Jet Launching and Magnetic Flux Supply

A black hole cannot power a strong electromagnetic jet with spin alone: coherent poloidal magnetic flux must first reach and remain near the horizon. That supply is a transport problem spanning the outer boundary, turbulent disk, plunging region, and magnetized funnel. Inward advection competes with outward turbulent diffusion and reconnection; local dynamos can build loops without supplying a persistent net polarity; and sufficiently accumulated flux pushes back on the accretion flow in a magnetically arrested state. The central lesson is causal: horizon flux is an evolved quantity set by disk thickness, transport coefficients, topology, boundary supply, and runtime, not an adjustable afterthought to the Blandford–Znajek power formula.

Required background. Classical Field Theory supplies magnetic-flux conservation and the induction equation.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon diagnostic surface; Relativistic Matter and Compact Objects supplies relativistic plasma and thermodynamic closure; and Fluid and Kinetic Theory supplies MHD transport, turbulence, and conservation laws.

Advection, diffusion, and the origin of net flux

In resistive MHD the magnetic field evolves according to

$$\partial_t\boldsymbol B =\boldsymbol\nabla\times(\boldsymbol v\times\boldsymbol B) -\boldsymbol\nabla\times(\eta_m\boldsymbol\nabla\times\boldsymbol B), \qquad \boldsymbol\nabla\cdot\boldsymbol B=0.$$

For a thin disk threaded by a large-scale vertical field, a height-averaged flux function $\Psi(r,t)$ is advected inward with an effective speed $v_{\rm adv}$ and diffuses outward as inclined field crosses the disk. A useful local estimate compares

$$|v_{\rm adv}|\sim\frac{\nu_t}{r}, \qquad v_{\rm diff}\sim\frac{\eta_t}{H}, \qquad \frac{|v_{\rm adv}|}{v_{\rm diff}} \sim {\rm Pm}\frac{H}{r},$$

where $H/r$ is disk thickness and ${\rm Pm}=\nu_t/\eta_t$ is the turbulent magnetic Prandtl number. For a very thin disk with ${\rm Pm}\sim1$, the simple vertical average predicts rapid outward slippage. The governing flux equation and its global field coupling are derived in Lubow, Papaloizou, and Pringle 1994, §2, pp. 236–237, PDF, with steady transport results in their §§4–5, pp. 238–240.

That estimate is not a theorem about all thin disks. The low-density surface layers can move magnetic flux faster than the density-weighted mass velocity, and the vertical profiles of stress and resistivity matter. A systematic thin-disk expansion shows how the vertical problem changes both advection and diffusion rates Guilet and Ogilvie 2012, §§2–5, pp. 2099–2116. Winds, anisotropic turbulence, magnetic back-reaction, and non-ideal effects add further transport channels. Consequently, quoting $H/r$ and a scalar ${\rm Pm}$ is a baseline calculation, not a complete closure.

Topology is equally important. A local dynamo can make poloidal loops of alternating polarity while preserving zero signed flux through the chosen open disk surface if its boundary supplies none. The signed flux through any closed surface is always zero by $\boldsymbol\nabla\cdot\boldsymbol B=0$, even in a MAD; it cannot distinguish magnetic states. Loops may feed a transient jet, cancel by reconnection, or reverse polarity. A long-lived, high-flux state needs coherent flux in the initial or outer-boundary data, or an explicitly modelled global route that separates and transports the polarities. Supplying flux through a boundary is legitimate when its upstream source and time dependence are declared.

Horizon accumulation and magnetic arrest

Let $r_g=GM$ and define one common dimensionless horizon-flux measure by

$$\phi_{\rm BH} =\frac{\Phi_{\rm BH}} {\sqrt{\langle\dot M\rangle r_g^2}}, \qquad \Phi_{\rm BH}=\frac12\int_H |B^r|\sqrt{-g}\,d\theta\,d\phi.$$

Here $B^r={}^*F^{rt}$ is the coordinate magnetic-flux density in Gaussian units and horizon-penetrating coordinates. The averaging interval, hemisphere factor, flux element, and positive inward mass-rate convention are part of this definition. As coherent flux accumulates, magnetic pressure near the hole eventually becomes comparable to the inflow’s ram pressure. Accretion then proceeds intermittently through streams and magnetic interchange while excess flux escapes: this is a magnetically arrested disk, or MAD. A weak-net-flux, turbulence-dominated configuration is often called SANE. These are dynamical regimes, not labels inferable from a single image.

The three-dimensional GRMHD calculation of Tchekhovskoy, Narayan, and McKinney 2011, §§1–3, pp. L79–L82 exhibits flux saturation, intermittent accretion, and an outflow energy budget exceeding the rest-mass inflow for a rapidly spinning model. Its result is conditional on the supplied coherent field, spin, torus, floors, grid, and boundaries. A multi-code comparison demonstrates why agreement should be judged through resolution-dependent diagnostics rather than visual similarity alone Porth et al. 2019, §§2.1, 4–6.

There is no universal numerical value of “the” MAD saturation flux. The measured $\phi_{\rm BH}$ changes with flux normalization, disk thickness, spin, thermodynamics, domain, magnetic topology, and averaging interval. More importantly, a large horizon flux does not close the causal chain by itself. A claimed jet calculation must separately close mass, magnetic-flux, energy, and angular-momentum budgets and identify how numerical density floors load the funnel.

