Jets from Accretion Disk Dynamos: Consistent Quenching Modes for Dynamo and Resistivity

We solve the time-dependent, resistive MHD equations with the PLUTO code ³ (Mignone et al. 2007) version 4.3 in spherical coordinates (R, θ, ϕ) assuming axisymmetry. We refer to (r, z) as cylindrical coordinates. The code solves the integral form of the MHD conservation laws through a high-order finite volume method. To the standard PLUTO MHD equations, we have added a mean-field dynamo term.

The set of primitive variables consist of, respectively, the plasma density ρ, the flow velocity v , the gas pressure p, and the magnetic field B .

The set of conservation laws consists of the conservation of mass,

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{v}})=0,\end{eqnarray} \tag{ 1 }$$

momentum,

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho {\boldsymbol{v}}}{\partial t}+{\rm{\nabla }}\cdot \left[\rho {\boldsymbol{v}}{\boldsymbol{v}}+\left(p+\displaystyle \frac{{\boldsymbol{B}}\cdot {\boldsymbol{B}}}{2}\right){\mathsf{I}}-{\boldsymbol{B}}{\boldsymbol{B}}\right]+\rho {\rm{\nabla }}{{\rm{\Phi }}}_{{\rm{g}}}=0,\end{eqnarray} \tag{ 2 }$$

total energy,

$$\begin{eqnarray}&&\displaystyle \frac{\partial e}{\partial t}+{\rm{\nabla }}\cdot \left[\left(\displaystyle \frac{\rho {v}^{2}}{2}+\displaystyle \frac{{\rm{\Gamma }}}{{\rm{\Gamma }}-1}p+\rho {{\rm{\Phi }}}_{{\rm{g}}}\right){\boldsymbol{v}}+{\boldsymbol{E}}\times {\boldsymbol{B}}\right]={{\rm{\Lambda }}}_{\mathrm{cool}},\end{eqnarray} \tag{ 3 }$$

and magnetic field,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}+{\rm{\nabla }}\times {\boldsymbol{E}}=0.\end{eqnarray} \tag{ 4 }$$

The total energy density is computed, from the primitive variables, as

$$\begin{eqnarray}&&e=\displaystyle \frac{p}{{\rm{\Gamma }}-1}+\displaystyle \frac{\rho {v}^{2}}{2}+\displaystyle \frac{{B}^{2}}{2}+\rho {{\rm{\Phi }}}_{{\rm{g}}},\end{eqnarray} \tag{ 5 }$$

with polytropic index Γ = 5/3, while the electric field is defined by (see Krause & Rädler 1980)

$$\begin{eqnarray}&&{\boldsymbol{E}}=-{\boldsymbol{v}}\times {\boldsymbol{B}}+\eta \cdot ({\rm{\nabla }}\times {\boldsymbol{B}})-\alpha \cdot {\boldsymbol{B}}\end{eqnarray} \tag{ 6 }$$

where the first is the ideal term, the second is the diffusive term, and the third is the dynamo term. The gravitational potential Φ_g is provided by a central object mass M, i.e., Φ_g = − GM/R.

Previous studies (see, e.g., Casse & Ferreira 2000a, 2000b; Zanni et al. 2007; Tzeferacos et al. 2013) have shown that the nonideal heating and cooling processes may play a relevant role in the context of jet formation. Here, for the sake of simplicity, the cooling term Λ_cool is set to be equal to the nonideal (i.e., diffusive and dynamo) contribution of the electric field in the energy equation, as in Sheikhnezami et al. (2012), Stepanovs & Fendt (2014):

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{c}={\rm{\nabla }}\cdot \{[\eta \cdot ({\rm{\nabla }}\times {\boldsymbol{B}})-\alpha \cdot {\boldsymbol{B}}]\times {\boldsymbol{B}}\}.\end{eqnarray} \tag{ 7 }$$

We choose to neglect the nonideal heating process in accordance with previous studies of jet launching from resistive disks (Casse & Keppens 2002).

The tensors α and η describe the effects of the mean-field dynamo and the magnetic diffusivity on the magnetic field evolution (see Sections 2.5 and 2.6).

Since the PLUTO code solves the MHD equation in their adimensional form, intrinsic physical scales are not involved. We apply code units, for which every scale (time, length, energy, etc.) is normalized to its initial value at the innermost disk radius R_in at the midplane. Therefore, velocities are normalized to v_k,in, the Keplerian speed at R_in. As a consequence, the time is normalized to t_in = R_in/v_k,in. Thus, 2π t_in corresponds to one disk revolution at R_in. For the sake of simplicity, all times that are shown without a normalization constant are measured in units of t_in.

The computational domain ranges in the radial direction between R ∈ [1, 100]R_in, and in the angular direction, between θ ∈ [10⁻⁸, π/2 − 10⁻⁸] ≃ [0, π/2] (where θ = 0 corresponds to the jet axis and θ = π/2 corresponds to the disk midplane). In order to be able to cover a large extension of the numerical grid, we apply stretched cells in the radial direction (ΔR = RΔθ), while the spacing in the angular direction is equidistant. We adopt a resolution of [N_R × N_θ ] = [512 × 256] through the whole paper. This gives a resolution throughout the paper of 32 cells per geometrical disk height H. We found that a lower resolution may lead to numerical issues, especially in the case of strong interplay between the magnetic field and the dynamo in the innermost accretion disk regions.

The spatial reconstruction is performed employing the piecewise parabolic method for spherical coordinates (Mignone 2014), while the time integration is achieved through a third-order Runge–Kutta scheme. The flux is computed through a Harten-Lax-van Leer-contact (HLLC) Riemann solver (Toro et al. 1994; Li 2005). ⁴

The solenoidality of the magnetic field is preserved through an upwind constrained transport (UCT) method (Londrillo & del Zanna 2004). In particular, as in Mattia & Fendt (2020a, 2020b), we employ a combination of the UCT-HLL for the ideal and resistive components, while the dynamo is added through an arithmetic average (see Mignone & Del Zanna 2021 for a more detailed study on the upwind constrained transport schemes). In order to preserve the stability, we adopt a Courant–Friedrichs–Levy time stepping with ${CFL}=0.4\lt 1/\sqrt{{N}_{\dim }}$.

The initial conditions are identical to those applied in Mattia & Fendt (2020a, 2020b). For the sake of clarity, we summarize the fundamental steps. The simulation starts with a very weakly magnetized disk. The initial magnetization, defined as the ratio between the thermal and the magnetic pressure both measured at the disk midplane, is ${\mu }_{\mathrm{in}}={B}_{\mathrm{in}}^{2}/2{p}_{\mathrm{in}}={10}^{-5}$.

Therefore, we recover the initial disk structure by assuming an equilibrium between the thermal pressure gradients, gravity, and centrifugal forces (Stepanovs et al. 2014; Fendt & Gaßmann 2018),

$$\begin{eqnarray}&&{\rm{\nabla }}p+\rho {\rm{\nabla }}{{\rm{\Phi }}}_{{\rm{g}}}-\displaystyle \frac{\rho {v}_{\phi }^{2}}{R}({{\boldsymbol{e}}}_{R}\sin \theta +{{\boldsymbol{e}}}_{\theta }\cos \theta )=0.\end{eqnarray} \tag{ 8 }$$

This equation can be solved by assuming that all the hydrodynamical variables X scale as power laws of the radius R, i.e., $X={X}_{0}{R}^{{\beta }_{X}}{F}_{X}(\theta )$, where X₀ is the corresponding variable evaluated at the inner disk radius, while F_X is the angular dependence. We assume a polytropic gas, p ∝ ρ^Γ, and that all characteristic velocities scale as the Keplerian velocity, ${\beta }_{{v}_{\phi }}=-1/2$. Under these assumptions, the radial dependence of the hydrodynamical variables is determined by β_ρ = − 3/2 and β_p = − 5/2 (Blandford & Payne 1982).

