic manufactured flux series, applies a declared $t\ge10{,}000\,GM/c^3$ averaging window, and writes the numeric convergence table as sane_mad_diagnostic.csv. This small script tests definitions and table generation; it does not solve GRMHD.

from pathlib import Path
import numpy as np

RESOLUTIONS = np.array([96, 144, 192, 288])
rows = []
previous = np.nan

for ntheta in RESOLUTIONS:
    time = np.linspace(0.0, 20_000.0, 8 * ntheta + 1)
    phi = (
        48.0
        - 2500.0 / ntheta**2
        + 1.2 * np.sin(2.0 * np.pi * time / 700.0)
        + 0.45 * np.sin(2.0 * np.pi * time / 157.0)
    )
    window = time >= 10_000.0
    mean_phi = np.trapezoid(phi[window], time[window])
    mean_phi /= time[window][-1] - time[window][0]
    change = np.nan if np.isnan(previous) else abs(mean_phi - previous)
    residual = 0.12 * (96.0 / ntheta) ** 2
    rows.append((ntheta, mean_phi, change, residual))
    previous = mean_phi

output = np.asarray(rows, dtype=float)
header = "Ntheta,mean_phi,change_from_previous,energy_residual"
np.savetxt(Path("sane_mad_diagnostic.csv"), output, delimiter=",",
           header=header, comments="", fmt=["%d", "%.4f", "%.4f", "%.4f"])
for row in rows:
    print(f"{row[0]:3d}  phi={row[1]:.4f}  "
          f"change={row[2]:.4f}  residual={row[3]:.4f}")

Create an isolated environment and rerun it with:

python3.12 -m venv .venv-sane-mad
.venv-sane-mad/bin/python -m pip install numpy==2.1.3
.venv-sane-mad/bin/python sane_mad_diagnostic.py

The script produces the four rows above. Before adapting it to simulation dumps, replace the manufactured phi array with $\Phi_{\rm BH}(t)/\sqrt{\langle\dot M\rangle r_g^2c}$, using one fixed mean mass rate from the same averaging window. Normalization by the instantaneous $\dot M(t)$ is a different diagnostic: averaging a ratio is not the ratio of averages. Retain the raw numerator and denominator so both choices can be compared.

Failure modes and boundaries

The most dangerous numerical failures are floor-generated jet power, polar-axis artifacts, inadequate azimuthal extent, under-resolved MRI turbulence, outer radii not in inflow equilibrium, and boundary-supplied flux masquerading as transport. Report the fraction of energy and mass introduced by floors, the quality factors resolving MRI wavelengths, and the radial range that passes an inflow-time test. Repeat the diagnostic after shifting the averaging window; a classification that vanishes under a reasonable shift is not robust.

Detailed spectra and instrument likelihoods lie beyond this page’s boundary. Radiation, Spectra, and Variability develops radiative transfer and data comparison, while Magnetospheres, Jets, and Outflows develops flux supply and jet extraction. This page establishes the fluid-state comparison and its reproducible diagnostics.

Classify a flux history

A run uses the convention above and has time-averaged $\Phi_{\rm BH}=30\sqrt{\langle\dot M\rangle r_g^2c}$ during its first window and $48\sqrt{\langle\dot M\rangle r_g^2c}$ during a later inflow-equilibrated window. The later interval also shows interchange and stable shell-integrated energy flux. What are the two $\phi_{\rm BH}$ values, and which interval can support a MAD classification?

Solution

Direct substitution gives $\phi_{\rm BH}=30$ and $48$. The first value is below the common saturation range in this convention and, without sustained arrested dynamics, does not establish MAD. The later interval lies in the usual empirical range and adds the essential dynamical and budget evidence, so it can support a MAD classification.

The conclusion still depends on the declared flux convention, adequate duration and resolution, floor control, and demonstrated inflow equilibrium. If the later value were only a transient spike or used a different hemisphere normalization, the numerical value alone would not be decisive.

References

• Narayan, Ramesh, Aleksander Sądowski, Roger F. Penna, and Akshay K. Kulkarni. “GRMHD Simulations of Magnetized Advection-dominated Accretion on a Non-spinning Black Hole: Role of Outflows.” Monthly Notices of the Royal Astronomical Society 426 (2012): 3241–3259. DOI.
• Porth, Oliver, et al. “The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project.” Astrophysical Journal Supplement Series 243 (2019): 26. DOI.
• Prather, Ben S., et al. “iharm3D: Vectorized General Relativistic Magnetohydrodynamics.” Version 3.7 (2021). Software archive.
• Tchekhovskoy, Alexander, Ramesh Narayan, and Jonathan C. McKinney. “Efficient Generation of Jets from Magnetically Arrested Accretion on a Rapidly Spinning Black Hole.” Monthly Notices of the Royal Astronomical Society: Letters 418 (2011): L79–L83. DOI.