Executable transport benchmark

Before a global calculation is interpreted, its induction update should reproduce controlled advection–diffusion. The following teaching benchmark evolves one periodic Fourier mode,

$$\partial_t\Psi+v\partial_x\Psi =\eta\partial_x^2\Psi, \qquad \Psi(x,0)=1+0.1\sin x,$$

on $0\le x<2\pi$. The exact solution is $\Psi=1+0.1e^{-\eta t}\sin(x-vt)$. The environment used for the values below is CPython 3.9.6 with NumPy 2.0.2. The numerical method is a method-of-lines discretization with centered second-order differences and a classical fourth-order Runge–Kutta integrator.

import numpy as np

L, velocity, eta, t_end = 2*np.pi, -0.35, 0.04, 1.0

def rhs(psi, dx):
    d1 = (np.roll(psi, -1) - np.roll(psi, 1)) / (2*dx)
    d2 = (np.roll(psi, -1) - 2*psi + np.roll(psi, 1)) / dx**2
    return -velocity*d1 + eta*d2

def run(n):
    dx = L/n
    x = np.arange(n)*dx
    psi = 1 + 0.1*np.sin(x)
    mean0 = psi.mean()
    dt_bound = min(0.4*dx/abs(velocity), 0.2*dx**2/eta)
    steps = int(np.ceil(t_end/dt_bound))
    dt = t_end/steps
    for _ in range(steps):
        k1 = rhs(psi, dx)
        k2 = rhs(psi + 0.5*dt*k1, dx)
        k3 = rhs(psi + 0.5*dt*k2, dx)
        k4 = rhs(psi + dt*k3, dx)
        psi += dt*(k1 + 2*k2 + 2*k3 + k4)/6
    exact = 1 + 0.1*np.exp(-eta*t_end)*np.sin(x-velocity*t_end)
    l2 = np.sqrt(np.mean((psi-exact)**2))
    amp = 2*abs(np.fft.rfft(psi-psi.mean())[1])/n
    amp_error = abs(amp - 0.1*np.exp(-eta*t_end))
    return steps, l2, abs(psi.mean()-mean0), amp_error

for n in (64, 128, 256):
    print(n, *run(n))

Grid $N$	Steps	$L_2$ error	Mean-flux drift	Mode-amplitude error
64	21	$3.8242\times10^{-5}$	$0$	$3.0859\times10^{-6}$
128	84	$9.5637\times10^{-6}$	$0$	$7.7164\times10^{-7}$
256	333	$2.3911\times10^{-6}$	$0$	$1.9292\times10^{-7}$

Successive $L_2$ errors fall by a factor of four, the expected second-order spatial convergence, while the periodic conservative stencil preserves the mean to floating-point precision. The script produces one row per grid, including the error and conservation diagnostics. To rerun, save the executable block as flux_transport_benchmark.py, create an isolated environment, and execute:

python3.9 -m venv .venv
.venv/bin/python -m pip install numpy==2.0.2
.venv/bin/python flux_transport_benchmark.py

This is a verification problem for one term in the causal chain, not a claimed production simulation of a relativistic accretion disk or jet. It has no Kerr geometry, turbulence, vertical structure, dynamo, reconnection model, mass loading, or horizon. A production GRMHD claim must additionally pin the code commit, initial torus and field, equation of state, grid, atmosphere and floors, inner and outer boundaries, runtime, flux surfaces, and analysis scripts. It should refine the disk scale height and current sheets, move the outer boundary, extend through inflow equilibrium, vary supplied flux, and demonstrate closed flux and energy budgets.

Failure modes and interpretation

Three adversarial checks are especially useful. First, reverse or remove the boundary-supplied net flux: if a strong steady jet persists unchanged, the supposed supply mechanism is not controlling the result. Second, vary floors and funnel magnetization caps: power carried by floor-injected matter is not clean electromagnetic extraction. Third, compare runtimes at radii inside and outside inflow equilibrium; an unevolved outer torus cannot establish a steady global supply.

Numerical reconnection can also imitate physical diffusion, and a grid that resolves the MRI quality factor may still underresolve a thin current sheet. Convergence must therefore include $\phi_{\rm BH}(t)$, accretion rate, signed and unsigned flux, energy partition, opening angle, and closure residuals—not only a time-averaged jet efficiency. The result is conditional: given the declared topology, transport, and boundaries, the calculation supplies a measured horizon flux. It is not proof that nature supplies the same flux.

Exercises

Transport competition. A disk has $H/r=0.05$ and ${\rm Pm}=1$. Using the simple vertically averaged estimate, compare inward advection with outward diffusion. What ${\rm Pm}$ would make the two speeds equal, and why is that number not a universal threshold?

Solution

The estimate gives $|v_{\rm adv}|/v_{\rm diff}\sim{\rm Pm}(H/r)=0.05$, so diffusion is about twenty times faster than advection. Equality in this closure requires ${\rm Pm}\sim r/H=20$. The value is not universal because surface-layer velocities, vertical profiles of diffusivity and stress, magnetic back-reaction, winds, anisotropy, and global field geometry alter both effective speeds. The calculation is a diagnostic of the simple closure, not a criterion that overrides a resolved transport model.

References

• Guilet, Jérôme, and Gordon I. Ogilvie. “Transport of Magnetic Flux and the Vertical Structure of Accretion Discs—I. Uniform Diffusion Coefficients.” Monthly Notices of the Royal Astronomical Society 424 (2012): 2097–2117. DOI. Open preprint.
• Lubow, Stephen H., John C. B. Papaloizou, and James E. Pringle. “Magnetic Field Dragging in Accretion Discs.” Monthly Notices of the Royal Astronomical Society 267 (1994): 235–240. DOI. Open PDF.
• Porth, Oliver, et al. “The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project.” The Astrophysical Journal Supplement Series 243 (2019): 26. DOI.
• 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.