In order to recover the explicit value of F_X (θ), we need to set two parameters, both evaluated at the inner disk radius: the disk density ρ₀ and the thermal disk height, defined as the ratio between the isothermal sound speed and the Keplerian speed = c_s /v_ϕ ∣_θ=π/2. We set ρ₀ = 1 and = 0.1.

Once the density and pressure distributions are recovered, we use the combination of the radial and angular component of Equation (8) in order to compute the angular dependence of the toroidal velocity (as in Zanni & Ferreira 2009, 2013 but neglecting the viscous terms).

Above the disk, we define a hydrostatic corona,

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{\rm{c}}} & = & {\rho }_{{\rm{c}},\mathrm{in}}{R}^{1/(1-{\rm{\Gamma }})},\\ {p}_{{\rm{c}}} & = & {p}_{{\rm{c}},\mathrm{in}}{R}^{{\rm{\Gamma }}/(1-{\rm{\Gamma }})}=\displaystyle \frac{{\rm{\Gamma }}-1}{{\rm{\Gamma }}}{\rho }_{{\rm{c}},\mathrm{in}}{R}^{{\rm{\Gamma }}/(1-{\rm{\Gamma }})},\end{array}\end{eqnarray} \tag{ 9 }$$

with ρ_c,in = 10⁻³ ρ₀. At the transition between the accretion disk and corona, the disk pressure equals the coronal pressure, involving a jump in density.

As pointed out by Stepanovs & Fendt (2014), we define as the geometrical disk height the region where density and toroidal velocity decrease significantly. Since through all the paper we assume = 0.1, we adopt a linear approximation H = 2r , which is able to reproduce with great accuracy the relation between the thermal and the geometrical disk height. The study of the influence that the disk height has on the dynamo-generated magnetic field will be the subject of our future works.

All the simulations are initialized with a weak purely radial magnetic field defined by the vector potential (Stepanovs et al. 2014)

$$\begin{eqnarray}&&{\boldsymbol{B}}={\rm{\nabla }}\times A{{\boldsymbol{e}}}_{\phi }={\rm{\nabla }}\times \left[\displaystyle \frac{{B}_{{\rm{p}},\mathrm{in}}}{r}\exp \left(-8{\left(z/H\right)}^{2}\right)\right]{{\boldsymbol{e}}}_{\phi }.\end{eqnarray} \tag{ 10 }$$

The strength of the initial poloidal magnetic field is determined by ${B}_{{\rm{p}},\mathrm{in}}=\sqrt{2{p}_{0}{\mu }_{\mathrm{in}}}$, where μ_in = 10⁻⁵ is the initial magnetization along the disk midplane. By construction, the initial toroidal magnetic field B_ϕ vanishes.

The physical evolution is heavily determined by the choice of boundary conditions. A "wrong" choice of boundary conditions may easily lead to nonconsistent or unphysical scenarios (see, e.g., a recent review on these issues by Boneva et al. 2021).

The boundary conditions we adopt here are a slightly revised version of the approach in Stepanovs et al. (2014), Fendt & Gaßmann (2018), and are reported in Table 1 for convenience. Standard symmetry conditions are applied along the disk midplane and along the rotational axis.

Equatorial symmetry (i.e., in this case, symmetry for the component B_θ and antisymmetry for the components B_R and B_ϕ ) conditions do have certain consequences. Concerning the magnetic field, it imposes that the toroidal magnetic field does vanish along the equatorial plane. Thus the poloidal field intersects the midplane perpendicularly. With our approach, we constrain the disk midplane to the equatorial plane; we expect similar effects also from a bipolar approach, unless a large-scale bipolar asymmetry will build up. In Fendt & Gaßmann (2018), the bipolar setup lead to some disk warping; however, on the large scales, the field remained symmetric. This implies that across the disk midplane (in that case, different from the equatorial plane of the grid) the toroidal field changes its sign, leading to a vanishing toroidal field over there—similar to our present setup. For the choice of the sign of α (in both papers), a large-scale flux emerged.

The situation may be different for mean-field dynamo of the opposite sign. This may be relevant in particular for bipolar simulations considering both hemispheres. Rekowski et al. (2000), Bardou et al. (2001) have shown that a change of sign for the α (applied to both hemispheres) leads to a quadrupolar large-scale magnetic field geometry. These results, obtained by a semianalytical treatment, have been confirmed by numerical simulations of a scalar mean-field dynamo, finding either dipolar (negative α, upper hemisphere) or quadrupolar fields (positive α, upper hemisphere; Fendt & Gaßmann 2018). This effect can, obviously, not be investigated if only one hemisphere is considered. However, such an investigation would be beyond the aim of the present paper, which emphasizes to apply more consistent dynamo and quenching models. A different polarity of the dynamo would also be in contrast with the analytical model of Rüdiger et al. (1995), and we thus choose to not investigate further the polarity of α.

Another reason to constrain ourselves to one hemisphere is just simplicity. We intend to investigate the pure dynamo effect that should not be affected by the dynamics of the disk midplane.

In fact, our prescription for the α and η distribution is fixed in space (and centered on the midplane). ⁵ Allowing for disk warping would require determining the physical disk midplane for each time step and updating for the α and η distribution consequently. This has not been done previously, and it is beyond the scope of the paper.

Both the inner and outer radial boundaries are separated into a disk (θ > π/2 − 3) and a coronal (θ < π/2 − 3) region. The extent of the inner disk boundary is somewhat broader than the initial disk height. The reason is that the disk, especially in case of strong accretion, may slightly inflate, but all material delivered by disk accretion must be able to be absorbed by the boundary.

Along both the disk and the coronal inner boundary, we prescribe v_θ = 0. For the disk part, the density, pressure, and toroidal velocity are extrapolated from the domain quantities by a power law. The radial velocity is also computed by extrapolating a power law; however we allow only for negative radial velocity.

The density and pressure for the inner coronal boundary condition are set to their initial value, but multiplied by a factor ρ_c,in (e.g., p_c,in for the pressure, see Equation (9)), and describing a fixed radial inflow of v_R = 0.1. Together, this allows for a more stable evolution between the disk and coronal boundary (compared to Stepanovs et al. 2014; Mattia & Fendt 2020a, 2020b) as strong density gradients may appear at the interface.

We prescribe a poloidal magnetic field boundary condition along the boundary, which is defined by satisfying the divergence-free condition. For that, we need to prescribe the θ-component (only). At the inner disk boundary, we prescribe a fixed inclination of the poloidal magnetic field, with an angle

$$\begin{eqnarray}&&\varphi =70^\circ {\left[1+\exp \left(-\displaystyle \frac{\theta -45^\circ }{15^\circ }\right)\right]}^{-1},\end{eqnarray} \tag{ 11 }$$

where φ is the angle between the magnetic field and the initial disk surface.

At the inner coronal boundary, we require the conservation of the magnetic flux. The B_ϕ component is recovered by extrapolating a power law in the radial direction ∝ R⁻¹.

For the outer boundary conditions, the density, pressure and toroidal magnetic field are computed by imposing a power-law extrapolation, while the three velocity components follow a zero-gradient prescription. However, in addition we require that there is no radial inflow into the coronal region and no outflow from the disk region. Finally, the component B_θ is computed assuming a power-law ∝ R⁻¹, while the radial magnetic field is computed satisfying the solenoidality condition.

