Dynamical Properties of Magnetized Low-angular-momentum Accretion Flows around a Kerr Black Hole

Most astounding astrophysical events in the Universe are driven by accretion onto compact objects like black holes and neutron stars (Frank et al. 2002). Depending on the observational requirements, different kinds of astrophysical accretion flows have been proposed—e.g., spherically symmetric Bondi flows (Bondi 1952), geometrically thin accretion disks (Novikov & Thorne 1973; Shakura & Sunyaev 1973), advection-dominated accretion flows (ADAFs; Narayan & Yi 1995), multitransonic accretion flows (Fukue 1987; Chakrabarti 1989), and super-Eddington accretion flows (Abramowicz et al. 1988), etc. With the most advanced general-relativistic magnetohydrodynamic (GRMHD) simulations, large numbers of studies have been performed to understand different kinds of flows (for reviews, see Davis & Tchekhovskoy 2020; Mizuno 2022). In the last few decades, some efforts have been made to understand low-angular-momentum accretion flows in a pseudo-Newtonian, axisymmetric (2D), hydrodynamic approach targeting the formation of standing as well as oscillating shocks (e.g., Molteni et al. 1994, 1996a, 1996b; Ryu et al. 1995; Lanzafame et al. 1998; Proga & Begelman 2003; Chakrabarti et al. 2004; Giri et al. 2010; Okuda & Molteni 2012; Okuda 2014; Okuda & Das 2015; Okuda et al. 2019, 2022; Singh et al. 2021). However, so far, not much effort has been devoted to understanding low-angular-momentum flows in a GRMHD framework that is related to multitransonic accretion flows.

Initially, Hawley et al. (1984a, 1984b) developed a general-relativistic hydrodynamic (GRHD) framework to study relativistic 2D accretion flows around black holes. They showed that the pressure-supported torus does not form for flows with angular momentum lower than that of marginally stable angular momentum. Recently, several authors have shown the possibility of multitransonic flows and the presence of standing/oscillatory shocks in 2D GRHD accretion flows (Kim et al. 2017, 2019; Suková et al. 2017; Palit et al. 2019). In 3D GRHD simulations, Suková et al. (2017) show the formation of oscillatory and expanding shocks. However, recent 3D GRHD simulations do not show the signatures of such standing shocks (Olivares et al. 2023). Nonetheless, Olivares et al. (2023) presented some local density jumps related to shocks in their simulations, which do not appear in global pictures. These results suggest that there are still some unexplored regions of the parameter space, and a systematic study of low-angular-momentum flows is needed. This study focuses on studying the accretion flows at different angular momentum limits to see how the flow properties change with them.

All of the models in the simulation library for Sgr A* and most of the models for M 87* use variations of the same initial conditions: a rotation-supported torus seeded with a weak poloidal magnetic field (Fishbone & Moncrief 1976; hereafter, FM torus). Undoubtedly, these models successfully constrain the parameters of these supermassive black holes, such as the black hole mass, accretion rate, inclination, and spin (Collaboration et al. 2019, 2021, 2022, 2024). However, initializing simulations with a finite amount of matter in the torus leads to a drop in the matter content and mass accretion rate over time. Moreover, a stable FM torus solution does not exist for low-angular-momentum flows. In such flows (even in hydrodynamics), the gas pressure and lower centrifugal force are not enough to hold the torus in a stable structure against gravity. Accordingly, in this study, we employ the FM torus setup by changing the parameters from the ones used in earlier GRMHD simulations to achieve a steady state in low-angular-momentum accretion flows.

There is one more important piece of the puzzle: the GRMHD simulations show more highly variable light curves than the observed light curve for Sgr A* (Murchikova & Witzel 2021; Collaboration et al. 2022; Murchikova et al. 2022; Wielgus et al. 2022). Although it is still not fully explained, so-called wind-fed accretion models are somehow preferred for accretion flows around Sgr A* (e.g., Murchikova et al. 2022; Ressler et al. 2023). It is widely accepted that the winds of approximately 30 massive stars orbiting on the parsec scale may fuel Sgr A* (Quataert 2004; Cuadra et al. 2008; Ressler et al. 2018). Such models are not rotation-supported and have low angular momentum (Ressler et al. 2018). Additionally, the hotspot/flaring studies of Sgr A* suggest that a magnetically arrested disk (MAD; e.g., Dexter et al. 2020; Porth et al. 2021; Scepi et al. 2022) around a rotating black hole is usually associated with a strong jet (e.g., Tchekhovskoy et al. 2011). On the contrary, there is no direct evidence for a jet in Sgr A^*; morphological and kinematical studies suggest outflowing material or a weak jet from Sgr A* (for discussions, see Royster et al. 2019; Yusef-Zadeh et al. 2020). With all these enigmas, it is worth exploring low-angular-momentum accretion flows and testing their relevance for Sgr A*.

In Section 2, we briefly discuss the numerical setup, and in the following Sections 3, 4, and 5, we elaborately discuss the results obtained from our simulation models. Finally, in Section 6, we summarize and discuss the perspectives of our results.