As we have shown recently (Mattia & Fendt 2020b), the nonisotropic disk dynamo model of Rüdiger et al. (1995) proved to have some considerable advantages, e.g., the reduced number of parameters required to describe the dynamo tensor and the greater stability (compared to a scalar dynamo model) when the initial magnetic field has a nonzero vertical component. Therefore, in this paper, we apply the dynamo tensor derived by Rüdiger & Kichatinov (1993), Rüdiger et al. (1995) within the thin-disk approximation, thus with negligible nondiagonal components,

$$\begin{eqnarray}&&\alpha =({\alpha }_{R},{\alpha }_{\theta },{\alpha }_{\phi })=-[{\bar{\alpha }}_{0}\circ {\bar{q}}_{\alpha }]{c}_{s}{F}_{\alpha }(z)\end{eqnarray} \tag{ 12 }$$

where the symbol ◦corresponds to the element-wise product of two vectors, and with the adiabatic sound speed c_s at the disk midplane. The vector ${\bar{\alpha }}_{0}$ determines the strength of the dynamo tensor, and F_α (z) describes the vertical profile of the alpha-effect. Naturally, we also need to assume a sufficiently high disk ionization. The vector ${\bar{q}}_{\alpha }$ is a generic dynamo-quenching function. The specific form of the dynamo quenching will be described in Section 2.7.

As pointed out by Rüdiger et al. (1995), the effects of the rotation on the turbulence can be described by the Coriolis number,

$$\begin{eqnarray}&&{\rm{\Omega }}* \,=\,2{\rm{\Omega }}{\tau }_{c}\end{eqnarray} \tag{ 13 }$$

where Ω is the frequency of revolution, and τ_c is the turbulence correlation time, which can be recovered only by direct simulations (see, e.g., Gressel 2010; Nauman & Blackman 2015; Gressel & Pessah 2015, 2022).

The exact connection between the local shearing box simulations and the large-scale mean-field dynamo is still unclear, since the values found by the local approach (Ω* ≲ 1) are ≃10 smaller than the ones required in order to recover the amplitude of the mean-field dynamo. Future multiscale dynamo simulations will hopefully solve this problem.

In order to investigate the effect of strong and weak dynamos, we select three values of the Coriolis number. With Ω* = 10, we investigate the strong dynamo regime, while Ω* = 5 refers to the moderate dynamo regime, and Ω* = 1 is the weak dynamo regime. For comparison, all of our simulations are performed in all three regimes.

The explicit form of ${\bar{\alpha }}_{0}$ is in cylindrical coordinates (Rüdiger et al. 1995; Mattia & Fendt 2020b):

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{0,r} & = & \displaystyle \frac{1}{2{{\rm{\Omega }}}^{* 3}}\left({{\rm{\Omega }}}^{* 2}+6-\displaystyle \frac{6+3{{\rm{\Omega }}}^{* 2}-{{\rm{\Omega }}}^{* 4}}{{\rm{\Omega }}* }\arctan {\rm{\Omega }}* \right)\\ {\alpha }_{0,z} & = & \displaystyle \frac{1}{2{{\rm{\Omega }}}^{* 3}}\left(-\displaystyle \frac{10{{\rm{\Omega }}}^{* 2}+12}{1+{{\rm{\Omega }}}^{* 2}}+\displaystyle \frac{2{{\rm{\Omega }}}^{* 2}+12}{{\rm{\Omega }}* }\arctan {\rm{\Omega }}* \right)\\ {\alpha }_{0,\phi } & = & {\alpha }_{0,r}.\end{array}\end{eqnarray} \tag{ 14 }$$

In the thin-disk approximation, we can safely assume α_0,R ≃ α_0,r ; however, for the sake of consistency, all the dynamo vector components are transformed into spherical coordinates.

The profile function F_α (z) confines the dynamo within the accretion disk

$$\begin{eqnarray}{F}_{\alpha }(z)=\left\{\begin{array}{lc}\sin \left(\pi \displaystyle \frac{z}{H}\right) & z\leqslant H\\ 0 & z\gt H\end{array}\right..\end{eqnarray} \tag{ 15 }$$

A sinusoidal function (as in Bardou et al. 2001) is adopted instead of a linear function (as in Rüdiger et al. 1995; Rekowski et al. 2000) in order to avoid sharp discontinuities between the disk and the coronal area.

The magnetic diffusivity tensor is assumed to have a diagonal structure. As in Zanni et al. (2007), Sheikhnezami et al. (2012), we adopt an α-prescription

$$\begin{eqnarray}&&\eta =({\eta }_{R},{\eta }_{\theta },{\eta }_{\phi })={\bar{\eta }}_{0}{v}_{A}{{HF}}_{\eta }(z)\end{eqnarray} \tag{ 16 }$$

where v_A is the Alfvén speed at the disk midplane, H denotes the geometrical disk height, and F_η (z) describes the vertical profile of the magnetic diffusivity. As pointed out by Stepanovs & Fendt (2014), all the diffusivity models that we investigate can be represented as

$$\begin{eqnarray}&&{\eta }_{\phi }={\alpha }_{\mathrm{ss}}{c}_{s}{{HF}}_{\eta }(z)\end{eqnarray} \tag{ 17 }$$

where c_s is the adiabatic sound speed at the disk midplane. The diffusivity profile is anisotropic, following

$$\begin{eqnarray}&&{\eta }_{R}={\eta }_{\theta }={\eta }_{\phi }\displaystyle \frac{{\eta }_{0,R}}{{\eta }_{0,\phi }}\end{eqnarray} \tag{ 18 }$$

(Ferreira & Pelletier 1995). The quantity α_ss represents the dimensionless parameter measuring the strength of turbulence (Shakura & Sunyaev 1973). Implicitly, the magnetic diffusivity is assumed to have a turbulent nature, caused by the MRI (Balbus & Hawley 1991). We apply

$$\begin{eqnarray}&&{\alpha }_{\mathrm{ss}}={\eta }_{0,\phi }\sqrt{\displaystyle \frac{2\mu {| }_{\pi /2}}{{\rm{\Gamma }}{\mu }_{0}}}\end{eqnarray} \tag{ 19 }$$

where μ∣_π/2 is the magnetization derived along the disk midplane. However, the magneto-rotational instability can be triggered by both the poloidal and the toroidal field (Fromang 2013). As the latter vanishes along the midplane by construction, we adopt the following prescription, if not specified otherwise,

$$\begin{eqnarray}&&{\alpha }_{\mathrm{ss}}={\eta }_{0,\phi }\sqrt{\displaystyle \frac{2{\mu }_{D}}{{\rm{\Gamma }}{\mu }_{0}}}\end{eqnarray} \tag{ 20 }$$

where the magnetization μ_D is defined by the average total magnetic pressure over the geometrical disk height and by the midplane gas pressure. This approach has also been adopted by Stepanovs et al. (2014), although a different diffusivity model was applied.

For the strength and the anisotropy of the diffusivity tensor, we follow Kitchatinov et al. (1994), Rüdiger et al. (1995), Rekowski et al. (2000), Mattia & Fendt (2020b),

$$\begin{eqnarray}\begin{array}{rcl}{\eta }_{0,R} & = & \displaystyle \frac{3}{4{{\rm{\Omega }}}^{* 2}}\left[1+\left(\displaystyle \frac{{{\rm{\Omega }}}^{* 2}-1}{{\rm{\Omega }}* }\right)\arctan {\rm{\Omega }}* \right]\\ {\eta }_{0,\theta } & = & {\eta }_{0,R}\\ {\eta }_{0,\phi } & = & \displaystyle \frac{3}{2{{\rm{\Omega }}}^{* 2}}\left[-1+\left(\displaystyle \frac{{{\rm{\Omega }}}^{* 2}+1}{{\rm{\Omega }}* }\right)\arctan {\rm{\Omega }}* \right].\end{array}\end{eqnarray} \tag{ 21 }$$

This is the same model as in Kitchatinov et al. (1994), Mattia & Fendt (2020b), although the terms are arranged differently ⁶ for the sake of clarity.

Quite different models for the anisotropic diffusivity have been employed in the last decades (see, e.g., Casse & Ferreira 2000a; Ferreira & Casse 2013). In our approach, the strength of anisotropy is not an independent quantity, but depends directly on the Coriolis number. In the low dynamo regime, Ω* → 0, the diffusivity is quasi-isotropic. The anisotropy becomes larger when the Coriolis number reaches higher values.

We apply a profile function

$$\begin{eqnarray}{F}_{\eta }(z)=\left\{\begin{array}{ll}1 & z\leqslant H\\ \exp \left[-2{\left(\displaystyle \frac{z-H}{H}\right)}^{2}\right] & z\gt H\end{array}\right.\end{eqnarray} \tag{ 22 }$$

that allows a smooth decrease of the diffusivity outside the disk region.

The dynamo action can be understood as a macroscopic effect of the local magneto-rotational instability, which results in an additional hyperbolic term in the induction equation. As the presence of a strong large-scale magnetic field is able to suppress the MRI, the same will happen to the dynamo process in the accretion disk as soon as the dynamo-amplified magnetic field becomes strong enough. Dynamo action will then be quenched.

2.7.1. Diffusive Dynamo Quenching

The diffusive dynamo quenching (DDQ) model has been proposed by Stepanovs & Fendt (2014) in the context of jet-launching simulations from resistive accretion disk and by Stepanovs et al. (2014) in order to saturate the dynamo amplification of the magnetic field. With this approach, no direct quenching on the dynamo or the magnetic diffusivity is applied.

Instead, the infinite exponential increase of the magnetic field is prevented by a strong increase in the magnetic diffusivity. In contrast to the "standard" diffusivity models (e.g., Casse & Keppens 2002; Zanni et al. 2007) applied in nonideal simulations of jet-launching regions, the quantity α_ss is defined as follows,

$$\begin{eqnarray}&&{\alpha }_{\mathrm{ss}}={\eta }_{{0}_{\phi }}\sqrt{\tfrac{2}{{\rm{\Gamma }}}}{\left(\displaystyle \frac{{\mu }_{D}}{{\mu }_{0}}\right)}^{2}.\end{eqnarray} \tag{ 23 }$$

The main advantage with this choice of quenching mode is that it avoids the accretion instability (Campbell 2009), which may suppress the jet-launching process and, in addition, is prone to numerical issues. On the other hand, the strong dependence of α_ss on the magnetization may lead to unphysically high values of the magnetic diffusivity.

2.7.2. Standard Dynamo Quenching

So far there is no general consensus about how to calculate the critical magnetization value for the quenching. Since the turbulence that is causing the turbulent dynamo effect is supposed to be a consequence of the MRI, the saturation of the dynamo action should most probably depend on the relative magnetic field strength at the disk midplane.

Moreover, a quenching based on the disk magnetization (see, e.g., Vourellis & Fendt 2021) is in agreement with the fact that the MRI is excited by both the poloidal and the toroidal magnetic field. ⁷ Thus, we start with the most simple approach for an isotropic quenching model (henceforth standard dynamo quenching, hereafter SDQ; Ivanova & Ruzmaikin 1977; Brandenburg & Subramanian 2005; Moss et al. 2016) that is basically depending on the disk magnetization,

$$\begin{eqnarray}&&{q}_{\alpha }=\displaystyle \frac{1}{1+{\mu }_{D}/{\mu }_{0}}.\end{eqnarray} \tag{ 24 }$$

We point out that such a global—thus nonlocal quenching prescription is not easy to fully parallelize in the code. As investigated by Vourellis & Fendt (2021), a weak parallelization in the θ-direction (i.e., a parallelization with only a few number of cores in the θ-direction) overcomes this problem with only a little additional computational costs. For a local quenching (using the local magnetization value on the grid cell, see Stepanovs et al. 2014; Tomei et al. 2020), which is straightforward to parallelize, the main idea of turbulence generation in the disk midplane in combination with the generation of a large-scale magnetic flux gets somehow lost. Also, the dynamo process itself becomes unstable when every grid cell applies a different strength of the dynamo—again a conflict with the aim of generating a large-scale magnetic flux.

2.7.3. Nonisotropic Dynamo Quenching

As for the strength of the mean-field dynamo, also the feedback of the magnetic field on the dynamo cannot always be approximated with a single scalar function. Thus, similar to the definition of an anisotropic dynamo tensor, the quenching of the dynamo effect is also tensorial, thus acting in a different strength on the dynamo tensorial components.

Here, we consider a feedback model (nonisotropic dynamo quenching, hereafter NDQ) that follows from an analytical study of turbulence (Rüdiger & Kichatinov 1993), which has elaborated different quenching functions for different components of the α-tensor. It does not only depend on the strength but also on the orientation of the dynamo-amplified magnetic field.

For our purpose, the suppression of the mean-field dynamo is first computed in cylindrical coordinates (Rüdiger & Kichatinov 1993) and then converted to spherical coordinates,

$$\begin{eqnarray}\begin{array}{rcl}{q}_{{\alpha }_{r}} & = & {\psi }_{\phi }+\displaystyle \frac{15}{8}\displaystyle \frac{{B}_{z,{\rm{D}}}^{2}}{{B}_{{\rm{D}}}^{2}}{\psi }_{p}\\ {q}_{{\alpha }_{z}} & = & {\psi }_{z}-\displaystyle \frac{15}{16}\displaystyle \frac{{B}_{z,{\rm{D}}}^{2}}{{B}_{{\rm{D}}}^{2}}{\psi }_{p}\\ {q}_{{\alpha }_{\phi }} & = & {\psi }_{\phi }\end{array}\end{eqnarray} \tag{ 25 }$$

where we have defined

$$\begin{eqnarray}\begin{array}{rcl}{\psi }_{p} & = & \displaystyle \frac{1}{4{\beta }^{4}}\left[{\beta }^{2}-5+\displaystyle \frac{2{\beta }^{2}}{3\left(1+{\beta }^{2}\right)}\right.\\ & & +\left.\displaystyle \frac{4{\beta }^{4}\left(3{\beta }^{2}-1\right)}{3{\left(1+{\beta }^{2}\right)}^{3}}+\displaystyle \frac{5+{\beta }^{4}}{\beta }\arctan \beta \right],\\ {\psi }_{z} & = & \displaystyle \frac{15}{64{\beta }^{4}}\left[{\beta }^{2}-3+\displaystyle \frac{8{\beta }^{4}}{3{\left(1+{\beta }^{2}\right)}^{2}}\right.\\ & & +\left.\displaystyle \frac{3+{\beta }^{4}}{\beta }\arctan \beta \right],\\ {\psi }_{\phi } & = & \displaystyle \frac{15}{32{\beta }^{4}}\left[1-\displaystyle \frac{4{\beta }^{2}}{3{\left(1+{\beta }^{2}\right)}^{2}}-\displaystyle \frac{1-{\beta }^{2}}{\beta }\arctan \beta \right].\end{array}\end{eqnarray} \tag{ 26 }$$

The quenching parameter β is defined as $\beta =\sqrt{{\mu }_{D}/{\mu }_{0}}$. The dependence of the quenching components on the magnetization is shown in Figure 1.

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Essentially, the quenching depends on the disk magnetization as ∝ μ^−3/2 for the radial and toroidal component, while follows ∝ μ^−1/2 for the θ-component. Therefore, we can expect a more sudden and rapid saturation of the magnetic field at lower magnetization. We point out that $0\leqslant {q}_{{\alpha }_{z}}\leqslant 1$, regardless of the values of μ_D or B_z,D .