This work investigates low-angular-momentum accretion flows with a set of 3D ideal GRMHD simulations using the GRMHD code BHAC (Porth et al. 2017; Okuda et al. 2019) in Modified Kerr–Schild (MKS) coordinates. The code BHAC assumes static spacetime, which means the mass of the central black hole is much larger than that of the surrounding matter. Thus, we neglect the self-gravity of the accretion flows in the simulations. We utilize a spherical polar grid denoted by (r, θ, ϕ), with a logarithmic grid spacing in the radial direction (r, from 90% of the event horizon up to r = 2500 r_g ) and a linear spacing in the polar (θ) and azimuthal (ϕ) directions. The simulations are conducted in a generalized unit system where G = M_BH = c = 1. Here, G, M_BH, and c represent the universal gravitational constant, the mass of the black hole, and the speed of light, respectively. Distances and times are expressed in units of r_g = GM_BH/c² and t_g = GM_BH/c³, respectively. To perform the simulations, we consider an effective resolution (320 × 128 × 128, with two static mesh refinement levels), where maximum resolutions are concentrated within ±45° from the equatorial plane with r ≤ 100 r_g . With this, the minimum uniform grid size (in MKS, two levels) in the radial direction is ${\rm{\Delta }}{s}_{\min }=0.02392$ and the maximum is ${\rm{\Delta }}{s}_{\max }=0.04784$. In linear scale, the minimum radial grid size is ${\rm{\Delta }}{r}_{\min }\sim 0.028692\,{r}_{g}$ at the inner edge. The grid size increases with the increasing radius. For example, at r = 100 r_g , the radial grid size is Δr ∼ 2.42072 r_g . For the sake of generalization, we consider a spinning black hole with Kerr parameter a = 0.9375. All simulations are performed up to t = 10,000 t_g .

This study does not intend to come up with a new, complicated way to simulate low-angular-momentum flows. For example, the wind-fed model (Ressler et al. 2018) or the perturbative simulation model (Olivares et al. 2023) are already being used for simulating low-angular-momentum flows around a black hole. We want to keep the spirit of the widely adopted rotation-supported FM torus (Fishbone & Moncrief 1976) and modify the parameters minimally, so that we can achieve steady-state solutions even for the lowest angular momentum. Accordingly, we set the initial density considering the FM torus distribution, with the inner edge at ${r}_{\min }=6\,{r}_{g}$ and the density maximum at ${r}_{\max }=15\,{r}_{g}$. These combinations give a density distribution with $\rho /{\rho }_{\max }\to 6.626\times {10}^{-4}$ for r → ∞ at the equatorial plane. Therefore, we have enough mass within the computational domain (r < 2500 r_g ) to reach a quasi-steady state. Note that for the usual torus setup (e.g., ${r}_{\min }=6\,{r}_{g}$, ${r}_{\max }=12\,{r}_{g}$ or ${r}_{\min }=20\,{r}_{g}$, ${r}_{\max }=40\,{r}_{g}$) with finite volume, the mass in the torus drained out before reaching a quasi-steady state for low-angular-momentum cases. The initial density distribution for the simulation models is shown in Figure 1. The figure shows that we allow accretion from a much wider direction as compared to the usual torus solutions. Additionally, we supply the angular momentum (λ₀) of the flow as a fraction of the Keplerian angular momentum (${ \mathcal F }$) at the ${r}_{\max }$ position, where λ₀ is the maximum angular momentum of the flow (see the Appendix for the detailed profile). In most cases, we choose the angular momentum to be less than the marginally stable angular momentum of the flow for the given Kerr parameter, i.e., λ_ms = 2.3752. Here we would like to mention that the marginally bound angular momentum for the given Kerr parameter is λ_mb = 2.5. We expect that, depending on the given angular momentum, the initial density at Figure 1 will be redistributed and reach a quasi-steady accretion state after time evolution. Further, depending on the transport of the angular momentum, we expect to have a distribution of angular momentum at the quasi-steady state for the given λ₀. To study the impact of the angular momentum on the magnetic field configuration, we supply a poloidal single-loop magnetic field considering a nonzero azimuthal component of the vector potential, which is given by

$$\begin{eqnarray}&&{A}_{\phi }\propto \max (\rho /{\rho }_{\max }-{10}^{-3},0).\end{eqnarray} \tag{ 1 }$$

To set the strength of the magnetic field, we choose the minimum ratio between the gas pressure and the magnetic pressure (plasma-β) to be ${\beta }_{\min }=100$. Following our motivations, we devised some simulation models, which we describe in Table 1. The radial profiles of other relevant quantities are shown in the Appendix. Here, we choose one model without any magnetic field for comparison (MOD5). Due to the presence of a very high volume of matter within the simulation domain, we could reach a quasi-steady state even for the lowest-angular-momentum case (MOD1). However, if we keep on evolving them for a longer time, we expect to see a lowering and vanishing of the accretion rate at the event horizon. This is because we do not supply any matter from outside. We catch on to the fact that this property may look like runaway instability (Abramowicz et al. 1983), but that is not the case, as we neglect self-gravity in this work, which is crucial for exciting runaway instability (e.g., Masuda & Eriguchi 1997; Korobkin et al. 2013). The goal of the current work is not to explore these properties, so therefore we do not evolve them farther. Additionally, the BHAC code is under development for considering self-gravity; we may be able to test such instability with the current framework in the future. Note that model MOD6 is a usual high-angular-momentum torus with ${ \mathcal F }=1$.