In order to investigate a set of different simulations, the comparison of dimensionless parameters, which do not have a strict dependence on the initial parameter space, plays a key role for understanding the physical evolution. In the context of the mean-field dynamo simulations, the dynamo number ${ \mathcal D }$ plays an essential role when it comes to understanding the efficiency of the dynamo process,

$$\begin{eqnarray}&&{ \mathcal D }=\displaystyle \frac{{\alpha }_{\phi }{\rm{\Omega }}{H}^{3}}{{\eta }_{\phi ,{\rm{D}}}^{2}}.\end{eqnarray} \tag{ 27 }$$

The dynamo number is the product of the azimuthal Reynolds number

$$\begin{eqnarray}&&{{ \mathcal R }}_{{\rm{\Omega }}}=R\left|\displaystyle \frac{d{\rm{\Omega }}}{{dR}}\right|\displaystyle \frac{{H}^{2}}{{\eta }_{\phi ,{\rm{D}}}}\approx | {\rm{\Omega }}| \displaystyle \frac{{H}^{2}}{{\eta }_{\phi ,{\rm{D}}}},\end{eqnarray} \tag{ 28 }$$

which defines the balance between magnetic diffusion and Ω-effect, and the magnetic Reynolds number

$$\begin{eqnarray}&&{{ \mathcal R }}_{\alpha }={\alpha }_{\phi }H/{\eta }_{\phi ,{\rm{D}}},\end{eqnarray} \tag{ 29 }$$

which defines the balance between magnetic diffusion and α-effect. Both processes contribute to the amplification of the magnetic field. While the component α_ϕ is related to the amplification of the poloidal components, the Ω-effect amplifies the toroidal magnetic field component.

Note, however, that our approach considers as well an α-effect for the toroidal field component, thus considering formally an α²Ω dynamo. Through the different tensorial components, both the toroidal and poloidal components of the magnetic field are amplified. However, in our case, the α-effect for the toroidal field is always less efficient than the Ω-effect, due to the strong shear that is present in a Keplerian accretion disk (in comparison to, e.g., a star). We therefore can neglect this effect for our further discussion, and consider only the α_ϕ for the magnetic Reynolds number, and, thus, also for the Dynamo number.

High dynamo numbers imply the possibility of an efficient dynamo process, low numbers, and vice versa. The critical dynamo number, separating the two regimes, depends on the details of the model setup (see, e.g., Stepinski & Levy 1988, 1990; Torkelsson & Brandenburg 1994) and has to be found by applying a series of parameter runs. For example, Brandenburg & Subramanian (2005) find a critical dynamo number ${ \mathcal D }\lesssim 10$, below which the magnetic field cannot be amplified. By connecting galactic dynamo simulations with accretion disk simulations, Boneva et al. (2021) find ${ \mathcal D }\lesssim 7$.

The size of the system is denoted by H, here represented by the disk height. By construction, the dynamo components vanish at z = H. We therefore compute the quantity α_ϕ at z = H/2.

Another key parameter in disk simulations as well as in jet-launching simulations is the turbulence parameter α_ss, which parameterizes the strength of disk turbulence, i.e., the disk turbulent viscosity (Shakura & Sunyaev 1973). On one hand, it represents a direct link between the disk magnetization and the disk diffusivity (see Equation (20)), and on the other hand, it can be recovered both from observations and direct simulations. As pointed out by King et al. (2007), observational evidences show that a range 0.1 < α_ss < 0.4 is required to provide a good description of the behavior of fully ionized, thin accretion disks. Nevertheless, the numerical simulations of direct turbulence recover values that are an order of magnitude below the observational value.

The primary effect of the dynamo tensor is the amplification of the magnetic field. However, below a critical value of the Coriolis number, even in the presence of a dynamo effect, the magnetic field is not amplified or is only weakly amplified. As a direct consequence, for example a fast and collimated outflow cannot be launched.

We identify a critical value for the Coriolis number as the one where the dynamo timescale is longer than the diffusion timescale (Dyda et al. 2018; Fendt & Gaßmann 2018),

$$\begin{eqnarray}&&{\tau }_{\alpha }=\displaystyle \frac{H}{\alpha }\gt \displaystyle \frac{{H}^{2}}{\eta }={\tau }_{\eta }.\end{eqnarray} \tag{ 30 }$$

Note that the quenching methods may act in quite a different way, as the initial strength of diffusivity for the diffusive quenching is ∼2–3 orders of magnitudes weaker than the one from the standard quenching. This difference strongly reflects on the existence of a critical Coriolis number.

When applying the diffusive quenching method, we have recovered a critical value (i.e., the value below which the magnetic field is not immediately amplified) of the initial Coriolis number ${{\rm{\Omega }}}_{C}^{* }\simeq 0.15$ (Mattia & Fendt 2020b). On the other hand, when applying the standard quenching method, or the nonisotropic quenching methods that we have developed, a critical value of the Coriolis number (in order to determine whether the initial dynamo can amplify the magnetic field) ${{\rm{\Omega }}}_{C}^{* }\simeq 2$ is found.

This difference can be seen in Figure 3, where we show the time evolution of the poloidal magnetic energy from radius R = 10 to the end of the domain (R_out = 100), while applying a Coriolis number of Ω* = 1 (dotted lines). As expected, the poloidal magnetic energy of the diffusive quenching method, which is shown in Figure 3, is amplified stronger and more rapidly than for the cases of the other quenching models.

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Moreover, the field amplification is preceded by a short decrease. The reason behind this is that, at t = 0, the diffusive timescale is shorter than the dynamo timescale. Thus, the magnetic field is diffused away, leading to a decrease in the magnetization and, therefore, of the magnetic diffusivity. Once the magnetic diffusivity has decreased, the dynamo timescale becomes again shorter than the diffusive timescale.

When the Coriolis number is higher than its critical value (i.e., in all the cases considered but the ones defined by the dotted red and green lines in Figure 3), the amplification of the magnetic field occurs almost instantly. Thus, when the magnetic field increases, also the dynamo quenching increases, suppressing the dynamo action and lowering the amplification of the magnetic field. In order to disentangle the impact of the quenching methods, we applied Coriolis numbers of Ω* = 5 and Ω* = 10. We find that the field amplification is faster and also stronger when the diffusive quenching model is applied (see Figure 3).

We notice that the magnetic energy that originates from dynamo action by employing the diffusive quenching method and a Coriolis number Ω* = 1 is approximately the same of the one obtained by the standard quenching method and Ω* = 10 until t = 3000. Since the DDQ model does not involve a suppression of the dynamo tensor, the magnetic field saturates as soon as the diffusivity is strong enough to counterbalance the dynamo effect. On the other hand, the standard quenching method features both an increase of the magnetic diffusivity and a decrease of the dynamo. Thus, the same Coriolis number will lead to a different strength of the disk poloidal field depending on the feedback model.

However, as shown in Figure 4, the presence or the absence of the dynamo-inefficient zones from the early stages (Mattia & Fendt 2020a) plays a key role in the jet structure and evolution. We find that a low Coriolis number leads to the formation of dynamo-inefficient zones regardless of the quenching model, in agreement with Mattia & Fendt (2020b). On the other hand, we also find that the dynamo-inefficient zones are present for large Ω*, which we did not find in our previous works. The formation of these zones may be connected with the increase of grid resolution that is coupled with the HLLC Riemann solver that we now apply, which together provides a better resolution of the disk substructures (probably smeared out by the more diffusive HLL solver).