Zoom In Zoom Out Reset image size

Table 1. The Explicit Values of the Angular Momentum Fraction ${ \mathcal F }$, Specific Angular Momentum (λ₀), Corresponding Percentage of λ_ms, and Magnetic Field Status for Different Models

Model	${ \mathcal F }$	λ0	% of λms	Mag. Field
MOD1	0.2	0.92	38.7	Yes
MOD2	0.3	1.38	58.0	Yes
MOD3	0.4	1.84	77.4	Yes
MOD4	0.5	2.30	96.7	Yes
MOD5	0.5	2.30	96.7	No
MOD6	1.0	4.60	193.4	Yes

Download table as:  ASCIITypeset image

In these simulations, we use a piecewise parabolic reconstruction scheme, the total variation diminishing (TVD) Lax–Friedrichs approximate Riemann solver, an upwind constrained transport scheme (to preserve the divergence-free constraint of the magnetic field), and a second-order Runge–Kutta time-stepping scheme (for details, see Porth et al. 2017; Okuda et al. 2019). Here, we reiterate the boundary conditions explicitly. We prohibit inflow at the inner radial boundary. Scalar and radial vector components are symmetric at the polar axis, while azimuthal and polar vector components are antisymmetric. Additionally, periodic boundary conditions are imposed along ϕ for all quantities. Our simulation setup does not allow matter inflow at the outer edge of the simulation domain. However, due to a larger simulation domain and a modified initial density distribution, our simulations could reach a steady state to study accretion flow properties around black holes.

In this section, we investigate the impacts of low-angular-momentum flows on time-series properties. To do this, we plot the accretion rate ($\dot{M}$) and the normalized magnetic flux (${\phi }_{\mathrm{BH}}/\sqrt{\dot{M}}$) calculated at the horizon (following Porth et al. 2017) for different models in Figure 2 (panels (a) and (b), respectively). In the figure, the black, red, magenta, blue, and green lines correspond to MOD1-4 and MOD6, respectively. They are calculated for ${ \mathcal F }=0.2,0.3,0.4,0.5$, and 1.0, respectively.

Zoom In Zoom Out Reset image size

In Figure 2(a), we observe distinct changes in the feature of the accretion rate profile with the decrease of angular momentum. For the case of MOD6 (magenta), the angular momentum is 193.4% of λ_ms and λ₀ > λ_mb; also, the flow is gravitationally bound (${ \mathcal E }=-{{hu}}_{t}\lt 1$; see the Appendix for details). Therefore, the flow cannot cross the marginally stable orbit unless it loses some angular momentum or gains energy via turbulence in the torus driven by magnetorotational instability (MRI; Balbus & Hawley 1991, 1998; see Section 4.1 for more details). Note that for this model, the minimum angular momentum at the inner edge (${\lambda }_{\min }\sim 3.44$) is also greater than both λ_ms as well λ_mb. As a result, we observe accretion across the horizon after the simulation time t > 250 t_g , and the value gradually increases and reaches a quasi-steady state. However, for the other models, the angular momentum is lower than that of the marginally stable angular momentum. Therefore, the flow directly plunges into the horizon, and we see a very high value of the accretion rate immediately. For model MOD4 (blue), the angular momentum is slightly lower than that of λ_ms. As a result, the pressure (thermal and magnetic) developed close to the black hole could still give us an accretion rate profile qualitatively similar to model MOD6 with variability. Nonetheless, it is clear from the plots that they are quantitatively very different. On the contrary, other models have very low angular momentum as compared to λ_ms, and therefore we see a very smooth accretion rate profile without any variability signatures. To study it quantitatively, we calculate the variability magnitude, $\delta \dot{m}/\langle \dot{m}\rangle \,={\left(\max (\dot{m})-\min (\dot{m})\right)}_{\delta t}/\langle \dot{m}\rangle$, with $\dot{m}={\mathrm{log}}_{10}\dot{M}$, for models MOD1, MOD2, MOD3, MOD4, and MOD6 within the simulation time t = 8000–10,000 t_g . They are obtained as 0.24%, 0.25%, 0.24%, 24.0%, and 168.0%, respectively. In terms of the magnitude of the accretion rate, a low-angular-momentum flow can fall onto the black hole faster, due to a weaker centrifugal barrier. Accordingly, we observe higher values of the accretion rate for lower-angular-momentum cases, and they have quite similar profiles throughout the simulation time.

Similar to the accretion rate, we see distinct features in the normalized magnetic flux profiles at different angular momentum limits in panel Figure 2(b). For a high-angular-momentum flow (model MOD6), the magnetic flux grows with simulation time. However, due to the limited resolution, the magnetic flux starts to drop after the simulation time t > 2000 t_g . For model MOD4, initially we observe a decrease in the magnetic flux. However, as time passes, it increases again and reaches a steady value. On the contrary, for the cases with lower angular momentum (MOD1-3), the magnetic flux decreases monotonically with time, and the profiles look quite similar to them. Overall, due to the longer infall time than MRI growth time, higher-angular-momentum cases can accumulate more magnetic flux near the event horizon. Finally, we can conclude that all our models are in the regime of standard and normal evolution (SANE); they do not have enough magnetic flux to become MAD regimes in this simulation time (e.g., Tchekhovskoy et al. 2010; Mizuno et al. 2021; Porth et al. 2021).

Next, we compare the time-series properties of the flows with and without magnetic fields. To do that, in Figure 3, we present the accretion rates for models MOD4 (with magnetic fields) and MOD5 (without magnetic fields). The accretion rate profiles seem qualitatively similar; however, the flow properties may still be different, which we will study in the next section. Note that for model MOD4, the magnetic flux around the event horizon is only about ${\phi }_{\mathrm{BH}}/\sqrt{\dot{M}}\sim 0.2$ at the end of the simulation. This implies that the magnetic field for the model is very weak, and we accordingly expect similar results for both models. We will study the impacts of strong magnetic fields on low-angular-momentum flows in our upcoming studies.

Zoom In Zoom Out Reset image size

The time-series analysis suggests that the flow properties for MOD1, MOD2, and MOD3 are quite similar. Accordingly, for the sake of better explanations in the later part of the paper, we call them very-low-angular-momentum cases and consider MOD1 as a representative case for them. Further, we call MOD4 and MOD6 intermediate-angular-momentum and high-angular-momentum cases, respectively.

In this section, we investigate the detailed flow properties of low-angular-momentum flows. To do that, in Figure 4, we present the time-averaged density distribution for a slice at ϕ = 0° within the simulation time t = 8000–10,000 t_g for different models. Panels (a)–(d) correspond to models MOD1, MOD2, MOD4, and MOD6, respectively. Remember that these models are arranged in an increasing trend of the magnitude of the angular momentum (see Table 1). The panels show clear differences in the density distribution while increasing angular momentum. For very low angular momentum (MOD1; panel (a)), we observe a high-density conical region close to the black hole. The conical shape of the disk essentially indicates a vertically pressure-supported structure, where the pressure gradient balances the gravitational attraction normal to the disk. In such a case, the disk height can be calculated as $H\propto \sqrt{{p}_{g}/\rho }\times {r}^{3/2}$ with r ≫ 1 (Shakura & Sunyaev 1973; Frank et al. 2002). In the case of temperature, p_g /ρ = Θ is proportional to r⁻¹, and the disk height becomes H ∝ r. In later sections, we demonstrate that for very-low-angular-momentum cases, Θ ∝ r⁻¹ (see Section 5 for details). With an increase in angular momentum, the opening angle of the cone increases (MOD2; panel (b)). However, when the angular momentum is comparable to marginally stable angular momentum (MOD4; panel (c)), we do not observe any conical shape. Instead, we see a high-density clump of matter around the black hole, indicating the presence of turbulent flows. On the contrary, for very high angular momentum (MOD6; panel (d)), the density distribution is much smoother than MOD4. However, we do not observe a clear conical shape for MOD6, as seen in the very-low-angular-momentum models (MOD1, MOD2).

Zoom In Zoom Out Reset image size

The difference in the density distribution can be comprehended by observing it on the equatorial plane for different simulation models, which we show in Figure 5. In this figure, we plot the density distribution at the simulation time t = 10,000 t_g . The panels are distributed exactly in the same fashion as in Figure 4. We note that with the increase in angular momentum, the radius of the high-density compact region close to the black hole increases, which suggests competition between the gravitational pull and centrifugal force. For MOD4 (panel (c)), we see spiral density structures in the accretion flow. Earlier low-angular-momentum GRHD simulations also reported similar spiral structures or density jumps associated with local shock transitions (Olivares et al. 2023). This indicates that our simple adaptation of FM torus could seemly reproduce earlier results from a much more complex setup. These kinds of structures are usually observed in the near-horizon region of a MAD flow (e.g., Dexter et al. 2020; Porth et al. 2021; Ripperda et al. 2022; Scepi et al. 2022). They are useful for understanding the flaring events in the light curve of Sgr A* (Dexter et al. 2020; Ripperda et al. 2022; Scepi et al. 2022); however, the timing properties cannot be reproduced properly with MAD models. However, in this case, we observe spiral structures slightly far from the horizon. This may help in understanding the timing properties of Sgr A* consistently. Note that this flow is not in the MAD regime. The magnetic flux around the horizon is about ${\phi }_{\mathrm{BH}}/\sqrt{\dot{M}}\sim 0.2$. Therefore, we do not have strong jet activity, which is also expected from Sgr A* (Royster et al. 2019; Yusef-Zadeh et al. 2020). To confirm these claims, we need explicit radiation calculations from general-relativistic radiation transfer (GRRT) codes, which we plan to do in subsequent studies. Finally, for the usual high-angular-momentum flows (MOD6, panel (d)), we see a more seminal flow structure than that in low-angular-momentum flows.

Zoom In Zoom Out Reset image size

Next, we investigate the mass flux distribution to study inflow/outflow activities with the increase in angular momentum. Figure 6 shows the time-averaged mass flux $(\sqrt{-g}\rho {u}^{r})$ distribution for a slice at ϕ = 0° within the simulation time t = 8000–10,000 t_g for different models. In the panels, the black solid line corresponds to $\sqrt{-g}\rho {u}^{r}=0$. In panels (a) and (b), we do not observe any bluish regions, indicating no outflows in the very-low-angular-momentum ranges. Similar to the density distributions, we see a distinct conical structure for mass flux as well. However, for MOD4 (panel (c)), we observe inflow as well as outflow regions, showing the transport of angular momentum. Interestingly, we do not observe bipolar outflows. On the contrary, the outflow region is toward the equatorial region. Moreover, the inflow is also tilted with respect to the equatorial plane. This suggests the formation of a tilted disk close to the black hole due to the shearing instabilities in the inflow–outflow boundary. We will discuss this in more detail subsequently. Finally, for the usual high-angular-momentum model, the angular momentum needs to be transported from all the regions through MRI for accretion to take place. Accordingly, we observe the bluish region in a more or less symmetric manner. We also observe bipolar outflows and jets around the polar axis. Similarly, in Figure 7, we show the mass flux on the equatorial plane analogously to Figure 6. The panels in the figure also suggest that the very-low-angular-momentum flow is seminal in nature. With the increase in angular momentum, we observe inflow–outflow boundaries on the equatorial plane. This feature is quite similar for MOD4 and MOD6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