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Finally, the results obtained by applying the NDQ model show no difference from the SDQ model for the low Coriolis number. However, for higher Coriolis numbers, the different suppression of the dynamo in the NDQ model (${\alpha }_{\phi }\propto \,{\mu }_{D}^{-3/2}$, while ${\alpha }_{\phi }\propto \,{\mu }_{D}^{-1}$ in the SDQ model) leads to a different saturation of the magnetic field. More specifically, we find that the magnetic field, in the NDQ model, is saturated toward a lower disk magnetization than the one obtained by the SDQ model.

The dynamo number is traditionally used to indicate the strength and efficiency of dynamo activity. A high dynamo number indicates an efficient dynamo, thus leading to strong field amplification. Vice versa, a low dynamo number indicates that a dynamo cannot act efficiently anymore, and the magnetic field generated has reached its saturation value—either established by a strong magnetic diffusivity, thus by diffusive quenching (DDQ), or by suppressing the dynamo activity itself, thus by direct quenching of the dynamo-alpha (SDQ, NDQ). The critical dynamo number (see Section 2.8) differentiates the two regimes.

Our simulations (see Figure 5) confirm earlier predictions, of Brandenburg & Subramanian (2005), Boneva et al. (2021), that (i) the field amplification does not occur for ${ \mathcal D }\lesssim 10$ (red in Figure 5), while (ii) the amplification occurs when the dynamo number supersedes its critical value because the magnetic diffusivity decreases (blue in Figure 5).

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Moreover, we find that the critical value of the dynamo number does not depend on the feedback model applied or the Coriolis number that is given. This suggests, essentially, that the amplification and the saturation of the magnetic field are very general properties of the mean-field dynamo approach, and do not depend on certain modeling details. Note that the exact value of the critical dynamo number can be influenced by the numerical resolution applied and the numerical algorithms adopted (see, e.g., Stepinski & Levy 1988, 1990; Torkelsson & Brandenburg 1994).

The main differences between the quenching models we apply are the absence of dynamo-inefficient zones for low values of the Coriolis number and the feedback models that imply a suppression of the dynamo. A possible explanation is that a magnetic field reversal can be maintained only if the dynamo does not vanish. If the magnetic field is not constantly amplified by the mean-field dynamo (because of the quenching), the field reversal zones are able to reconnect and be diffused away. This is not possible if the dynamo is not suppressed. However, since the SDQ model and the NDQ model lead to a suppression of the dynamo (and not in the diffusivity), the dynamo-inefficient zones are more likely to be suppressed. We also point out that the presence of dynamo-inefficient zones, which are not strictly related to the component α_ϕ , is still possible. In this regard, the dynamo number may require a different definition considering all the tensorial components of the dynamo.

On the other hand, the turbulence parameter α_ss (Shakura & Sunyaev 1973) does not explicitly depend on the dynamo tensor, but only on the magnetic diffusivity. Therefore it can be applied as a useful tool in order to understand the evolution of the magnetic diffusivity once the magnetic field amplification takes place. As we can see from Figure 6, the feedback models that include the suppression of the dynamo and the DDQ model show several differences with regard for how the turbulence parameter depends on the Coriolis number once the magnetic field is saturated. In particular, we find that the results applying the DDQ model show a unique dependence on the Coriolis number.

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As pointed out in the previous section, for the DDQ model, the magnetic diffusion is the only process that is able to saturate the mean-field dynamo. Because of the different amplification of the magnetic field for different Coriolis numbers, different in both the strength of the magnetic field and the timescale of amplification, the disk magnetization saturates to different levels (see Figure 3). Because of the strong, i.e., quadratic, dependence of the diffusivity η on the disk magnetization μ, in the DDQ model, this dependence is reflected in the fact that also the α_ss is found to depend on μ.

The different saturation levels for the magnetic field strength play a key role in the dynamical evolution of the accretion disk and the subsequent jet-launching process. This is demonstrated in Figure 7 showing the accretion and the ejection rate of the launching area. The accretion rate is computed by integrating the net radial mass flux through the disk at R = 7,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{acc}}(R)=2\pi R{\int }_{\pi /2}^{\pi /2-{\theta }_{S}}\rho {v}_{R}{Rd}\theta ,\end{eqnarray} \tag{ 31 }$$

with the disk opening angle defined as ${\theta }_{S}\equiv \arctan (2H/r)$. Similarly, the ejection rate is computed by integrating between R_in and R = 7 along the disk surface,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{eje}}(R;{\theta }_{S})={\int }_{{R}_{\mathrm{in}}}^{R}\rho {v}_{\theta }(\tilde{R})2\pi \tilde{R}d\tilde{R}.\end{eqnarray} \tag{ 32 }$$

The interrelation between the accretion rate and the Coriolis number confirms, regardless of the feedback mode, previous results obtained by Fendt & Gaßmann (2018). In particular, we find that a stronger dynamo leads to a stronger accretion rate. We investigated the impact of the magnetic field topology on the accretion process (i.e., the presence or the absence of dynamo-inefficient zones) in detail previously (Mattia & Fendt 2020a).

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This influence is also confirmed by comparing the SDQ model for Ω* = 10 with the DDQ model for Ω* = 1. Here, despite a similar amplification of the poloidal magnetic field (see Figure 3), the presence (DDQ model, Ω* = 1) or absence (SDQ model, Ω* = 10) of the dynamo-inefficient zones plays a key role in the accretion of material.

On the other hand, the mass ejection, acting on much shorter timescales, shows less-pronounced differences for the variety of quenching methods or the different values of the Coriolis number. However, we observe that lower Ω* generally leads to a higher ejection-to-accretion ratio, which is in good agreement with the findings of Mattia & Fendt (2020b). In case of a strong dynamo, the isotropic quenching models show that <50% of the accreted material is ejected. This is in good agreement with the previous resistive jet-launching simulations that do not consider dynamo action, but start from a prescribed large-scale magnetic field (see, e.g., Zanni et al. 2007; Sheikhnezami et al. 2012). For lower Coriolis numbers, accretion requires more time to be established because of the slower amplification of the magnetic field. On the other hand, the nonisotropic quenching model shows that almost all the matter accreted is ejected. This is a direct consequence of the feedback model that leads to a saturation of the magnetic field at lower magnetizations (see Figure 3). As a consequence, the magnetic diffusivity, which depends on the disk magnetization, is less amplified than in the isotropic feedback models, leading to a lower accretion. For this reason, during certain periods of time, the ejection–accretion rate (defined as ${\dot{M}}_{\mathrm{ej}}/{\dot{M}}_{\mathrm{accr}}$) may actually exceed unity. This may imply that disk volume regions of very low mass or density may be present for some time until they are replenished from the mass reservoir at larger disk radii.

We observe a particularly interesting period at time ≈3500 for the (SDQ) model with high Coriolis number Ω* = 10. A sudden drop in the accretion and, consequently, in the ejection rate appears. When looking at the strength of the mean-field dynamo ϕ-component as a function of radius, we notice that at t = 3500 it becomes significantly stronger than immediately before or after.

We find that the reason behind this sudden change is a small decrease in the toroidal magnetic field strength, which seems to amplify the dynamo because of the smaller quenching term (which is related to the magnetic field strength; see Figure 8). In particular, a weaker magnetic field (and therefore a weaker magnetization) leads to a lower quenching and therefore a stronger amplification of the magnetic field. In addition, a low magnetic field strength triggers a lower magnetic diffusivity, which, in turn, curbs the accretion process, because it implies a lower diffusivity. Once the magnetic field is amplified again by the dynamo, the system goes back to a more stable configuration.

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All our models, which consider feedback of the magnetic field on the dynamo, lead to a quasi-steady, saturated state. Therefore, our dynamo simulations should retrieve, qualitatively, the essential jet properties found by the simulations that do not invoke a dynamo process (see, e.g., Tzeferacos et al. 2009; Murphy et al. 2010; Stepanovs & Fendt 2016), regardless of the method for the dynamo feedback.