At the end, we study the differences between magnetized and unmagnetized flows. To do that, we show the density (panels (a) and (b)) and mass flux (panels (c) and (d)) distributions at the equatorial plane in Figure 8, where models MOD4 and MOD5 correspond to magnetized and unmagnetized flows, respectively. Qualitatively, we see quite similar features in the magnetized and unmagnetized flows. This is due to the presence of a very weak magnetic field. However, in the density distribution, we observe that the spiral filaments are more prominent but compact for the magnetized flows than those of the unmagnetized flows. This is because, due to the magnetized flow, the angular momentum transport is efficient. Therefore, the inner part has lower angular momentum for magnetized flows than that for unmagnetized flows. As a result, the mass flux distributions in magnetized flows show slightly higher values than those of the unmagnetized flows (see the dark red and blue colors in panel (c)). These differences may be more drastic with the increase in magnetic field strength. In the next section, we discuss in more detail the differences in the flow properties in terms of vertically integrated quantities.

Zoom In Zoom Out Reset image size

4.1. Development of Turbulent Structure

In all the simulation models, we observe that the flow becomes turbulent with the increase in angular momentum. In this section, we want to understand the detailed process of its formation and how the spiral filaments develop in the accretion flow. We note that the turbulent structure is present in magnetized as well as unmagnetized flows. Therefore, we expect the source of it to have a nonmagnetic origin. However, MRI could still influence the turbulence in the flow. In Figure 9, we show the time evolution of the grid-averaged MRI quality factor for the fastest-growing modes: 〈Q〉 = (1/3)(〈Q_r 〉 + 〈Q_θ 〉 + 〈Q_ϕ 〉) (see Takahashi 2008; Porth et al. 2019 for the detailed calculation), where the averaging is performed within r < 100 r_g near the equatorial plane within ±π/5. The figure suggests that depending on the angular momentum, the fastest-growing MRI wavelength grows differently. For the highest angular momentum, the wavelength is longer and can be resolved with a higher value of 〈Q〉. However, with a decrease in angular momentum, the infall timescale becomes shorter than the MRI growth timescale. As a result of the lower angular momentum, the MRI quality factor becomes low. Note that to resolve the MRI adequately, at least 〈Q〉 ≳ 6 is needed (Sano et al. 2004). Except for the MOD6 model, the accretion process can naturally happen, as the angular momentum is lower than that of the marginally stable angular momentum and transits to a quasi-steady/steady state at later simulation time. In this case (MOD6), the MRI quality factor becomes ∼10 at the end of the simulation, where the accretion is driven by the resolved MRI. Therefore, the qualitative results from our simulation models are not expected to alter with the increase in resolution. Since, with resolution, the magnetic pressure will grow due to well-resolved MRI, we may expect quantitatively similar results at slightly lower angular momentum as compared to the current values mentioned in this work. Moreover, for high resolution, the numerical dissipation will be reduced. Along with the extra magnetic pressure at high resolution, we expect to see a reduction in the overall accretion rate. We plan to do such simulations in the future and report elsewhere.

Zoom In Zoom Out Reset image size

Finally, to understand the origin of the turbulent nature of the accretion flows in models MOD4 and MOD5, we show the distribution of the mass flux $(\sqrt{-g}\rho {u}^{r})$ and specific angular momentum (λ = − u_ϕ /u_t ) in panels (a) and (b) of Figure 10, respectively, during the initial phase of the simulation at t = 1500 t_g for model MOD4. For the intermediate-angular-momentum case, gravity and centrifugal force are more or less equal at the innermost stable circular orbit (ISCO). Accordingly, the increase in pressure close to the black hole could push some outflow in a bipolar direction for an intermediate range of angular momentum. As a result, we observe the development of a shear between the inflow and outflow, which we can clearly see in panel (a) of Figure 10. The black solid line (u^r = 0) in Figure 10(a) indicates the inflow–outflow boundary. The velocity shear at the boundary triggers the excitation of instability and turbulence. Due to the turbulence developed by the shear instability, angular momentum is transported outward. That further enhances the outflow close to the black hole. The consequences can be seen in the angular momentum distribution. We observe islands of high angular momentum in the outflow region in panel (b) of Figure 10. In Figure 10(b), the solid lines show the contour of λ = λ_ms. This outflow does not have enough kinetic energy to leave the system (i.e., it is bounded). Thus, it falls back on the accretion flow far from the black hole. This recurrent process destabilizes the seminal flow near the equatorial plane and, finally, we see a spiral filament structure in the accretion flow. In the very-low-angular-momentum cases (MOD1–MOD3), the centrifugal force is always much smaller than that of gravity. Accordingly, pressure is not enough to push outflow and shear instability is never triggered. As a result, we do not observe any spiral structure. Nonetheless, there is a possibility of having such a structure even in a low-angular-momentum flow with an increase in magnetic pressure. This is something that we plan to study in the future.

Zoom In Zoom Out Reset image size

In this section, we study the influence of angular momentum on the time-averaged vertically integrated structure of the accretion flow close to the black hole. To do that, we calculate the average quantities 〈X〉 following

$$\begin{eqnarray}&&\begin{array}{l}\langle X\rangle =\displaystyle \frac{\displaystyle \int \sqrt{-g}{Xd}\theta d\phi }{\displaystyle \int \sqrt{-g}d\theta d\phi },\end{array}\end{eqnarray} \tag{ 2 }$$

where X is a time-averaged quantity between t = 8000–10,000 t_g . Following Equation (2), we plot the averaged value of the density (〈ρ〉), gas pressure (〈p_g 〉), flow temperature (〈Θ = p_g /ρ〉), specific entropy (〈κ〉), and plasma-β (〈β〉) as a function radius in panels (b)–(f) of Figure 11, respectively. In panel (a) of Figure 11, we show vertically integrated $\int \sqrt{-g}\rho {u}^{r}$ (without vertical averaging) to study the inflow/outflow equilibrium of the models. In the panels, the black, blue, and green lines correspond to MOD1 (very low angular momentum), MOD4 (moderate angular momentum), and MOD6 (high angular momentum), respectively. Additionally, we also plot the same quantities for the unmagnetized model (MOD5) with the dotted blue lines for comparison.

Zoom In Zoom Out Reset image size

In panel (a) of Figure 11, we show an inflow/outflow equilibrium diagram for different angular momentum limits. In the lowest-angular-momentum case, the value of the equilibrium of the mass flux is the highest (MOD1). Subsequently, in the higher-angular-momentum case, it becomes lower. This suggests that low-angular-momentum flows can reach the event horizon without any barrier. We also note that the equilibrium reaches up to r ∼ 400 r_g , ∼100 r_g , and ∼2 0 r_g for models MOD1, MOD4, and MOD6, respectively, although the simulation time is the same, i.e., t = 8000–10,000 t_g . Accordingly, for the following discussions, we will only compare results within r < 20 r_g for consistency.

In panel (b) of Figure 11, we observe that the nature of the density profile changes significantly with the increase of angular momentum. Interestingly, the intermediate range of angular momentum has the highest density, with a radius of r < 20 r_g . Additionally, high-angular-momentum flows have the lowest density close to the black hole. To quantify the differences, we fit the density with a power-law profile, $\rho \propto {r}^{{\alpha }_{\rho }}$. We find that the values of the power-law index α_ρ are α_ρ = − 1.54 (∼ − 3/2), −1.00, and −0.06 (∼0), for models MOD1, MOD4, and MOD6, respectively. Similar to panel Figure 11(b), the averaged pressure also shows similar behavior in Figure 11(c). We observe maximum pressure in the intermediate range of angular momentum. The power-law index that could fit ${p}_{g}\propto {r}^{{\alpha }_{p}}$ is obtained as α_p = −2.49 (∼ − 5/2), −1.74, and −0.71 for models MOD1, MOD4, and MOD6, respectively. Similar to the pressure and density, the temperature profiles also show maximum values for the intermediate range of angular momentum. However, for the temperature profile, the nature of the plots is quite similar, although we see some changes in the power law ${\rm{\Theta }}\propto {r}^{{\alpha }_{{\rm{\Theta }}}}$, where α_Θ = −1, −0.75, and −0.75 for the very-low-, intermediate-, and high-angular-momentum cases, respectively.

Subsequently, in panels (e) and (f) of Figure 11, we show entropy (〈κ〉) and plasma-β (〈β〉) profiles for different limits of angular momentum. Figure 11(e) shows that higher angular momentum has a higher entropy of the flow. The solutions are also not isentropic; the entropy increases as the flows proceed toward the black hole. In the very-low-angular-momentum case, the entropy throughout is more or less constant. Therefore, we expect the properties of the flows for the very-low-angular-momentum cases to be similar to isentropic semi-analytic solutions (Bondi 1952; Chakrabarti 1989). Similarly, the strength of the magnetic field is higher for higher-angular-momentum flows. Since the power-law indices for density and pressure are different at different angular momentum limits, the power-law for entropy is also different. We do not calculate them explicitly here. However, it is interesting to check the power-law indices for plasma-β ($\beta \propto {r}^{{\alpha }_{\beta }}$). We find that α_β = 1.66 (∼5/3), 0.54 (∼1/2), and 1.31 (∼4/3) for models MOD1, MOD4, and MOD6, respectively.

Earlier, such power-law indices were often used to model accretion flows around a black hole using self-similarity. The power-law indices for the very-low-angular-momentum case show extraordinary similarities to the values proposed by Narayan & Yi (1995) for ADAF solutions. Subsequently, Blandford & Begelman (2004) also calculated the power-law indices, considering a more general scenario involving outflows. They proposed that the density and pressure must satisfy ρ ∝ r^n−3/2 and p_g ∝ r^n−5/2 for self-similarity in the axisymmetric assumption, where n depends on the outflows. By comparing the power-law indices, we find that n = 0 matches exactly for the very-low-angular-momentum case. n = 0 is also reported for low-angular-momentum, magnetized, and transonic accretion flows (Mitra et al. 2022). Recently, Olivares et al. (2023) have performed similar calculations for transonic relativistic hydrodynamic simulations. They reported density variations ρ ∝ r^−3/2. Note that here we have only shown plots for MOD1; nonetheless, similar to this model, MOD2 and MOD3 also follow n ∼ 0 for both density and pressure. This is reasonable, because these models do not have any outflows and the flow properties are also axisymmetric (see panels (a) and (b) of Figures 4–7). These conditions do not hold for the intermediate- or the high-angular-momentum flows. Accordingly, such self-similarity does not hold. With such self-similarity, we expect α_Θ = –1, and we observe its deviation with the increase of angular momentum. A similar trend is also reported in Olivares et al. (2023). It has been shown that for MAD models, density varies as ρ ∝ r⁻¹, which, interestingly, is exactly the same for the intermediate-angular-momentum flow model, although it is in the SANE regime. The wind-fed hydrodynamic simulations done by Ressler et al. (2018) show density variations ρ ∝ r⁻¹, which are exactly the same as our intermediate-angular-momentum case.