In Mattia & Fendt (2020b), we discovered a unique numerical correlation between the Coriolis number Ω* (and therefore the dynamo strength) and the asymptotic jet speed. In this paper, we now make a further step in this regard and investigate this interrelation for different feedback models. For this purpose, we select certain magnetic flux surfaces (thus contours of the vector potential, respectively poloidal magnetic field lines), and compute the local disk magnetization and the poloidal velocity of the corresponding outflow along that field line.

We do this for each of the code outputs (uniformly distributed and separated by Δt = 100), starting from t = 700, i.e., the time when a jetted outflow is already formed and has reached the outer boundary, until the final time step of each simulation. The results are shown in Figure 9, where the two panels present, respectively, the values obtained for radii 1.5 < R < 5 and 5 < R < 10.

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As we can see, the radial distance makes a difference concerning the smoothness of the interrelation. For radii 5 < R < 10, the interrelation looks very well defined, while for smaller radii this correlation is partially broken. At small radii, we notice the presence of vertical columns, i.e., zones with similar magnetization that exhibit a variety in jet speed. This is mainly due to the time evolution of these parameters: both the disk magnetization as well as the jet speed may vary in time for the same radial distance, during the same simulation. Such variations occur more frequently in the inner disk region because of different timescales involved.

Our understanding is the following. The small temporal variations of the dynamo lead to suppression and amplification of the magnetic field in the inner disk region (as seen on the left region of Figure 8). As a consequence, small differences in the magnetic field can lead to different outflow speed in the inner launching region. Such differences are leading to internal shock within the jet, affecting the outflow. The small variations in the inner disk magnetic field are also able to affect the ratio between the poloidal and the toroidal magnetic field, which can be related to the jet speed and collimation. This is a result of the magneto-centrifugal acceleration involved.

On the other hand, at larger radii, the system has reached saturation toward a steady state, leading to a more narrow interrelation. At even larger radii, R > 10, the low magnetization and the slow rotation lead to a weak disk magnetization and, therefore, to a slower outflow speed. We point out that, at these large radii, the disk has accomplished only a few revolutions, and the whole inflow–outflow structure has not yet settled into a quasi-steady state.

However, as a key result, despite showing differences in both the disk magnetization and the jet speed, the different feedback models we have examined show in general a unique trend. They all follow a very similar relation between the two quantities, suggesting that this relation of the jet speed versus disk magnetization does not depend on the dynamo process, the diffusivity model, or the quenching method. It simply confirms the general relation between these leading inflow–outflow parameters that have been discovered previously (Stepanovs & Fendt 2016; Mattia & Fendt 2020b).

Still, the feedback of the magnetic field on the dynamo action plays a key role for the saturated disk magnetization. This holds true, in particular, because of the more efficient suppression of the dynamo, and the NDQ model reaches the saturation of the magnetic field already at lower magnetization levels, about one order of magnitude below.

For the feedback of the magnetic field on the dynamo—the dynamo quenching—we follow Rüdiger & Kichatinov (1993), as described in Section 2.7.3. As in Section 2.6, the anisotropic components of the magnetic diffusivity can be computed directly in spherical coordinates.

Here we apply the quenching model following Kitchatinov et al. (1994), Rüdiger et al. (1994),

$$\begin{eqnarray}&&\eta =({\eta }_{R}{q}_{{\eta }_{R}},{\eta }_{\theta }{q}_{{\eta }_{\theta }},{\eta }_{\phi }{q}_{{\eta }_{\phi }}),\end{eqnarray} \tag{ 33 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{q}_{{\eta }_{R}} & = & \displaystyle \frac{3}{2{\beta }^{2}}\left[-\displaystyle \frac{1}{1+{\beta }^{2}}+\displaystyle \frac{1}{\beta }\arctan \beta \right],\\ {q}_{{\eta }_{\theta }} & = & {q}_{{\eta }_{R}},\\ {q}_{{\eta }_{\phi }} & = & \displaystyle \frac{3}{8{\beta }^{2}}\left[\displaystyle \frac{{\beta }^{2}-1}{{\beta }^{2}+1}+\displaystyle \frac{{\beta }^{2}+1}{\beta }\arctan \beta \right].\end{array}\end{eqnarray} \tag{ 34 }$$

The dependence of the diffusivity quenching components on the magnetization is shown in Figure 10. We point out that the dependence of magnetic diffusivity on the disk magnetization is also determined by the disk turbulence parameter α_ss. Here, we model this applying ${\alpha }_{\mathrm{ss}}\propto \,\sqrt{{\mu }_{{\rm{D}}}}$ (as in Equation (20)).

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Mean-field dynamo models applying a diffusivity quenching (as in Rüdiger et al. 1994) have been applied in the context of galactic dynamos (Schultz et al. 1994; Elstner et al. 1996), although the quenching model was never coupled with a nonisotropic diffusivity. Here, because of the rapid disk rotation and the strong magnetization needed for jet launching, we have included both anisotropic effects. In the limits of slow rotation and weak magnetization, the diffusivity tensor becomes isotropic. When either a rapid disk rotation or a strong magnetization becomes relevant, the isotropy of the diffusivity tensor is broken. This is the first time that such modeling, with a higher degree of more self-consistency, has been applied in the context of jet-launching simulations from accretion disks.

In order to investigate the effects of the feedback of the magnetic field on the diffusivity, we have chosen to focus on the case Ω* = 1 as a reference simulation.

In Figure 11, we show the time evolution of the density and poloidal magnetic field of this simulation. We see that the saturation of the magnetic field occurs on a short timescale, i.e., already until t = 1000, corresponding to ≃50 revolutions of the inner disk. At this point in time, the magnetic energy is amplified by an order of magnitude, while the magnetic field lines are already opened up to a radius R = 70 in the outflow region.

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Since our approach considers an α²–Ω dynamo, the toroidal magnetic field is amplified by the disk rotation and by the radial dynamo component (Mattia & Fendt 2020a). Although the dynamo components are quenched by the magnetic field, the Ω-effect still acts on the toroidal magnetic field component. This, combined with the shorter timescale of the Ω-effect (compared to the amplification of the poloidal field through the dynamo action), leads to a faster and stronger amplification of the toroidal magnetic field.

Note that our initial condition is that of a purely radial field. However, as in Fendt & Gaßmann (2018), we point out that the dynamo-amplified magnetic field should not depend on the initial conditions.

The formation of a magnetic loops (rooted at foot points of different radius in the accretion disk) strongly indicates that, at least in the early evolutionary stages, the launching mechanism for this initial outflow is that of a tower-jet, thus a magnetic-pressure-driven outflow (Lynden-Bell & Boily 1994; Lynden-Bell 1996). This is further supported by the inclination of the magnetic field toward the disk surface, as being not favorable for a magneto-centrifugal driving of the outflow. As the system evolves, the magnetic loops diffuse outwards, more and more poloidal field lines break up, and the magnetic field geometry reaches the inclination required for a Blandford–Payne-like outflow. The system evolves further until a quasi-steady state is reached. At this point, the system consists of a highly magnetized accretion disk and a super-Alfvénic disk wind, which evolves into a high-speed outflow. The Alfvén surface is located at ≈5 thermal disk scale heights above the disk surface.

4.2.1. Amplification of the Magnetic Field

The amplification of the poloidal magnetic energy within the accretion disk is shown in Figure 12. The amplification of the magnetic field is stronger in the innermost accretion disk regions, because of the combination of the Ω- and α-effects, in agreement with Fendt & Gaßmann (2018).

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The poloidal magnetic energy is amplified by about 2 orders of magnitude and occurs, mostly, within t = 3000. By comparing the red line of Figure 12 with the dotted green line of Figure 3, we notice that the feedback on the diffusivity has a very minor impact on the disk poloidal magnetic field until t = 4000. However, the oscillations in the magnetic energy at t ∼ 5000, t = 6000, t = 8000, and t ∼ 12,000 are consequences of this novel diffusivity feedback model (see Section 4.2.3).

We observe a different evolution compared to our previous models without feedback on diffusivity (Mattia & Fendt 2020b). The magnetic energy does not undergo any intermittent decrease before the magnetic field reaches a quasi-steady state. We think that this difference can be explained by a combination of effects that rely on the chosen feedback model. At first, the quenching of the dynamo leads to a lower disk magnetic diffusivity, just because of the lower magnetization level at which the magnetic field saturates. Then, because of the lower magnetic diffusivity, the disk mass loss is less compared to a diffusive quenching model (where the dynamo tensor is not explicitly suppressed); therefore the sound speed (which affects both dynamo and the diffusivity) shows no decrease. Moreover, the magnetic diffusivity is suppressed to an even lower level, just because of the feedback model (quenching of turbulent magnetic diffusivity).

At t ≃ 20,000, the magnetic field has saturated out to a radius R ≲ 50. The field still continues to be amplified in the outer disk region, as this part of the disk has performed so far only few rotations and is not yet in dynamic equilibrium. Also, since the α-effect is less efficient here, it takes more time to saturate the dynamo. Nevertheless, the evolution of the outer disk can weakly affect the disk evolution, most likely by triggering episodic ejections (see Section 4.2.3).

4.2.2. Disk Structure

Because of the magnetic field amplification and the evolution of the magnetic field structure due to dynamo activity, also the disk structure evolves through time. This is partly due to the change of forces acting on the disk material, but also due to the change of angular momentum balance and the subsequent redistribution of disk material.

At time t = 50,000, the magnetic field has saturated in most parts of the disk (of the size we investigate). We may therefore investigate the disk structure by looking at the profiles of its midplane quantities. In Figure 13, we display the radial profile of some of the leading MHD quantities measured at the disk midplane and compare them with an idealized radial self-similar solution of the steady-state MHD equations (see Blandford & Payne 1982). We also compare with the power-law approximation (obtained by performing a linear fit on the logarithmic radial profiles) for each quantity.

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We see that the disk kinematics remain unaffected by the dynamo action as the disk rotation remains Keplerian, i.e., ${\beta }_{{v}_{\phi }}=\,$−1/2. The density profile power-law index changes, however, from β_ρ = −3/2 to β_ρ = −5/4, as the mass is mostly accreted from the inner disk, and the whole disk loses only very little mass.

We find that both the sound speed and the Alfvén speed show very small deviations from the self-similar solution. The power-law indexes obtained are, respectively, ${\beta }_{{c}_{s}}\approx \,$ −4/9 and ${\beta }_{{v}_{A}}\approx \,$ −5/9. The radial dependence of the magnetization that is obviously strongly affected by dynamo action can be recovered by computing the ratio of the Alfvén speed versus sound speed, which corresponds to computing the difference in the respective power-law indexes,

$$\begin{eqnarray}&&{\beta }_{\mu }={\beta }_{{v}_{A}}-{\beta }_{{c}_{S}}=-\displaystyle \frac{1}{9}.\end{eqnarray} \tag{ 35 }$$

We find a very good agreement with the results obtained by Stepanovs & Fendt (2014), Stepanovs et al. (2014), suggesting that the properties of the saturated state do not depend (or depend only weakly) on the quenching model for dynamo action and diffusivity.

4.2.3. Intermittent Ejection

In order to investigate the similarities and the differences between the CTQ and the dynamo quenching methods, we have performed simulation runs applying different Coriolis numbers Ω*.

The results are shown in Figure 15. In the left panel, we see the time evolution of the poloidal disk magnetic energy, which is essentially the dynamo-generated field amplification by the α-effect. We notice that, for a higher Coriolis number, the differences between the NDQ model and the CTQ model are only little. This finding is interesting as it may sound counterintuitive—would not one expect that a stronger magnetization is leading to a stronger quenching on the magnetic diffusivity, and therefore to more differences when compared with the model without the consistent quenching?

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However, note that a stronger dynamo implies also a stronger feedback on the dynamo (as shown in Section 3). Because of the feedback (quenching) on the diffusivity, in combination the α-effect and Ω-effect lead to a stronger amplification of the toroidal magnetic field. ⁸ Since the dominating launching mechanism at the early evolutionary stages of jet launching is the toroidal pressure-dominated launching (tower jet), the quenching on the dynamo and the diffusivity is mainly triggered by the toroidal field.

As a result, the quenching of the diffusivity plays a minor role in the amplification and saturation of the poloidal field. Therefore, despite the quenching of magnetic diffusivity, which would be thought to lead to a higher dynamo number, we find that these numbers are almost identical as shown in Figure 15 (right panel) for the case Ω* = 10 at t = 10,000. This is consistent with the usual understanding that the dynamo number is a useful tool in order to characterize the onset of field amplification for a dynamo (in the disk).

On the other hand, although working in the accretion disk, the quenching on the diffusivity plays a key role also for the jet dynamics. The interrelation between the disk magnetization and the jet speed for the new quenching setup shows substantial differences from those by applying only a quenching on the dynamo (see Figure 16). While we find a clear correlation between the disk magnetization and the jet speed in Section 3.4 above, the simulations with the consistent feedback, thus the quenching of diffusivity, show almost no correlation between these two quantities. On the other hand, the simulations with the CTQ and high Coriolis number show an outflow with an extremely high degree of collimation (see Figure 17). This lack of correlation is also present at larger radii.

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We recognize that at t = 10,000 the magnetic loops, which are the main driver for the magnetic tower at initial evolutionary stages, are diffused outwards, and the outflow that is launched from the inner disk is now driven by the magneto-centrifugal mechanism, i.e., launched sub-Alfvénic and subsequently supersedes in the Alfvén and the fast magnetosonic speed. However, because we also have strong amplification of the toroidal field, the collimation process is very rapid (in terms of spatial scales). The Alfvén surface is rather close to the disk surface where the jet is launched. As a consequence, magneto-centrifugal acceleration can happen only along a short distance, and the final speed remains rather low.

This is in particular true for the simulations with higher Coriolis number (and therefore stronger amplification of the magnetic field). The Alfvén surface moves farther down to the disk surface. This is a very interesting effect. These jets are less efficient concerning the Blandford-Payne acceleration mechanism (when compared with jet launched by dynamo disks with different feedback models, as in, e.g., Stepanovs et al. 2014; Fendt & Gaßmann 2018; or the previous section of this paper), but very efficient concerning the Blandford-Payne collimation mechanism. Note that the latter is indeed a consistent result as the material along collimated field lines cannot be accelerated magneto-centrifugally anymore.

In summary, we find that our consistent feedback model that quenches also the disk diffusivity strongly impacts the launching process. This holds true in particular for a weak dynamo (low Coriolis number Ω* = 1), because of the magnetic field reversals that are induced in the jet-launching region and also the intermittent disruption of the jet followed by the production of a flare. However, even for high Coriolis numbers, when the differences are less pronounced, the jet structure is severely affected by this new feedback model, as the acceleration decreases and collimation increases. One may hypothesize about a critical or optimal Coriolis number for the dynamo that can produce the fastest jets. However, this issue needs a further detailed analysis, which is beyond the scope of this paper.

So far we have investigate only thin Keplerian disks (i.e., neglecting the nondiagonal components of the dynamo tensor). The differences we find by applying a small Coriolis number (representing a small effect of the rotation on the turbulence) suggest that we may expect even more structured and probably unstable jets that are produced by thicker accretion disks.