Finally, comparing the solid and dotted lines in all the panels, we investigate the flow properties between magnetized and unmagnetized flows. We have already mentioned that due to the weak nature of the magnetic field, the differences in all the profiles are minimal. Although we observed slightly lower values for density, pressure, temperature, and entropy for unmagnetized flows, we expect quite similar power-law indices for magnetized and unmagnetized flows. At the same time, it will be interesting to study them by gradually increasing the magnetic field strength. We plan to do such a study in the future.

5.1. Sonic State of the Models

Finally, we study the Mach number variation with different angular momentum, which gives the sonic state of the flow. Here, we calculate the radial velocity ${\tilde{v}}_{\mathrm{rad}}$ in a corotating frame in the Boyer–Lindquist coordinates, following ${\tilde{v}}_{\mathrm{rad}}^{2}={\gamma }_{\phi }^{2}{u}^{r}{u}_{r}/(-{u}_{t}{u}^{t})$, where ${\gamma }_{\phi }^{2}$ is the Lorentz factor for rotation, i.e., ${\gamma }_{\phi }^{2}=1/(1+{u}_{\phi }{u}^{\phi }/{u}_{t}{u}^{t})$ (e.g., Dihingia et al. 2018a). Additionally, we calculate the sound speed as ${a}_{s}^{2}={\rm{\Gamma }}{p}_{g}/(e+{p}_{g})$, where e and Γ = 4/3 are the internal energy and adiabatic index of the flow, respectively. Figure 12 shows the Mach number (M_a ) on the equatorial plane for models MOD1 (very low angular momentum; black solid line), MOD4 (intermediate angular momentum; blue solid line), MOD5 (unmagnetized intermediate angular momentum; dotted blue line), and MOD6 (high angular momentum; solid green line). The stars on the curve correspond to the sonic point (sonic surface in 3D visualization) for the flows, where the flows make a transition from subsonic (M_a < 1) to supersonic (M_a > 1). The plots suggest that with the decrease in angular momentum, the location of the sonic point moves far from the black hole. The locations of the sonic points for MOD1, MOD4, and MOD6 are obtained as r_c = 4.06 r_g , 2.16 r_g , and 1.97 r_g , respectively. Note that the ISCO for this Kerr black hole is r_ISCO = 2.04 r_g . Thus, with high angular momentum, a sonic point resides inside the ISCO radius, whereas with a decrease in angular momentum, a sonic point may form outside the ISCO radius. For unmagnetized flows, the difference in the Mach number profile is minimal. These kinds of solutions contain only a single sonic point; however, it is possible to have two sonic points in a single solution if there is a standing shock in the solution (e.g., Fukue 1987; Chakrabarti 1989). Although our goals for this work are separate from these investigations, it is nonetheless important to carry them out in the future. Accordingly, we plan to do more parameter=space surveys in our upcoming studies, to investigate whether shock solutions are possible in 3D GRMHD simulations or not.

Zoom In Zoom Out Reset image size

In this study, we have focused on investigating the flow properties at different limits of angular momentum. Primarily, we have considered very-low-angular-momentum, intermediate-angular-momentum, and high-angular-momentum cases and have performed 3D GRMHD simulations around the Kerr black hole with moderate resolution. We employ the initial conditions of the FM torus to perform our simulations. Interestingly, we could reproduce results from a much more complicated simulation setup for low-angular-momentum accretion flows (e.g., Ressler et al. 2018; Olivares et al. 2023). Here, we list our major findings in point-wise order:

• 1.  
  We found that a lower-angular-momentum accretion flow has a higher saturated accretion rate. A higher-angular-momentum flow has a higher variability magnitude $(\delta \dot{m}/\langle \dot{m}\rangle )$. For the variability of very-low-angular-momentum flows, the magnitude tends to zero or the accretion rate profiles become absolutely smooth. Similarly, the normalized magnetic flux ($\phi /\sqrt{\dot{M}}$) does not saturate for very-low-angular-momentum cases. For the intermediate-angular-momentum case, the normalized magnetic flux saturates to lower values as compared to the high-angular-momentum case.
• 2.  
  By lowering the angular momentum to an intermediate value of ∼97%λ_ms, we observe accretion flows with distinct spiral filaments. These are absent if the angular momentum is very low or very high. The high-density region close to the black hole becomes compact with a decrease in angular momentum.
• 3.  
  We found that angular momentum impacts the onset of outflows/jets from the accretion flow. In the intermediate-angular-momentum range, we observe outflows more toward the equatorial plane. However, for very-low-angular-momentum cases, outflows/jets also cease to exist. In such cases, we observe pure inflows.
• 4.  
  The flow properties of different angular momentum fit with different power-law indices. In general, we can fit density and pressure with ρ ∝ r^n−3/2 and p_g ∝ r^n−5/2 (Blandford & Begelman 2004) for very-low-angular-momentum axisymmetric cases without any outflows with n = 0. However, with the increase in angular momentum, the accretion develops nonaxisymmetric flows. Therefore, such simple self-similarity does not hold for intermediate- as well as high-angular-momentum cases.
• 5.  
  In our study, we also investigated the sonic properties of the simulation models. We found that for high-angular-momentum flows, the sonic point or sonic surface resides inside the ISCO radius. With a decrease in angular momentum, the sonic point or sonic surface can also exist outside the ISCO radius.
• 6.  
  Throughout our study, we did not find significant differences between magnetized and unmagnetized low-angular-momentum flows. All the timing and flow properties look quite similar in both cases. This is due to the fact that we restrict ourselves within the weak limits of a magnetic field. In our upcoming studies, we will explore strong-magnetic-field limits.

In light of the above findings, it is interesting to note that the intermediate-angular-momentum case is significantly different from other cases. This case has spiral filaments that are quite similar to those seen in MAD flows (e.g., Porth et al. 2021; Begelman et al. 2022), but they are seen slightly far from the black hole. Such spiral filaments could possibly be connected to the flaring activities in Sgr A* (e.g., Porth et al. 2021; Ripperda et al. 2022; Scepi et al. 2022). However, MAD flows are known to have very powerful jets, which have not clearly been confirmed in the case of Sgr A* (Royster et al. 2019; Yusef-Zadeh et al. 2020). Therefore, there are lots of ambiguities regarding the nature of the accretion flows around Sgr A*. Interestingly, in the intermediate-angular-momentum flows, we do not see any bipolar outflows or jets. At the same time, there is still uncertainty in explaining the observed light curve of Sgr A* using the conventional SANE or MAD models (Murchikova & Witzel 2021; Collaboration et al. 2022; Murchikova et al. 2022; Wielgus et al. 2022). Accordingly, we suggest that the intermediate-angular-momentum flow case could be a reasonable alternative for explaining the accretion flows around Sgr A*. We should mention that for such a study, we would need to perform GRRT calculations, which we plan to do in our upcoming studies.

There have been lots of semi-analytical and numerical (axisymmetric and pseudo-Newtonian) studies that show low-angular-momentum flows can help explain the radiative (luminosity, spectra, etc.) and timing properties (quasiperiodic oscillations, flaring, etc.) in black hole X-ray binaries as well as active galactic nuclei, taking into account standing/oscillating shock solutions with multiple sonic points (e.g., Chakrabarti & Titarchuk 1995; Chakrabarti et al. 2004; Das et al. 2014; Aktar et al. 2015; Dihingia et al. 2018b, 2019a, 2019b, 2020; Chakrabarti 2018; Okuda et al. 2019; Okuda et al. 2022). However, with global 3D GRMHD simulations, it has never been tested. Due to magnetohydrodynamics, many rich phenomena may appear, such as reconnection-driven turbulence, MAD configurations, jets, and magnetized winds, etc. (e.g., Nathanail et al. 2019; Vourellis et al. 2019; Dihingia et al. 2021; Ripperda et al. 2022; Jiang et al. 2023). The semi-analytical expectation may or may not hold. To capture these phenomena, higher-resolution simulations than the current study are needed, which we plan to test in our future studies.

This research is supported by the National Key R&D Program of China (No. 2023YFE0101200), the National Natural Science Foundation of China (grant No. 12273022), and the Shanghai Municipality orientation program of Basic Research for International Scientists (grant No. 22JC1410600). The simulations were performed on the TDLI-Astro cluster in the Tsung-Dao Lee Institute, Pi2.0, and the Siyuan Mark-I clusters in the High-Performance Computing Center at Shanghai Jiao Tong University. This work has made use of NASA's Astrophysics Data System (ADS). We appreciate the thoroughness of the thoughtful comments provided by the anonymous reviewers, which have improved the manuscript.

In this section, we show the radial and vertical distributions of some initial quantities, which are crucial to determining the evolution state of our simulation models. In Figure 13, we show the radial distributions of ${ \mathcal E }=-{{hu}}_{t}$, ${ \mathcal L }={{hu}}_{\phi }$, and the specific angular momentum (λ = − u_ϕ /u_t ) for different simulation models in panels (a), (b), and (c), respectively. In panels (d), (e), and (f), we show the same, but a vertical distribution only for model MOD6, since we follow the FM torus (Fishbone & Moncrief 1976) for the initial conditions, Accordingly, the profiles of ${ \mathcal E }$, ${ \mathcal L }$, and λ are not constant throughout. The panels (a) and (d) suggest that the flow is always less than unity $({ \mathcal E }\lt 1)$; that is, the flow is gravitationally bound. The value of ${ \mathcal L }$ scales with the choice of ${ \mathcal F }$. Finally, the specific angular momentum (λ) also has the same scaling as ${ \mathcal L }$, and the maximum values of each case reach λ₀ as r ≫ 1.

Zoom In Zoom Out Reset image size

In the right panels of Figure 13, we show a vertical distribution of the same quantities as the left panels. For these panels, we only show a vertical distribution at radius r = 20 r_g for model MOD6. We observe that the values of ${ \mathcal E }$ and ${ \mathcal L }$ decrease far from the equatorial plane initially, but finally increase slightly. On the contrary, the value of the specific angular momentum always decreases far from the equatorial plane (panel (c)). These features are similar at different radii and for other models. Accordingly, we do not show them here explicitly.